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The statistical validity of using ratio variables in human kinetics research Liu, Yuanlong 1999

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THE STATISTICAL VALIDITY OF USING RATIO VARIABLES IN HUMAN KINETICS RESEARCH by Yuanlong Liu B.Sc, Inner Mongolia Teacher's University, 1982 M.P.E, The University of British Columbia, 1993  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES (School of Human Kinetics) We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRI1ISHTJOLUMBIA September 1999 ©Yuanlong Liu 1999  UBC  Special Collections - Thesis Authorisation Form  http://www.library.ubc.ca/spcoll/thesauth.htmi  I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n .  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver, Canada  1 of 1  9/20/99 9:40 PM  ii  ABSTRACT There were two main purposes of this investigation. The first was to examine the validity and reliability of commonly used ratio variables in human kinetics research, and to evaluate four ratio score models used to deflate the effect of the denominator. The second was to use computer simulation procedures to investigate the effect of using ratio variables on the circularity assumption of the covariance matrix and type I error rates in RM  A N O V A tests. It is shown that a suitable common deflation model for all ratio  variables may not exist, and different models should be used to derive an appropriate deflation model in empirical research. The results indicate that high reliability of the component variables does not necessarily result in high reliability of the transformed ratio variable. Thus, when a ratio variable is used the reliability should be examined based on the ratio variable data. It is recommended that five criteria be used to evaluate and compare the validity of deflation models: (a) zero correlation between a derived ratio variable and the denominator variable, (b) no curvilinear relationship between the derived ratio and the denominator in the scatterplots, (c) equality of the estimated expected value of the model and calculated mean of the derived ratio data, (d) high R , and (e) high 2  reliability of the derived ratio data. Simulation results show that the characteristics (e i, x  £x2,  V i/V 2, and p i 2 ) of the two component variables strongly affect e i/ 2 and the type I X  x  X  x  X  x  error rate. In the condition £ i=£ 2, the magnitude of e i/ 2 is virtually the same as that of x  X  X  x  E i and e 2, regardless the level of p i 2 and V i/V 2. When £ i<1.0 and e 2=1.0, e i/ 2 x  X  x  X  X  x  x  X  x  tends to have smaller magnitudes when p i 2 and V i/V 2 are high, and greater x  magnitudes when p i x  x 2  and V i / V x  X  x  X  are low. £ i/ 2 exhibited the greatest bias and the A  x2  x  X  largest standard deviation, resulting in a serious inflation of type I error rate in the  X  condition V i/V =0.5, regardless of the conditions e i, e 2, and p i . If homogeneity of x  x2  x  X  x  x2  the denominator variable (small V ) and large sample size are present, it may reduce the x2  likelihood of bias in e / and protect the type I error rate. A  x]  x2  iv  TABLE OF CONTENTS Abstract Table of Contents List of Tables List of Figures Acknowledgements  ii iv vii ix x  CHAPTER I. Introduction Introduction Purpose of the Study  II. Literature Review Statistical Characteristics of Ratios The Distribution of Ratio Variables Variance, Covariance, and Correlation of Ratios Reliability of Ratio Variables Ratio Variable Issues in the Literature Using Ratio Variables for the Purpose of Deflation Ratio Variables in Regression Analysis Ratio Variables in Analysis of Variance Ratio Variables and Unsolved Issues in Human Kinetics Research Deflation Issues Reliability Issues Issues in R M A N O V A with Ratio Data  III. Methods and Procedures Ratio Selection Deflation Model Evaluation The Models Controlling for the Denominator Effect Reliability of a Ratio Computer Simulation Investigation on Ratio Issues Overview of the Simulation Conditions and Investigations Preparing Population Covariance Matrices Data Generation Overview of Methods Used to Answer Questions  IV. Results and Discussion: Characteristics of Ratio Variables Ratio Variables Used for Deflation Purposes Empirical Data Set 1: Vo2max/Body Mass  1 1 6  8 8 8 12 13 15 15 17 20 22 23 25 25  28 28 29 29 32 35 37 39 45 49 51  52 52 52  V  Simple Ratio Model Alternate Models Empirical Data Set 2: D L C O / V A Simple Ratio Model Alternate Models A Summary of the Evaluation of the Deflation Models  53 60 69 70 74 76  Ratio Variables Used Not for Deflation Purposes Empirical Data Set 3: Nondominant Torque/Dominant Torque Empirical Data Set 4: Waist Girth/Hip Girth A Discussion on the Reliability of the Ratio Variable: Data Sets 3 and 4 A Summary of the Evaluation of the Ratio Variables Not for Deflation  V. Characteristics of Ratio Variables as A Function of the Component Variables: A Simulation Study  x  x  X  x  X  The Effect of e i,V i/V x  x  x2  x2  x  Sample Characteristics of e i / Condition 1 (e i=e =1.0)  X  x  and p i  x 2  x]  on the Magnitude of  x  x 2  x2  Condition 2 (e i=e =0.7) x  P( i/x2)ij X  88 90 90 93  105  x2  Condition 3 ((E i=0.7, E =1.0) x  X  102 103  A  x  86  88  /X  The Effect of £ j,V i/V and p i 2 on the Magnitude of e / 2 Magnitude of e i/ 2 When the Ratio Components Meets the Assumption of Circularity Magnitude of £ i/ 2 When the Ratio Components Violate the Assumption of Circularity X  80  88  Population Characteristics of e i 2 x  76 77 79  105  x2  Summary of Characteristics of e / 2 and e i/ 2 Type I Error Rate Expected effects of Characteristics of Xi and X on Type I Error Rates for X 1 / X 2 Monte Carlo Simulation investigation on  111 112  A  xj  X  x  X  Type I Error Rate  2  112 114  Sample Estimates e i / 2 and Type I Error Rates A  x  X  119  VI. A Summary of the Investigation  128  References  136  VI  Appendices  142  Appendix A: Approximating  p ii, 2j x  X  Using the Attenuation Correlation  Function  142  Appendix B: Covariance Matrices Used to Generate the Population Data  145  Appendix C: Accuracy of the Estimation Formulas (Mean, Sd, and Correlation)  156  Appendix D: Correlation Matrices of the Component and Ratio Variable Populations  159  vii  LIST OF TABLES  Table  2- 1.  Ratio Variables in Human Kinetics Research  24  3- 1.  Data Matrix of the Component Variables  40  3-2.  Covariance Matrix of Raw Score Population for Simulation  46  3- 3.  Computer Simulation Conditions  50  4- 1.  Means and Standard Deviations of Body Mass, Raw Score Vo2max, and Vo2max/kg: Data Set 1  54  4-2.  Evaluation of the Simple Ratio Model: Data Set 1  55  4-3.  Evaluation of the Alternate Models: Data Set 1  61  4-4.  Means and Standard Deviations of Va, Raw score Dlco, and Dlco/Va: Data Set 2  71  4-5.  Evaluation of the Simple Ratio Model: Data Set 2  72  4-6.  Evaluation of the Alternate Models: Data Set 2  75  4-7.  Means, Standard Deviations, and Reliability Coefficients of the Torques and Nondominant/Dominant Ratio (Pronation)  4-8.  Means, Standard Deviations, and Reliability Coefficients of the Torques and Nondominant/Dominant Ratio (Supination)  4-9.  78  78  Means, Standard Deviations, and Reliability coefficients of the Waist and Hip Girth, and Waist/Hip Ratio  81  4-10.  Correlations in Data Set 3 and Data Set 4  83  4-11.  Coefficients of Variation in Data set 3 and Data Set 4  83  4-12.  Reliability of the Ratio Variable in Data Set 3 under Different Coefficient of Variation Conditions  84  Vlll  5-1.  Population Values of 8 i/ x  X  2  for the Ratio Variable Under  Different Component Variable Conditions 5-2.  Intertrial Correlations of Ratio Variables for Selected Conditions of the Component Variables  5-3.  96  X  x  The Correlation Matrices of Ratio Variable populations 97  (Pxlx2=0.7)  5-5.  Comparison of the Correlation Matrices of the Ratio 100  Variable under Selected Conditions 5-6.  Distribution Characteristics of e i / 2 A  X  x  (e i=e =1.0, u.i=75, u. =100) x  5-7.  x2  x  104  x2  Distribution Characteristics of  A  £ i/ 2 x  X  (Exi=£x2=0.7, p, i=75, |a =100) x  5-8.  94  The Correlation Matrices of the Component and Ratio Variable Populations (e i=e 2=1.0)  5-4.  89  106  x2  Distribution Characteristics of  A  £ i/ 2 x  X  (e =0.7, e =1.0, u. =75,1^=100)  109  5-9.  Empirical Type I Error Rates  116  5-10.  Effect of e i / 2 and p 2 on Type I Error Rates  118  5-11.  A Summary of the Effect of V i / V  xl  x2  xl  A  x  X  xtx  x  x2  on the Type I  Error Rates (p i 2=0.7, a=.05, n=15) x X  126  ix  LIST OF FIGURES  Figure  4-1.  The Scatterplots of the Simple Ratio and the L R M Ratio: Data Set 1  4-2.  58  The Scatterplots of the NLRM1 Ratio and NLRM2 Ratio: Data Set 1  4-3  63  Comparison of the Fit of the Three Deflation Models: Data Set 1  4- 4.  66  The Scatterplots of the Simple Ratio, LRM, NLRM1 And NLRM2 Ratios: Data Set 2  5- 1. 5-2.  A  E i/x2 x  A  e i x  73  in the Conditions 8 i=E 2=1.0 and e i=e 2=0.7 x  107  X  x  X  in the Condition 8 i=0.7 and 8 2=1.0  / x 2  110  X  x  5-3.  Mean of epsilon estimates with +/- 2sd for e i=s 2=1.0, e i 2=1.0 121  5-4.  Mean of epsilon estimates with +/- 2sd for e i=e 2=0.7, e i/ =0.7 122  5- 5.  Mean of epsilon estimates with+/-2sd for E i=0.7 and e 2=1.0  6- 1.  The E i / 2 values and the empirical type I error rates  X  x  X  x  x  x  x  x  X  /X  x2  124  X  in the condition e =0.7 and £ =1.0 ((X=0.05, n=45) x)  x2  142  X  ACKNOWLEDGEMENT  At the moment of the completion of this dissertation, I would like to express my sincere appreciation to many individuals I am indebted. Without their trust, guidance, and support, this endeavor would not have been completed. Foremost, I owe a great debt of gratitude to my supervisor Dr. Schutz. His patience, persistence, insight and support were felt in each of the steps during my many years of study under his guidance. His time, effort and thorough knowledge were very much needed and appreciated, and also his example as a scientist has left a lifelong impression. I am also indebted to my research committee member Dr. Petkau. His thoughtful feedback and support were always provided in a constructive manner. I am deeply grateful to Dr. Coutts, who served as a committee member for both my master and doctoral program, for sharing his expertise and enthusiasm to my study. Thanks to Dr. Berkey, who hired me as a faculty member in Western Michigan University, for her support and understanding at the final stage of my dissertation. I am grateful to Ms. Bacon, Dr. Kramer, Dr. Sonnenschein, and Dr. Taunton allowed me to process their valuable data for my study. I would also like to acknowledge my cheering squad of friends: Park, Cliff, Lisa, Mona, Terry, Bill, and those who have assisted me at different stages of this research. Finally, and very importantly, I would like to thank my family. My mother and father, sister and brothers educated and encouraged me to face the challenge of life. Their moral support is always appreciated. My wife, Teresa Huang, married me while I was struggling for my student life. Since we have survived this, the rest should be easy. As I look at this long list, I feel humbled and blessed, because it presents much energy and trust which each of you invested in me. Thank you!  1  CHAPTER I. INTRODUCTION  Introduction R a t i o variables have a l o n g history o f use i n different d i s c i p l i n e s related to h u m a n kinetics research (e.g., b i o m e c h a n i c s , exercise e p i d e m i o l o g y , p h y s i c a l e d u c a t i o n , sport p h y s i o l o g y , and sport p s y c h o l o g y ) . A ratio v a r i a b l e is a c o m p o s i t e variable that consists o f a numerator v a r i a b l e and a denominator v a r i a b l e . F o r e x a m p l e , the m o m e n t o f force i n b i o m e c h a n i c s is u s u a l l y expressed as a ratio v a r i a b l e w h i c h consists o f the m e a s u r e d m o m e n t o f force d i v i d e d b y b o d y mass, and the V o 2 m a x i n p h y s i o l o g y is u s u a l l y expressed as a ratio v a r i a b l e w h i c h consists o f the measured V o 2 m a x d i v i d e d b y b o d y mass. P e r capita i n c o m e and other variables i n s o c i o l o g y are ratios w i t h p o p u l a t i o n size as the d e n o m i n a t o r to c o m p a r e different countries. W a i s t and hip girth and n o n d o m i n a n t and d o m i n a n t strength ratio are p o p u l a r l y used i n the e p i d e m i o l o g y and health fields, and m a n y sport statistics are expressed as a ratio o f successful attempts d i v i d e d b y the total n u m b e r o f attempts (e.g., basketball s h o o t i n g percentage). R a t i o variables are p o p u l a r l y used i n different areas o f h u m a n k i n e t i c s research, and they can be c l a s s i f i e d b y the purpose o f use. R a t i o variables are t y p i c a l l y used for s c a l i n g data, w i t h the v a r i a b l e o f p r i m a r y interest b e i n g the numerator v a r i a b l e . D i v i d i n g b y the d e n o m i n a t o r is an attempt to express the numerator on a per-unit o f the d e n o m i n a t o r basis i n order to r e m o v e or deflate the effects o f the d e n o m i n a t o r ' s v a r i a t i o n f r o m the numerator's v a r i a t i o n , and full deflation w o u l d y i e l d a zero c o r r e l a t i o n between the ratio variable and its d e n o m i n a t o r variable. F o r example, heavier people can lift heavier weights partially because they have greater body mass. Often it is desirable to compare individuals with respect to strength independent o f (or at least m i n i m i z i n g the effect  2  of) body mass. In O l y m p i c weightlifting competitions, this is done by forming homogeneous groups with respect to body mass, and individuals are compared only to others o f similar body mass. In other situations the formation o f homogeneous groups is either not possible or not desirable. F o r example, in schoolboy weightlifting competitions there often are not enough competitors to form different weight classes. In this situation, a ratio variable, weight lifted d i v i d e d by body mass, is computed and used to declare an overall champion (Keller, 1977). In many research studies, it is desirable to examine changes i n strength, or between-group differences i n strength, independent o f body mass changes on group differences. T h e ratio variable, strength/body mass, is used in such situations to "deflate" the effect o f body mass. T h e t e r m " s c a l i n g " , " e l i m i n a t i n g " , " r e m o v i n g " and " d e f l a t i n g " are e q u i v a l e n t terms used i n the literature, but the t e r m " d e f l a t i n g " is used through this dissertation for c o n s i s t e n c y and f a m i l i a r i t y i n the f i e l d . R a t i o variables d e r i v e d for deflation purposes (attempting to r e m o v e the effect o f the d e n o m i n a t o r variable from the v a r i a t i o n o f the numerator variable) serve as dependent variables i n m a n y studies i n different d i s c i p l i n e s . R a t i o variables are also used as measurement variables where the purpose is not to derive a denominator-free variable. In this case, a ratio variable has some s p e c i a l aspect not a c c o u n t e d for b y the numerator or d e n o m i n a t o r variables, either i n d i v i d u a l l y or a d d i t i v e l y . F o r e x a m p l e , the ratio o f w a i s t and hip girths is treated as a c o m b i n e d measurement variable, c a l l e d the w a i s t - t o - h i p ratio ( W H R ) . It is used to describe b o d y shape, and has been s h o w n to be v e r y useful i n e p i d e m i o l o g y and obesity studies to e x a m i n e m o r b i d i t y , stroke, heart disease, breast cancer, and other diseases ( D a r r o c h & M o s i m a n n , 1985; S o n n e n s c h e i n , K i m , Pasternack, & T o n i o l o , 1993).  3  Since the statistical consequences of using ratios were first mentioned by Pearson (1897), there has been a recognition of both the theoretical and practical limitations of using ratio variables (e.g., Anderson & Lydic, 1977a, 1977b; Albrecht, 1978; Albrecht, Gelvin, & Hartman, 1993; Atchley, Gaskins, & Anderson, 1976; Firebaugh & Gibbs, 1985; Kronmal, 1993; Long, 1980; MacMillan & Daft, 1980; Packard & Boardman, 1988; Schuessler, 1974). For deflation purposes, some researchers have indicated that the transformation of data into ratios may produce misleading interpretations due to the residual correlation between the ratio variable and its denominator (Atchley et al, 1976; Firebaugh & Gibbs, 1985; Kronmal, 1993; Long, 1980; Schuessler, 1974). Others have argued that a correctly used ratio variable does not create "spurious" correlations and misleading results (Corruccini, 1995; Kasarda & Nolan, 1979; Kritzer, 1990; MacMillan & Daft, 1980). The focus on ratio issues has mostly been on the correlation between a ratio variable and its component variables, and the choice of the most valid deflation models (Albrecht et al, 1993). If the purpose for using a ratio variable is to deflate the effects of the denominator's variation from the numerator, the main issue is whether a simple ratio variable can fully deflate the denominator effect, as evidenced by a zero correlation between a ratio variable and its denominator variable. Albrecht and his colleagues (1978, 1993), and others, examined the statistical characteristics of ratios and the methods to fully control the denominator effect under different regression models. The results indicated that correctly formulated adjusted ratios do control for the correlation between a ratio variable and its denominator, but could introduce denominator-related changes in variances of the ratio variable (i.e., the variance of a ratio variable decreases as the value  4  o f the d e n o m i n a t o r variable increases). If the hypothesis to be tested has i n i t i a l l y been f o r m u l a t e d i n terms o f a ratio v a r i a b l e , some researchers ( C o r r u c c i n i , 1995; K u h & M e y e r , 1955) have argued that the question o f spurious correlation between a ratio and its c o m p o n e n t variables does not arise. A c t u a l l y , for those ratio variables (e.g., w a i s t / h i p ratio), although a zero c o r r e l a t i o n is not an issue, the statistical consequences o f u s i n g such variables s h o u l d be e x a m i n e d because the r e l i a b i l i t y o f the ratios is affected b y v a r i a t i o n o f the t w o c o m p o n e n t variables. H o w e v e r , the r e l i a b i l i t y issue o f u s i n g ratio variables is u s u a l l y neglected b y researchers. A l t h o u g h i n practice there m a y be t w o different purposes for u s i n g ratio variables, the consequences o f u s i n g ratio variables i n statistical inferential analyses are the same. It can be s h o w n that the v a r i a t i o n and c o v a r i a t i o n o f the c o m p o n e n t variables strongly affect the statistical characteristics o f a d e r i v e d ratio variable i n d e s c r i p t i v e and inferential analyses. S o m e o f the issues r e g a r d i n g the v a l i d i t y o f u s i n g ratio variables have been noted i n the literature. F o r e x a m p l e , the issues o f u s i n g ratio variables i n c o r r e l a t i o n and linear regression analyses have been d i s c u s s e d for a l o n g time. F r i e d l a n d e r (1980), and m a n y others, p o i n t e d out that a linear regression m o d e l w i t h a ratio as the dependent variable or the independent variable w o u l d not appropriately c o n t r o l for the effect o f the denominator, and therefore c o u l d y i e l d m i s l e a d i n g results. It has been i n d i c a t e d that a r e c i p r o c a l c o m p o n e n t o f the d e n o m i n a t o r s h o u l d be i n c l u d e d i n the regression m o d e l . H o w e v e r , some other issues c o n c e r n i n g the use o f ratio variables have not been r e s o l v e d . In h u m a n k i n e t i c s research, ratio variables are c o m m o n l y used i n both descriptive and inferential statistics. W h e n a ratio variable is d e r i v e d for deflation purposes, a  5  residual correlation between the ratio and its d e n o m i n a t o r v a r i a b l e m a y seriously affect the v a l i d i t y o f the ratio variable. H o w e v e r , it appears that this is rarely e x a m i n e d , and the i m p a c t o f the d e n o m i n a t o r on related ratio variables s h o u l d be investigated e m p i r i c a l l y i n all studies. In a d d i t i o n to the p r o b l e m o f a non-zero c o r r e l a t i o n between a ratio and its denominator, the use o f a ratio variable c o u l d introduce denominator-related distortions o f variances. In some studies i n h u m a n k i n e t i c s research, the d e n o m i n a t o r related variance distortion i n a ratio variable m a y not be an issue because o f the s m a l l range o f the d e n o m i n a t o r v a r i a b l e i n the data, w h i l e i n others it m a y be a serious issue because o f the large range o f the d e n o m i n a t o r variable. F o r n o n - d e f l a t i o n type ratio variables, g i v e n that the ratio v a r i a b l e has content v a l i d i t y , i n f o r m a t i o n about the r e l i a b i l i t y o f the ratio variable and v a r i a t i o n effect o f the c o m p o n e n t variables i n the inferential analysis needs to be k n o w n . H o w e v e r , such i n f o r m a t i o n is u s u a l l y m i s s i n g i n h u m a n k i n e t i c s research publications. Researchers i n h u m a n k i n e t i c s often use ratio variables i n inferential analyses, e s p e c i a l l y i n the analysis o f variance w i t h repeated measures ( R M A N O V A ) .  It is  apparent that the characteristics o f a ratio variable i n the repeated measures w i l l be strongly i n f l u e n c e d b y the characteristics o f the numerator and d e n o m i n a t o r variables. H o w the characteristics o f the c o m p o n e n t variables affect the type I error rate i n R M A N O V A is still not k n o w n . A n i n v e s t i g a t i o n o f these issues has not yet been done for the c o m m o n l y used ratio variables i n h u m a n k i n e t i c s research.  6  Purpose of the Study A p r i m a r y purpose o f this study was to investigate whether ratio variables are appropriately used i n the h u m a n k i n e t i c s research d o m a i n . F o r ratio variables used for deflation purposes, the v a l i d i t y o f the ratio variable w a s a m a i n c o n c e r n . A s i m p l e ratio variable and adjusted ratio variables were i n v e s t i g a t e d a n d c o m p a r e d u s i n g variables c o m m o n l y used i n o u r f i e l d . T h e v a l i d i t y o f deflation m o d e l s for each ratio v a r i a b l e w a s evaluated. B e c a u s e the r e l i a b i l i t y o f a measurement v a r i a b l e is one o f the most i m p o r t a n t considerations i n any e m p i r i c a l research, this study e x a m i n e d the r e l i a b i l i t y properties o f ratio variables. T h e r e l i a b i l i t y o f a ratio v a r i a b l e is m o r e c o m p l e x than the r e l i a b i l i t y o f a r a w score variable because it is affected b y several factors, s p e c i f i c a l l y , the r e l i a b i l i t y o f the numerator and d e n o m i n a t o r variables, the c o r r e l a t i o n between the t w o c o m p o n e n t variables, and the relative v a r i a t i o n o f the t w o variables. T h e s e effects are r a r e l y addressed i n the literature. T h e r e l i a b i l i t y o f c o m m o n ratio measures i n h u m a n k i n e t i c s research needs to be e x a m i n e d for p o s s i b l e bias as a result o f these effects. T h e s e c o n d purpose was to investigate h o w the R M A N O V A test u s i n g a ratio variable violates the assumptions o f this statistical procedure. S i n c e a ratio v a r i a b l e Y* = X i i / X i is a c o m b i n e d v a r i a b l e w h i c h contains numerator a n d d e n o m i n a t o r c o m p o n e n t s , 2  the c o v a r i a n c e structure o f ratio variables for trial i and j , Y j = X n / X 2 i a n d Y j = X 1 / X 2 J say, is different f r o m that o f the raw scores. T h e c o v a r i a n c e structure o f ratio variables i n R M A N O V A is affected b y not o n l y the c o v a r i a n c e s o f the c o m p o n e n t variables across trials, but also the c o v a r i a n c e between the numerator and d e n o m i n a t o r variables w i t h i n each trial. T h e s e effects m a y cause serious v i o l a t i o n s o f some o f the assumptions (e.g.,  circularity), and the violation of the circularity assumption may affect type I error rates. Although the impact of the violation of the circularity assumption has been discussed extensively when a raw score is used, there is little discussion on the effects of the component variables in RM  A N O V A using ratio variables, and it remains an unresolved  issue. In summary, there is insufficient information in the related literature to provide appropriate guidance for human kinetics researchers to resolve the following issues: 1. The validity and reliability of a number of commonly used ratio variables based on different deflation models in human kinetics research. 2. The effect of using ratio variables on the structure of the covariance matrix and type I error rates in RM  A N O V A analysis.  Therefore, the purpose of this investigation was to seek answers to the above questions, and to provide a guide to the correct use of ratio variables. In doing so, the most popularly used ratio variables were selected from a survey of the field of human kinetics. Then, the issues addressed in the previous section were investigated. First, the deflation models were compared and the best deflation model was identified for the selected variables. Second, the reliability and validity of the ratio variables were evaluated. Third, the impact of using ratio variables in RM  A N O V A was investigated  under different conditions of the variation and correlation between the numerator and denominator variables. The results were intended to provide guidance for more appropriate use of ratio variables in human kinetics research.  8  CHAPTER II. LITERATURE REVIEW  Statistical Characteristics of Ratios The Distribution of Ratio Variables O n e o f the basic assumptions o f the statistical m o d e l s c o m m o n l y used i n h u m a n k i n e t i c s is that, i n the p o p u l a t i o n , the dependent variable is distributed n o r m a l l y . A question o f interest i n the literature i s , g i v e n the distribution characteristics o f the c o m p o n e n t variables that are used to f o r m a ratio dependent variable, what is the distribution o f the ratio variable? It is noted that the n o r m a l i t y assumption i n most statistical procedures is not critical unless sample sizes are s m a l l (e.g., n<15 i n R M A N O V A procedures).  The  general statistical properties o f ratios o f r a n d o m n o r m a l variables have been e x a m i n e d b y a n u m b e r o f statisticians (Fieller, 1932; K e n d a l l & Stuart, 1969; M a r s a g l i a , 1965; Schneeberger & F l e i s c h e r , 1993). F o r e x a m p l e , it c a n be s h o w n that the d i s t r i b u t i o n o f the ratio o f t w o independent standard n o r m a l r a n d o m variables is a C a u c h y d i s t r i b u t i o n ( K e n d a l l & Stuart, 1969). T h e density function is \  dy  df(y)=  ^  -oo<y<oo  2  71(1+  y  (1)  )  where Y=Z]/Z . In this function the ratio o f Z\ and Z is treated as a n e w independent 2  2  variable and the effect o f Z\ and Z o n the ratio cannot be seen. 2  T h e general f o r m o f the density function o f the ratio o f t w o r a n d o m variables can be d e r i v e d . If/(Xi, X ) is the j o i n t density o f (Xi, X ), then y/(y), 2  2  the density o f the ratio  Y=Xi/X is g i v e n by: 2  Y(y)  = {_/(V*2>* )K| 2  r f X  2  -oo<y<+oo  (2)  R a t i o distributions under bivariate n o r m a l c o n d i t i o n s have e x a m i n e d a n a l y t i c a l l y ( F i e l l e r , 1932; Schneeberger & F l e i s c h e r , 1993), and also through the use o f s i m u l a t i o n procedures (Friedlander, 1980). T h e density function o f the bivariate n o r m a l d i s t r i b u t i o n for X i and X is g i v e n b y Schneeberger and F l e i s c h e r (1993): 2  /(*! ,x ) = 2  1 :  ,  exp  1 2(1-P  -2p  -x  2  12  cr.  X (  2 1 2  -  2  )  !- + ( cr  2  >-)  2  (3)  a,  A s w e can see f r o m (2) and (3), the d i s t r i b u t i o n o f the ratio o f c o n t i n u o u s variables X ] and X is affected b y the f o l l o w i n g factors: 2  M x i , M-x  a  : t 2  n  e  p o p u l a t i o n means,  x p ° x : the p o p u l a t i o n standard deviations, 2  P i : the p o p u l a t i o n correlation o f X i and X . 2  2  T h e d i s t r i b u t i o n o f the ratio variable d e r i v e d f r o m (2) and (3) has infinite m o m e n t s w h e n the c o m p o n e n t variables can take negative values, but this is u s u a l l y not the case i n h u m a n k i n e t i c s because the c o m p o n e n t variables used are u s u a l l y p o s i t i v e and fall w i t h i n a certain range. F i e l l e r (1932) p o i n t e d out that the l i m i t a t i o n o f the c o m p o n e n t variables to the p o s i t i v e quadrant m a y change the m o m e n t s o f the ratio d i s t r i b u t i o n to finite values w i t h o u t h a v i n g a v i s i b l e effect on the appearance o f the d i s t r i b u t i o n . U s i n g the aforementioned functions and a c o m p u t e r s i m u l a t i o n , F r i e d l a n d e r (1980) demonstrated that w h e n the c o r r e l a t i o n o f the c o m p o n e n t variables is h i g h and the coefficients o f variation are large, there c o u l d be a considerable d e v i a t i o n o f the ratio d i s t r i b u t i o n f r o m n o r m a l i t y . T h e s i m u l a t i o n also s h o w e d that w h e n the c o r r e l a t i o n  10  between the numerator and d e n o m i n a t o r increases, the mean and standard d e v i a t i o n o f the ratio variable tend to decrease. B e c a u s e o f the c o m p l e x relationship between a ratio and its c o m p o n e n t variables, the effect o f the characteristics o f the c o m p o n e n t variables (e.g., ( i ^ ( i , a i, o , x 2  x  x2  and  Pxix2) on the d i s t r i b u t i o n characteristics o f a ratio variable has attracted the attention o f m a n y researchers i n different d i s c i p l i n e s . F o r e x a m p l e , Schneeberger and F l e i s c h e r (1993) s h o w e d different ratio d i s t r i b u t i o n shapes under v a r i o u s c o n d i t i o n s on X i and X . 2  H i n k l e y (1969) discussed the d i s t r i b u t i o n o f the ratio o f t w o correlated n o r m a l r a n d o m variables, and K o r h o n e n and N a r u l a (1989) p r o p o s e d a m e t h o d to c o m p u t e the d i s t r i b u t i o n o f the ratio o f the absolute values o f t w o correlated n o r m a l r a n d o m variables. Issues relating to the n o r m a l i t y o f ratio variables were also d i s c u s s e d b y S a m u e l and N o r m a n (1985, 1986), and A r n o l d and B r o c k e t t (1992). S u m m a r i z i n g these studies, the distribution o f a ratio v a r i a b l e is affected b y the c o v a r i a t i o n and the d i s t r i b u t i o n characteristics o f the numerator and d e n o m i n a t o r variables. W h e n the numerator variable is a p p r o x i m a t e l y n o r m a l and the d e n o m i n a t o r is less variable or non-stochastic, the ratio variable is still a p p r o x i m a t e l y n o r m a l . W h e n the variation o f the c o m p o n e n t variables is large, and the c o r r e l a t i o n between the c o m p o n e n t variables is h i g h , considerable d e v i a t i o n o f the ratio d i s t r i b u t i o n f r o m n o r m a l i t y m a y occur. F o r some ratio variables used i n h u m a n k i n e t i c s , the d e n o m i n a t o r m a y have large v a r i a t i o n and the c o m p o n e n t variables m a y be strongly s k e w e d . W h e n a ratio variable is d e r i v e d f r o m the c o m p o n e n t variables w h i c h have large v a r i a t i o n (e.g., V = r j / u > 0 . 2 , V refers to the coefficient o f variation defined as the standard d e v i a t i o n d i v i d e d by the  11  mean) or are s t r o n g l y s k e w e d , the ratio variable m a y be less s k e w e d but the d i s t r i b u t i o n remains n o n - n o r m a l (Friedlander, 1980). A l o g transformation is a m e t h o d to m a k e the ratio data m o r e n o r m a l l y distributed, h o w e v e r , this o p t i o n m a y lack appeal because o f the d i f f i c u l t y o f interpretation i n some situations. U s i n g a c o m p u t e r s i m u l a t i o n , A t c h l e y et a l . (1976) demonstrated h o w changes i n the coefficients o f v a r i a t i o n o f t w o bivariate n o r m a l l y distributed variables ( V i = a i / u , i , x  x  V 2=C7 2/|J.2) affect the skewness and the kurtosis o f a ratio variable. T h e results s h o w e d X  X  that the d i s t r i b u t i o n o f the ratio variable w a s strongly s k e w e d to the right w h e n the variation o f the d e n o m i n a t o r was r e l a t i v e l y greater than that o f the numerator.  For  e x a m p l e , the coefficient o f skewness reached 2.06 w h e n V i / V 2 = 0 . 1 , a n d as V i / V x  X  x  x 2  approached 1.0 the skewness o f the d i s t r i b u t i o n o f the ratio v a r i a b l e decreased to 0 . 1 . A d d i t i o n a l l y , w h e n the v a r i a t i o n o f the d e n o m i n a t o r was r e l a t i v e l y greater than that o f the numerator, the kurtosis o f the ratio variable was v e r y h i g h (e.g., the coefficient o f the kurtosis>10.0 w h e n V i / V < 0 . 2 ) . W h e n V i / V x  x 2  x  x 2  a p p r o a c h e d 1.0 the kurtosis o f the ratio  variable decreased m o r e q u i c k l y than skewness. T h e kurtosis also became slight w h e n V i / V 2 > 0 . 3 . B o t h skewness and kurtosis values w e r e unaffected (i.e., r e m a i n e d near x  X  zero) for c o n d i t i o n s i n w h i c h V i / V > 1 . 0 . x  x 2  In s u m m a r y , the d i s t r i b u t i o n o f a ratio v a r i a b l e is strongly affected b y the relative variation o f its c o m p o n e n t variables. A l t h o u g h the c o m p o n e n t variables m a y be n o r m a l l y distributed, as the coefficient o f v a r i a t i o n o f the d e n o m i n a t o r v a r i a b l e ( V 2 ) increases X  relative to that o f the numerator, the d i s t r i b u t i o n o f the ratio variable becomes s k e w e d a n d l e p t o k u r t i c . O b v i o u s l y , it is not appropriate to assume that ratio variables meet the n o r m a l i t y assumption based on k n o w l e d g e o f the n o r m a l i t y o f the c o m p o n e n t variables.  12  Variance, Covariance, and Correlation of Ratios It is more complex to evaluate the variances, covariances, and correlations of ratio variables than that of raw score variables because a ratio variable is a combination of two component variables. The variances and covariances of the component variables strongly affect the statistical properties of ratio variables. Pearson (1897) first addressed this issue, and developed a formula to approximate the correlation between ratios using classic measurement theory and Taylor's approximation method (Formula (7)). The variance and covariance of a ratio variable can also be approximated based on Pearson's procedure (Friedlander, 1980). The following are the formulas for the approximations to the mean, variance, covariance, and correlation:  =E&--\a-P ,v y ^ x  x  +V<)  x  (4)  ,2  < ,  = V a i A « ^ ( V : + Vl - 2p, V V ) 2 f*x,  2  x  x  (5)  X2  A  a  x  = Cov(^,^-)  x  c'l,, !, , . 1  v A  r  V 2  VV  « ^L^(p  X  x  A  V 4  II II i"2 ^ 4  2  X  +p V  X  4  2  xx  4  V  x  1 3  x  1  -p V V xx  3  2 3  x  V V )  -p  x  2  x  xx  3  1 4  1  x  4  (6) _  „  1 __3 \  I ^  -"i*  Here V i , V , V , and V x  x2  x3  mean) for variables X  u  3  *l3 X  ^  2  x  x 4  "^l-^  **23 X  x  l  x  l  x  -2p 7 y Jv ] +vl XA  x  xi  x  ^ 24 X  X  2  x  4  (n\  x  -2p y y v  x  Xt  are coefficients of variation (standard deviation divided by  X , X , and X , and p 2  •*!4  3  4  x ] x 2  ,pi ,p, , x  x3  x  x4  p  x2x  3 , Px2x4, and  p  x 3 x 4  are  correlation coefficients between the component variables. By replacing a component with unity in the formula, various ratio correlations can be derived. For example, the  13  c o r r e l a t i o n between X 1 / X 2 and X can be estimated b y t a k i n g X 3 = X 2 a n d X 4 = l . It s h o u l d 2  be noted that the f o r m u l a s are a p p r o x i m a t i o n s , a n d they are accurate o n l y w h e n the coefficients o f v a r i a t i o n for a l l c o m p o n e n t variables are s m a l l e n o u g h to render c u b i c and higher order v a r i a t i o n terms o f the T a y l o r series n e g l i g i b l e . C h a y e s (1971) s h o w e d that P e a r s o n ' s a p p r o x i m a t e f o r m u l a became i n c r e a s i n g l y inaccurate w h e n the c o e f f i c i e n t o f v a r i a t i o n o f the c o m p o n e n t variables i n the f o r m u l a was greater than 0.15. T h e r e f o r e , the a c c u r a c y o f u s i n g the formulas for h u m a n k i n e t i c s measures w i t h greater v a r i a t i o n becomes an issue. F o r m u l a s (4) to (7) s h o w that the statistical characteristics o f a ratio v a r i a b l e c a n be v e r y different f r o m that o f the c o m p o n e n t variables. F i r s t , F o r m u l a s (4) to (7) s h o w that the statistical characteristics o f a d e r i v e d ratio v a r i a b l e are c o m p l e x functions o f the characteristics o f the c o m p o n e n t variables. S e c o n d , b y k n o w i n g the characteristics o f the c o m p o n e n t variables, researchers can a p p r o x i m a t e the characteristics o f a ratio v a r i a b l e u s i n g these f o r m u l a s . T h i r d , F o r m u l a s (4) to (7) are a p p r o x i m a t i o n s and the a c c u r a c y o f these f o r m u l a s depends upon the properties o f the c o m p o n e n t variables. In some situations these a p p r o x i m a t i o n s m a y be acceptable, w h i l e i n other situations they m a y not be acceptable due to large variation i n the c o m p o n e n t variables.  Reliability of Ratio Variables W h e n e v e r a measurement is a d m i n i s t e r e d to subjects i n a study, researchers w o u l d l i k e to have some assurance that the measurement scores c o u l d be r e p l i c a t e d i f the same i n d i v i d u a l s were measured again under s i m i l a r c i r c u m s t a n c e s . O n e aspect o f this r e p l i c a t i o n is the degree o f c o n s i s t e n c y i n the i n t e r - i n d i v i d u a l v a r i a b i l i t y , often referred to the r e l i a b i l i t y o f a measure. B a s e d o n c l a s s i c a l measurement theory (observed score =  14  true score + error), the degree of reliability in a two trial situation is often expressed by a Pearson correlation coefficient, ranging from 0.00 to 1.00. For the situations with three or more trials, the intraclass correlation provides a more valid approximation of the reliability. The intraclass correlation for the within-subject factor in the R M A N O V A model, assuming compound symmetry of the covariance matrix, is defined as rint=(c 7i )/(cfn + Cf ). The estimate of the intraclass correlation from data can be obtained r  2  2  2  e  as rint=(MS -MSres)/MS , where MS is the mean square for subjects and MS A  s  s  S  res  is the  mean square for error (details of the intraclass correlation are shown in the chapter on methods and procedures). Because the correlation and variation of the numerator and denominator variables affect the reliability of a ratio variable, the evaluation of reliability for a ratio variable in terms of the properties of the components is more complex than for a raw score variable. Holzinger claimed that the reliability of a ratio variable could be approximated from Pearson's correlation estimation Formula (7) (Cronbach, 1941):  (8)  v -2 y y v ] 2  x l  where p i i and p x  x  X  22 X  x  Xi+  x  are the reliability coefficients of the numerator and denominator  variables, and V i and V x  PxiX  x 2  are the coefficients of variation of the component variables.  This formula is derived by substituting X i for X 3 and X for X in Formula (7). 2  4  However, Formula (8) has never been used in the literature. The validity of Formula (8) for the approximation of reliability is suspect, because substituting X i for X and X for 3  2  X4 in Formula (7) and assuming that the variable Y=Xi/X correlates with itself would 2  15  result in p =1.0. Thus, this formula may not be appropriate to estimate the reliability of yy  a ratio variable. Evaluating the reliability of ratio variables is important in any study, and it is affected by both the numerator and denominator variables. However, the reliability of ratio variables in the literature is rarely examined.  Ratio Variable Issues In the Literature The controversy over the validity of using ratio variables has a long history. The issue has been commented on by statisticians, sociologists, economists, political scientists, geologists, zoologists, and others (e.g., Albrecht, 1978; Chayes, 1949; Chayes, 1971; Firebaugh & Gibbs, 1985; Kronmal, 1993; MacMillan & Daft, 1980; Pearson, 1897, 1910; Schuessler, 1974; Yule, 1910). The discussion in this dissertation focuses on the validity of using ratio variables for deflation purposes, and the validity of using ratio variables in R M ANOVA.  Using Ratio Variables for the Purpose of Deflation The literature indicates that a simple ratio in which a measurement variable is divided by another measurement variable may not always adequately deflate the denominator effect, as evidenced by the residual correlation between the ratio and its denominator variable. For example, Pearson (1897), Chayes (1949), Atchley (1976), and Albrecht (1978) warned that a simple ratio may be inadequate for the purpose of deflation. Pearson showed that the correlation between a ratio variable and its denominator is usually not zero, even when there is no correlation between the component variables. Based on Pearson's formula (Formula (7)), Albrecht (1978)  16  d e r i v e d the pattern o f correlation between the ratio ( Y = X i / X 2 ) and its denominator  (X2)  under different c o m b i n a t i o n s o f the coefficient o f variation (VX1/VX2). In general, the v a l u e o f p 2 is a l w a y s less than p i, 2 and o n l y approaches p i , y>X  x  X  x  V x i / V x . W h e n V x i / V x < 1.0, p , 2 is negative, and w h e n p ^ 2  2  y  x  X  x 2  x 2  for high values o f approaches zero, p , 2 y  X  r a p i d l y approaches -1.0. Therefore, one s h o u l d not assume that a s i m p l e ratio w o u l d have zero correlation w i t h its denominator. T h e m o r e interesting question is h o w to interpret the correlation p 2 g i v e n the yx  difference between p  and p i  y j X 2  x  j X 2  . M a c M i l l a n and D a f t (1979, 1980) gave a reasonable  explanation for the case p , 2=0. T h e y s h o w e d that p y  X  2  X  y j X 2  s h o u l d be zero, not w h e n X  f  and  are unrelated, but w h e n X) and X 2 are related i n a special w a y (a linear relationship  between X i and X  2  w i t h a zero intercept). p , 2 is sometimes r e c o m m e n d e d as a measure y  X  of "nonproportionality" ( F i r e b a u g h & G i b b s , 1985; K a s a r d s & N o l a n , 1979). I f p , y  x 2  equals zero, it suggests that there is a linear relationship between X i and X 2 w i t h a zero intercept, h o w e v e r , i f p , 2 is not zero, one cannot evaluate the degree o f y  X  n o n p r o p o r t i o n a l i t y b y p , 2 alone. Sources o f n o n p r o p o r t i o n a l i t y i n c l u d e a non-zero y  X  y-intercept, non-linearity, or both. T o o v e r c o m e the inadequacy o f u s i n g a s i m p l e ratio as a deflation m e t h o d , various adjusted deflation m o d e l s have been r e c o m m e n d e d ( A l b r e c h t et a l , 1993; A t c h l e y , 1978; G e l v i n & A l b r e c h t , 1985). In order to get full deflation, it is necessary to derive a n e w scaled variable i n w h i c h the variation is not affected b y the variation o f the denominator variable. T h e m a i n statistical criterion for full deflation is that the adjusted ratio Yadj s h o u l d have a zero correlation w i t h its c o n t r o l variable (the denominator variable i n a s i m p l e ratio). A l b r e c h t et al. pointed out that deficiencies i n s i m p l e ratios  can be alleviated by incorporating regression coefficients describing the bivariate relationship between the numerator and denominator variables (they called these two variables measurement and control variables). Four deflation models were suggested by Albrecht to determine the best ratio variable that can fully remove the effect of the denominator: (a) a simple ratio model: Y = X i / X 2 , (b) an intercept adjusted ratio model, Y = ( X i - a ) / X , where a is a constant, (c) an allometrically adjusted ratio model, 2  Y = X i / X \ where k is a constant, and (d) a fully adjusted ratio model, Y = ( X i - a ) / X , k  2  2  which includes both an intercept and exponent parameter. A full description of these models is provided in Chapter 3. Albrecht et al. (1993) indicated that correctly using the adjusted models does control for the correlation between a ratio variable and its denominator variable. However, the major drawback of using these adjusted ratio models is that it could introduce denominator-related distortions of variances (i.e., the variance of a ratio variable Y or Y dj decreases as the value of the denominator X 2 increases). The a  reduction of variance in Y j as the denominator X increases is most marked for aa  2  allometrically adjusted and fully adjusted ratios. The degree of the distortion of variance also depends on the range of the denominator variable in the empirical data, and the distortion may be serious for some data sets and trivial for others. Therefore, the validation of a ratio variable in a study should include not only a ratio model selection to achieve p , 2=0, but also a ratio model evaluation to examine the degree of distortion of y X  variance.  Ratio Variables in Regression Analysis The validity of using ratio variables in linear regression and correlation analysis has been extensively discussed in the literature (Albrecht et al, 1993; Belsley, 1972;  18 Bollen, 1979; Chilton, 1982; Corruccini, 1995; Firebaugh et al, 1985; Friedlander, 1980; Kronmal, 1993; MacMillan et al., 1984, 1980; Pendleton, 1984, Schuessler, 1974). The validity issue has been summarized and clarified by Firebaugh and Gibbs' research (1985). Firebaugh and Gibbs discussed ratio issues for the cases in which all variables appearing in the regression equation are divided by a common variable, or one of the independent variables also appears as the denominator of the dependent variable. A summary of their results follows. Given that the raw score regression model is the true model, a ratio score regression model in which both the dependent variable and one of the independent variables are divided by another variable would generally be appropriate only if the reciprocal of the denominator variable is included as an independent variable. The only exception is the case where the numerator is directly proportional to the denominator (Friedlander, 1980). It is also indicated that even if a regression equation does include the reciprocal term, it may produce inefficient estimates if the deflation causes a heteroscedastic error term. Friedlander indicated that the partial correlation of two ratio variables with a common denominator variable, adjusting for the reciprocal of the denominator, is the same as the partial correlation of the numerator variables, adjusting for the denominator. Based on a linear relationship between the original dependent variable, the common deflator variable and the independent variables in the regression equation, three regression models involving ratio variables were considered in Kronmal's research (1993): (a) both independent and dependent variables are divided by a common variable, (b) only the dependent variable is a ratio, and (c) only one of the independent variables is  19  a ratio. It was confirmed that including the reciprocal of the denominator variable in the case (a) could solve the problem of possible biases, given the error term in the equation is not heteroscedastic. Using Neyman's example (1952), Kronmal (1993) illustrated the serious spurious correlation between two ratios when both the independent and dependent variables in the regression equation are divided by a common variable, but the original independent variable and dependent variable do not have a linear relation. For the case (b), analytic results and an empirical example suggested that a problem would occur when only the dependent variable is a ratio. Even though the numerator of the ratio is uncorrected with the original independent variable, the ratio dependent variable could be significantly correlated with the independent variable through its relationship to the denominator of the ratio. For the case (c), using a ratio as an independent variable in a regression equation often results in a better predictor than using only the numerator or the denominator variable in the equation, but is usually not a better model than one which includes both components. Firebaugh and Gibbs (1985) also concluded that ratio variables can be used in regression analyses without apology in sociology if the following rules governing the use of ratio variables are followed: (a) use regression analysis rather than correlation analysis when analyzing ratio variables, except when assessing whether X i and X are 2  proportional, (b) avoid mixed methods (i.e., part ratio, part component), (c) strive for a reliable measure of the denominator variable, because measurement error in the denominator is likely to cause considerable bias, (d) include 1/X as a regressor (X is the 2  2  denominator in the ratio variable), unless it can be safely assumed that the Y-intercept is zero.  20  Ratio Variables in Analysis of Variance As mentioned in previous sections, a ratio variable is often used in inferential analyses (often R M ANOVA) in human kinetics research. Therefore, we are especially interested in the statistical consequences of using ratio variables in such an analysis. Unfortunately, there appears to be no publications discussing the issue of using ratio variables in R M ANOVA, although, Anderson and Lydic did look at ratio variables in the randomized groups A N O V A model (Anderson et al. 1977a, 1977b). Bivariate normal observations X and Y were generated by computer, and a Monte Carlo simulation procedure was designed to investigate whether the statistical properties induced by ratio transformations can produce a loss of sensitivity (power) in a two-group random-subject and fixed-treatment A N O V A (although the comparing power for two different model may not be valid, see discussion in the following section). They also tried to characterize the circumstances under which the statistical analysis of ratio data is appropriate. Three different models were used in their simulations. First, an A N O V A model ignoring the covariate: (9) Second, an A N C O V A model: (10) Third, for the ratios, an A N O V A model using the measured dependent variable divided by the covariate as the new dependent variable: x lx =n Xl]  2l]  xnj: numerator variable; X2y: denominator variable;  I- r, t (>  :j  (11)  21  \x: the grand m e a n ; TJ: a constant associated w i t h they'th treatment; eij: the e x p e r i m e n t a l error associated w i t h the ith subject under the jth treatment. T h e data were generated based o n f o l l o w i n g c o n d i t i o n s : 1. T h e correlations between the numerator and d e n o m i n a t o r variables (p=0.0, 0.25, 0.50, 0.75, 0.90, 0.99). 2. Treatment effects (x=0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1.0). 3. M e a n c o n d i t i o n s (u, =10 and n = 5 , p. i=10 a n d n 2=10, p. i=5 a n d u. =10). xl  x 2  x  X  x  x2  4. C o m b i n a t i o n s o f the coefficient o f v a r i a t i o n between the numerator a n d d e n o m i n a t o r variables ( V i / V 2 = 0 . 5 , 1.0, 2.0). T h e variances o f the c o m p o n e n t variables x  X  2  2  were f i x e d for a l l the c o n d i t i o n s ( o i =a 2 =1). T w o groups were used a n d the sample x  X  size was 10 observations per group. T h e useful f i n d i n g o f this study was that the p o w e r o f the ratio A N O V A m o d e l was h i g h l y dependent u p o n the ratio o f the coefficients o f v a r i a t i o n o f the c o m p o n e n t variables. H o w e v e r , c o m p a r i s o n s a m o n g the three p r o p o s e d m o d e l s are not m e a n i n g f u l , because separate A N O V A m o d e l s w i t h different hypotheses were c o m p a r e d i n A n d e r s o n a n d L y d i c ' s study. T h e r a w score m o d e l , the A N C O V A m o d e l , a n d the ratio m o d e l actually tested different hypotheses.  In their t w o - g r o u p A N O V A d e s i g n , the n u l l  statistical hypothesis o f the r a w score m o d e l w o u l d be that n o difference exists between the raw score p o p u l a t i o n means i n the t w o groups. T h e n u l l hypothesis for the A N C O V A m o d e l is that no difference exists between the t w o p o p u l a t i o n means after these group means have been adjusted for differences between the groups o n the relevant covariate variable. In general, A N C O V A is m o r e p o w e r f u l than A N O V A because o f the s m a l l e r  22  error variance i n A N C O V A (this statement is true o n l y w h e n a g o o d covariate v a r i a b l e is used and the assumptions are met). Therefore, it is o b v i o u s that A N C O V A has greater p o w e r than A N O V A i n A n d e r s o n and L y d i c ' s study. T h e p o w e r o f the F-tests i n these three A N O V A m o d e l s i n this study are not comparable, but this study d i d show that the p o w e r w i t h i n the ratio A N O V A m o d e l is affected b y the coefficients o f variation o f the c o m p o n e n t variables. T h e study i n d i c a t e d that the ratio m o d e l p e r f o r m e d p o o r l y w h e n the coefficient o f variation o f the denominator variable was greater than the coefficient o f v a r i a t i o n o f the numerator variable. A l t h o u g h there are intensive discussions on the issue o f v i o l a t i n g the assumptions o f R M A N O V A for r a w score variables, there appears to be no p u b l i c a t i o n i n v e s t i g a t i n g ratio issues i n R M A N O V A . W h e n a ratio variable is used i n R M A N O V A , the c i r c u l a r i t y a s s u m p t i o n o f the variance and c o v a r i a n c e m a t r i x for the within-subject factor is affected not o n l y b y the v a r i a t i o n and c o v a r i a t i o n o f the numerator variable over trials but also b y that o f the d e n o m i n a t o r variable. W h e n a ratio variable is used i n R M A N O V A , the variance and c o v a r i a n c e o f the d e n o m i n a t o r ( i n c l u d i n g the c o m m o n denominator c o n d i t i o n ) o v e r trials m a y seriously affect the p o p u l a t i o n characteristics o f the c o v a r i a n c e m a t r i x (i.e., affect the c i r c u l a r i t y o f the p o p u l a t i o n c o v a r i a n c e m a t r i x i n the within-subject factor).  Ratio Variables and Unsolved Issues in Human Kinetics Research R a t i o variables are c o m m o n l y used i n h u m a n kinetics research, and are used i n almost all o f the areas i n the h u m a n kinetics d o m a i n (e.g., f r o m p h y s i c a l education to  23  p h y s i o l o g y ) . T a b l e 2-1 s h o w s h o w ratio variables are used i n a n u m b e r o f different d i s c i p l i n e s . In these studies, some ratio variables are used for deflation purposes a n d some are not, a n d u s u a l l y serve as measurement variables i n inferential analyses, especially in A N O V A .  T h e analysis o f ratio variables w i t h A N O V A procedures is  c o m m o n i n h u m a n k i n e t i c s research, for e x a m p l e , between 1992 and 1995, 6 2 % o f the articles p u b l i s h e d i n the R e s e a r c h Q u a r t e r l y for E x e r c i s e and Sport used A N O V A or M A N O V A , a n d 3 9 % o f these articles used ratio variables. A l t h o u g h ratio variables are w i d e l y used, the v a l i d i t y a n d r e l i a b i l i t y issues o f u s i n g these ratio variables are r a r e l y a c k n o w l e d g e d a n d i n v e s t i g a t e d i n h u m a n k i n e t i c s research. T h e f o l l o w i n g are the m o s t important issues r e g a r d i n g the use o f ratio variables i n our f i e l d .  Deflation Issues In h u m a n k i n e t i c s research, as s h o w n above, m a n y ratio variables are used for deflation purposes. T h a t i s , u s i n g ratio variables is an attempt to express the numerator on a per-unit o f the d e n o m i n a t o r basis i n order to r e m o v e the effects o f the d e n o m i n a t o r ' s variation f r o m the numerator.  W h e n the i n i t i a l purpose o f u s i n g a ratio v a r i a b l e is  deflation, it has been s h o w n that t w o assumptions (homogeneous variances o f a ratio variable o v e r the range o f the d e n o m i n a t o r v a r i a b l e , and z e r o c o r r e l a t i o n between the ratio v a r i a b l e and its denominator) s h o u l d be e x a m i n e d . I f the assumptions are v i o l a t e d by the s i m p l e ratio v a r i a b l e , other alternative deflation m o d e l s s h o u l d be c o n s i d e r e d . R e c e n t l y , the v a l i d i t y o f the s i m p l e ratio V o 2 m a x score ( V o 2 m a x / b o d y mass) has been e x a m i n e d and c o m p a r e d w i t h alternative deflation ratios ( H e i l , 1997; N e v i l l , 1994; V a n d e r b u r g h & K a t c h , 1996; W e l s m a n , A r m s t r o n g , N e v i l l , W i n t e r , and K i r b y , 1996). T h e m o d e l Y = X i / X 2 was preferred for V o 2 m a x i n some o f these studies a n d the k  magnitudes o f k reported v a r i e d f r o m 0.41-0.80. H o w e v e r , a systematic i n v e s t i g a t i o n o f  24  Table 2-1 Ratio Variables in Human Kinetics Research  Variable Definition  Discipline  Examples  Pedagogy  Total energy expenditure of lesson is defined by the energy expenditure in a class divided by the body weight (kcal/kg).  McKenzie, Y . L . et al. (1995). A randomized group A N O V A was used for two groups.  Exercise Epidemiology  Weight (kg) divided by height squared (m).  Hofstetter, C. R. et al. (1991).  Total cholesterol divided by high-density lipoprotein cholesterol (a stronger indicator of potential coronary risk).  Rimmer, J, H . et al. (1994). t-test  Psychology  The Weber Ratio - just noticeable difference (JND) divided by the point of subjective equality (PSE: the location or distance perceived as equal to the standard on 50% of the trials).  Meeuwsen, H.J. et al. (1992). M A N O V A was used.  Physiology  Ventilatory Threshold (ml/kg*min). Relative Fat  Hughes, R. A . et al. (1991).  (%)• Mean power output/body weight. Work in 30s/body weight.  Hill, D. W . etal. (1992).  Education  Defining intelligence quotient as IQ = 100 * mental age (MA) / chronological age (CA) and educational quotient as E Q = 100 * educational age (EA) / C A .  Friedlander, L J . (1980).  Biomechanics  The vertical, anteroposterior, and mediolateral ground reaction force variables divided by body mass.  Schot, P.K. et al. (1992). RM MANOVA  Antero-postero ground reaction force and vertical ground reaction force.  Sanderson, D. J. et al. (1993). R M A N O V A  Growth & Development  Waist/Hip ratio.  Sonnenschein, E . G. et al. (1993).  Management  The administration size divided by the total organization size.  Friedlander, L J . (1980).  Sociology  The urban population divided by the total population. The number of deaths divided by the total population. The births divided by the population.  Bollen, K . A . et al. (1979). Schuessler, K . (1974).  25  the deflation issue has not been done, and the h o m o g e n e o u s variance a s s u m p t i o n was not e x a m i n e d i n these studies. A s s h o w n i n T a b l e 2 - 1 , s i m p l e ratio data are still c o m m o n l y used i n analyses w i t h o u t a n y v a l i d a t i o n .  Reliability Issues B e c a u s e data are v i r t u a l l y a l w a y s m e a s u r e d w i t h error, the c o n s i d e r a t i o n o f the r e l i a b i l i t y o f a measurement variable is c r u c i a l for a n y study. W h e n a ratio v a r i a b l e is u s e d i n an analysis, the r e l i a b i l i t y issue b e c o m e s more important due to the c o m p l e x i t y o f c o m b i n i n g t w o variables i n one. S u r p r i s i n g l y , the r e l i a b i l i t y issue i s rarely addressed i n o u r f i e l d and needs to be investigated.  Issues in R M ANOVA with Ratio Data A s m e n t i o n e d i n p r e v i o u s sections, R M A N O V A w i t h ratio data is a c o m m o n l y used technique i n h u m a n kinetics research. Issues o f u s i n g r a w scores as a measurement variable i n R M A N O V A have been discussed e x t e n s i v e l y i n the literature. S c h u t z and G e s s a r o l i (1987) have g i v e n a c o m p r e h e n s i v e r e v i e w o n this topic w i t h e m p i r i c a l e x a m p l e s i n o u r f i e l d . F o r the o n e - w a y R M A N O V A , the assumptions o f c i r c u l a r i t y , a n o r m a l d i s t r i b u t i o n o f the measurement variable, a n d h o m o s c e d a s t i c i t y o f the error variances are r e q u i r e d f o r the exact F statistics o f the treatment effect to be v a l i d . In practice, A N O V A procedures are very robust w i t h respect to v i o l a t i o n s o f the n o r m a l i t y a s s u m p t i o n , but very sensitive to v i o l a t i o n o f the assumptions o f c i r c u l a r i t y a n d h o m o s c e d a s t i c i t y o f the error variances. E o m a n d S c h u t z (1993) s h o w e d that the inflation i n T y p e I error rates i n the n a i v e F tests w h e n e<1.0 is not, as c o m m o n l y assumed, m e r e l y a function o f the p o p u l a t i o n e p s i l o n value. Rather, it is a f u n c t i o n o f the nature o f the relative magnitudes o f the p o p u l a t i o n values and the s a m p l e estimate o f  26  e p s i l o n . T h a t is, a large T y p e I error rate is associated w i t h an overestimated e p s i l o n . O n the other hand, w h e n the p o p u l a t i o n e p s i l o n is underestimated, the T y p e I error rate is c o n s i d e r a b l y s m a l l e r than the c o r r e s p o n d i n g n o m i n a l l e v e l . H o w e v e r , the effects o f u s i n g ratio score variables on the assumptions i n R M A N O V A are h a r d l y n o t i c e d and d i s c u s s e d i n our f i e l d . A s s h o w n i n F o r m u l a (4), (5), (6) and (7), the variances and covariances o f ratio variables are affected b y characteristics o f the numerator and d e n o m i n a t o r variables. E v e n though there are the f o r m u l a s a v a i l a b l e to a p p r o x i m a t e the variance and c o v a r i a n c e o f ratio variables, there are few p u b l i c a t i o n s that investigate the i m p a c t o f u s i n g ratio variables i n R M A N O V A . T h e f o l l o w i n g are s o m e o f the issues that need to be investigated. F i r s t o f a l l , w e do not k n o w to what extent the ratio transformation affects the c i r c u l a r i t y assumption o f the ratio populations i n R M A N O V A .  F o r e x a m p l e , i f the raw  score p o p u l a t i o n s meet the c i r c u l a r i t y assumption, w e do not k n o w h o w changes i n the variance and c o v a r i a n c e o f the c o m m o n denominator affect the c i r c u l a r i t y property i n R M A N O V A u s i n g ratio variables. S e c o n d , w h e n an inference test is c o n d u c t e d f r o m a s a m p l e o f ratio data based on the R M A N O V A , it is not clear h o w changes i n the variances and covariances o f the c o m p o n e n t variables affects the s a m p l i n g characteristics o f the variance and c o v a r i a n c e m a t r i x o f the ratio v a r i a b l e . In s u m m a r y , w h e n ratio variables are used i n h u m a n k i n e t i c s research, the f o l l o w i n g questions s h o u l d be addressed: 1. W h a t is the purpose o f u s i n g ratio variables i n the study? I f it is for deflation purposes, has an appropriate deflation m o d e l been c o n s i d e r e d ? 2. D o e s the ratio v a r i a b l e meet the r e l i a b i l i t y c r i t e r i o n i n the study?  27  3. H o w does the ratio variable affect the assumptions i f a R M A N O V A procedure is to be used?  28  CHAPTER III. METHODS AND PROCEDURES  Variable Selection A s s h o w n i n p r e v i o u s chapters, ratio variables are c o m m o n l y used i n almost a l l areas o f h u m a n k i n e t i c s research. F o r the purposes o f this study, four o f the most frequently used ratio variables f r o m several d i s c i p l i n e s w i t h i n the f i e l d were selected and studied, but the p r i n c i p l e s d e v e l o p e d based on these four variables are a p p l i c a b l e to a l l ratio variables. P u b l i c a t i o n and research databases c o m m o n to the f i e l d were searched for relevant studies that used ratio variables i n repeated measurement situations, and letters were sent to researchers requesting their data. T h e r e were three c r i t e r i a used to choose appropriate studies and data sets for this research. F i r s t , the study h a d to be i n the h u m a n kinetics research d o m a i n , and the ratio variable used i n the study had to be a c o m m o n l y used v a r i a b l e . S e c o n d , the data sets i n c l u d e d the ratio variable and one o f its c o m p o n e n t variables, or both the numerator v a r i a b l e and the d e n o m i n a t o r v a r i a b l e , so that the ratio v a r i a b l e c o u l d be d e r i v e d . T h i r d , the ratio v a r i a b l e data (both the numerator and d e n o m i n a t o r variables) were repeatedly measured. D a t a sets u t i l i z i n g four c o m m o n l y used ratio variables m e e t i n g the above c r i t e r i a were obtained and a n a l y z e d . T h e r e were t w o data sets i n w h i c h the ratio variable was used for deflation purposes. T h e first data set was o b t a i n e d f r o m a study c o n d u c t e d at the U n i v e r s i t y o f B r i t i s h C o l u m b i a (Taunton, 1992). T h e ratio variable used i n this study was V o 2 m a x / k g and the sample size was 52. T h e s e c o n d data set was f r o m B a c o n ' s research (1997). T h e diffusing capacity o f the lungs for carbon m o n o x i d e ( D L c o ) d i v i d e d  29  b y a l v e o l a r v e n t i l a t i o n v o l u m e ( V A ) was the ratio v a r i a b l e ( D L C O / V A ) used for deflation purposes, and 13 subjects w e r e i n B a c o n ' s study. T h e other t w o sets o f ratio data selected c o n t a i n e d ratio v a r i a b l e s that were not used for the purpose o f deflation. O n e o f these t w o data sets w a s f r o m a project c o n d u c t e d b y K r a m e r , N u s c a , B i s b e e , M a c D e r m i d , K e m p , and B o l e y (1994). T h e n o n d o m i n a n t / d o m i n a n t forearm strength ratio was the ratio variable used, a n d 4 3 subjects were i n the project. T h e other ratio data set w a s the w a i s t and h i p girth ratios o b t a i n e d f r o m the study c o n d u c t e d b y S o n n e n s c h e i n , K i m , Pasternack, and T o n i o l o (1993), a n d the s a m p l e s i z e was 1851. T h e s e four data sets served as a basis for the i n v e s t i g a t i o n i n this dissertation.  Deflation Model Evaluation The Models In h u m a n k i n e t i c s research, a s i m p l e ratio o f t w o c o m p o n e n t variables is c o m m o n l y used for the purpose o f deflation. A s s h o w n i n the literature r e v i e w chapter, a s i m p l e ratio m a y not adequately e l i m i n a t e the d e n o m i n a t o r effect, a n d alternative m o d e l s need to be c o n s i d e r e d . T h e v a l i d i t y o f the four deflation m o d e l s used i n literature w a s evaluated for each o f the ratio variables chosen for the purpose o f deflation i n this study.  Simple ratio model.  B a s e d o n c l a s s i c measurement theory, a v a r i a b l e c a n be  expressed as a p o p u l a t i o n m e a n w i t h a error term. A s i m p l e ratio Y = X i / X 2 can be presented as Y j = X n / X 2 i = H + e* (where p. is the p o p u l a t i o n mean for the ratio v a r i a b l e Y , a n d e; is the error t e r m for subject i ; i = l , 2 , . . . , N ) . In the literature, a l i n e a r regression function o m i t t i n g the intercept is c o m m o n l y u t i l i z e d to represent the r e l a t i o n s h i p between the expected mean o f the s i m p l e ratio and its c o m p o n e n t variables ( A l b r e c h t , et a l , 1 9 9 3 ;  30  A l l i s o n , Paultre, G o r a n , P o e h l m a n , & H e y m s f i e l d , 1995; N e v i l l , R a m s b o t t o m , & W i l l i a m , 1992), i.e.; X„ =p X t  Where X i  i t  t  2 l t  +e  (i=l,2,...,n;t=l,2,...,k)  i t  (12)  is the numerator v a r i a b l e , X i is the d e n o m i n a t o r variable. T h e s i m p l e ratio 2  t  score i n trial t for a subject i based on this m o d e l can thus be presented as Y = X i i / X 2 i t = p \ + e i t , w h e r e the slope p\ i n F o r m u l a (12) serves as the expected p o p u l a t i o n it  t  m e a n p. , and e t  lt  is the error term (e; = e i / X i ) . O c c a s i o n a l l y , o r d i n a r y least-squares l i n e a r t  t  2  t  regression based o n E q u a t i o n (12) has been used to fit the c o m p o n e n t v a r i a b l e data (separately for each trial t) a n d obtain the estimated slope ( (3 ) o f the m o d e l , a n d p \ used A  A  t  as the estimate o f the e x p e c t e d value o f the s i m p l e ratio (e.g., N e v i l l et a l . , 1992). T h e weakness o f u s i n g this m o d e l is that the error term en is related to the d e n o m i n a t o r variable e^ =ei /X2i . T h i s i m p l i e s that the expected v a l u e o f e^ g i v e n X 2 i is still z e r o but t  t  t  its variance varies as a f u n c t i o n o f X j ( A l l i s o n , et a l . , 1995). A l t h o u g h u s i n g this m o d e l 2  t  c o u l d alter the error d i s t r i b u t i o n , it has been used b y researchers to estimate the e x p e c t e d mean o f the s i m p l e ratio v a r i a b l e . A l l i s o n stated: " h a v i n g been alerted to the fact that t a k i n g ratios alters the error d i s t r i b u t i o n , the case for c a r e f u l l y c h e c k i n g the residuals for h o m o s c e d a s t i c i t y b e c o m e s even more c o m p e l l i n g " . Therefore, a g r a p h i c a l c r i t e r i o n w a s i n c l u d e d i n this study to i d e n t i f y the v i s i b l e d i s t o r t i o n o f the v a r i a t i o n o f the ratio variable.  Linear regression model (LRM). T h i s m o d e l is an intercept adjusted ratio model: Xi i t =P j X it+at+eit (ad  )t  2  (13)  31  Where a is the intercept and p\ dj)t is the slope of the regression equation for trial t, the t  a  other terms are the same as defined in previous section. Based on the model, the ratio for each subject is represented by the transformed form Y( dj)it=(Xiit-a )/X2it=P(adj)t+eit, where a  t  eu is the error term (ei =ei /X i ). This linear regression model was used to fit the t  t  2  t  component variable data (separately for each trial t) to obtain the estimates of the slope, A  P(adj)t andA a t in Equation (13).  A  p\ dj)t serves as the estimate of the expected value of the a  adjusted ratio variable. The central issue hinges on the desire for a simple ratio (or an adjusted ratio variable based on the L R M model) which is independent of (i.e., uncorrelated with) the denominator. Allison (1995) indicated that under bivariate normality and a linear relationship between the component variables, a population intercept (a ) equal to zero is a necessary and sufficient condition for the population t  correlation (p , 2) to be zero. Therefore, in the model LRM, subtracting the regression y  X  intercept a was necessary in order to meet the zero correlation criterion. As with the t  simple ratio model, the weakness of the L R M model is that the error distribution is related to the denominator variable due to ej =ei /X2i . t  t  t  Nonlinear regression model 1 (NLRM1). This model has the form of an exponent adjusted ratio: Xlit=P(adj)tX2it eit k  (14)  Nevill, Ramsbottom, and Williams (1992) indicated that the multiplicative error term is a particularly attractive feature of this model. It accommodates a spread in the subjects' Xiit and X it values when plotted against each other, provided these scores diverge at a 2  constant proportion to each other—and most component variables used for deflation purposes appear to exhibit this pattern. This model has been used by numerous  32 researchers in our field (e.g., Nevill, 1994;Vanderburgh & Katch, 1996; Welsman, Armstrong, Nevill, Winter, & Kirby, 1996). In this study, Equation (14) was fitted to the component variable data to obtain the estimated parameters in the model and p\ dj)t serves as the estimate of the expected value A  a  of the adjusted ratio variable. The nonlinear regression option with Levenberg-Marquardt algorithm in SPSS was used for the fitting process (Norusis, M. J , 1997). The adjusted ratio for each subject is presented as Y(adj)it=X /X2itk=P(adj)teit. lit  Nonlinear regression model 2 (NLRM2). This model is a fully adjusted ratio model of the form: Xiit=P ) X it eit+a  (15)  k  (adj  t  2  t  This model, which was suggested by Huxley (1972) and used by Albrecht et al. (1993), has both the intercept a and constant k. t  The model fitting procedure for Equation (15)  was similar to that used for the NLRM1 using the nonlinear regression option with Levenberg-Marquardt algorithm in SPSS. Equation (15) was fitted to the component variable data to derive the estimates of the three parameters (a, k, and p\ j)t) in the model, aa  and p\ j)t serves as the estimate of the expected value of the adjusted ratio variable. The aa  ratio (3(adj)it for subject i in trial t is presented by the equation Y(adj)it (Xiit-a)/X2itk=P( dj)tej . =  a  t  Controlling for the Denominator Effect The empirical data were fitted to the different deflation models mentioned above, then the simple ratio variable and the adjusted ratio variables derived from the deflation models were evaluated with the following criteria:  33  1. Statistical c r i t e r i o n . T h e linear relationship between the d e r i v e d ratio variable (the s i m p l e ratio variable Y or the adjusted ratio variable Y j ) and the denominator a d  variable ( X ) is zero or nearly so. T h e Pearson correlation coefficient between the 2  d e r i v e d ratio variable and X was c a l c u l a t e d to ascertain the strength o f the linear 2  relationship. 2. G r a p h i c a l c r i t e r i o n . T h e relationship between Y and X t  2  t  can be plotted as a  h o r i z o n t a l l i n e (i.e., the least-squares regression line has a slope o f zero). 3. A l g e b r a i c c r i t e r i o n . T h e estimate o f the expected value (3 is equal to the A  t  e m p i r i c a l mean o f the ratio variable. F o r e x a m p l e , i n a s i m p l e ratio m o d e l (XiJt=p\X it+ej ), the expected mean i n the ratio variable data Y t = E ( X i j / X j ) / n s h o u l d be 2  t  t  equal to the estimated parameter A P t .  2  t  It is noted that i n the s i m p l e ratio m o d e l the  expected value w o u l d be E(Yj )=E[Pt+ejt/X 2 jt]= p\+E[eit/X t]. G i v e n that the error term e^ t  2i  has mean zero (E(ei )=0), the expected v a l u e E [ e j / X i ] = 0 p r o v i d e d e; and X j t  t  2  t  t  2  are  t  u n c o r r e c t e d (in this case, E ( e i / X ) = E ( e j ) E ( l / X i ) = 0 ) . Therefore, the expected value o f t  2 i t  t  2  t  Y;t is equal to p\- In practice, the estimate o f the expected value A P t s h o u l d be equal to the e m p i r i c a l mean Y . t  4. R . H i g h R i n the deflation m o d e l . 2  2  5. R e l i a b i l i t y . T h e r e l i a b i l i t y o f the d e r i v e d ratio data was assessed b y u s i n g an intraclass correlation approach, the details o f w h i c h are s h o w n i n next section. W h e n evaluating the second c r i t e r i o n , b y graphing Y or Y jj)t and X t  (a(  2  t  in a  scatterplot, w e can e x a m i n e the i m p a c t o f denominator related distortions o f variances o f a ratio variable. A distortion o f the variances as the value o f the d e n o m i n a t o r increases m a y affect the v a l i d i t y o f the deflation m o d e l s . B i v a r i a t e plots and the algebraic criterion  34  are simple but powerful supplements that reveal the distortion of the variances of ratio variables. The plots and algebraic criterion safeguard against situations where a correlation is low despite a strong curvilinear relationship between X  2t  and Y or Y dj) . A t  (a  t  valid deflation model for the ratio variable should meet all of these criteria. The traditional A N C O V A model (based on an assumed linear relationship among the raw score variables) could also be used to test the differences between levels of the independent variable after dependent variable scores are adjusted for differences associated with one or more covariates. However, the A N C O V A model may not be a good model for deflation purposes. Vanderburgh (1998) has shown that the traditional A N C O V A model is not a most parsimonious and plausible model for scaling (deflation) purposes and nonlinear regression is better than the A N C O V A model. That is, in traditional A N C O V A using non-transformed data, the group difference is assumed to be additive, not multiplicative, thus for group comparisons, the difference between groups must be an additive constant. In a nonlinear regression model, however, a more plausible multiplicative difference is assumed which allows for an increasing group difference as the "covariate" becomes larger (Nevill et al., 1992). The other weakness which makes the A N C O V A method lack appeal is that if the assumption of parallel slopes is violated, a more complex method (Johnson-Neyman follow-up method, Johnson & Neyman, 1936) has to be used for the group comparisons. Therefore, the A N C O V A model was not included in the examination of the deflation models in this study. Based on the criteria mentioned above, the simple ratio variable and other adjusted ratio variables were computed and compared with each of the empirical data sets. By comparing the different ratio variables, I was able to find an appropriate ratio for  35  each o f the chosen data sets. H o w the best deflation m o d e l s h o u l d be d e v e l o p e d was discussed.  Reliability of a Ratio W h e n a ratio v a r i a b l e is used, the r e l i a b i l i t y o f the ratio variable is c r u c i a l for the v a l i d i t y o f the study. U n r e l i a b l e repeated measurements o f a ratio variable i n an inferential analysis c o u l d s e r i o u s l y affect the v a l i d i t y o f the study. Therefore, the r e l i a b i l i t y o f the four ratio variables is investigated i n this research. A s argued i n p r e v i o u s sections, the r e l i a b i l i t y issues are m o r e c o m p l e x for ratio variables because a ratio v a r i a b l e is a c o m b i n e d v a r i a b l e w h i c h consists o f t w o c o m p o n e n t variables. It was s h o w n i n F o r m u l a (8) that the r e l i a b i l i t y o f a s i m p l e ratio variable is affected not o n l y b y the r e l i a b i l i t y o f the c o m p o n e n t variables but also b y the variances, c o v a r i a n c e s , and the c o r r e l a t i o n o f the c o m p o n e n t variables. E v e n w h e n the c o m p o n e n t variables are r e l i a b l e , it does not a u t o m a t i c a l l y guarantee that the d e r i v e d ratio variable has appropriate r e l i a b i l i t y . It is e x p e c t e d that a ratio v a r i a b l e is less r e l i a b l e than its c o m p o n e n t variables i n repeated measurements because both the numerator and d e n o m i n a t o r variables are sources o f l a c k o f r e l i a b i l i t y . A researcher m a y ask h o w the r e l i a b i l i t y o f ratio data can be a p p r o x i m a t e d f r o m r e l i a b i l i t y o f the c o m p o n e n t variable data. I f the assumptions o f the f o r m u l a are met, the r e l i a b i l i t y o f s i m p l e ratio variables w i t h test-retest measurement data can be evaluated b y P e a r s o n ' s approach ( F o r m u l a (7)) based on the c o r r e l a t i o n o f the c o m p o n e n t variables. T h e most important assumption for u s i n g this f o r m u l a is that the coefficients o f v a r i a t i o n for a l l the c o m p o n e n t variables are s m a l l enough ( V < 0 . 1 5 ) so that the c u b i c and h i g h e r  36  order v a r i a t i o n terms i n the T a y l o r series are n e g l i g i b l e . T h e variation and the c o r r e l a t i o n o f the c o m p o n e n t variables m a y also strongly affect the v a l i d i t y o f u s i n g F o r m u l a (7). W i t h repeated measurement data i n v o l v i n g m o r e than t w o repeated observations (e.g., trials), the intraclass correlation coefficient can be used to compute the r e l i a b i l i t y o f the observed scores. A c c o r d i n g to c l a s s i c measurement theory, r e l i a b i l i t y is d e f i n e d as 2  2  the ratio o f true score variance (a ) to o b s e r v e d score variance ( o n  n  2  + o~ ). U s i n g the e  additive m o d e l w i t h the assumption that subjects are a r a n d o m variable and the t i m e (trial) factor is a f i x e d variable (the most c o m m o n situation i n h u m a n k i n e t i c s research), the intraclass c o r r e l a t i o n coefficient can be estimated f r o m the mean squares i n a o n e - w a y R M A N O V A test ( W i n e r , 1991):  MS, - MS  =  mt  e  MS +(k-l)MS ' s  K  )  e  where M S s is the m e a n square due to subjects, and M S e is the mean square due to error, k is the n u m b e r o f trials. In our f i e l d , the m e a s u r e d scores for several trials are u s u a l l y averaged to increase the r e l i a b i l i t y . T h e intraclass c o r r e l a t i o n for the m e a n o f m e a s u r e d scores i n several trials is c o m p u t e d as:  _ MS. - MS, MS,  (  '  T h e s e formulas have not i n c l u d e d the variance due to the treatment, because it is not a part o f the true variance or the total variance i n the r e l i a b i l i t y evaluations. B o t h the t w o formulas appear i n p u b l i c a t i o n s to report the r e l i a b i l i t y . In s u m m a r y , u s i n g the above procedures (i.e., ratio variable selection, deflation m o d e l e v a l u a t i o n , and r e l i a b i l i t y evaluation), w e e x a m i n e d the e m p i r i c a l ratio variable data and investigated issues related to the v a l i d i t y o f ratio variables used i n our f i e l d . F o r  37  the selected ratio variables, this study evaluated the v a l i d i t y o f the four deflation m o d e l s . A most r e l i a b l e and v a l i d deflation m o d e l was suggested for each o f the selected e m p i r i c a l data sets.  Computer Simulation Investigation of Ratio Issues T h e v a l i d i t y issues o f some ratio variables used i n our f i e l d were addressed b y e x a m i n i n g four e m p i r i c a l data sets a c c o r d i n g to the criteria s h o w n i n the above section. H o w e v e r , the g e n e r a l i z a b i l i t y o f those f i n d i n g s is l i m i t e d because the procedures w e r e based o n specific e m p i r i c a l samples. F o r e x a m p l e , h o w the v a r i a t i o n and c o r r e l a t i o n o f the c o m p o n e n t variables affect the v a l i d i t y o f u s i n g ratio variables i n R M A N O V A was not e x a m i n e d . In fact, w e cannot k n o w h o w different c o m b i n a t i o n s o f v a r i a t i o n o f the c o m p o n e n t variables affect the v a l i d i t y o f a ratio v a r i a b l e i n R M A N O V A b y i n v e s t i g a t i n g l i m i t e d e m p i r i c a l studies. B e c a u s e R M A N O V A is a c o m m o n l y u s e d technique i n o u r f i e l d , the v a l i d i t y o f u s i n g ratio variables i n R M A N O V A requires further i n v e s t i g a t i o n . S i m u l a t e d p o p u l a t i o n data o f ratio score variables were used to investigate characteristics o f ratio variables and the extent to w h i c h they m a y l e a d to a v i o l a t i o n o f assumptions i n the o n e - w a y R M A N O V A .  T h e effects o f the relative v a r i a t i o n ,  c o v a r i a t i o n , and correlation between the numerator and d e n o m i n a t o r variables were e x a m i n e d under different c o n d i t i o n s . T h e parameters o f the ratio variables for generating the p o p u l a t i o n data were based on the statistical characteristics o f the selected c o m p o n e n t variables. B y v a r y i n g these c o n t r o l l e d c o n d i t i o n s for a broad range o f the parameters, we  38  investigated h o w certain c o n d i t i o n s o f the c o m p o n e n t variable distributions l e a d to v i o l a t i o n o f the assumptions o f R M A N O V A w h e n a ratio variable was used. T h e effects o f u s i n g ratio variables on the type I error rate i n the o n e - w a y R M A N O V A m o d e l (treatment f i x e d and subjects random) were investigated u s i n g M o n t e C a r l o s i m u l a t i o n procedures. It is w e l l k n o w n that v i o l a t i o n o f the c i r c u l a r i t y assumption results i n inflation i n the type I error rate, and that the inflation can be a l l e v i a t e d b y u s i n g the e p s i l o n correction procedures (Greenhouse & G e i s s e r , 1959; H u y n h & F e l d t , 1976). In this study, the effect o f the ratio transformation on the c i r c u l a r i t y property o f the p o p u l a t i o n c o v a r i a n c e m a t r i x and the consequences o f u s i n g s a m p l e data o f ratio variables i n a R M A N O V A test were e x a m i n e d . F u r t h e r m o r e , the effect o f u s i n g ratio variables o n the type I error rate i n o n e - w a y R M A N O V A test was investigated. In general, b y u s i n g s i m u l a t i o n procedures, it was e x p e c t e d that the f o l l o w i n g questions c o u l d be answered: 1. I f the c o m p o n e n t variables p r o d u c e a c o v a r i a n c e m a t r i x that meets the c i r c u l a r i t y assumption i n R M A N O V A , does the c o v a r i a n c e m a t r i x o f the ratio variable c o m p r i s e d o f these components also e x h i b i t c i r c u l a r i t y ? 2. If the c o m p o n e n t variables produce a c o v a r i a n c e m a t r i x that violates the c i r c u l a r i t y assumption, under what c o n d i t i o n s do the ratio variables p r o d u c e a c o v a r i a n c e m a t r i x w h i c h leads to less or greater v i o l a t i o n o f the assumption? 3. W h a t is the s a m p l i n g d i s t r i b u t i o n o f the 8 estimate (a measure o f the degree to w h i c h the c i r c u l a r i t y assumption is v i o l a t e d ; see b e l o w for a c o m p l e t e d e f i n i t i o n ) w h e n a ratio variable is used i n R M A N O V A , and h o w does this affect the type I error rate o f the F test?  39  Overview of the Simulation Conditions and Investigations T o investigate the issues addressed above, a single group a n d five trials ( g = l , k=5) R M A N O V A i n different c o n t r o l l e d c o n d i t i o n s w a s d e s i g n e d for the i n v e s t i g a t i o n . R a w score p o p u l a t i o n s o f the c o m p o n e n t variables were first generated based u p o n c o n t r o l l e d c o n d i t i o n s . B e c a u s e the r a w score variables i n o u r f i e l d have p o s i t i v e values and l i e w i t h i n a certain finite range (e.g., V o 2 , H e i g h t , W e i g h t , a n d w a i s t a n d h i p girths), a generated r a w score p o p u l a t i o n w a s a large p o s i t i v e p o p u l a t i o n . T a b l e 3-1 s h o w s the design a n d layout o f the p o p u l a t i o n data o f the c o m p o n e n t variables. X H J is defined as the numerator variable a n d X i j as the d e n o m i n a t o r v a r i a b l e , i is the subject n u m b e r a n d j 2  is the trial n u m b e r ( X j a n d X are used through this dissertation to represent the 2  numerator a n d d e n o m i n a t o r variables). T h e ratio score variables ( Y = X i / X ) were d e r i v e d 2  b y transformations f r o m the r a w score populations. T h e f o l l o w i n g sections describe the c o n d i t i o n s under w h i c h the s i m u l a t i o n s were c o n d u c t e d .  Box's Epsilon (e). B o x ' s e, a measure o f the degree o f departure f r o m c i r c u l a r i t y i n a p o p u l a t i o n c o v a r i a n c e m a t r i x , is a function o f the variances a n d covariances i n the p o p u l a t i o n m a t r i x . B o x (1954) defined 8 as:  (k-l)(J Zo ,-2kJ al y  J  jJ  j  + ko) 2  2  (18) Where k: the order o f the c o v a r i a n c e m a t r i x (number o f trials), Gjj: the mean o f the variances (diagonal elements),  40  T a b l e 3-1 D a t a M a t r i x o f the C o m p o n e n t V a r i a b l e s Numerator Trials  Denominator  T  T  T  T  Xiii  X112  Xu3  X114  X115  s  X121  X122  X]23  X124  St  Xni  Xii  Xii3  5/v  XiNl  XiN2  XiN3  2  3  T  T  T  T  X211  X212  X213  X214  X215  X125  X221  X222  X223  X224  X225  X)i4  Xii5  X2H  X2i2  X2i3  X2i4  X2i5  X]N4  XiN5  X N1  X2N2  X2N3  X2N4  X2N5  4  5  2  3  4  s  Subject  2  2  2  41  a : the grand m e a n o f a l l elements i n the c o v a r i a n c e m a t r i x , Oj.: the mean o f j t h r o w o f the c o v a r i a n c e m a t r i x , Ojj': an i n d i v i d u a l entry o f the c o v a r i a n c e m a t r i x (j, j ' = l , 2 , . . . , k ) . W h e n the p o p u l a t i o n data m a t r i x meets the c i r c u l a r i t y assumption, 8 = 1.0; otherwise, 8 < 1.0, w i t h a m i n i m u m o f l / ( k - l ) , where k is the n u m b e r o f repeated trials i n this study (the order o f the c o v a r i a n c e m a t r i x ) . E p s i l o n is used to correct a p o s i t i v e bias i n the n a i v e F test b y adjusting the degrees o f freedom b y an amount p r o p o r t i o n a l to e. T h a t i s , the c o m p u t e d F statistic for the repeated measures factor i n a repeated measures design is distributed as F [ ( k - l ) e , ( n - l ) ( k - l ) e ] . In general, the characteristics o f a p o p u l a t i o n c o v a r i a n c e m a t r i x are u n k n o w n and are estimated b y a sample c o v a r i a n c e m a t r i x . C o l l i e r , B a k e r , M a n d e v i l l e , and H a y s (1967) s h o w e d that r e p l a c i n g the components i n F o r m u l a (18) w i t h those estimated f r o m a s a m p l e c o v a r i a n c e m a t r i x and c o m p u t i n g an estimate o f e p s i l o n ( E ) to a  adjust the degree o f freedom results i n a r e l a t i v e l y robust F-test for reasonable s a m p l e sizes o f 15 or larger. T h a t is, the n a i v e F statistic is a p p r o x i m a t e l y distributed as F [ ( k - 1 ) 8 , ( n - l ) ( k - l ) e ] . W e w i l l refer to this procedure for testing as the e-adjusted F A  A  A  test. H u y n h and F e l d t (1976, 1978) subsequently s h o w e d that the s - a d j u s t e d test is A  negatively b i a s e d and w o u l d m a k e the F test too conservative. T h e bias is most serious w h e n a p o p u l a t i o n 8 is greater than 0.75, e s p e c i a l l y w h e n the sample size is s m a l l . T h e y r e c o m m e n d e d a less biased estimate ~e:  n(k-])i-2 E  ~  {k-l)[n-l-(k-l)e] (19)  42  F o r any values o f n and k, ~£ >=  e, and the e q u a l i t y holds w h e n £ = l / ( k - l ) . T h e upper A  b o u n d o f ~£ was set to unity, though it c o u l d be greater that unity. In this study, there were t w o £ c o n d i t i o n s for the t w o c o m p o n e n t variables generated. T h e values o f E were set to p r o v i d e v i o l a t i o n ( £ = 0 . 7 0 ) , a n d n o v i o l a t i o n (E=1.0) o f the c i r c u l a r i t y assumption (e.g., H e r t z o g & R o v i n e , 1985; H u y n h , 1978). T h e t w o £ c o n d i t i o n s for the c o m p o n e n t variables resulted i n four c o m b i n a t i o n s for the ratio populations (i.e., £ i = 1 . 0 , £ = 1 . 0 ; £ i = 1 . 0 , £ = 0 . 7 ; £ i = 0 . 7 , £ = 1 . 0 ; £ i = 0 . 7 , £ 2 = 0 . 7 ) . x  x 2  x  x 2  x  x 2  x  X  K n o w i n g the £ value i n a r a w score p o p u l a t i o n and s y s t e m a t i c a l l y v a r y i n g the correlation and relative variances o f the c o m p o n e n t variables a l l o w e d us to investigate the i m p a c t o f u s i n g the ratio variable on the c i r c u l a r i t y assumption o f the c o v a r i a n c e matrices. F o r £ = 1 . 0 o f the c o m p o n e n t v a r i a b l e p o p u l a t i o n , the p o p u l a t i o n variances a n d the c o r r e l a t i o n between the repeated trials w e r e constant, and the values w e r e set based u p o n the waist a n d h i p girth variables i n S o m m e n s c h e i n , K i m , Pasternack, a n d T o n i o l o ' s study (1993). F o r the £ = 0 . 7 0 c o n d i t i o n , i n each p o p u l a t i o n data o f the c o m p o n e n t variables, the c o r r e l a t i o n between the repeated trials f o l l o w e d a s i m p l e x pattern i n w h i c h the adjacent trials w e r e m o r e h i g h l y correlated than the trials w h i c h w e r e farther apart (the details o f c o r r e l a t i o n c o n d i t i o n s are d i s c u s s e d later). G i v e n the characteristics o f the c o m p o n e n t variables, the effect o f u s i n g ratio variables on the p o p u l a t i o n £ w a s evaluated b y c o m p a r i n g the £ values o f the ratio score p o p u l a t i o n s variables on the s a m p l i n g characteristics o f and standard d e v i a t i o n s o f the  A  £ i/ x  x 2  A  £ i/ 2 x  X  (£ i/ x  x 2  ).  T h e effect o f u s i n g ratio  was evaluated b y c o m p a r i n g the means  i n different s i m u l a t i o n c o n d i t i o n s . It can be  43  speculated that a s a m p l e c o v a r i a n c e m a t r i x is a l w a y s expected to violate the c i r c u l a r i t y a s s u m p t i o n to some degree, even w h e n the p o p u l a t i o n m a t r i x does not. B y i n v e s t i g a t i n g the e estimates o f samples d r a w n f r o m the ratio p o p u l a t i o n under different c o n d i t i o n s , w e c o u l d e x a m i n e the effect o f u s i n g ratio variables o n the s a m p l i n g b e h a v i o r o f the estimated e coefficient ( e i/x2)A  x  Population means of the component variables. T h e population means o f the c o m p o n e n t variables were set to u, ij=75.0 a n d p. j=100.0 for a l l treatment trials based o n x  x2  the waist a n d h i p girths i n S o n n e n s c h e i n , K i m , Pasternack, and T o n i o l o ' s study (1993). H e r e j denotes the j t h trial o f the repeated measurement experiment.  B y setting the  means to be constant o v e r trials, w e w e r e able to investigate h o w ratio variables w o u l d affect the type I error rates o f the R M A N O V A test i n different c o m b i n a t i o n c o n d i t i o n s o f the variances a n d covariances o f the c o m p o n e n t variables. PY1Y2, V Y I ,  and  V ^ .  A s s h o w n i n F o r m u l a s (4), (5), (6), and (7), b o t h the  correlations o f the c o m p o n e n t variables w i t h i n trials and o v e r trials, a n d the coefficients o f v a r i a t i o n o f the c o m p o n e n t variables p l a y an important r o l e i n d e t e r m i n i n g the ratio v a r i a b l e means, variances, c o v a r i a n c e s , and correlations. A m o n g the four types o f correlations between the numerator a n d d e n o m i n a t o r variables (i.e., p n 2 i , Pxii,x2j, Pxii.xij, x  ;X  Px2i,x2j, i , j = l , 2 , . . . , 5 ) , the most m e a n i n g f u l correlations i n p r a c t i c a l research are p n , 2 i , x  X  Pxii.xij, and p 2i,x2j- H e r e p n , 2 i is the c o r r e l a t i o n between the numerator a n d d e n o m i n a t o r X  x  X  variables i n the same trial (i), p i i , x i j is the c o r r e l a t i o n between the numerator variables i n x  different trials ( i , j ) , and p 2i,x2j is the c o r r e l a t i o n between the d e n o m i n a t o r variables i n X  different trials. T h e c o r r e l a t i o n p n 2 j (i * j ) is the correlation between the numerator x  iX  v a r i a b l e i n trial i a n d the d e n o m i n a t o r v a r i a b l e i n trial j . Its m a g n i t u d e was based on the  44 other c o r r e l a t i o n c o m p o n e n t s i n the c o r r e l a t i o n matrix (the details on c o m p u t i n g p n, 2j x  X  are s h o w n i n a subsequent section). T h e i n i t i a l c o r r e l a t i o n a m o n g the five trials was also based on the e m p i r i c a l data ( S o n n e n s c h e i n et a l , 1993). T h e m a i n objective o f the i n v e s t i g a t i o n was to e x a m i n e h o w the relative v a r i a t i o n and the c o r r e l a t i o n pattern o v e r trials between the numerator and d e n o m i n a t o r variables affect the characteristics o f the c o v a r i a n c e m a t r i x o f a ratio p o p u l a t i o n . Therefore, to c l e a r l y s h o w the m a i n effect i n the i n v e s t i g a t i o n , the magnitudes o f p ii,xij and p 2i,x2j were set to f i x e d values for the c o n d i t i o n o f E I=E 2=1.0. x  X  X  X  T h a t i s , for the £ i=e 2=1.0 c o n d i t i o n , the c o r r e l a t i o n between the t w o adjacent trials i n x  X  S o n n e n s c h e i n ' s data set (p n, ij=0.87 and p 2i,x2j=0.82, i?*j) was e m p l o y e d to f o r m the x  x  X  c o v a r i a n c e m a t r i x for the r a w score p o p u l a t i o n data w i t h equal variances and equal covariances o v e r the f i v e trials. T h e s e t w o values also served as a baseline for other e p s i l o n c o n d i t i o n s (i.e., p ii, i2=0.87 and p 2i, 22=0.82 i n the c o n d i t i o n s e i=0.7 and x  x  X  X  x  e =0.7). F o r the E=0.70 c o n d i t i o n , the pattern o f the correlations p n, ij and x2  x  x  p 2i,x2j X  f o l l o w e d a s i m p l e x pattern s h o w i n g the correlation between the first trial and subsequent trials decreasing e x p o n e n t i a l l y (i.e., p, p , p , p ...). T o investigate h o w the c o r r e l a t i o n 2  3  4  between the numerator and d e n o m i n a t o r affect the result o f u s i n g ratio variables i n R M A N O V A , the three levels o f the c o r r e l a t i o n between the numerator and d e n o m i n a t o r i n each trial (p n 2i) were chosen to represent a range o f l o w (.50), m e d i u m (.70), and h i g h x  X  (.90) relationships (e.g., A t c h l e y , 1976). T h e value o f  p u 2i was f i x e d i n a l l five trials i n x  X  each p o p u l a t i o n to reflect a fact that m a n y ratios i n h u m a n k i n e t i c s have a near constant c o r r e l a t i o n between the numerator and d e n o m i n a t o r variables i n a repeated measurement  e x p e r i m e n t ( w a i s t / h i p , weight/height, and n o n d o m i n a n t / d o m i n a n t strength).  Therefore,  this i n v e s t i g a t i o n was l i m i t e d to a f i x e d p n 2 i o v e r trials. x  V  x !  X  a n d V 2 are the coefficients o f v a r i a t i o n o f the numerator X i and d e n o m i n a t o r X  X 2 , r e s p e c t i v e l y , w h e r e the coefficient o f v a r i a t i o n is V  x  = a I/i . x  t  T h r e e relative  v a r i a t i o n situations were s i m u l a t e d ( V i / V = 2 . 0 , 1.0, 0.5). T h e three c o m b i n a t i o n s o f x  x 2  the coefficient o f v a r i a t i o n between the numerator a n d d e n o m i n a t o r variables w e r e d e s i g n e d to investigate the ratio effect a n d to c o v e r different relative variations between the numerator a n d d e n o m i n a t o r variables.  Preparing Population Covariance Matrices. T h e first step i n generating the p o p u l a t i o n data w a s to prepare the p o p u l a t i o n c o v a r i a n c e matrices o f interest. T a b l e 3-2 shows the format o f the p o p u l a t i o n c o v a r i a n c e m a t r i x w i t h five repeated measurement trials. B e c a u s e this c o v a r i a n c e m a t r i x is s y m m e t r i c a l , o n l y the upper right c e l l s are s h o w n i n T a b l e 3-2. T h e t w o s h a d o w e d parts are the s u b - c o v a r i a n c e matrices for the n u m e r a t o r a n d d e n o m i n a t o r variables, r e s p e c t i v e l y . T h e elements o f the t w o s h a d o w e d s u b - c o v a r i a n c e matrices w e r e d e r i v e d a c c o r d i n g to the d e f i n e d p o p u l a t i o n e p s i l o n (e j, i = l , 2), X  correlations o f the t w o c o m p o n e n t variables o v e r the repeated trials, a n d c o e f f i c i e n t o f v a r i a t i o n ( V i , and V x  x 2  ) for each o f the s i m u l a t i o n c o n d i t i o n s .  T h e u n s h a d o w e d part o f the m a t r i x contains the c o v a r i a n c e s between the t w o c o m p o n e n t variables i n the five measurement trials. T h e elements i n the d i a g o n a l o f the u n s h a d o w e d s u b - m a t r i x are the c o v a r i a n c e s between the c o m p o n e n t variables i n the same t r i a l . T h e s e c o v a r i a n c e s w e r e c a l c u l a t e d b y u s i n g the f o l l o w i n g equation: (20)  46  x.  X.  x.  X  X  X  ~> o U  o U  •k X  o  o  u  u  X.  X.  x  X  x  X  X  o o  o CJ  o  x.  |:1 -* 3  X >  8  >  -  X  IXS  >  ..iiiiM.:: -  c !!!©;! ro  s S o c  x.  X.  X.  X  X  X  x  X  o CJ  o O  o u  x.  x  X  X  X  o O  cj  u  o  o CJ  X.  x  x  x.  X  X  X  X  cj  u  CJ  o CJ  X.  X  x.  X.  IN  X  X  X. ro  ~> o  X  c o X  13  6  cB  x  X  >  a  c o  I  o o D  5 -  a s s a s  IS  CJ  6  g  x  3  c3  x  >  X  >  ¥ >  o o  GO  X  ¥ >  <+-(  o X  CO •a o  3  a)  "a  a  "a  5  —  X X ^>  CJ  3  x  I  o  d ftSiCJ: X  vm  x  •X  x  47  where p i i , 2 i denotes the correlation between the numerator and d e n o m i n a t o r variables i n x  X  the same trial. S i n c e the variances a n d the c o r r e l a t i o n between the numerator a n d d e n o m i n a t o r variables i n the same trial were defined f r o m the c o n t r o l l e d c o n d i t i o n s , the Cov(X  u  X)  i n the f i v e trials was easily d e r i v e d . T h e off-diagonal elements o f the  2i  u n s h a d o w e d sub-matrix are the c o v a r i a n c e s between the numerator and d e n o m i n a t o r variables i n different trials (trial i and j ) : Oxn,x2r=Cov(X X ) u  2J  = P  i , j = 1, 2 , . . . , 5  ^o a .  Xu<Xll  xU  x2  i*j  (21)  w h e r e Px.ii.x2j is the c o r r e l a t i o n between the n u m e r a t o r variable i n the ith trial a n d the d e n o m i n a t o r v a r i a b l e i n the j t h trial. T h i s c o r r e l a t i o n can not be obtained d i r e c t l y f r o m the defined c o n t r o l c o n d i t i o n s , and thus must be estimated. It is e x p e c t e d that the c o r r e l a t i o n between X i and X w i l l be l o w e r than that between X i and X t w o f o l d . F i r s t , i f X i and X  2  2  i n different trials (p iu2j, i ^ j ) x  i n the same trial. T h e rationale for this is  were perfectly r e l i a b l e , it m i g h t be expected that the  c o r r e l a t i o n between X i and X and X  2  2  2  i n different trials w o u l d be i d e n t i c a l to that between X i  i n the same t r i a l . H o w e v e r , the r e l i a b i l i t i e s o f X i and X  than unity, a n d this trial-to-trial u n r e l i a b i l i t y o f both X i and X correlations between X i and X  2  2  2  are c o n s i d e r a b l y less  w i l l affect (reduce) the  i n different trials. It c o u l d be h y p o t h e s i z e d that this  reduction w o u l d be a function o f the r e l i a b i l i t y o f X i and X . S e c o n d , the c o r r e l a t i o n 2  matrices o f the e m p i r i c a l data sets (e.g., K r a m e r ' s data, 1994, and S o n n e n s c h e i n ' s data, 1993) do s h o w that p n , j is less under the c o n d i t i o n i ^ j than i n the i=j c o n d i t i o n . T h e x  x2  relationships between p n, 2i and p ii,x2j ( i ^ j ) w e r e e x a m i n e d u s i n g t w o e m p i r i c a l data sets x  X  x  ( K r a m e r et a l , 1994, S o n n e n s c h e i n et a l , 1993), and the e x a m i n a t i o n suggested that an  appropriate a p p r o x i m a t i o n for the c o r r e l a t i o n between the numerator i n trial i a n d the d e n o m i n a t o r i n trial j c o u l d be presented as:  p  x i l  ,x  2 1  =p  J t  „.x ,v^"v^r  (  2  2  )  2  T h e estimated values o f the c o r r e l a t i o n p n, 2j ( i ^ j ) u s i n g F o r m u l a (22) were v e r y c l o s e to x  X  the e m p i r i c a l c o r r e l a t i o n values i n the c o r r e l a t i o n matrices i n these t w o data sets (the largest d e v i a t i o n w a s o n l y 0.02). A n a n a l y t i c approach based o n the attenuation c o r r e l a t i o n f o r m u l a ( N u n n a l l y , 1978, p p 2 1 5 - 2 2 1 ) also supported the v a l i d i t y o f F o r m u l a (22). T h e details o f the support based o n the attenuation c o r r e l a t i o n f o r m u l a are presented i n A p p e n d i x A . T o d e r i v e the c o v a r i a n c e s between the numerator and d e n o m i n a t o r i n different trials, w e r e p l a c e d p n 2 j i n F o r m u l a (21) w i t h F o r m u l a (22): x  a ,i, j = Cov(X X ) x  u  x2  2j  X  =p  7P^V *2*2 p  XiiXii  i,j = l,2,... ,5  ^ xu x2j a  a  i*j  (23)  B a s e d u p o n the d e f i n e d c o n d i t i o n s , F o r m u l a (23) was used to c a l c u l a t e the o f f - d i a g o n a l elements i n the u n s h a d o w e d part o f the c o v a r i a n c e m a t r i x . In order to generate the r a w score p o p u l a t i o n s for this i n v e s t i g a t i o n , each o f the elements i n the c o v a r i a n c e m a t r i x i n T a b l e 3-2 was d e f i n e d o r c a l c u l a t e d based upon the c o n t r o l l e d c o n d i t i o n s and the procedures s h o w n above. A T u r b o P A S C A L c o m p u t e r p r o g r a m d e v e l o p e d b y E o m (1993) w a s e m p l o y e d to construct the c o v a r i a n c e matrices for X i a n d X  2  (i.e., s h a d o w e d parts o f the p o p u l a t i o n m a t r i x i n T a b l e 3-2). W i t h the  s p e c i f i e d p o p u l a t i o n parameters (i.e., variances, correlations, and means), the c o m p u t e r p r o g r a m first constructed a p o p u l a t i o n c o v a r i a n c e m a t r i x w i t h an e p s i l o n equal to u n i t y ( e = l ) for the c o m p o n e n t variables. S e c o n d , each o f the matrices was further m a n i p u l a t e d  49  by varying the correlations between repeated trials for each of the component variables to reflect a simplex pattern, resulting in a population matrix possessing a desired epsilon value (8=0.70). Third, the covariances between X i and X (i.e., unshadowed part of the 2  population matrix in Table 3-2) were calculated individually following the procedures shown earlier. The covariance matrices based on the defined conditions for generating the raw score populations are shown in Appendix B.  Data Generation A FORTRAN program, which incorporated the Random Number Generator in IMSL (JMSL, 1991), was developed to conduct the data generation and the simulation. The Multivariate Random Number Generator (DRNMVN) routine in IMSL was used to generate raw score population data matrices according to the defined population covariance matrix in each of the experimental conditions shown in Table 3-3. Each raw score data matrix contained Nx2k data points (see Table 3-1), where N refers to the population size and k refers the number of treatment trials in the experiment (i.e., N=90,000 and k=5). The first five variables were the numerator variable ( X i ) over five treatment trials, and the last five variables were the denominator variable ( X ) over five 2  treatment trials. Therefore, there were 2k=10 measurements for each subject in each condition (five measurements for the numerator variable and five measurements for the denominator variable). The ratio population data were derived by dividing each value in the first five columns by the corresponding value of the denominator variable in the last five columns in the raw score data matrices.  50  Table 3-3 Computer Simulation Conditions e  M;  6x1 = 6x2=1.0 6xi = 6x2 = 0.7  Ox, = 75 ( i = 100 x2  Pxix2  v  0.9  Vxi=0.2V =0.1  0.7  Vxi=V =1.55  0.5  V i=0.1 V =0.2  x2  x2  8x1 = 0.7 < e = 1.0 x2  8x1=  1.0 >  8  x 2  =0.7  x  x2  51  Overview of Methods Used to Answer Questions I first investigated the characteristics o f the ratio variable p o p u l a t i o n s under different v a r i a t i o n and c o r r e l a t i o n c o n d i t i o n s o f the c o m p o n e n t variables. T h i s process p r o v i d e d e v i d e n c e to e x a m i n e h o w characteristics o f the c o m p o n e n t variables affected the c i r c u l a r i t y a s s u m p t i o n o f the c o v a r i a n c e matrices for the d e r i v e d ratio v a r i a b l e p o p u l a t i o n . B o x ' s E values o f the ratio score p o p u l a t i o n s i n different c o n d i t i o n s were c o m p u t e d to p r o v i d e i n f o r m a t i o n regarding the degree o f n o n - c i r c u l a r i t y o f the c o v a r i a n c e matrices o f the ratio variables. S e c o n d , the b e h a v i o r o f the sample estimates o f £ (e ) for ratio data across the s i m u l a t e d c o n d i t i o n s was e x a m i n e d . T h e F O R T R A N M o n t e C a r l o s i m u l a t i o n p r o g r a m was used for the purpose o f i n v e s t i g a t i n g the effect o f u s i n g ratio variables o n the s a m p l i n g characteristics o f e . A f t e r d r a w i n g samples o f n subjects f r o m the ratio score populations (n=15, 3 0 , 45), the F O R T R A N p r o g r a m c o n d u c t e d 2 0 0 0 o n e - w a y R M A N O V A tests i n each c o n d i t i o n for the ratio score data. T h e f o l l o w i n g sample estimates were c o m p u t e d : mean squares i n the F test, standard d e v i a t i o n o f the m e a n squares, m e a n o f £ , standard d e v i a t i o n o f £ , and type I error rate. B y c o m p a r i n g these s a m p l e estimates i n different c o n d i t i o n s , w e were able to investigate h o w the relative variance and correlation changes between the numerator and d e n o m i n a t o r variables affected the type I error rate o f the F test w i t h ratio variable data.  52  CHAPTER IV. RESULTS AND DISCUSSION: CHARACTERISTICS OF RATIO VARIABLES  In this chapter, the results emanating f r o m each o f the four e m p i r i c a l data sets are presented. First, I present detailed evaluation results for each o f the t w o data sets i n w h i c h the ratio variables w e r e used for the purpose o f d e f l a t i o n . T h e four ratio deflation m o d e l s are c o m p a r e d and evaluated based on the c r i t e r i a l i s t e d i n the methods and procedures chapter. T h e f i v e c r i t e r i a ; statistical, g r a p h i c a l , algebraic, R , and r e l i a b i l i t y , 2  are e x a m i n e d for each o f the deflation m o d e l s . T h e n , the best deflation m o d e l is identified for each o f the t w o data sets, and general issues related to the d e v e l o p m e n t o f an appropriate deflation m o d e l for ratio variables i n h u m a n k i n e t i c s research are discussed. S e c o n d , the statistical characteristics o f the other t w o sets o f ratio data (in w h i c h the ratio variables were not used for the purpose o f deflation) are evaluated, and the issue o f r e l i a b i l i t y o f u s i n g ratio variables is discussed.  Ratio Variables Used for Deflation Purposes Empirical Data Set 1: Vo2max/Body Mass These data were obtained f r o m a study c o n d u c t e d at the V a n c o u v e r G e n e r a l H o s p i t a l and the U n i v e r s i t y o f B r i t i s h C o l u m b i a (Taunton, 1992), w h i c h investigated the effectiveness o f exercise for healthy w o m e n aged 65 to 75 years. S e v e n t y - s i x subjects were r a n d o m l y assigned into a water-based exercise group, a land-based aerobic group or a c l a s s r o o m health p r o m o t i o n group (control group). E a c h group met for one hour, three times a week, o v e r a t w e l v e w e e k s p e r i o d . T h e variables o f interest were measured at  three time points (trials) d u r i n g the t w e l v e w e e k s (pre-test, m i d - p r o g r a m test, a n d post-test). T h e measured variables i n c l u d e d m a x i m a l fatigue, strength, b o d y c o m p o s i t i o n , f l e x i b i l i t y , balance, and V o 2 m a x , w i t h the latter measure b e i n g one o f the most important variables measured i n the study. T h e measured V o 2 m a x was d i v i d e d b y the b o d y mass i n order to express the V o 2 m a x on a per-unit o f b o d y mass basis, thus deflating the effects o f the b o d y mass variation f r o m the m e a s u r e d V o 2 m a x variable. T h e d e r i v e d ratio V o 2 m a x v a r i a b l e , V o 2 m a x / k g , then s e r v e d as a measurement v a r i a b l e i n a 3 x 3 (group x time) R M A N O V A to answer the research questions. T h e researchers' p r i m a r y interest w a s to investigate whether c a r d i o v a s c u l a r endurance, as m e a s u r e d b y V o 2 m a x per-unit o f b o d y mass, w a s affected b y the different t r a i n i n g techniques. T o evaluate the v a l i d i t y o f the V o 2 m a x / k g ratio v a r i a b l e and the alternative deflation m o d e l s i n these data, the i n i t i a l p l a n was to evaluate the three groups separately ( G i = w a t e r , n i = 2 7 ; G2=land, n2=25, G3=control, n = 2 3 ; k=3). H o w e v e r , as s h o w n i n 3  T a b l e 4 - 1 , the l a n d a n d c o n t r o l groups h a d a c o n s i d e r a b l e amount o f m i s s i n g data, e s p e c i a l l y i n the s e c o n d a n d t h i r d measurement points, due to subject drop out. B e c a u s e the F tests i n the 3 x 3 (group x time) m i x e d m o d e l A N O V A s h o w e d n o significant differences i n V o 2 m a x / k g a m o n g the groups a n d n o s i g n i f i c a n t groups b y t i m e interaction, the c o m p l e t e cases i n the three groups w e r e c o m b i n e d a n d a n a l y z e d together (n=52).  Simple Ratio Model T h e results reported i n T a b l e 4-2 are based on the s i m p l e ratio data. A s s h o w n i n T a b l e 4 - 2 , the c o r r e l a t i o n between V o 2 m a x ( X i , raw score) and b o d y mass ( X ) ranged 2  f r o m 0.46 to 0.54 i n the three repeated trials. T h i s result indicates that the v a r i a t i o n o f  54  co  co H  co  CN CN  53  ts  CN  CN  CN  CN  co  oo °) 3  CN  CO  oo  !£  o  O  S3  ^  Os  oo  CN  CN  H  ©  2  K  JO H  g  so  S  ©  o  CO  CN  r-  ^  O GO  CO  Q  3  bi oo  ?  x  s  r-  G  2  ^ ^ CN8 OO ^  >  ^ S c o  d,©'  2  C  CN  O  CN  ^ ^  co H  •2  B  CN  O  >  8 o o  3  S  CN  S3 >  R  £  oo  SO  ^  ^ g  ^  so Q © ^ q  GO GO C3  >  CO  H  O  r- CN c o in o ^ so *™H  GO  C  o  Q  Ii  S  CN  T3 O  H  m  CN  SO  O  co  r—(  • CN  IT) r~oo  ,—i  oo  l>  H^  ?o © ©  h  5 6  o S? '-'  £  5 ^  2  g, -  SO i—l  CO  SO ^  so O  1  2  °  CN  SO  ll  > <+-< o G CO  . so OS  CO CN CO  O  co o  o ^  00  |^  *n £  CN  in ~ V O  s o <—i  m  £ >n ~!  00 CN  1  «n CN r-W  co H  H  ^ ^ SO  CN CN  2  Os Os in  Os  CN  o-i _• O  CN  -a o PQ o  '> CO  ©?  ^  oo co  2 a  Pi  1  Fi  2  CN  ^  so —•< " ^  F-  SO  -H  n SO  oo 2 Q  <-»  PQ  §  GO  co oo  oo  SO  2  on  ~> CN ^  Os  SO  CO  so  SO  CN  r-W M  so  C  CO 1^ CN CN  ^o  r--  SO  © ii  ON  CN  l>  Os  SO  oo  e o  - H  so i—i  >  "fj  I-  8  CN  CO  GO  co  3  CN  S "  CUVO  3  5  6  CN  ll  c  CO CU CO 3 CN 5  6  'I  c  -  | CN U  o  13  'i  T a b l e 4-2 E v a l u a t i o n o f the S i m p l e R a t i o M o d e l : D a t a Set 1  Ratio  r  Model  X i VS X  r 2  Y vs X  Estimated Expected  2  Empirical Mean  V a l u e |3 A  Y=X,/X  2  Trial 1  0.48  -0.45  18.28  18.85  Trial 2  0.54  -0.61  19.57  20.22  Trial 3  0.46  -0.58  20.30  21.02  Reliability Coefficient XI  x  0.88  0.99  0.89  0.72  0.99  0.72  2  Y  Intraclass r Average Trials Single Trial  Y  56  b o d y mass does affect the variation o f the raw score V o 2 m a x variable. T h e heavier subjects tended to have greater V o 2 m a x raw scores. A n appropriate ratio variable c o u l d be used to f u l l y r e m o v e the effect o f b o d y mass f r o m the V o 2 m a x variable. T h e question o f interest w i t h these data is whether the s i m p l e ratio ( V o 2 m a x / k g ) c a n appropriately deflate the effect o f the b o d y mass f r o m the V o 2 m a x . A s s h o w n i n the methods and procedures chapter, one important c r i t e r i o n is that the c o r r e l a t i o n between the d e r i v e d V o 2 m a x / k g and b o d y mass s h o u l d be zero or n e a r l y so i f the s i m p l e ratio can appropriately deflate the b o d y mass f r o m the V o 2 m a x . T h i s c r i t e r i o n is not met, as the results (r ix2i i n T a b l e 4-2) s h o w substantial negative correlations (0.45-0.61) between the y  V o 2 m a x / k g ( Y ) and b o d y mass ( X ) i n a l l three trials. G i v e n a p o s i t i v e c o r r e l a t i o n 2  between the t w o c o m p o n e n t variables, u s i n g P e a r s o n ' s a p p r o x i m a t i o n function ( F o r m u l a (7)), A l b r e c h t (1978) s h o w e d that the correlation between a ratio and its d e n o m i n a t o r variable is negative w h e n the coefficient o f v a r i a t i o n o f the d e n o m i n a t o r v a r i a b l e is equal to or greater than that o f the numerator variable. A s can be seen i n T a b l e 4 - 1 , the coefficients o f v a r i a t i o n o f the b o d y mass are almost equal to or greater than that o f the r a w score V o 2 m a x i n a l l three trials, and the raw score V o 2 m a x and b o d y mass have p o s i t i v e c o r r e l a t i o n . T h u s , negative correlations between V o 2 m a x / k g and b o d y m a s s i n these data are to be expected. T h e g r a p h i c a l criterion e x p l a i n e d i n the p r e v i o u s chapter was d e s i g n e d to identify any v i s i b l e pattern o f a relationship between a ratio variable and its d e n o m i n a t o r variable, a n d any d e n o m i n a t o r related distortion o f the variances for the ratio variables. I f the s i m p l e deflation m o d e l is v a l i d , V o 2 m a x / k g s h o u l d have not o n l y a zero linear c o r r e l a t i o n but also zero c u r v i l i n e a r relation w i t h the b o d y mass. A s s h o w n i n the three scatterplots  57  i n the first panel o f F i g u r e 4 - 1 , a negative re la tio n sh ip pattern is c l e a r l y s h o w n i n a l l three trials. It c o n f i r m s that the s i m p l e ratio v a r i a b l e favors the lighter subjects and disadvantages the heavier subjects. T h a t is, the lighter subjects s h o w larger V o 2 m a x / k g than the heavier subjects. T h e r e is n o suggestion o f a c u r v i l i n e a r r e l a t i o n s h i p between the V o 2 m a x / k g a n d the b o d y mass i n any o f the three trials. I f there w e r e c u r v i l i n e a r r e l a t i o n s h i p between V o 2 m a x / k g and b o d y mass, the ratio V o 2 m a x / k g w o u l d not be v a l i d for the deflation purpose. H o w e v e r , this is not the case i n these data. T h e scatterplots, a n d the negative P e a r s o n c o r r e l a t i o n coefficients, indicate that the c o m m o n l y used ratio, V o 2 m a x / k g , over-deflates the d e n o m i n a t o r effect a n d m a y not be v a l i d for deflation purposes based on these t w o criteria. A s stated i n the p r e v i o u s chapter, a c o r r e c t l y f o r m u l a t e d ratio v a r i a b l e s h o u l d result i n v i r t u a l e q u a l i t y o f the estimate o f the expected value o f the ratio m o d e l and the c a l c u l a t e d mean average o f the e m p i r i c a l data. In the s i m p l e ratio m o d e l , the estimate o f the expected value ( H ) o f the V o 2 m a x / k g w a s c o m p u t e d f r o m the (3 coefficient i n the A  y  linear regression equation w i t h o u t an intercept (i.e., Xii =p\X2it+ej ). T h e e m p i r i c a l means t  t  i n the three trials were c o m p u t e d f r o m the ratio V o 2 m a x / k g data. T h e c a l c u l a t i o n s s h o w that the e m p i r i c a l means are consistently greater than the estimate o f the e x p e c t e d values i n the s i m p l e ratio m o d e l (see T a b l e 4-2). T h e over-estimation i n the three trials is between 0.57 to 0.72, about 4 % error relative to the actual m e a n o f the e m p i r i c a l data. T h e last c r i t e r i o n for e v a l u a t i n g the v a l i d i t y o f a ratio data is its r e l i a b i l i t y . T h e r e l i a b i l i t y o f these data was c a l c u l a t e d based on F o r m u l a s (16) a n d (17) s h o w n i n the methods a n d procedures chapter. T h e m e a n squares used i n F o r m u l a s (16) and (17) were d e r i v e d f r o m a three trials R M A N O V A test p e r f o r m e d on each o f the three dependent  o o  in in  ra 5 >. •o o CO  •c H  O C O C O ^ - C S I O C O C O ^ ' C M O C O C M C \ I < N C \ J C \ I T - T - I - T - 7 -  ( W U l ) CA  >. •o o  >» TJ O  m  13  m  o co  •c H  o O C O C O - ^ C M O O O C D M - C O O  6>(/XBUJ20A  oocDTfoaoooco-^CAio  (wan) Z A  o o  TJ O  03  •c  m  o O l O J I t N O O O O ^ N O n ( M ( M « « « ' - ' - < - r - r -  6>|/xeuizoA  SI  C O C O - t f O U O C O C D ' t f C M O  (lAIHl) LA  variables, the r a w score V o 2 m a x , b o d y mass, and s i m p l e ratio V o 2 m a x / k g . T h e n , the intraclass c o r r e l a t i o n coefficients ( r i i , r 2 2, % ) were c a l c u l a t e d based on the d e r i v e d x  x  X  X  m e a n squares o f the R M A N O V A tests. T h e intraclass correlations are i n c l u d e d i n reporting o f results. T h e d i s c u s s i o n o f r e l i a b i l i t y is based on the intraclass c o r r e l a t i o n for the mean o f a l l trials ( F o r m u l a (17), but the s i n g l e trial r e l i a b i l i t y ( F o r m u l a (16) is also reported as both forms are often presented i n the literature. B a s e d on the d e f i n i t i o n , the single trial r e l i a b i l i t y reported is the same as the average p a i r w i s e c o r r e l a t i o n o f the trials i n the data. T h e results (see T a b l e 4-2) s h o w that the b o d y mass data are v e r y r e l i a b l e (r 2x2=0.99), and the r e l i a b i l i t y o f the raw score V o 2 m a x data is also h i g h ( r X  x l x )  =0.88).  T h e h i g h r e l i a b i l i t y o f b o d y mass i n these data is expected because b o d y mass s h o u l d not be substantially affected i n the t w e l v e w e e k e x p e r i m e n t p e r i o d . T h e s i m p l e ratio d i d not reduce the r e l i a b i l i t y o f the ratio data, w i t h the r e l i a b i l i t y o f the s i m p l e ratio data b e i n g close to that o f the m e a s u r e d V o 2 m a x variables ( r i i = 0 . 8 9 ) . B e c a u s e the r e l i a b i l i t y o f y  y  the d e n o m i n a t o r v a r i a b l e is so h i g h , the r e l i a b i l i t y o f the d e r i v e d s i m p l e ratio v a r i a b l e is very close to the r e l i a b i l i t y o f the numerator v a r i a b l e . T h i s c o n f i r m s C r o n b a c h ' s d e d u c t i o n (1941). T h a t is, w h e n the d e n o m i n a t o r v a r i a b l e is h i g h l y reliable (e.g., r  x 2 x  2  close to unity), it can be p r o v e d that the r e l i a b i l i t y o f the ratio variable is c l o s e to the r e l i a b i l i t y o f the numerator variable. In s u m m a r y , for the evaluation o f the s i m p l e ratio i n D a t a Set 1, the ratio V o 2 m a x / k g does not appropriately deflate the effect o f b o d y mass f r o m the V o 2 m a x variable. A l t h o u g h the s i m p l e ratio has acceptable r e l i a b i l i t y , the s i m p l e ratio does not f u l l y deflate the effect o f b o d y mass. It o n l y reverses the relationship between the  60  measured V o 2 m a x and the b o d y mass f r o m p o s i t i v e to negative. B a s e d on the results o f e x a m i n i n g the c r i t e r i a for deflation purposes i n these data, the s i m p l e ratio does not meet the important c r i t e r i a o f full deflation, e s p e c i a l l y zero c o r r e l a t i o n between the ratio variable and the d e n o m i n a t o r variable, and e q u a l i t y o f c a l c u l a t e d e m p i r i c a l m e a n and the estimate o f the e x p e c t e d v a l u e . Therefore, it is c o n c l u d e d that the s i m p l e ratio does not appropriately deflate the effect o f the b o d y mass for the V o 2 m a x variable i n these data. A n appropriate alternate deflation m o d e l m a y be needed f o r this c o m m o n l y u s e d measure.  Alternative Ratio Models T h e s e data w e r e fitted to the three alternative deflation m o d e l s ( L R M , N L R M 1 , and N L R M 2 , see p31 for definitions). T h e results are s h o w n i n T a b l e 4 - 3 . A s s h o w n i n T a b l e 4 - 3 , the adjusted ratio V o 2 m a x variables have a near-zero correlation w i t h the d e n o m i n a t o r variable (body mass) i n a l l three alternative m o d e l s . F o r the m o d e l s L R M and N L R M 2 , the correlations between the adjusted ratio V o 2 m a x and the b o d y mass i n the three trials were not greater than 0.02. W h e n N L R M 1 is used, the correlations were as s m a l l as 0.01 i n a l l three trials. T h u s , i n this data it is c o n c l u d e d that a l l three m o d e l s meet the criterion o f zero c o r r e l a t i o n . S o m e researchers (e.g., A l b r e c h t , 1993) c l a i m e d that adjusted ratios e s p e c i a l l y w i t h the n o n l i n e a r m o d e l s ( N L R M 1 and N L M R 2 ) b r i n g about a serious denominator-related d i s t o r t i o n o f variances. T h i s d i s t o r t i o n o f variance w o u l d be i n d i c a t e d i n scatterplots o f the adjusted ratio v a r i a b l e versus the d e n o m i n a t o r v a r i a b l e . T h a t is, the adjusted ratio variable has less variance for greater values o f the d e n o m i n a t o r and higher variance for s m a l l e r values o f the d e n o m i n a t o r . A s s h o w n i n F i g u r e s 4-1  61  T a b l e 4-3 E v a l u a t i o n o f the A l t e r n a t e M o d e l s : D a t a Set 1  r Y vs X  Reliability 2  R  2  Intraclass r  Empirical Mean  Y  Estimated Parameters  Expected V a l u e |3 A  Y=(Xi-a)/X (LRM)  2  Trial 1  0.02  Trial 2  -0.01  Trial 3  -0.01  Y=X!/X (NLRMl) Trial 1 Trial 2 Trial 3  all Trials  Single Trial  0.83  0.62  a  k  0.229  9.22  9.23  619.53  0.293  8.84  8.84  732.98  0.214  8.24  8.23  826.05  0.233  148.63  148.67  0.289  198.41  198.38  -0.01  0.210  251.51  251.47  0.01  0.233  312.64  309.47  -367.80  0.39  k  2  Y=(X!-a)/X (NLRM2) Trial 1 Trial 2 Trial 3  0.01 -0.01  0.83  0.62  •  0.50 0.45 0.41  k 2  -0.02 -0.02  0.82  0.60  0.291  56.20  55.42  426.96  0.66  0.212  44.88  45.73  563.89  0.69  62  (panel 2) a n d 4 - 2 , the scatterplots o f the three ratios i n the three trials do not indicate v i s i b l e variance decreases w h e n the value o f the d e n o m i n a t o r v a r i a b l e ( b o d y mass) increases (although the s a m p l e size m i g h t be too s m a l l to v i s u a l l y r e v e a l any such effects). Therefore, b o d y mass related d i s t o r t i o n o f variance does not seem to be an issue for these three m o d e l s i n this data set. The  t h i r d c r i t e r i o n for the e v a l u a t i o n o f a deflation m o d e l is the e q u a l i t y o f the  estimated e x p e c t e d value and the e m p i r i c a l means. T h e c a l c u l a t i o n s (see T a b l e 4-3) indicate that the e x p e c t e d means are accurately estimated ( w i t h i n +/- 1% difference) b y all three m o d e l s . The  r e l i a b i l i t y o f the adjusted ratio variables is e x a m i n e d and s h o w n i n T a b l e 4 - 3 .  The  r e l i a b i l i t y o f the adjusted ratio V o 2 m a x variables was l o w e r than the r e l i a b i l i t y o f the  two  c o m p o n e n t variables (e.g., 0.83 c o m p a r e d to 0.88 a n d 0.99). T h e r e w a s n o substantial  difference i n the r e l i a b i l i t y a m o n g these three m o d e l s for these data. T h e r e l a t i v e l o w e r r e l i a b i l i t y for the adjusted ratio variables w a s expected, a n d for the f o l l o w i n g reasons. The L R M ratio is based o n a s i m p l e l i n e a r regression r e l a t i o n s h i p X n = a + p \ X 2 i + i t e  t  t  t  ;  where ej is the error t e r m and p serves as the e x p e c t e d value o f the adjusted ratio data. t  The  t  adjusted ratio v a r i a b l e ( L R M )  can be expressed as Yj =(Xij -a )/X2it=Pt+eit/X2it. O f t  t  t  interest i n the L R M m o d e l is the error t e r m e j / X j . H e t e r o s c e d a s t i c i t y o c c u r s w h e n the t  2  t  v a r i a n c e o f Y ; is not constant for a l l values o f X j , and the l e v e l o f heteroscedasticity t  2  t  w o u l d d i r e c t l y affect the r e l i a b i l i t y o f the data. T h e larger the range o f the v a r i a b l e X 2 H , the greater the heteroscedasticity o f the data, and the l o w e r the r e l i a b i l i t y o f Y . i t  The  heteroscedasticity o f the L R M m o d e l is v e r y sensitive to the range o f the d e n o m i n a t o r v a r i a b l e (Pedhazur, 1982). T h u s , the heteroscedasticity that is i n t r o d u c e d b y  63  >  TJ O  m  •c H  o o in o c r > C O  o  o  m C M  o C M  o  o  m  o  i-  o  m  i-  o o r -  o c o  o  o i n  (Hflld1N)£A  o  CO  o  OJ  i-  (ewtnN) EA  •o o m  13 •a H  o o o co  o in  CM  o o  CM  o in  o o  ••-  r-  o m  o o  o  c  o  o e  t  o  o  s  o i  o  D  o i  u  *  o  o w  n  o  <  -  o  tewunN) Z A  (iWHnN)ZA  co  V ) CO  1 (00  s  >  TJ  TJ O CO  o  o  •c H  o m  CM  o o  CM  o in  i-  o o  *-  U w a n N ) LA  PH  o m  o i  n  o  o  o  m  o o  i  o  n  o  o  o m  o  (ZIAIUIN) L A  o  i  o  n  o  the ratio transformation is the m a i n factor to reduce the r e l i a b i l i t y o f the adjusted ratio variable i n the L R M m o d e l . B e c a u s e the range o f the b o d y mass i n these data is r e l a t i v e l y s m a l l , the r e l i a b i l i t y o f the adjusted ratio v a r i a b l e is r e d u c e d , but o n l y s l i g h t l y , a n d the r e l i a b i l i t y o f Y under both the L R M and N L R M 1 m o d e l s is still reasonably strong (0.83). T h e N L R M 1 ratio is based on the relationship X i J t = Y t X i - e . A l l i s o n (1995) k  2  t  it  c l a i m e d that this approach m i g h t be quite desirable a n d appropriate. H o w e v e r , w h e n the adjusted ratio v a r i a b l e is used i n group c o m p a r i s o n s or R M A N O V A , it is v a l i d o n l y w h e n the exponent parameters o f the adjusted ratio v a r i a b l e i n the different trials are i d e n t i c a l . B e c a u s e the exponents are different i n the different trials, the units o f the adjusted ratio v a r i a b l e are different. F o r e x a m p l e , w h e n c a l c u l a t i n g the r e l i a b i l i t y o f the adjusted ratio v a r i a b l e , one c o u l d not calculate the r e l i a b i l i t y coefficient b y u s i n g the N L R M 1 m o d e l w i t h different parameters such as k i = 0 . 5 0 , k = 0 . 4 5 , and k = 0 . 4 1 i n the 2  3  three trials. Therefore, a c o m m o n exponent (k=0.45) w a s d e r i v e d by a v e r a g i n g the three exponent parameters (i.e., k=(ki+k +k3)/3). T h e c o m m o n k=0.45 was used to derive the 2  N L R M 1 ratio for each trial, a n d the r e l i a b i l i t y c a l c u l a t i o n s were based on those data. W h e n the c o m m o n exponent k is used i n the different trials, it increases the variances o f the adjusted ratio v a r i a b l e . B e c a u s e a c o m m o n k is used, the i n f l a t i o n o f the variances is m o r e serious i n the first a n d t h i r d trials. Therefore, it reduces the r e l i a b i l i t y o f the adjusted ratio variable i n the N L R M 1 m o d e l . F o r the same reason, a c o m m o n k has to be used i n the N L R M 2 m o d e l , thus r e d u c i n g the r e l i a b i l i t y o f the adjusted ratio data. A s e x p l a i n e d i n the methods and procedures chapter, the o r d i n a r y least squares linear regression o p t i o n i n S P S S was used to fit the L R M m o d e l . T h e n o n l i n e a r  65  regression o p t i o n w i t h a L e v e n b e r g - M a r q u a r d t a l g o r i t h m i n S P S S was used to fit the N L R M 1 and N L R M 2 m o d e l s . It has been s h o w n that the s i m p l e ratio over-adjusts the effect o f the d e n o m i n a t o r and is not an appropriate deflation m o d e l for these data. T o c o m p a r e the relative merits o f the three alternative m o d e l s , R , the p r o p o r t i o n o f variance accounted for b y the m o d e l , was e x a m i n e d . It indicates that the differences o f R the three adjusted ratios are t r i v i a l (see T a b l e 4-3). T h e magnitudes o f the R  2  among  obtained  u s i n g the N L R M 2 m o d e l are s l i g h t l y h i g h e r than those for the N L R M 1 m o d e l , and the L R M m o d e l has s l i g h t l y h i g h e r R than the other t w o n o n l i n e a r m o d e l s i n these data but difference is not substantial. R e s u l t s also s h o w that the r e l i a b i l i t y o f the three adjusted ratio data was v i r t u a l l y the same. T h e s e results suggest that there is n o substantial difference a m o n g the three m o d e l s i n their a b i l i t y to p r o v i d e a v a l i d ratio score for these data. It m a y s e e m contradictory that the three deflation m o d e l s w i t h different parameters fit these data about e q u a l l y w e l l . T h e e x p l a n a t i o n for this p h e n o m e n o n is that the regression lines diverge outside the range o f the o b s e r v e d data i n accordance w i t h the very different regression m o d e l s . F o r e x a m p l e , as s h o w n i n F i g u r e 4 - 3 , the fitted c u r v i l i n e a r regression lines o f the second trial i n D a t a Set 1 are v e r y c l o s e to l i n e a r i n the r e l a t i v e l y n a r r o w range o f these observed data. H o w e v e r , the n o n l i n e a r m o d e l s do s h o w the better fit than the l i n e a r m o d e l i n some other situations ( A l b r e c h t et a l . , 1993). G i v e n the s i m i l a r goodness o f fit i n the m o d e l s , the L R M and N L R M 1 m o d e l s m a y be more preferable than the N L R M 2 m o d e l for these data because the L R M and N L R M 1 m o d e l s are easier to use. I f the L R M m o d e l is used, the linear regression approach ( w h i c h is a v a i l a b l e i n most c o m m e r c i a l statistical software) can be used to fit  67  the data. A s s h o w n above, L R M a n d N L R M 1 have s u c c e s s f u l l y met the criteria a n d are s i m p l e r than N L R M 2 for deflation purposes i n D a t a Set 1, although w i t h s o m e r e d u c t i o n i n the r e l i a b i l i t y . T h e s i m p l e ratio m o d e l f a i l e d to meet these criteria. T h e r e f o r e , i f the r e l i a b i l i t y o f the t w o m o d e l s is acceptable, it m a y be suggested that L R M a n d N L R M 1 are statistically v a l i d and appropriate for the purpose o f deflation i n D a t a Set 1. E v e n though the L R M a n d N L R M 1 ratios have s i m i l a r v a l i d i t y based o n statistical criteria, p h y s i o l o g i s t s m a y have reason to choose one o v e r the other. In recent years, N L R M 1 has been used i n some h u m a n k i n e t i c s research (e.g., H e i l , 1997; N e v i l l , R a m s b o t t m , & W i l l i a m s , 1992; V a n d e r b u r g h & K a t c h , 1996). B a s e d o n the theory o f geometric s i m i l a r i t y , a c o m m o n v a l u e o f 0.67 for the k parameter w a s d e v e l o p e d for V o 2 m a x ( A s t r a n d , & R o d a h l , 1986; H e i l , 1997). H e i l (1997) p o i n t e d out that the 0.67 exponent appears to be m o r e appropriate for samples that are v e r y s i m i l a r w i t h respect to age, t r a i n i n g b a c k g r o u n d , b o d y height, etc. H o w e v e r , a s a m p l e that is h o m o g e n e o u s o n so m a n y measurement variables is hard to f i n d i n an e m p i r i c a l study. T h i s study supports that o f other researchers w h o have s h o w n that a c o m m o n magnitude o f the k parameter i n the N L R M 1 m o d e l for V o 2 m a x data is questionable (e.g., B e r g h , S j o d i n , F o r s b e r g , & S v e d e n h a g , 1 9 9 1 ; N e v i l l , 1994; W e l s m a n , A r m s t r o n g , N e v i l l , W i n t e r , & K i r b y , 1996). T h e d e r i v e d magnitudes o f k for V o 2 m a x reported i n the literature range f r o m 0.41 to 0.80, i n d i c a t i n g that it is a data dependent parameter, and therefore it is not appropriate to use a c o m m o n parameter k i n the N L R M 1 m o d e l for V o 2 m a x . It is suggested here that researchers s h o u l d fit a N L R M 1 m o d e l to their data and derive a sample-dependent k value for the adjusted ratio v a r i a b l e V o 2 m a x i n each study. A l t h o u g h the N L R M 2 m o d e l  68  is not suggested for this specific data set, it s h o u l d be c o n s i d e r e d b y researchers as a p o s s i b l e deflation m o d e l . W h e n alternative deflation m o d e l s are used, special c a u t i o n must be taken w h e n c o m p a r i n g the means o f the adjusted ratio variables a m o n g repeated trials or different groups, and w h e n interpreting the m e a n i n g o f the actual values o f the adjusted ratio variables. F i r s t , w h e n the n o n l i n e a r regression m o d e l s are used, group or trial c o m p a r i s o n s are appropriate o n l y w h e n the exponents (k) for different groups or trials are i d e n t i c a l . F o r e x a m p l e , i n D a t a Set 1, because the magnitudes o f the parameter k o f the N L R M l m o d e l i n the three trials are s l i g h t l y different (i.e., k i = 0 . 5 0 , k = 0 . 4 5 , and 2  k = 0 . 4 1 ) , the scale o f the adjusted V o 2 m a x o f N L R M l i n the three trials are different. 3  T h a t i s , the scales o f the adjusted V o 2 m a x i n the three trials are m l - k g ~ ' - m i n ~ , 0  ml-kg"  0 4 5  50  1  - m i n " ' , and m l - k g ^ ' - m i n " , r e s p e c t i v e l y . T h u s , one c o u l d not c o m p a r e the three 1  trial means o f the adjusted V o 2 m a x d e r i v e d f r o m the N L R M l m o d e l under the different scales. T o c o m p a r e the three group means, the scale o f the adjusted ratio variable i n a l l trials or groups s h o u l d be i d e n t i c a l . In these data, a c o m m o n k parameter o f N L R M l can be obtained b y the average o f the three k parameters o v e r the three trials (i.e., k=(k,+k +k )/3). 2  3  A v e r a g i n g different values o f the k parameter is not the o n l y m e t h o d for getting the same unit scale i n different trials or groups. I f different groups are c o m p a r e d (not the same group i n different trials), the data c o u l d also be p o o l e d and a c o m m o n k can be d e r i v e d b y fitting the N L R M l m o d e l to the p o o l e d data. H o w e v e r , w h e n the N L R M l m o d e l is used to fit the data i n repeated measurement data l i k e D a t a Set 1, the p o o l e d data m a y not be v a l i d for this purpose because it w o u l d violate the assumption o f  69  independence o f the error variance o v e r subjects. T h a t i s , i f the repeatedly measured data w e r e p o o l e d , there w o u l d be a between-subject error c o r r e l a t i o n because the repeated measurements for the same subject were treated as different subjects i n the p o o l e d data. A s e c o n d c a u t i o n is that n o n l i n e a r regression m o d e l i n g y i e l d s adjusted ratio variables that m a y not be as easy to interpret as the r a w score c o m p o n e n t variables due to the c o m p l e x scale (e.g., m l - k g "  0 4 5  -min  _ 1  i n D a t a Set 1). H o w e v e r , the relative differences  a m o n g groups or trials are c o m p a r a b l e . T h e actual g r o u p o r trial differences can be d e s c r i b e d d i m e n s i o n l e s s l y a n d expressed as a percentage difference. F o r e x a m p l e , the s a m p l e means o f the adjusted V o 2 m a x u s i n g N L R M 1 w i t h c o m m o n k=0.45 i n the first a n d s e c o n d trials are 184.72 a n d 198.41, r e s p e c t i v e l y . T h i s difference c a n be expressed by s a y i n g that based o n the N L R M 1 m o d e l the first trial h a d 9 3 % o f the b o d y mass adjusted V o 2 m a x o f the s e c o n d t r i a l . It s h o u l d also be noted that the percentage difference c o m p a r i s o n depends o n w h i c h m o d e l is b e i n g e m p l o y e d .  Empirical Data Set 2:  DLCD/VA  T h i s data set w a s f r o m a research project c o n d u c t e d i n the Sports M e d i c i n e C e n t e r at the U n i v e r s i t y o f B r i t i s h C o l u m b i a ( B a c o n , 1997). O n e o f the purposes o f the research was to investigate whether there was a change i n " d i f f u s i n g c a p a c i t y o f the lungs for c a r b o n m o n o x i d e " ( D L C O ) w i t h i n an o v u l a t o r y menstrual c y c l e i n r e g u l a r l y menstruating w o m e n . In order to deflate the v a r i a t i o n o f the i n s p i r e d v o l u m e s , w h i c h was measured as a l v e o l a r v e n t i l a t i o n v o l u m e ( V A ) , the ratio v a r i a b l e D L C O / V A w a s used i n the study. T h i s procedure o f deflation is frequently used and D L C O / V A is a c o m m o n v a r i a b l e i n s i m i l a r studies i n the related literature (e.g., C r a p o & Forster, 1989). H o w e v e r , u n l i k e the  70  V o 2 m a x / k g ratio measure, the statistical a n d v a l i d i t y issues o f this ratio have n e v e r been addressed i n the literature. T h e r e w e r e 13 subjects i n the study, and D L C O a n d V A w e r e measured at five points d u r i n g the menstrual c y c l e . T h e f i v e test points were early menstrual, late m e n s t r u a l , e a r l y f o l l i c u l a r , m i d c y c l e , a n d m i d luteal. A o n e - w a y R M A N O V A (n=13, k=5) w a s u s e d to analyze these data. T a b l e 4-4 shows the d e s c r i p t i v e characteristics o f the data.  Simple Ratio Model The c r i t e r i a to evaluate the v a l i d i t y o f the m o d e l w i t h these data are the same as i n D a t a Set 1. A s s h o w n i n T a b l e 4 - 5 , the correlations b e t w e e n the m e a s u r e d D L C O a n d V A v a r y f r o m 0.29 to 0.59 i n the f i v e measurement trials. T h i s indicates the p o s i t i v e relationship b e t w e e n V A a n d D L C O , a n d thus an appropriate deflation m o d e l c o u l d be used to deflate the effect o f V A f r o m the D L C O data. The results s h o w negative correlations b e t w e e n the s i m p l e ratio D L C O / V A a n d V A i n the five trials, i n d i c a t i n g that the s i m p l e ratio m o d e l over-adjusts the effect o f V A f r o m the D L C O data. T h e scatterplots o f F i g u r e 4-4 also s h o w that the subjects w h o have greater V A tend to have s m a l l e r D L C O / V A (this trend w a s not c l e a r l y s h o w n i n the first trial), but there is n o suggestion o f a p o s s i b l e c u r v i l i n e a r r e l a t i o n s h i p between the s i m p l e ratio D L C O / V A a n d V A . T h e estimated e x p e c t e d v a l u e a n d the e m p i r i c a l means are not the same o v e r the five trials. T h e intraclass c o r r e l a t i o n s h o w s that the s i m p l e ratio D L C O / V A has a r e l i a b i l i t y ( r = 0 . 9 0 ) w h i c h is a p p r o x i m a t e l y the same as its c o m p o n e n t s . yy  In s u m m a r y , although the s i m p l e ratio m o d e l meets some o f the criteria, the ratio D L C O / V A m a y not be an appropriate m o d e l for d e f l a t i o n i n these data because it overdeflates the effect o f V A f r o m D L C O A n alternative deflation m o d e l s h o u l d be c o n s i d e r e d .  71  H  c3  H  H  H o  P  o  03  Q  H  Q  u 6  m  CO  •n d  i—t  CO  co  d  d  CN  oo  CO  d  d  (N 00 CO  oo  CO  d  d  »—1  1-H  CO .—1  co 1—1  <N CO  CN  VO  co  1—1  CO  d  co  H  in  T — 4 CN CN  CO  d  co CO  m  H  o (N CN CN  oo CN co  CO  m oo  i—1  d  d  o  m  H  CO  CO  CN  IT)  1—I  d  H  Q o o oo  vo  os  r-  O  d  T(-  CN  m  1—1  d  0.10  in  CO  i-H  2.23  H  co oo  VO  ON  H  >  CO  co  21.69  3  oo  vq co  H  O  o  in  co  CO 1—1  d m  d  co i—1  CO i-H  £  C O  'I  Va  00  m  H  in  H  -t—>  00  H  CO 1—1  d  . VO  m  >n  CO  in  d  CN oo  in  1—1  1—1  00  i-H  m m  Os o  co  PS  1  CM  d  i—i  CO  '>  co Q  os  0.14  >  OS 00  0.85  H  OS  5.95  ca  >  c  d  d  d  ^  CO  00  c  Mean  T3  T a b l e 4-5 E v a l u a t i o n o f the S i m p l e R a t i o M o d e l : D a t a Set 2  Ratio  r  Model  X , vs X  r 2  Y vs X  Estimated Expected  2  Value Y=Xi/X Trial 1  A  Empirical Mean  p  2  0.58  -0.08  Trial 2  0.29  -0.59  3.81 3.80  Trial 3  0.59  -0.35  3.69  3.91 3.74  Trial 4  0.49  -0.49  3.58  3.66  Trial 5  0.50  -0.68  3.61  3.70  3.82  Reliability Coefficient XI  x  2  Y  Intraclass r A l l Trials Single Trial  0.90 0.62  0.93 0.71  0.90 0.64  Y  ^  • •  • I /*  •  CO  I-  I—I—I—h  ID in t  n  01  •  co  co  W  co  • •  < >  •  4 ••l-t-)-lco CM ••- o  CM 10 T- in o  a>  •  +  CO 4 i I I II  •  H—I—h  (O U) f  CO CM  H  h  CO CM •>-  O  c\jT-oo>oor--io  to  <  H—I—t-  (O Ul ^  CO CM  < > H—I—tto CO  CO CM  CO CM ••- o  O) GO I  CO CO  O) CO N (O U)  <  •  •  •  •  CM CO  •4. CO LO ^  CO CM  •  •  +4  CO CM i -  < >  • <>  _l—l_  O  O CO CM CO CM ••- i -  L__H CO CM  O CM t -  00  74  Alternative Ratio Models T h r e e alternative deflation m o d e l s were evaluated u s i n g the same procedures used w i t h D a t a Set 1. A s s h o w n i n T a b l e 4-6, w h e n the alternative deflation m o d e l s ( L R M , N L R M 1 , and N L R M 2 ) are used, the adjusted ratio D L C O variables have a near-zero c o r r e l a t i o n w i t h V A . T h e scatterplots i n F i g u r e 4-4 do not s h o w a c u r v i l i n e a r r e l a t i o n s h i p between the adjusted ratio v a r i a b l e and its d e n o m i n a t o r variable, and the estimated e x p e c t e d value and the e m p i r i c a l m e a n o f the data are v i r t u a l l y the same. W h e n the N L R M 2 m o d e l was fitted to the data for the second, t h i r d , and fifth trials, the estimate o f the k parameters were s i m i l a r to those for the N L R M 1 m o d e l i n these three trials. H o w e v e r , w h e n the R was used to evaluate goodness o f fit for the m o d e l s w i t h these data, as s h o w n i n T a b l e 4 - 6 , the N L R M 2 does not have a better fit than the N L R M 1 m o d e l i n the three trials. W h e n the N L R M 2 m o d e l was fitted to the data for the first trial and fourth trial, the estimate o f the k parameter was u n i t y , and the N L R M 2 m o d e l is the same as the L R M m o d e l i n the t w o trials. It suggests that the N L R M 2 m o d e l does not have a better fit than the L R M m o d e l i n the t w o trials. T h e R for the N L R M 1 and 2  N L R M 2 m o d e l s i n the other three trials d i d not substantially differ. T h e L R M and N L R M 1 m o d e l s c o u l d be suggested for these data because they are m o r e p a r s i m o n i o u s than the N L R M 2 m o d e l . It is noted that the s a m p l e size is s m a l l i n D a t a Set 2 (n=13). I f the sample size is s m a l l , it m a y result i n very different magnitudes o f the parameters i n the fitting m o d e l (especially for the n o n l i n e a r regression m o d e l s ) for s i m i l a r data i n different trials. F o r e x a m p l e , the magnitude o f the estimated parameter k i n the N L R M 1 m o d e l ranged f r o m 0.30 to 0.93 o v e r the five trials i n D a t a Set 2. T h e serious i n c o n s i s t e n c y o f the estimated  75  Table 4-6 Evaluation of the Alternate Models: Data Set 2 r Y vs X  2  Reliability Intraclass r  R  Estimated Expected Value(3  Empirical Mean Y  a  k  A  All Trials  Y=(X,-a)/X  2  (LRM)  Trial 1 Trial 2 Trial 3 Trial 4 Trial 5  -0.010 0.003 0.040 -0.010 0.005  Y=X,/X  0.90  Single Trial  0.62  0.34 0.10 0.35 0.24 0.25  3.51 1.16 2.37 1.88 1.30  3.51 1.16 2.37 1.89 1.30  0.62  0.34 0.10 0.35 0.24 0.25  4.34 13.23 7.13 8.46 11.39  4.34 13.23 7.12 8.46 11.39  0.60  0.35 0.09 0.36 0.24 0.25  3.51 13.03 7.81 1.88 12.60  3.51 13.02 8.01 1.88 13.02  1.76 15.52 8.00 10.22 13.97  k  :  (NLRM1)  Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Y=(X -a)/X 1  -0.0200 -0.0003 0.0300 -0.0100 0.0003  0.89  0.93 0.30 0.64 0.52 0.36  k 2  (NLRM2)  Trial 1 Trial 2 Trial 3 Trial 4 Trial 5  -0.010 -0.001 0.050 -0.010 0.040  0.88  1.76 0.22 -1.00 10.22 -1.00  1.00 0.30 0.61 1.00 0.35  76  parameters c o u l d reduce the r e l i a b i l i t y o f the adjusted ratio variable data because a c o m m o n k has to be used to obtain the ratio scores i n different trials, and the l o w r e l i a b i l i t y w i l l affect the v a l i d i t y o f u s i n g the deflation m o d e l . A larger sample size m a y be needed to reduce the i n c o n s i s t e n c y o f the parameters i n these data.  A Summary of the Evaluation of the Deflation Models F i r s t , one s h o u l d not assume that a s i m p l e ratio m o d e l is v a l i d for deflation purposes, but rather that the procedures d e v e l o p e d i n this study c o u l d be used to determine a best deflation m o d e l . S e c o n d , the v a l i d i t y o f a deflation m o d e l depends o n the sample size and the statistical characteristics o f the particular c o m p o n e n t variables used, and an o p t i m a l deflation m o d e l for a l l variables m a y not exist. Different deflation m o d e l s s h o u l d be used to fit each e m p i r i c a l data set for d e t e r m i n i n g the best deflation m o d e l . T h i r d , to derive a best deflation m o d e l , the f o l l o w i n g criteria c o u l d be considered: (a) zero c o r r e l a t i o n between the ratio variable and the d e n o m i n a t o r v a r i a b l e , (b) no c u r v i l i n e a r r e l a t i o n s h i p between the ratio variable and the d e n o m i n a t o r v a r i a b l e , (c) equality o f the estimated expected value o f the m o d e l and the e m p i r i c a l m e a n o f the ratio variable data, (d) h i g h R i n the deflation m o d e l , and (e) h i g h r e l i a b i l i t y o f the ratio 2  variable data.  Ratio Variables Used Not for Deflation Purposes A s p r e v i o u s l y m e n t i o n e d , i n h u m a n k i n e t i c s research, ratio variables are also used as a measurement variable o f direct interest, w i t h o u t any intent to derive a denominator-free variable. In this case, a ratio variable has some special aspect not accounted for b y the numerator or d e n o m i n a t o r variables i n d i v i d u a l l y . T h e m a i n c o n c e r n  77  w h e n u s i n g this k i n d o f ratio variables as a measurement variable is the r e l i a b i l i t y o f the ratio v a r i a b l e and the effect o f c o v a r i a t i o n o f the c o m p o n e n t variables i n a statistical analysis. In this section, t w o e m p i r i c a l studies were e x a m i n e d and the r e l i a b i l i t y issue o f u s i n g this type o f ratio variable was addressed. T h e effect o f c o v a r i a t i o n o f the c o m p o n e n t variables i n a statistical analysis was investigated, u s i n g s i m u l a t i o n procedures, i n C h a p t e r V .  Empirical Data Set 3: Nondominant Torque/Dominant Torque T h e s e data are f r o m a research project c o n d u c t e d b y K r a m e r , N u s c a , B i s b e e , M a c D e r m i d , K e m p , and B o l e y (1994). T h e study e x a m i n e d the r e l i a b i l i t y o f f o r e a r m pronation and s u p i n a t i o n i s o m e t r i c strength (torques) i n absolute units and n o n d o m i n a n t / d o m i n a n t ratios. T w e n t y - o n e m e n and 2 2 w o m e n were tested w i t h t w o repetitions i n t w o o c c a s i o n s on each o f t w o m a c h i n e s , a B T E w o r k s i m u l a t o r and a C y b e x d y n a m o m e t e r . T h e data w e r e c e i v e d f r o m the authors were c o l l e c t e d f r o m the B T E w o r k simulator. T o reduce the v a r i a b i l i t y o f the measurement, the m e a s u r e d strengths o f the t w o repetitions i n each o c c a s i o n were averaged. T h e n , the absolute torques o f the d o m i n a n t a r m and n o n d o m i n a n t arm, and the n o n d o m i n a n t / d o m i n a n t ratio o n the t w o occasions were c a l c u l a t e d . T h e r e l i a b i l i t y o f the c o m p o n e n t variables and the n o n d o m i n a n t / d o m i n a n t ratio was evaluated. T a b l e s 4-7 and 4-8 show the characteristics o f the ratio v a r i a b l e and the c o m p o n e n t variables for the forearm p r o n a t i o n and supination, r e s p e c t i v e l y . T h e results show that the absolute torques e x h i b i t e d large between subject v a r i a b i l i t y ( V > 0 . 4 0 ) , and h i g h r e l i a b i l i t i e s (0.94-0.98). H o w e v e r , w i t h the n o n d o m i n a n t / d o m i n a n t ratio v a r i a b l e  78  T a b l e 4-7 M e a n s , Standard D e v i a t i o n s , and R e l i a b i l i t y C o e f f i c i e n t s o f the T o r q u e s and N o n d o m i n a n t / D o m i n a n t R a t i o (Pronation) Pronation Torques  ND1  a  ND2  DI  D2  RATIO-1  RATIO-2  Mean  8.01  8.24  8.95  8.98  0.91  0.94  Sd  3.71  3.58  4.12  3.97  0.22  0.16  V  0.46  0.43  0.46  0.44  0.24  0.17  Reliability A l l Trials  0.97  0.94  0.68  Single Trial  0.94  0.89  0.51  ND1: The torque of the non-dominant arm in the first trial; ND2: The torque of the non-dominant arm in the second trial; D I : The torque of the dominant arm in the first trial; D2: The torque of the dominant arm in the second trial; RATIO-1: The non-dominant/dominant ratio in the first trial; RATIO-2: The nondominant/dominant ratio in the second trial. a  T a b l e 4-8 M e a n s , Standard D e v i a t i o n s , and R e l i a b i l i t y C o e f f i c i e n t s o f the T o r q u e s and N o n d o m i n a n t / D o m i n a n t R a t i o (Supination) Supination Torques  ND1  ND2  DI  D2  RATIO-1  RATIO-2  Mean  7.49  7.67  8.17  8.19  0.93  0.97  Sd  3.31  3.27  3.47  3.67  0.18  0.25  V  0.44  0.43  0.42  0.45  0.19  0.26  Reliability A l l Trials  0.95  0.98  0.75  Single Trial  0.90  0.96  0.61  79  the coefficients o f v a r i a t i o n were r e d u c e d c o n s i d e r a b l y (0.26 > V > 0.17), as were the r e l i a b i l i t i e s (0.68 and 0.75 for the average o f the t w o trials, and 0.51 and 0.61 for a single trial). T h e results s h o w that the ratio transformation substantially reduces the r e l i a b i l i t y i n these data e v e n though the r e l i a b i l i t y o f the c o m p o n e n t variables i s h i g h . A s s h o w n i n F o r m u l a (7) i n the literature r e v i e w chapter, the r e l i a b i l i t y o f ratio variables i n a single trial is affected b y not o n l y the coefficient o f v a r i a t i o n but also b y the w i t h i n and between trial correlations o f the c o m p o n e n t variables. Therefore, although the c o m p o n e n t variables are v e r y r e l i a b l e i n these data, the r e l i a b i l i t y o f the ratio v a r i a b l e reduces substantially because o f the effect o f the w i t h i n and between trial correlations o f the c o m p o n e n t variables. T h e details o f the d i s c u s s i o n o n h o w these factors affect the r e l i a b i l i t y o f the ratio v a r i a b l e are presented i n a f o l l o w i n g section. T h e results i n the D a t a Set 3 suggest that one s h o u l d not assume that ratio data are r e l i a b l e m e r e l y because the c o m p o n e n t variables are r e l i a b l e . T h e r e l i a b i l i t y o f the ratio data s h o u l d be c o m p u t e d w h e n e v e r the ratio transformation is c o n d u c t e d .  Empirical Data Set 4; Waist Girth/Hip Girth T h e s e data were obtained f r o m a study c o n d u c t e d i n the N e w Y o r k U n i v e r s i t y M e d i c a l C e n t e r ( S o n n e n s c h e i n , K i m , Pasternack, & T o n i o l o , 1993). T h e subjects were 35-65 year o l d w o m e n w h o v i s i t e d a breast screening c l i n i c and e n r o l l e d i n the N e w Y o r k U n i v e r s i t y W o m e n ' s H e a l t h S t u d y between 1985 and 1991 (n=14,290). T h e purpose o f the o r i g i n a l study was to investigate the relationship between endogenous h o r m o n e s and cancer. T h e l o n g i t u d i n a l data i n c l u d e d the f o l l o w i n g i n f o r m a t i o n : A self-administered questionnaire w h i c h gathered d e m o g r a p h i c , m e d i c a l , lifestyle, and r e p r o d u c t i v e  i n f o r m a t i o n , i n c l u d i n g current m e n o p a u s a l status and self-reported height and w e i g h t ; a self-administered semi-quantitative f o o d frequency questionnaire; and waist and h i p girths measured for 1,851 cohort m e m b e r s at each c l i n i c visit. The w a i s t / h i p ratio has recently b e c o m e a c o m m o n measure i n health science research, as it p u r p o r t e d l y reflects fat d i s t r i b u t i o n i n the b o d y . H i g h ratio scores are i n d i c a t i v e o f health r i s k s associated w i t h obesity. Therefore, the data used i n our study were the w a i s t and h i p girth measures. A l l 1,851 subjects had the first three l o n g i t u d i n a l measurements o f w a i s t and h i p girths, 693 subjects h a d the first four measurements, and o n l y 109 subjects had a l l five measurements.  T o m a x i m i z e the n u m b e r o f subjects i n the  e x a m i n a t i o n o f r e l i a b i l i t y , o n l y the measurement data i n the first three trials were i n c l u d e d i n this i n v e s t i g a t i o n . T a b l e 4-9 s h o w s the d e s c r i p t i v e statistics for the h i p , w a i s t and ratio, for a l l three trials. T h e coefficients o f v a r i a t i o n for the waist and h i p girths e x h i b i t e d very little change o v e r the three trials, r a n g i n g f r o m 0.15-0.16 for the waist girth and 0.10-0.11 for the h i p girth. It indicates that w a i s t girth is m o r e v a r i a b l e than h i p girth i n these data ( V i s t > V h i ) . T h e coefficients o f v a r i a t i o n o f the ratio v a r i a b l e (waist/hip) were w a  P  s o m e w h a t l o w e r , r a n g i n g f r o m 0.09-0.10 o v e r the three trials. T h e r e l i a b i l i t y o f the ratio variable (0.89 for a l l trials and 0.73 for a single trial) was still quite high although it was l o w e r than that o f the c o m p o n e n t variables.  A Discussion on the Reliability of the Ratio Variable: Data Sets 3 and 4 The results o f this study show that the w a i s t / h i p ratio v a r i a b l e i n D a t a Set 4 has m u c h higher r e l i a b i l i t y than the n o n d o m i n a n t / d o m i n a n t ratio i n D a t a Set 3, although the r e l i a b i l i t y o f the c o m p o n e n t variables are h i g h i n both data sets. A s s h o w n i n T a b l e s 4-7,  81  T a b l e 4-9 M e a n s , Standard D e v i a t i o n s , and R e l i a b i l i t y C o e f f i c i e n t s o f the W a i s t and H i p G i r t h , a n d Waist / H i p Ratio Waist  Hip  W/H Ratio  Tl  T2  T3  Mean  74.93  76.27  Sd  11.64  V  0.16  •  Tl  T2  T3  Tl  T2  T3  76.74  97.71  100.83  100.96  0.77  0.76  0.76  11.97  11.80  10.77  10.00  9.86  0.07  0.07  0.07  0.16  0.15  0.11  0.10  0.10  0.09  0.10  0.10  Reliability A l l Trials  0.96  0.94  0.89  Single Trial  0.89  0.84  0.73  82  4-8, and 4-9, the coefficients o f variation o f the c o m p o n e n t variables i n D a t a Set 4 w e r e m u c h s m a l l e r than i n D a t a Set 3. H o w e v e r , as s h o w n i n F o r m u l a (7), the r e l i a b i l i t y o f a ratio variable has a c o m p l e x relationship w i t h the coefficients o f v a r i a t i o n and the w i t h i n and between trial c o r r e l a t i o n o f the c o m p o n e n t variables. It is not i m m e d i a t e l y o b v i o u s w h y the r e l i a b i l i t y o f the ratio i n D a t a Set 4 is so m u c h higher than that o f the ratio (pronation) i n D a t a Set 3. T a b l e s 4-10 and 4-11 presents the c o m p a r a t i v e values for a l l the statistics that c o u l d contribute to this effect. It can be seen that a l l correlations are higher i n D a t a Set 3, and yet the r e l i a b i l i t y o f the ratio is l o w e r . It was speculated that the higher r e l i a b i l i t y i n D a t a Set 4 c o u l d be due to either the l o w e r coefficients o f v a r i a t i o n , or the m u c h larger difference o f the coefficients o f v a r i a t i o n between the numerator and d e n o m i n a t o r variables. T o e x p l o r e these p o s s i b i l i t i e s , D a t a Set 3 (the pronation variable o n l y ) was u s e d to create three a d d i t i o n a l data c o n d i t i o n s b y v a r y i n g the magnitudes o f the coefficients o f variation. T h e r e l i a b i l i t y o f the ratio variable was then a p p r o x i m a t e d for each c o n d i t i o n b y a p p l y i n g F o r m u l a (7) (a v a l i d a t i o n o f the accuracy o f F o r m u l a s (4), (5), and (7) is given i n A p p e n d i x C ) . T h e w i t h i n and between trial correlations were set to the o r i g i n a l values i n D a t a Set 3 (as i n T a b l e 4-10) for a l l the three a d d i t i o n a l c o n d i t i o n s . T h e three sets o f the coefficients o f variation were set based on the values i n D a t a Set 4 to reflect the f o l l o w i n g situations; (a) smaller, but a l l equal, values, (b) numerator value larger than d e n o m i n a t o r value, and (c) numerator value s m a l l e r than denominator value. T a b l e 4 - 1 2 shows these values. In C o n d i t i o n 1, although the coefficient o f variation for both the numerator and the d e n o m i n a t o r variables was reduced f r o m a p p r o x i m a t e l y 0.45 to 0.16, the results s h o w  Table 4-10 C o r r e l a t i o n s i n D a t a Set 3 a n d D a t a Set 4 Correlation Components  Xiand X  Correlation Data 3  Data 4  Difference  2  trial 1  r i1x21  0.85  0.83  0.02  Xiand X2 trial 2  Tx12x22  0.92  0.80  0.12  Xi trial 1 a n d trial 2  1x11x12  0.94  0.89  0.05  x trial 1 a n d trial 2  Tx21x22  0.89  0.84  0.05  Tx11x22  0.88  0.71  0.17  Tx12x21  0.84  0.79  0.05  x  2  X i trial 1 a n d X  2  trial 2  X i trial 2 a n d X  2  trial 1  T a b l e 4-11 Coefficients o f V a r i a t i o n i n D a t a Set 3 a n d D a t a Set 4 Coefficient o f Variation Data 3  Data 4  Difference  0.46  0.16  0.30  0.16  0.27  Numerator trial 1  v  trial 2 Denominator trial 1  v  x l 2  0.43  v  x 2  ,  0.46  0.11  0.35  x 2 2  0.44  0.10  0.34  Vyl v Reliability  0.24  0.09  0.15  0.17  0.10  0.07  ^single  0.51  0.73  -0.22  trial 2  v  x  n  Ratio trial 1 trial 2 Ratio  y ?  84  Table 4-12 R e l i a b i l i t y o f the R a t i o V a r i a b l e i n D a t a Set 3 under Different C o e f f i c i e n t o f V a r i a t i o n Conditions Data  Condition 1  Condition 2  Condition 3  3  v =v  v „ v  0.46 0.43  0.16 0.16  0.16 0.16  0.11 0.11  trial 1 trial 2  v , v  0.46 0.44  0.16 0.16  0.11 0.10  0.16 0.16  Ratio  Reliability  0.51  0.50  0.72  0.60  Set  xl  x2  V i>V x  x 2  v ,<v x  x2  Numerator trial 1 trial 2  x  x l 2  Denominator x2  x 2 2  Tsingle  N o t e . A l l correlation values are identical to those listed under D a t a Set 3 i n T a b l e  4-10.  85  that the r e l i a b i l i t y o f the d e r i v e d ratio variable d i d not increase ( r j i = 0 . 5 0 ) . T h i s result s  ng  e  is e x p e c t e d i f w e set V n = V 2 i = V i 2 = V 2 2 , F o r m u l a (7) is s i m p l i f i e d to: x  (PJC x Px -M1-M2  X  '  X  x )"(P, ,  +  VF  x  2\ 22  x  +P  x  x )  X  ' \\ 22  x  2\ \2  x  x  x  P~P^-P,^ )  2  2  (24) In F o r m u l a (24), the coefficients o f v a r i a t i o n are c a n c e l l e d out, a n d the r e l i a b i l i t y o f the ratio variable becomes o n l y a f u n c t i o n o f the w i t h i n and between trial correlations o f the numerator and d e n o m i n a t o r variables. T h e four coefficients o f v a r i a t i o n i n D a t a Set 3 w e r e a p p r o x i m a t e l y equal (0.43<V<0.46), and thus the s i m i l a r v a l u e ( r i i = 0 . 5 1 ) for the S  ng  e  r e l i a b i l i t y o f the ratio v a r i a b l e w i t h those data. T h i s indicates that the r e l i a b i l i t y o f the ratio variable is not affected b y the coefficient o f v a r i a t i o n i f the coefficients o f v a r i a t i o n o f the c o m p o n e n t variables are the same. In C o n d i t i o n 2, the coefficient o f v a r i a t i o n o f the numerator variable was higher than that o f the d e n o m i n a t o r v a r i a b l e , the values b e i n g set e q u a l to those o f trials 1 a n d 2 o f D a t a Set 4. A s s h o w n i n T a b l e 4 - 1 2 , the r e l i a b i l i t y o f the ratio variable increased f r o m 0.51 to 0.72, a value v e r y c l o s e to the o b s e r v e d ratio r e l i a b i l i t y o f D a t a Set 4 (0.73). In C o n d i t i o n 3, the coefficient o f v a r i a t i o n o f the numerator v a r i a b l e was l o w e r than that o f the d e n o m i n a t o r v a r i a b l e , a n d the r e l i a b i l i t y o f the ratio v a r i a b l e s t i l l increased b e y o n d the 0.51 value o f D a t a Set 3, but o n l y to 0.60. G i v e n that the w i t h i n and between trial correlations o f the c o m p o n e n t variables d i d not change, the results o f these t w o c o n d i t i o n s suggest that the r e l i a b i l i t y o f a ratio variable is h i g h i f the coefficients o f variation o f the c o m p o n e n t variables are different, and the effect is most p r o n o u n c e d w h e n the numerator variable has the h i g h e r coefficient o f v a r i a t i o n . A c o m p a r i s o n o f D a t a Set 4 a n d C2 is o f interest. C2 and D a t a Set 4 have i d e n t i c a l  86  coefficients o f v a r i a t i o n , but differ i n the values o f the correlations ( C has correlations 2  equal to those o f D a t a Set 3). A s can be seen i n T a b l e 4 - 1 0 , a l l o f the correlations o f D a t a Set 3 are greater than those o f D a t a Set 4, some b y as m u c h as 0.17. H o w e v e r , this has v e r y little effect on the r e l i a b i l i t y o f the ratios, as they are almost i d e n t i c a l (0.72 i n C o n d i t i o n 2 and 0.73 i n D a t a Set 4). T h e results indicate that the unequal coefficients o f variation o f the numerator and d e n o m i n a t o r variables result i n the higher r e l i a b i l i t y o f the ratio variable i n D a t a Set 4 than i n D a t a Set 3.  A Summary of the Evaluation of the Ratio Variables not for Deflation In h u m a n k i n e t i c s research, ratio variables s i m i l a r to those i n D a t a Set 3 and D a t a Set 4 are c o m m o n l y used, h o w e v e r little or no c o n s i d e r a t i o n is g i v e n to the r e l i a b i l i t y o f these d e r i v e d measures.  T h e results o f this study have c l e a r l y s h o w n that even i f the  c o m p o n e n t variables are h i g h l y r e l i a b l e the d e r i v e d ratio variable data m a y have l o w r e l i a b i l i t y . T h e consequences o f such reduction i n r e l i a b i l i t y c o u l d s e r i o u s l y affect the v a l i d i t y o f e m p i r i c a l research, because unless the repeated measurements are r e l i a b l e research d e c i s i o n s based on these data m a y be questionable. F o r e x a m p l e , l o w r e l i a b i l i t y results i n an increased error term (for the R M effect) i n a R M A N O V A , thus r e d u c i n g statistical p o w e r . A s was s h o w n w i t h D a t a Sets 3 and 4, the r e l i a b i l i t y o f the ratio variables d e r i v e d f r o m c o m p o n e n t measures are affected b y the coefficient o f v a r i a t i o n , w i t h i n and between trial correlations between the numerator and d e n o m i n a t o r variables (see T a b l e s 4-10 and 4 - 1 1 ) . Therefore, w h e n e v e r a ratio variable is used, researchers s h o u l d compute the r e l i a b i l i t y o f the d e r i v e d ratio scores, and not assume that strong reliabilities i n the c o m p o n e n t measures a u t o m a t i c a l l y lead to h i g h r e l i a b i l i t y i n the ratio  87  measures.. I f the r e l i a b i l i t y o f the ratio variable is too l o w , the v a l i d i t y o f the research m a y be c o m p r o m i s e d and an alternative approach s h o u l d be considered. F o r e x a m p l e , i n the case o f n o n d o m i n a n t / d o m i n a n t ratio, the strength o f the n o n d o m i n a n t and d o m i n a n t arms m a y have to be a n a l y z e d separately.  88  CHAPTER V. CHARACTERISTICS OF RATIO VARIABLES AS A FUNCTION OF T H E COMPONENT VARIABLES: A SIMULATION STUDY  T h i s chapter reports o n h o w the characteristics o f the c o m p o n e n t variables affect the c i r c u l a r i t y c o n d i t i o n o f the c o v a r i a n c e m a t r i x and type I error rates w h e n a ratio v a r i a b l e is used i n o n e - w a y R M A N O V A .  A s m e n t i o n e d i n the methods a n d procedures  chapter, I first generated the p o p u l a t i o n s o f the numerator a n d d e n o m i n a t o r variables b a s e d o n the c o n t r o l l e d characteristics o f the t w o c o m p o n e n t variables (see T a b l e 3-3). T h e data for each o f the 9 0 , 0 0 0 cases w i t h i n a p o p u l a t i o n c o n s i s t e d o f a n u m e r a t o r a n d d e n o m i n a t o r v a r i a b l e for each o f the five repeated observations (hereafter referred to as trials). T h e ratio v a l u e , X i / X 2 i ( i = l , 9 0 0 0 0 , i = l , . . . , 5), was then c o m p u t e d for each o f the f i v e p a i r e d X n a n d X 2 i values for each case. T h e effects o f the characteristics o f the c o m p o n e n t variables o n the m a g n i t u d e o f e p s i l o n (e), the measure o f the extent to w h i c h the c o v a r i a n c e m a t r i x o f the ratio v a r i a b l e departs f r o m c i r c u l a r i t y , were e x a m i n e d for each o f the transformed ratio score p o p u l a t i o n s . T h e n , a M o n t e C a r l o s i m u l a t i o n procedure was used to investigate the s a m p l i n g characteristics o f e i / 2 under different A  x  X  s a m p l e sizes. F i n a l l y , the effect o f u s i n g ratio variables o n the type I error rate was investigated.  Population Characteristics of The Effect of  £ i/ 2 x  X  V*i/V 2 and p i 2 on the Magnitude of e^^i  Eyj,  V  Y  V  T a b l e 5-1 presents the values o f £ i/ 2 for each o f the 33 s i m u l a t e d ratio variable x  X  p o p u l a t i o n s . T h e values o f £ i/ 2 for the c o v a r i a n c e m a t r i x o f the ratio p o p u l a t i o n data are x  X  89  T a b l e 5-1 P o p u l a t i o n V a l u e s o f e^mj for the R a t i o V a r i a b l e X j / X ? under Different Variable  Component  Conditions V /V x l  x 2  =2.0  V /V x l  x 2  =1.0  V /V =0.5 x l  x 2  Exi=Ex2=l-0 Pxix2=0.9  1.00  1.00  1.00  Pxix2=0.7  1.00  1.00  1.00  p x 1x2=0.5  1.00  1.00  1.00  Pxix2=0.9  0.70  0.73  0.72  Pxix2=0.7  0.70  0.71  0.72  Pxix2=0.5  0.70  0.71  0.72  Pxix2=0.9  0.59  0.87  0.86  Pxix2=0.7  0.64  0.91  0.96  Pxix2=0.5  0.70  0.91  0.99  P x 1X2=0.7  0.78  0.81  0.71  Pxix2=0.5  0.99  0.84  0.72  £xi=£ 2=0.7 X  e i = 0 . 7 Ex2=1.0 X  £xi=1.0 £ =0.7 X 2  90  presented for each o f the three c o n d i t i o n s o f e q u a l i t y / i n e q u a l i t y o f the coefficients o f v a r i a t i o n o f the c o m p o n e n t variables, four e p s i l o n ( e i , e 2) c o n d i t i o n s , and three levels o f x  c o r r e l a t i o n between X i and X  2  X  (p i 2). x  X  Magnitude of e i/^ when the Ratio Components Meet the Assumption of Y  Circularity T h e results i n T a b l e 5-1 s h o w that the data i n the transformed ratio score p o p u l a t i o n meet the a s s u m p t i o n o f c i r c u l a r i t y w h e n the p o p u l a t i o n s o f the numerator and d e n o m i n a t o r variables meet the a s s u m p t i o n (i.e., the p o p u l a t i o n e o f the ratio v a r i a b l e equals unity e i / x  x 2  = 1.0, w h e n e i = e x  x2  = 1.0). T h i s indicates that w h e n both the  p o p u l a t i o n e values o f the c o m p o n e n t variables equal unity (i.e., perfect c i r c u l a r i t y ) , the ratio transformation does not affect the c i r c u l a r i t y o f the c o v a r i a n c e m a t r i x , a n d the h o m o g e n e i t y o f c o v a r i a n c e c o n d i t i o n s t i l l h o l d s i n the p o p u l a t i o n o f a ratio variable. T h i s h o m o g e n e i t y h o l d s regardless o f the l e v e l s o f c o r r e l a t i o n and the relative v a r i a t i o n between the numerator and d e n o m i n a t o r variables (i.e., p i 2 a n d V i / V 2 ) . x  Magnitude of  e i/ 7. v  Y  x  X  X  when the Ratio Components Violate the Assumption of  Circularity A s s h o w n i n T a b l e 5-1, w h e n the p o p u l a t i o n c o v a r i a n c e matrices o f the numerator a n d d e n o m i n a t o r variables have the same degree o f v i o l a t i o n o f c i r c u l a r i t y ( e i = £ x  0.7), the e i / x  x 2  x 2  =  o f the ratio p o p u l a t i o n c o v a r i a n c e m a t r i x does not deviate substantially  f r o m the 8 o f the c o m p o n e n t v a r i a b l e p o p u l a t i o n c o v a r i a n c e matrices. T h e value o f £ i / x  x2  increased s l i g h t l y (from 0.70 to 0.72) w h e n the coefficient o f v a r i a t i o n o f the numerator variable decreased relative to the d e n o m i n a t o r v a r i a b l e (i.e., V i / V x  x 2  c h a n g e d f r o m 2.0 to  91  0.5), regardless o f the levels o f the correlation (p i 2)- T h e c o r r e l a t i o n between the x  X  numerator and d e n o m i n a t o r variables, p i 2, has n o substantial effect on e i x  X  x  / x 2  (except  £xi/x2 was s l i g h t l y higher w h e n V i / V = 1 . 0 , p i 2=0.9) i n the c o n d i t i o n £ i = £ 2 = 0.7. x  x 2  x  X  X  x  In the £ i = 0 . 7 a n d £ 2 = 1 . 0 c o n d i t i o n , the value o f £ i/ 2 increased c o n s i d e r a b l y as x  X  x  X  the coefficient o f v a r i a t i o n o f the numerator v a r i a b l e decreased relative to the d e n o m i n a t o r v a r i a b l e . F o r e x a m p l e , w h e n p i 2=0.9 a n d V i / V 2 decreased f r o m 2.0 to x  0.5, the value o f £ i x  /x2  X  X  x  increased f r o m 0.59 to 0.86. In the other t w o p i x  e x h i b i t e d a s i m i l a r pattern o f change. A d d i t i o n a l l y , the p o p u l a t i o n £ , / x  x 2  conditions,  £ ] X  / X  2  is affected b y the  x 2  correlation between the numerator a n d d e n o m i n a t o r variables. A s s h o w n i n T a b l e 5-1, w h e n the p i x  x 2  decreased f r o m 0.9 to 0.5, the value o f £ i/ 2 increased, e s p e c i a l l y w h e n the x  X  coefficients o f v a r i a t i o n o f the c o m p o n e n t variables were not e q u a l . T o s y s t e m a t i c a l l y e x a m i n e the characteristics o f e i / , a n d to understand h o w the x  t w o factors, p i x  and V i / V  x 2  x  x 2  x 2  , affect £ i/ 2, the c o n d i t i o n £ i=1.0 and £ 2=0.7 w a s also x  X  x  X  i n c l u d e d i n the £ i/ 2 i n v e s t i g a t i o n . W h e n this c o n d i t i o n was i n c l u d e d i n this study, it was x  X  a c k n o w l e d g e d that the d e n o m i n a t o r variable u s u a l l y does not s h o w a greater v i o l a t i o n o f the a s s u m p t i o n o f c i r c u l a r i t y than the numerator v a r i a b l e w i t h h u m a n kinetics research data. F o r e x a m p l e , w h e n a ratio variable is used for deflation i n our f i e l d , o n l y the numerator v a r i a b l e is e x p e c t e d to be affected b y the treatment, and the v a r i a t i o n o f the d e n o m i n a t o r is u s u a l l y not affected (e.g., V o 2 m a x / k g ) . F o r those ratio variables not used for deflation purposes, the numerator and d e n o m i n a t o r variable u s u a l l y have a s i m i l a r variance pattern (e.g., w a i s t / h i p girth, n o n d o m i n a n t / d o m i n a n t strength). In a d d i t i o n , the s i m u l a t i o n p r o g r a m was unable to generate the p o p u l a t i o n data w h e n p i 2=0.9, because x  X  92  the d e r i v e d c o v a r i a n c e m a t r i x for the ratio v a r i a b l e w a s not p o s i t i v e definite.  Therefore,  the subsequent i n v e s t i g a t i o n o n the s a m p l i n g characteristics o f e i/ 2 w i l l not i n c l u d e this X  x  c o n d i t i o n , a n d the results reported for the c o n d i t i o n e i = 1 . 0 a n d e 2=0.7 are based o n o n l y X  x  t w o c o r r e l a t i o n c o n d i t i o n s ( p i 2 = 0 . 7 and p i 2 = 0 . 5 ) . x  X  x  X  In the c o n d i t i o n e i = 1 . 0 and e = 0 . 7 , the results s h o w that £ i / x  x2  x  decreased w h e n  x 2  the ratio o f the coefficients o f variation o f the c o m p o n e n t variables decreased f r o m 2.0 and 1.0 to 0.5 w h e n p i 2 = 0 . 7 , and decreased constantly f r o m V i / V = 2 . 0 to 0.5 for x  X  x  x 2  Pxix2=0.5. T h e pattern o f e i/ 2 over the three v a r i a t i o n c o n d i t i o n s is the opposite f r o m the X  x  pattern i n the £ i = 0 . 7 a n d £ 2=1.0 c o n d i t i o n . T h e results also s h o w that e i/ 2 i n c r e a s e d X  x  X  x  w h e n p i 2 decreased i n a l l three v a r i a t i o n c o n d i t i o n s . T h i s effect o f p i 2 o n E i / 2 w a s x  X  x  X  x  X  the same i n both e c o n d i t i o n s ( e = 0 . 7 a n d e = 1 . 0 , and £ i = 1 . 0 a n d e = 0 . 7 ) . x]  x2  x  x2  G i v e n the c o v a r i a n c e structure a n d the interrelationship o f the c o m p o n e n t variables s h o w n i n the methods and procedures chapter, the results indicate that the p o p u l a t i o n e o f a ratio variable ( e v 2 ) is a f u n c t i o n o f e i and e 2, the c o r r e l a t i o n between x  X  X  x  the c o m p o n e n t variables ( p [ 2 ) , and the relative v a r i a t i o n o f the c o m p o n e n t variables x  X  ( V i / V 2 ) . I f the matrices o f the c o m p o n e n t variables have n o v i o l a t i o n , or the same x  X  degree o f v i o l a t i o n , o f the assumption o f c i r c u l a r i t y ( e i = e ) , the results indicate that the x  x2  magnitude o f the e i/ 2 is v i r t u a l l y the same as that o f e i and e , regardless the l e v e l o f x  p  x l x 2  and V i / V x  x 2  .  X  x  x2  H o w e v e r , w h e n the c o m p o n e n t variables have different degrees o f  v i o l a t i o n o f the a s s u m p t i o n o f c i r c u l a r i t y , one can not assume that the structure o f the c o v a r i a n c e m a t r i x o f the ratio variable is a s i m p l e function o f the c o m p o n e n t variables. T h e results i n d i c a t e that E i / 2 has a negative r e l a t i o n s h i p w i t h p i 2 and V ] / V 2 , w h e n x  X  x  X  X  X  93  o n l y the c o v a r i a n c e matrix o f the numerator variable violates the assumption o f c i r c u l a r i t y . T h a t is, £ i 2 tends to have s m a l l e r magnitudes w h e n p i x  /X  x  h i g h , and greater magnitudes w h e n p i x  x 2  and V  x ]  /V  x 2  x 2  and V i / V x  x  negative relationship w i t h p x l x 2 but a p o s i t i v e relationship w i t h V i / V x  xi  x  X  x  are  are l o w . If o n l y the c o v a r i a n c e  m a t r i x o f the d e n o m i n a t o r variable violates the assumption o f c i r c u l a r i t y , £ i /  The Effect of e , V i/V 2 and p i  x 2  x 2  x 2  has a  .  on the Magnitude of P( i/ 2)y  x2  X  X  Researchers i n h u m a n kinetics u s u a l l y measure raw score variables i n repeated measurement experiments and have some k n o w l e d g e o f the magnitudes o f the correlation between variables i n repeated trials. H o w e v e r , the correlation between t w o ratio variables i n a repeated measures design (i.e., p( i/x2)ij, i ^ j ) m a y not be the same as the X  correlation between raw score variables (i.e., P( i)ij or P( )ij, i ^ j ) . In this section, I report X  x2  how certain factors affect the correlation between ratio variables i n repeated  measurement  situations. G i v e n the designed covariance structure and the interrelationship o f the c o m p o n e n t variables, T a b l e 5-2 presents the inter-trial correlations o f the ratio variable as a function o f £ i and £ , the correlation between the c o m p o n e n t variables ( p i ) , and the x  x 2  x  x 2  relative v a r i a t i o n o f the c o m p o n e n t variables ( V i / V ) . T h e unshaded panels o f T a b l e x  x 2  5-2 s h o w the assigned inter-trial correlations o f the c o m p o n e n t variables for the different £ i and £ , and V i / V x  x 2  x  x 2  c o n d i t i o n s . In the c o n d i t i o n s £ x l =£ x 2 =0.7, £ i=0.7 and £ x 2 =1.0, x  and £ i=1.0 and £ 2=0.7, the correlations for the variable(s) for w h i c h £ i=0.7 v a r i e d f r o m x  X  X  high for adjacent trials correlation to l o w for the most distant trials (i.e., p u, i5). T h e x  x  94  Table 5-2 Intertrial C o r r e l a t i o n s o f R a t i o V a r i a b l e s f o r S p e c i f i c C o n d i t i o n s o f the C o m p o n e n t Variables  V i/V x  Input:  =2.0  VxlA^ 2=1.0  VxiA^ =0.5  0.87  0.87  0.85  0.82  0.81  0.83  o.sy  0.S4  0.87  0.S3  0.87  0.83  X 2  x  x2  8x1=8x2=1.0  Py: N u m e r a t o r Denominator Ratio:  Input: 6x2=8x2=0.7 Pij: N u m e r a t o r Denominator  0.87-0.66  0.87-0.60  0.85-0.60  0.82-0.49  0.79-0.45  0.83-0.45  0.88-0.70  0.82-0.52  0.S0-0.42  0.87-0.66  0.87-0.60  0.85-0.60  Ratio: Pxii,>_:i=0.7  Input:  6x1=0.7, e 2=1.0 X  Py: N u m e r a t o r Denominator  0.82  0.81  0.83  Ratio: 0.S7-0.64  Input:  0.83-0.73  0.S0-0.S6  8xi=1.0, 6x2=0.7  Pij: N u m e r a t o r Denominator  0.87  0.87  0.85  0.82-0.49  0.79-0.45  0.83-0.45  0.SS-0.94  O.S2-0.56  0.S0-0.40  Ratio: PMI ^I=0.7  95  c o r r e l a t i o n m a t r i x for these c o n d i t i o n s a p p r o x i m a t e d a s i m p l e x pattern. T h e shaded panels are the c a l c u l a t e d inter-trial correlations o f the d e r i v e d ratio variables for the data in a f i v e trials design p o p u l a t i o n for v a r y i n g c o n d i t i o n s o f e j and £ , and the relative x  x 2  m a g n i t u d e s o f the coefficients o f v a r i a t i o n o f the c o m p o n e n t variables. T h e full c o r r e l a t i o n matrices for the c o n d i t i o n £ i = £ = 1 . 0 are presented i n T a b l e 5-3 and the x  x 2  c o r r e l a t i o n matrices for the c o n d i t i o n p i 2 = 0 . 7 i n the other e p s i l o n c o n d i t i o n s are x  X  presented i n T a b l e 5-4. T h e other c o r r e l a t i o n matrices are i n A p p e n d i x D . In general, as s h o w n i n shaded panels o f T a b l e 5-2, the relative v a r i a t i o n o f the n u m e r a t o r and d e n o m i n a t o r variables affects the correlations a m o n g the ratio variables i n the f i v e trials ( p i j , w h e r e y i = x i i / x i and i ^ j ) . That i s , p ; j decreased w h e n V i / V y  y  y  2  decreased i n the c o n d i t i o n e = e = 1 . 0 . V a r y i n g p i xl  x  x2  x 2  y  x  x 2  f r o m 0.9 to 0.5 had n o substantial  effect o n the magnitude o f the correlations i n the p i j matrices. T h i s was true for a l l three y  y  £ c o n d i t i o n s ( E = £ = 0 . 7 , £ i = 0 . 7 and e = 1 . 0 , and £ i = 1 . 0 and e = 0 . 7 ) , thus o n l y the x 1  x 2  x  x  x2  x2  results for the p i=0.7 c o n d i t i o n are presented i n T a b l e 5-4 and s u m m a r i z e d i n the shaded x  panels o f T a b l e 5-2. W h e n the c o m p o n e n t variables have the same degree o f v i o l a t i o n o f the a s s u m p t i o n o f c i r c u l a r i t y (i.e., e i = £ = 0 . 7 ) , and the correlations f o l l o w a s i m p l e x pattern, x  x 2  the c o r r e l a t i o n matrices o f the ratio variable e x h i b i t a decreasing pattern but do not f o l l o w the s i m p l e x pattern e x a c t l y for a l l V j / V x  c o n d i t i o n s . F o r e x a m p l e , i f the correlations  x 2  f o l l o w the s i m p l e x pattern (given V i / V = 2 . 0 and p i = 0 . 8 8 ) , the m a g n i t u d e s o f the x  x 2  y  correlations s h o u l d f o l l o w the pattern: p i 3 = p i y  y  y  2 y2  y2  =0.77, p i 4=p i y  y  y  3 y 2  =0.68,  and  P i 5 = P i 2 = 0 . 6 0 . H o w e v e r , i n the c o n d i t i o n £ = £ = 0 . 7 , the m a g n i t u d e o f the 4  y  y  y  y  x l  x 2  io c- r- tCO CO 00 CO O  O  O  r- r » rCO CO CO O  O  * r- c>, co co  ro ro ro ro CO CO CO CO  o o  O  O rH  o o  O rH  ^  O  c00  o o  O  rH  O O rH  ro ro ro  o  CO CO CO  o  O  O  O  O  O rH  o o  O O rH  •H o o o *>, CO CO o  O rH  ro O 00 o rH  in  in  O  o o  O O rH  • t  Q. O  H  O  o o o CO CO CO  ro ro O >, 00 00 o  o o  '—'  d o  O  o o o o CO CO CO CO  o o  o  o  00  o  O  rH  in "' o M  •Sri  H H d H OrorororoO 00 00 00 00 O CO OO CO CO o  CN CN C N C N or^c^-r^t^o 00 00 00 00 O 00 00 00 C 0 O O O  3 j!  rH O  CN CN CN O 00 o oo oo  Q. O  O O  rH  O  O  ^ O  O  • n O O O O  J3 H X 00  r- r— i o 00 00 00 o  rH O  H H O 00 CO O  O  O  «  ro ro ro O 00 00 00 O  rtHO 00 00  o o  o  o  ro ro ro  o  rH  00 CO  ro ro 00 00  o o  r^r^t^r~ o CO CO CO CO O O O O O  rH  3 r- r- o It CO CO o ttOOH  O  t-  o  00  o  O  rH  o o  o O  o  rH  O rH  QL O  ON O CO O O ON  o  o  r~ r~ o oo oo o  ON ON O >, CO 00 O Q. O  O rH  O O O CO 00 O O  O  o 00  Sco  O i-l  o o  O rH  rH  O rH  o  o o  o  H  iH  o  d o  Q. O  H  ro O CO o O  rH  in  o  o  in in o co oo o O rH  in co  o o  O  rH  o o  O O O O r H  ON ON Ol o o  in cn 'in o 00 00 00 o  3 ji{  O  ON co cn r> c- c- r-  ON  O O O O r H  ; n ^ ) o ^ CO CO o  Ol  o 00  in in in in o co co co co o  o o  o 00 00 CO o  t- O CO o  rH  «d< CO CO CO CO  ^  o o  o o  r H O O O O r H  o 00 00 CO o  3 °  CO  a:  O O O O  ON ON cn o 00 00 00 o  ro  <  1 1  [- r~ r~ c- o CO CO CO CO O  o o  r H O O O O r H  ^  o o  rH O CO o  o o  o i o i ai cf\ CO CO CO CO  o o  O  o o  Q. O  o  11  O  o  00 OO 00 o Q . O O O rH  a o o H CN O CO o  O  >j  ^ ro ro O >, 03 CO O  o  cn cn o t— co  ro ro ro ro 00 00 00 00  •r> o  O  Q . O O O r H , _ , 0 0 0  O i-l  * r- r>, CO 00  CN CN ' 00 00 <  O t-H  O  rH  o  o  JJ O N cn o >, r- p- o Q. O  O rH  ON O t- o O ON  rH  rH  97  T a b l e 5-4 T h e C o r r e l a t i o n M a t r i c e s o f R a t i o V a r i a b l e P o p u l a t i o n s (p*_ix2=0.7) e i=e 2=1.0 x  X  e =e =0.7 xl  e =0.7, E =1.0  X2  XL  X2  E I=1.0, e =0.7 X  X2  V /V =2.0 XL  X2  0.88 0.87 0.87 0.87 1.00 0.87 0.87 0.87 1.00 0.87 0.87 1.00 0.87 1.00  w  x2  1.00 0.88 0.77 0.74 0.70 1.00 0.88 0.78 0.74 1.00 0.88 0.77 1.00 0.87 1.00  1.00 0.87 0.74 0.69 0.64 1.00 0.88 0.74 0.69 1.00 0.87 0.74 1.00 0.86 1.00  1.00 0.88 0.91 0.94 1.00 0.88 0.92 1.00 0.88 1.00  0.94 0.92 0.89 0.89 1.00  =1.0  0.83 0.83 0.83 1.00 0.83 0.83 1.00 0.83 1.00  0.83 0.83 0.83 0.83 1.00  1.00 0.82 0.71 0.62 0.52 1.00 0.82 0.71 0.63 1.00 0.82 0.71 1.00 0.80  1.00 0.83 0.79 0.76 1.00 0.83 0.79 1.00 0.83 1.00  0.73 0.76 0.78 0.81 1.00  1.00 0.82 0.76 0.73 0.56 1.00 0.82 0.76 0.58 1.00 0.82 0.61 1.00 0.65 1.00  1.00 0.80 0.82 0.84 1.00 0.80 0.82 1.00 0.80 1.00  0.86 0.84 0.82 0.80 1.00  1.00 0.80 0.58 0.47 0.40 1.00 0.73 0.59 0.47 1.00 0.77 0.56 1.00 0.71 1.00  1.00  V /V =0.5 XL  X2  0.80 0.80 0.80 1.00 0.80 0.80 1.00 0.80 1.00  0.80 0.79 0.79 0.80 1.00  1.00 0.80 0.60 0.50 1.00 0.74 0.61 1.00 0.77 1.00  0.42 0.50 0.57 0.71 1.00  98  correlations ( p i 2 = 0 . 8 8 , p y  y  y]y  3=0.77, p i = 0 . 7 4 , and p y  y 4  y l y 5  = 0 . 7 0 ) decreased s l o w e r than  w o u l d be the case i n a s i m p l e x pattern. T h e results s h o w t w o different patterns i n the c o r r e l a t i o n matrices o f the ratio variables for the t w o c o n d i t i o n s £ i = 0 . 7 and E 2=1.0, a n d e i = 1 . 0 and £ 2 = 0 . 7 (see the t w o x  x  X  X  shaded panels f r o m the b o t t o m o f T a b l e 5-2). In the c o n d i t i o n e i = 0 . 7 a n d e 2=1.0, the x  X  decreasing c o r r e l a t i o n pattern i n the c o r r e l a t i o n m a t r i x disappeared w h e n the d e n o m i n a t o r v a r i a b l e h a d the h i g h e r relative v a r i a t i o n ( V i / V = 0 . 5 ) . In this case the correlations x  x 2  a m o n g the ratio variables actually increased s l i g h t l y as the distance between trials increased: p i 2 = 0 . 8 0 , p y  y  y]y  3 = 0 . 8 2 , p i = 0 . 8 4 , a n d p i = 0 . 8 6 . In the c o n d i t i o n £ i = 1 . 0 y  y 4  y  x  y 5  and 8 2=0.7 the decreasing c o r r e l a t i o n pattern also disappeared, but w h e n the numerator X  variable (not the d e n o m i n a t o r variable) h a d the h i g h e r relative v a r i a t i o n ( V i / V 2 = 2 . 0 ) . x  X  T h i s study p r o v i d e s some details o f the effect o f the characteristics o f c o m p o n e n t s on the c o r r e l a t i o n between ratio variables. T h e results indicate that both the r e l a t i v e v a r i a t i o n a n d the relative £ l e v e l o f the t w o c o m p o n e n t variables affect the c o r r e l a t i o n s a m o n g ratio variables i n a repeated trials d e s i g n . W h e n the v a r i a t i o n o f the d e n o m i n a t o r variable increases relative to the numerator v a r i a b l e (i.e., V i / V x  x 2  decreases), the  c o r r e l a t i o n a m o n g the ratio variables decreases i n the c o n d i t i o n s E i=e 2=1.0 a n d x  X  £ x i = £ x 2 = 0 . 7 . If b o t h the numerator a n d d e n o m i n a t o r variables meet the a s s u m p t i o n o f c i r c u l a r i t y (E I=E 2=1.0), a constant c o r r e l a t i o n pattern is s h o w n i n the c o r r e l a t i o n m a t r i x X  X  of the ratio variables. If both the c o r r e l a t i o n matrices o f the numerator a n d d e n o m i n a t o r variables are d e s i g n e d to f o l l o w the s i m p l e x pattern and have the same l e v e l o f v i o l a t i o n o f the a s s u m p t i o n o f c i r c u l a r i t y ( E = E = 0 . 7 ) , a decreasing c o r r e l a t i o n pattern, s i m i l a r but X1  x2  99  not i d e n t i c a l to a s i m p l e x pattern, is s h o w n i n the correlation m a t r i x o f the ratio variables. In the other t w o e c o n d i t i o n s , the pattern o f the correlation m a t r i x o f the ratio v a r i a b l e is affected b y the c o m b i n e d factor o f relative level o f v a r i a t i o n and e l e v e l o f the c o m p o n e n t variables. If o n l y the numerator v a r i a b l e violates the a s s u m p t i o n o f c i r c u l a r i t y , a pattern o f i n c r e a s i n g correlations is s h o w n w h e n V i / V = 0 . 5 . I f o n l y the x  x 2  d e n o m i n a t o r variable violates the a s s u m p t i o n o f c i r c u l a r i t y , a pattern o f i n c r e a s i n g correlations is s h o w n w h e n V i / V 2 = 2 . 0 . x  X  T o e x a m i n e h o w the characteristics o f the c o m p o n e n t variables affect the c o r r e l a t i o n pattern o f the ratio variable, the c o m p a r a t i v e values for a l l the statistics and the c o r r e l a t i o n matrices o f the c o m p o n e n t variables that c o u l d contribute to the c o r r e l a t i o n pattern o f the ratio v a r i a b l e for f i v e related c o n d i t i o n s are presented i n T a b l e 5-5. T h e first t w o c o l u m n s show the c o r r e l a t i o n matrices o f the t w o c o m p o n e n t variables and the t h i r d c o l u m n s h o w s the c o r r e l a t i o n m a t r i x for the transformed ratio v a r i a b l e , and the e i/ 2 v a l u e associated w i t h that m a t r i x . x  X  In C o n d i t i o n s 1 and 2, the t w o c o m p o n e n t variables have the same c o r r e l a t i o n pattern, and the numerator has a h i g h e r v a r i a t i o n relative to the d e n o m i n a t o r ( V i / V 2 = 2 . 0 ) . A s s h o w n i n T a b l e 5-5, w h e n p x  X  x ] x  2 decreases f r o m 0.9 to 0.5, the  c o r r e l a t i o n m a t r i x o f the ratio variable s t i l l e x h i b i t s a decreasing pattern, but to a lesser degree. T h e c o m p a r i s o n between C o n d i t i o n s 2 and 3 shows that w h e n the d e n o m i n a t o r has the constant correlation pattern and large v a r i a t i o n ( V / V 2 = 0 . 5 ) the decreasing x l  X  pattern o f the correlation disappears for the ratio variable. T h e c o m p a r i s o n o f the c o r r e l a t i o n matrices i n C o n d i t i o n s 1, 2 and 3 indicates that the c o r r e l a t i o n pattern o f the ratio variable is c l o s e to that o f the c o m p o n e n t variable w h i c h has larger relative  100  T a b l e 5-5 C o m p a r i s o n o f the C o r r e l a t i o n M a t r i c e s o f the R a t i o V a r i a b l e under S e l e c t e d C o n d i t i o n s  C o r r e l a t i o n X nj  Conditionl: 1.00  Correlation X y  e i=0.7, e =1.0, p i = 0 . 9 , V / V = 2 . 0 x  x)  x  0.87  0.76  0.71 0.66  1.00  0.87  0.76 0.71  1.00  0.87 0.75 1.00  x2  x l  1.00  e i/x =0.59  x 2  x  0.82 0.82  0.82 0.82  1.00  0.82  0.82 0.82  1.00  0.82 0.82  0.85  1.00  e ,=0.7, e =1.0, p x  x]  0.87  0.76  0.71 0.66  1.00  0.87  0.76 0.71  1.00  0.87 0.75 1.00  Condition3: e i=0.7, x  1.00  0.76  0.66 0.60  0.85  0.76 0.68  Condition4: e i=1.0, 1.00  0.87 0.87  1.00  0.87  0.87 0.87  1.00  0.87 0.87 1.00  1.00  x)  0.82 0.82  0.82 0.82  1.00  0.82  0.82 0.82  1.00  0.82 0.82  xlx2  x l  1.00  0.83  0.83 0.83  1.00  0.83  0.83 0.83  1.00  0.83 0.83  0.85  0.85 0.85  1.00  0.85  0.85 0.85  1.00  0.85 0.85  0.75 0.71  1.00  0.87 0.75 0.85  x2  0.80 0.80  0.81 0.82  1.00  0.81 0.81  0.80 1.00  0.81 0.81 1.00  0.80 1.00  0.57 0.49  1.00  0.80  0.63 0.57  1.00  0.80 0.66  1.00  0.87  0.87  0.88 0.88  1.00  0.87  0.87 0.88  1.00  0.87 0.87 1.00  0.87 1.00  =0.5, V / V = 0 . 5  1.00  0.71 0.66  0.87  x2  0.82 0.65  x l  0.75  1.00  £xi/ =0.99  1.00  0.85  1.00  x 2  1.00 0.76  xlx2  0.87  1.00  =0.5, V / V = 2 . 0  1.00  x2  x  0.83  x l  0.82  e i/ =0.99  1.00  xi  0.87 0.71  1.00  x 2  0.87  =1.0, e =0.7, p  1.00  xlx2  1.00  =0.5, V / V = 0 . 5  1.00  0.87  0.71 0.64  1.00  1.00  x  1.00  x  0.90  £ i/ =0.70  1.00 0.83  e i =0.7, p  0.64 0.58  1.00  x 2  0.83  0.87  Condition5: e  1.00  0.85 0.75 1.00  x  x l  1.00  1.00  1.00  =0.5, V / V = 2 . 0  1.00 0.82  0.85  1.00  xlx2  1.00  x  0.89 0.72  1.00  0.85  e i = 1.0, p  1.00  2  1.00 0.82  1.00.  Condition^:  Correlation Y y  2  E i/ =0.72  x 2  x  0.83  0.65  0.55 0.47  1.00  0.78  0.65 0.55  1.00  0.80 0.63  0.85  1.00 0.75  1.00  1.00  1.00  x2  0.80 0.61  0.51 0.45  1.00  0.75  0.62 0.52  1.00  0.78 0.60 1.00  0.72 1.00  101  v a r i a t i o n . T h a t i s , i n C o n d i t i o n s 1 and 2, the numerator has the larger v a r i a t i o n , and the c o r r e l a t i o n pattern o f the ratio v a r i a b l e is c l o s e r to the decreasing pattern o f the numerator. In C o n d i t i o n 3 the c o r r e l a t i o n pattern o f the ratio v a r i a b l e is m o r e s i m i l a r to the d e n o m i n a t o r due to the larger v a r i a t i o n o f the d e n o m i n a t o r v a r i a b l e . T h e s e results seem to indicate that the c o r r e l a t i o n pattern o f the ratio v a r i a b l e is m o s t l y dependent o n the relative v a r i a t i o n o f the c o m p o n e n t variables ( V i / V 2 ) . T o c o n f i r m that the x  X  c o r r e l a t i o n pattern o f the ratio v a r i a b l e is m o r e s i m i l a r to that o f the c o m p o n e n t v a r i a b l e w h i c h has a large coefficient o f v a r i a t i o n , the t w o c o n d i t i o n s under the c o n d i t i o n e i = 1 . 0 x  a n d e = 0 . 7 were also e x a m i n e d ( C o n d i t i o n s 4 a n d 5 i n T a b l e 5-5). In these t w o x2  c o n d i t i o n s , the c o r r e l a t i o n matrices o f the c o m p o n e n t variables have the opposite pattern f r o m C o n d i t i o n 3. C o n d i t i o n 4 results s h o w that the decreasing pattern o f the ratio variable does not appear w h e n the numerator has a constant c o r r e l a t i o n pattern w i t h large v a r i a t i o n . In C o n d i t i o n 5, w h e n the d e n o m i n a t o r has larger v a r i a t i o n a n d a decreasing c o r r e l a t i o n pattern, the results s h o w that the ratio v a r i a b l e also e x h i b i t s a decreasing c o r r e l a t i o n pattern w h i c h is s i m i l a r to that o f the d e n o m i n a t o r . T h e s e results c o n f i r m that the c o r r e l a t i o n pattern o f the ratio v a r i a b l e is m o s t l y affected b y the c o m p o n e n t v a r i a b l e w h i c h has the larger coefficient o f v a r i a t i o n . A s s h o w n above, the c o r r e l a t i o n a m o n g the d e r i v e d ratio variables o v e r t i m e is m o r e c o m p l e x than the c o m p o n e n t variables. T h e e l e v e l a n d relative v a r i a t i o n o f the c o m p o n e n t variables p l a y important role i n the c o r r e l a t i o n o f ratio variables i n a repeated measurement design. Therefore, one s h o u l d not a u t o m a t i c a l l y assume that the ratio variable w o u l d s h o w the same c o r r e l a t i o n pattern as the c o m p o n e n t variables. E s p e c i a l l y , i n the situation e i < 1 . 0 a n d e = 1 . 0 (a c o m m o n case i n h u m a n k i n e t i c s research) and the x  x2  102  d e n o m i n a t o r variable has higher v a r i a b i l i t y , the d e r i v e d ratio variable w o u l d not e x h i b i t a decreasing correlation pattern o v e r time.  Sample Characteristics of £ i/ 2 A  x  X  S i n c e £ was i n t r o d u c e d b y B o x (1954), it has been used as a correction factor i n a R M A N O V A to c o n t r o l for probable inflation i n type I error rates brought about b y heterogeneity o f c o v a r i a n c e . F o r any c o v a r i a n c e m a t r i x , the upper and l o w e r l i m i t s o f £ are 1.0 and l / ( k - l ) , where k is the n u m b e r o f trials. B e c a u s e the p o p u l a t i o n £ is u n k n o w n i n practice, it is u s u a l l y estimated f r o m a sample c o v a r i a n c e m a t r i x (using the same equation used to c o m p u t e E , but a p p l y i n g it to a s a m p l e c o v a r i a n c e m a t r i x ) . It can be s h o w n that g i v e n £ = 1 . 0 this estimate o f £ ( E ) a l w a y s e x h i b i t s some degree o f d o w n w a r d a  bias, and this bias becomes larger w i t h i n c r e a s i n g k. E o m (1993) e x a m i n e d the distribution characteristics o f E . G i v e n n o r m a l i t y , £ = 1 . 0 , and k=5 i n the p o p u l a t i o n , a A  M o n t e C a r l o s i m u l a t i o n s h o w e d that the means o f E (2,000 samples) w i t h n=15, 30, 45 A  were 0.77 (Sd=0.08), 0.87(0.05), and 0.91(0.04), respectively. F o r £ = 0 . 7 and k=5, the means o f £ w i t h n=15, 30, 45 were 0.61(Sd=0.09), 0.66(0.08), and 0.67(0.06), A  respectively ( E o m , 1993). R a t i o variables are c o m m o n l y used i n R M A N O V A i n our f i e l d , h o w e v e r , the s a m p l i n g characteristics o f £ for a covariance m a t r i x d e r i v e d f r o m ratio scores, A  A  £ i/ 2, x  X  are not w e l l understood. H o w the relative variation and correlation o f the numerator a n d d e n o m i n a t o r variables affect  A  £ i/ 2 x  X  is u n k n o w n . Therefore, one o f the purposes i n this  study was to s y s t e m a t i c a l l y e x a m i n e the s a m p l i n g characteristics o f  A  £ i/ x  x 2  .  W i t h the  103  aforementioned characteristics o f 8 o f the c o m p o n e n t variables and the characteristics o f the ratio score v a r i a b l e populations i n m i n d , u s i n g M o n t e C a r l o s i m u l a t i o n procedures, we n o w investigate the s a m p l i n g characteristics o f  A  E i/ 2 x  X  for different c o n d i t i o n s .  T h e M o n t e C a r l o s i m u l a t i o n was c o n d u c t e d u s i n g a five-trial (k=5) repeated measures design w i t h three sample sizes (n=15, 30, 45) and 2,000 repetitions f r o m each o f the s i m u l a t e d p o p u l a t i o n data sets ( N = 9 0 , 0 0 0 ) . T h e s i m u l a t i o n results are s u m m a r i z e d i n T a b l e s 5-6, 5-7, and 5-8. These three tables s h o w the d i s t r i b u t i o n characteristics o f A  E i/ 2 x  X  for each o f the populations.  Condition 1 (8x1=8x2=1.0) T a b l e 5-6 c l e a r l y s h o w s that e i / 2 is a d o w n w a r d biased estimator and the degree A  x  X  o f bias is v i r t u a l l y unaffected b y the c o r r e l a t i o n between the numerator and d e n o m i n a t o r variables. F o r e x a m p l e , for the c o n d i t i o n n=15 and V i / V = 2 . 0 , the m e a n o f x  A  x 2  E i/ x  was  x 2  0.76-0.77 under a l l three correlation c o n d i t i o n s . H o w e v e r , e i / 2 is affected b y the A  x  X  coefficients o f v a r i a t i o n o f the numerator and d e n o m i n a t o r variables. T h e m e a n o f £ i / A  x  x 2  decreases w h e n the coefficient o f v a r i a t i o n o f the d e n o m i n a t o r variable increases relative to the numerator variable. F o r e x a m p l e , w h e n the sample size was 15 and V  x )  /V  x 2  decreased f r o m 2.0 to 0.5, the mean o f e i 2 decreased f r o m 0.76 to 0.68. A d d i t i o n a l l y , A  x  / X  the estimator became m o r e variable, as e v i d e n c e d b y the increase i n the standard deviation of s i A  x  / x 2  f r o m 0.08 w h e n V , / V = 2 . 0 to 0.12 w h e n V i / V = 0 . 5 . T h e effect o f x  x 2  x  x 2  this, and o f the characteristics o f the c o m p o n e n t variables, on type I error rates i n a R M A N O V A is d i s c u s s e d i n a later section.  104  T a b l e 5-6 D i s t r i b u t i o n C h a r a c t e r i s t i c s o f e w ? (Exj=e«2 l-0< M-XJ=75. 0^2=100) A  =  1  Vx,/Vx =2.0 Sd  x  2  Pxlx2  0.9  Sample Size  xl/x2 mean  A£  0.5  fc  Apxl/x2  Vxl/V = x2  Sd  fc  mean  .0 xl/x2 " £xl/x2  e  A  Ap  Vxi/V =0.5 Sd x2  mean  e  xl/x2 "  Apxl/x2 fc  15  .76  (.08)  .24  -75  (.08)  .25  .68  (.12)  .32  30  86  (06)  14  .85  (.06)  .15  .78  (.11)  .22  ')()  ( 04)  .83  (.10)  .17  45 0.7  xl/x2 Acxl/x2  E  i 04)  15  .76  .08)  .24  .75  . os i  .25  .69  (.11)  .31  30  .86  (.05)  .14  .85  (.06)  .15  .78  (.11)  .22  45  90  i 04)  NO  1.05.  .83  l . ID)  .17  15  .77  i.  .23  .74  (-09)  .26  (.11)  .31  30  .87,  (.05)  .13  .84  (-06)  .16  .79  (.10)  .21  45  .91  (.04)  .(W  .M  . (IS,  .83  (.()'))  .17  08)  Note. The population value of e i 2 is 1.0 for all conditions. x  /X  I^iliii!  105  T h e s a m p l e size effect on £ i / 2 is o b v i o u s : w h e n the s a m p l e s i z e increases, the A  x  X  bias o f e i / 2 decreases. T h i s r e d u c t i o n is a p p r o x i m a t e l y 0.10 w h e n n increases f r o m 15 A  x  X  to 3 0 , a n d 0.04 w h e n n increases f r o m 3 0 to 4 5 . In general, as expected, the s a m p l e size effect o n e i / 2 o f the ratio v a r i a b l e data is the same as for the c o m p o n e n t variables (raw A  x  X  score data) reported i n literature.  Condition 2 (£^=£^=0.7) In this c o n d i t i o n , the c o v a r i a n c e matrices o f both the n u m e r a t o r a n d d e n o m i n a t o r variables e x h i b i t e d moderate v i o l a t i o n s o f the a s s u m p t i o n o f c i r c u l a r i t y a n d the intertrial c o r r e l a t i o n f o l l o w e d the s i m p l e x pattern for both the numerator and d e n o m i n a t o r variables. T a b l e 5-7 s u m m a r i z e s the s i m u l a t i o n results for this c o n d i t i o n , a n d reveals the d o w n w a r d bias i n £ i / 2 - T h e £ i / 2 h a d the greatest bias w h e n V i / V 2 = 0 . 5 , but the A  A  x  X  x  X  x  X  l e v e l o f V i / V 2 h a d less effect here than i n the E = E = 1 . 0 c o n d i t i o n (see F i g u r e 5-1). x  X  X 1  x 2  The standard d e v i a t i o n o f £ i / 2 i n c r e a s e d s l i g h t l y w h e n V 2 increased, but also less than the A  x  X  X  increase s h o w n i n the £ i = £ 2 = 1 . 0 c o n d i t i o n . G i v e n £ = £ = 0 . 7 and the s i m p l e x pattern x  X  x l  x 2  o f the intertrial c o r r e l a t i o n for the c o m p o n e n t variables, the c o r r e l a t i o n between the n u m e r a t o r a n d d e n o m i n a t o r variables (p ix2) does not seem to have an effect o n the x  s a m p l i n g characteristics o f £ i / 2 A  x  X  Condition 3 (e i=0.7, £ 7=1.0) Y  Y  In this c o n d i t i o n , o n l y the c o v a r i a n c e m a t r i x o f the numerator c o m p o n e n t v i o l a t e d  106  3 10  ON  ro  —i  o  Os •: O  8  ~  2  so  v o -  o  CO <  o  os  ,\  r~  o  K CD  ' "3  <2 3  o o  ir-,  o  OS  SI  IS  in r-  <? s  II  o  .CO  OO  in  —i  o  o O  IO  O O I T )  r ' l S O  i n S O  m  O —i  so  ,00' 0  O ~>  o  ! o  S O  i/~ o  r~ o  Os  ir  r~  o  o  f  NO::;  O s  >r,  o  o  O  :g  OS  so  ,o„  —i  NO  o f-  CN so  so so  ,00 , [so  CD  Os  v.  c«-  o  o  00 so  !  —1  o  o  Os  loo  i  r-' d ll X  CO II  3  9  <  >  wl  oo  CO  II  o  « to  O  O  O  O  o  <  T3 C/5  K CO  o  ro  SO  so  sO  o  os o  —i  so  oo o  irs so  ,so , ,\ os  ,o  r~  ,sq  O  oo O  rH  I/",  so  so  ,o  V-» oo  •c  CO  r-  m  .c U c o  -a 3  •a  r v  C J  o m  a bo O S  o  IO I-H  d  O  m  U">  TJ-  <0  —i  d  O  rn  U"> ^j-  108  the assumption o f c i r c u l a r i t y . That is, the intertrial correlations o f the numerator variable f o l l o w e d the s i m p l e x pattern, w h i l e those for the d e n o m i n a t o r were constant for a l l pairs of trials. In a p r e v i o u s section, it was s h o w n that both p i x  and V i / V  x 2  x  affect the  x 2  p o p u l a t i o n 8 o f the ratio variable w h e n £ i = 0 . 7 and e = 1 . 0 . In this c o n d i t i o n , e i / x  to have a s m a l l e r m a g n i t u d e w h e n p when p i x  and V i / V  x 2  x  population e i / x  x 2  x ! x 2  x2  and V i / V x  x 2  x  tends  are h i g h , and a greater magnitude  are l o w . W i t h the aforementioned characteristics o f the  i n m i n d , w e n o w e x a m i n e the characteristics o f £ i / A  x 2  x 2  x  A s can be seen i n T a b l e 5-8, the bias o f e i A  x  / x 2  x 2  i n this c o n d i t i o n .  b e c o m e s more p r o n o u n c e d w h e n  the coefficient o f v a r i a t i o n o f the denominator variable is h i g h ( V i / V = 0 . 5 ) . F o r x  e x a m p l e , the m e a n o f e i / A  x  x 2  x 2  was lowest (0.53-0.61) and the d e v i a t i o n e i x  - e i/ A  / x 2  x  x 2  was  o n l y 0.06-0.09 for the V i / V = 2 . 0 c o n d i t i o n w h e n n=15, whereas the m e a n o f s i /  was  A  x  x 2  x  x 2  slightly larger (0.60-0.69) but the deviation was m u c h larger (0.26-0.30) for the V i / V = 0 . 5 c o n d i t i o n . F i g u r e 5-2 illustrates this trend. x  x 2  In contrast to C o n d i t i o n s 1 and 2, the magnitude o f the correlation between the numerator variable and the denominator variable affects the magnitude o f £ i / 2 i n A  x  C o n d i t i o n 3. A s s h o w n i n T a b l e 5-8, w h e n the c o r r e l a t i o n p i x  between £ i / A  x  x 2  and £ i / x  x 2  x 2  X  decreases, the difference  tends to increase, although b y o n l y a s m a l l amount (a difference  of 0.03-0.04 between the p i = 0 . 9 and p i = 0 . 5 c o n d i t i o n s w h e n n=15). T h e r e appears x  to be no V i / V x  x 2  by p i x  x 2  x  x2  interaction effect, as the bias o f £ i / A  x 2  x  increases, regardless the l e v e l o f the correlation p i . x  x 2  x 2  increases w h e n the V  x  2  109  as  so  as  {.as o  C I  —  -oo.:  o  cn  —  — CM  O  o  II t> vo  tO to <  o o  II II  v o  o  CN  IS  <a l>  II  jo a  O O  VO O  o  o  —i  oc r~  oo  'O oo  Os vo  :p-  —<  oo  CN  —i  t—o  O CN  .I-H  Os  r--  \o  Os O  oo O  ,o  •*  <-> r-  C\ r--  oo oo  Os O  m O  co lO  o  o o  O  o  o  o  o  —'  oo  r--  oo  CN !  00» O  vO  Ov  ON  c-J|  cn  30  vo r-  co  r-' ©'  3 S x EO t0 <  4 II  3  I" a  O  o  o  r -  t  CN  o  o  to  o  O  \  r  O  -  so !©-;  o  o  r -  o  r n  r-» CO  C  CN  vo  VO  vo  vo  o  m  >/> -  in  O  m  Tl-  O cn  m  CO  -H  CO  Tf  v o o  cn  co O  •n  i  -VO  •c CO  i  3 CO  u 00 «  c o  va  -C  -Q to eg -j-i  H Q  2 K  Q.  -H  Fi gure 5-2. £ i / A  x  x 2  i n the c o n d i t i o n e |—0.7 a n d £ 2—1.0. X  X  Ill Summary of Characteristics of e i A  x  / x 2  and e i / x  x 2  M o n t e C a r l o s i m u l a t i o n procedures were used to investigate h o w the characteristics o f the numerator and d e n o m i n a t o r variables affect the c i r c u l a r i t y o f the ratio score p o p u l a t i o n c o v a r i a n c e m a t r i x (the effect o n the p o p u l a t i o n e i/ 2) a n d o f the X  x  s a m p l i n g characteristics o f £ i / A  x  x 2  i n the different c o n d i t i o n s . A s i n d i c a t e d i n the methods  a n d procedures chapter, the i n v e s t i g a t i o n was based o n a ratio v a r i a b l e w i t h a specific c o v a r i a n c e structure that is s i m i l a r to that o b s e r v e d i n h u m a n k i n e t i c s research. T h e ratio v a r i a b l e p o p u l a t i o n was constructed f r o m p o p u l a t i o n s o f the c o m p o n e n t variables w i t h specific d i s t r i b u t i o n characteristics. T h a t i s , the numerator and d e n o m i n a t o r variables were restricted to a bivariate n o r m a l d i s t r i b u t i o n , constant w i t h i n - t r i a l c o r r e l a t i o n between the numerator and d e n o m i n a t o r variables ( p i = 0 . 9 , 0.7, 0.5), a n d constant x  x 2  intertrial c o r r e l a t i o n (e j=1.0, i = l , 2 ) o r a s i m p l e x pattern o f the intertrial correlations (e i x  x  <1.0). F o l l o w i n g s u m m a r y o f the f i n d i n g s are based o n the above c o n d i t i o n s . 1. T h i s study indicates that i f the c o m p o n e n t variables meet the a s s u m p t i o n o f c i r c u l a r i t y ( e i = e = 1 . 0 ) , the c o v a r i a n c e m a t r i x o f a ratio v a r i a b l e also meets the x  x2  a s s u m p t i o n ( e i / = 1 . 0 ) , and the d o w n w a r d bias o f e i / A  x  x2  x  x 2  has a pattern s i m i l a r to what is  k n o w n to h o l d for r a w score variables. In the c o n d i t i o n e = e = 0 . 7 , e i xl  x2  x  same as that o f the c o m p o n e n t variables, and the d o w n w a r d bias o f  A  E  x  ]  /  / x 2  x  2  is v i r t u a l l y the is less than i n  the c o n d i t i o n e i = e = 1 . 0 . T h e p o p u l a t i o n e v a l u e o f a ratio variable is affected b y the x  x2  relative variation o f the c o m p o n e n t variables ( V ] / V ) w h e n o n l y the numerator variable X  x 2  has a moderate v i o l a t i o n o f the a s s u m p t i o n o f c i r c u l a r i t y (e.g., e i = 0 . 7 and e = 1 . 0 ) . T h e x  x2  c o v a r i a n c e m a t r i x o f a ratio score p o p u l a t i o n tends to have a m o r e h o m o g e n e o u s  112  c o v a r i a n c e structure a n d h i g h e r bias l e v e l o f  A  e w h e n the coefficient o f v a r i a t i o n o f the  numerator variable decreases. 2. T h e c o r r e l a t i o n between the numerator and d e n o m i n a t o r variables does not affect the p o p u l a t i o n 8 n o r the s a m p l i n g characteristics o f e i / A  x  x 2  i f the t w o c o m p o n e n t  variables do not v i o l a t e the a s s u m p t i o n o f c i r c u l a r i t y (i.e., e i=e 2=1.0), or i f they have the X  x  same moderate degree o f v i o l a t i o n e i=e 2=0.7. H o w e v e r , i f the numerator variable X  x  violates the a s s u m p t i o n o f c i r c u l a r i t y and the d e n o m i n a t o r v a r i a b l e does not (e.g., e i = 0 . 7 x  and e = 1 . 0 ) , E / 2 increases w h e n the c o r r e l a t i o n between the numerator and x2  x ]  X  d e n o m i n a t o r v a r i a b l e ( p i x 2 ) decreases, e s p e c i a l l y w h e n V x  ^ V 2 . T h e bias o f e i / 2 is A  x ]  X  x  X  m o r e serious i n the c o n d i t i o n e i = 0 . 7 a n d E 2=1.0 than i n the c o n d i t i o n s e i = e = 1 . 0 a n d X  x  8xi=8 2=0.7, especially when p X  A s s h o w n above, e i / x  x 2  is affected b y a c o m b i n a t i o n o f the characteristics o f the  c o m p o n e n t variables ( V i / V 2 , p i x 2 , 8 i , and e 2). x  x2  2 is l o w .  x ] x  A  x  X  x  X  x  I f a ratio v a r i a b l e is used, g i v e n certain  m e a n differences, and magnitudes o f e i and e 2 o f the c o m p o n e n t v a r i a b l e s , the results x  X  indicate that the m a g n i t u d e o f e i / 2 w o u l d be substantially affected b y V i / V 2 a n d p i 2 A  x  X  x  T h u s , it w o u l d affect the type I error rates i n the R M A N O V A .  X  x  X  T h e details o f these  effects are s h o w n i n the f o l l o w i n g sections.  Type I Error Rates Expected Effects of Characteristics of Xi and Xg on Type I Error Rates for Xi/X? In the p r e v i o u s t w o sections, the effect o f u s i n g ratio variables o n the m a g n i t u d e of 8 i / 2 has been presented. B e c a u s e n o n c i r c u l a r i t y (e<1.0) affects the type I error rate i n x  X  113  a o n e - w a y R M A N O V A ( W i n e r , 1991), the i m p l i c a t i o n s o f u s i n g ratio variables o n type I error rates can be i n d i c a t e d b y l o o k i n g at the values o f E i/ 2 i n the different c o n d i t i o n s . x  X  B a s e d o n the characteristics o f £ i/ 2, this section discusses h o w the c o r r e l a t i o n a n d x  X  relative v a r i a t i o n between the numerator a n d d e n o m i n a t o r variables affect the type I error rate i n a o n e - w a y R M A N O V A . G i v e n that the numerator a n d d e n o m i n a t o r variables have n o v i o l a t i o n , or the same l e v e l o f v i o l a t i o n , o f the a s s u m p t i o n o f c i r c u l a r i t y (e.g., E I=E 2=1.0, or £ i = £ = 0 . 7 ) , X  X  x  x 2  bivariate n o r m a l d i s t r i b u t i o n , and the restricted c o v a r i a n c e structure i n this study, it seems that u s i n g a ratio v a r i a b l e results i n no change i n the p o p u l a t i o n v a l u e o f £ i/ 2. T h a t is x  X  £xi/x2=l-0 w h e n E I=E 2=1.0, a n d E \/ 2~0-7 w h e n E I=E 2=0.7, regardless the l e v e l s o f p i 2 X  and V i / V x  x 2  X  x  X  X  X  x  X  (see T a b l e 5-1). T h u s , i f the c o v a r i a n c e matrices o f both the c o m p o n e n t  variables do not v i o l a t e the a s s u m p t i o n o f c i r c u l a r i t y , w e m a y c o n c l u d e that the e m p i r i c a l type I error rate o f a ratio v a r i a b l e i n a o n e - w a y R M A N O V A w o u l d be c l o s e to the n o m i n a l l e v e l . N o substantial effect o n the type I error rate is e x p e c t e d f r o m the c o r r e l a t i o n between the numerator and d e n o m i n a t o r variables ( p i ) a n d the r e l a t i v e x  x 2  v a r i a t i o n o f the c o m p o n e n t variables ( V i / V 2 ) . If the c o m p o n e n t variables have v i o l a t e d x  X  the a s s u m p t i o n o f c i r c u l a r i t y at the same l e v e l (e.g., E I = E = 0 . 7 ) , the i n f l a t i o n o f the type X  x2  I error rate u s i n g a ratio v a r i a b l e is e x p e c t e d to be equal to that o f its c o m p o n e n t s , but Pxix2 and V i / V 2 w o u l d have little a d d i t i o n a l effect. T h e results o f this study suggest that x  X  i f the numerator and d e n o m i n a t o r variables have the same l e v e l o f v i o l a t i o n o n c i r c u l a r i t y (£xi=£x2), Pxix2 and V i / V x  x 2  w o u l d have no effect o n the type I error rate i n R M A N O V A  test w i t h the ratio v a r i a b l e data. H o w e v e r , i f o n l y one o f the c o m p o n e n t variables violates the a s s u m p t i o n o f  114  c i r c u l a r i t y (e.g., e i = 0 . 7 a n d e 2=1.0, or e = 1 . 0 and e 2=0.7), x  X  x]  X  p  x ) x 2  and V  x )  /V  x 2  o f the  c o m p o n e n t variables affect the level o f v i o l a t i o n o f the assumption o f c i r c u l a r i t y for the ratio variable (see T a b l e 5-1). T h e f o l l o w i n g statements are based on the c o n d i t i o n £ i = 0 . 7 a n d e 2=1.0. T h e results indicate that e i/ 2 has l o w values w h e n p i 2 is h i g h a n d x  X  x  X  x  X  V i / V = 2 . 0 , a n d h i g h values w h e n p ] 2 is l o w a n d V i / V 2 = 0 . 5 . Therefore, i f o n l y the x  x 2  X  X  x  X  numerator v a r i a b l e violates the assumption o f c i r c u l a r i t y i n a ratio R M A N O V A , it is expected that there is a greater e m p i r i c a l type I error rate w h e n V > V 2 , a n d a s m a l l e r x )  X  e m p i r i c a l type I error rate w h e n V i < V 2 . A d d i t i o n a l l y , the higher the c o r r e l a t i o n x  X  between the numerator a n d d e n o m i n a t o r variables, the greater the e m p i r i c a l type I error rate.  Monte Carlo Simulation Investigation on Type I error Rates T h e i n f l a t i o n o f the type I error rate for the F test o n the trials effect i n a o n e - w a y R M A N O V A is related to the l e v e l o f the v i o l a t i o n o f c i r c u l a r i t y i n a p o p u l a t i o n c o v a r i a n c e m a t r i x w h i c h is measured b y 8. In practice, the p o p u l a t i o n e p s i l o n 8 is u s u a l l y not k n o w n a n d it has to be estimated b y the sample e p s i l o n e . H o w e v e r , the e p s i l o n A  estimate e based o n a sample is a b i a s e d estimator o f p o p u l a t i o n e. Therefore, the A  characteristics o f the d i s t r i b u t i o n o f the s a m p l e e need to be investigated to s h o w the A  i m p a c t o f e o n the type I error rate i n a R M A N O V A . A  A s s h o w n i n p r e v i o u s sections, the 8 i/ 2 estimate e i / 2 , is a b i a s e d estimator a n d A  x  the bias is affected b y p i 2 and V i / V x  X  x  x 2  X  x  X  , e s p e c i a l l y w h e n o n l y one o f the c o m p o n e n t  variables violates the assumption o f c i r c u l a r i t y . In this M o n t e C a r l o study, I i n v e s t i g a t e d  115  h o w the interactive characteristics o f the numerator a n d d e n o m i n a t o r variables affect the e m p i r i c a l type I error rates w h e n u s i n g ratio variables i n a o n e - w a y R M A N O V A design (k=5). T a b l e 5-9 s u m m a r i z e s the general pattern o f type I error rates for the three £ c o n d i t i o n s across the l e v e l s o f a (.01, .05, .10), V i a n d V 2 , and p i 2 - T h e values i n x  X  x  X  each b l o c k i n T a b l e 5-9 were the o b s e r v e d type I error rates based on the average o f the o b s e r v e d type I error rates o f the three s a m p l e sizes (n=15, 3 0 , 45) and 2,000 repetitions f r o m each o f the s i m u l a t e d p o p u l a t i o n data sets ( N = 9 0 , 0 0 0 ) . T h e general pattern o f the e m p i r i c a l type I error rates was r e p e t i t i v e l y demonstrated across the three c o r r e l a t i o n c o n d i t i o n s (p ix2=-9, .7, .5) a n d r e l a t i v e x  v a r i a t i o n c o n d i t i o n s w h e n £ i = £ 2 = 1 . 0 (see T a b l e 5-9). T h e c o r r e l a t i o n p i x  X  x  x 2  does not  have substantial effect o n the type I error rate i n the c o n d i t i o n £ i = £ 2 = 1 . 0 , w i t h the x  X  differences b e t w e e n the o b s e r v e d type I error rates o v e r the three p i 2 c o n d i t i o n s never x  X  e x c e e d i n g 0 . 0 0 3 . A d d i t i o n a l l y , there is n o consistent pattern o f the difference between the three c o r r e l a t i o n c o n d i t i o n s . T h e r e is a slight decrease i n the type I error rate w h e n the v a r i a t i o n o f the numerator variable decreases.  T h e largest d i s c r e p a n c y o c c u r s i n the  c o n d i t i o n s p i = 0 . 9 , oc=0.05 and p i 2 = 0 . 5 , a = 0 . 1 0 , where the type I error rate under x  x  x 2  X  c o n d i t i o n V i / V = 0 . 5 is 0.008 less than it is under V i / V = 2 . 0 . T h i s difference x  x 2  x  x 2  represents a 1 5 % a n d 8% difference, respectively. H o w e v e r , these differences are s m a l l and indicate that there is n o substantial d e v i a t i o n o f the e m p i r i c a l type I error rate f r o m the n o m i n a l l e v e l w h e n a ratio v a r i a b l e is used i f the c o m p o n e n t variables do not violate the a s s u m p t i o n o f c i r c u l a r i t y .  A s s h o w n i n T a b l e 5-9, the type I error rate is inflated w h e n there are heterogeneous c o v a r i a n c e s o f the c o m p o n e n t variables (i.e., £ i = £ = 0 . 7 , E i = 0 . 7 a n d x  x 2  x  116  T a b l e 5-9 E m p i r i c a l T y p e I E r r o r Rates V /V =2.0 x l  e  e  oc=  .01  V /V =1.0  x 2  x l  V /V =0.5  x 2  x l  .10  .01  x 2  .05  .10  .01  .05  .05  .10  . 010 . 010 . 010  . 053 . 051 . 051  . 097 . 099 .100  . 012 . Oil . 010  . 048 . 051 . 049  . 099 . 098 . 099  . 008 . 009 . 010  . 045 . 044 . 044  .094 . 093 . 092  . 020 . 021 . 022  .064 . 066 . 066  .116 . 116 . 115  . 021 .023 . 023  . 067 . 067 . 065  . 114 . 112 . 114  . 019 . 020 . 019  . 060 . 066 . 066  .113 .115 . 112  . 030 . 024 . 022  . 073 . 072 . 067  .123 .118 . 112  . 015 . 013 . 014  . 059 . 057 . 055  . 113 . 105 . 104  . 014 . 010 . 010  . 060 . 049 . 045  . Ill . 096 . 093  x l - x 2 - l •0 e  Pxlx2= • 9 Pxlx2= • 7 Pxlx2 = • 5  Ex2=0.7 Pxlx2 = • 9 Pxlx2= • 7 Pxlx2= • 5  0.7 e =1.0 x2  Pxlx2= - 9 Pxlx2= • 7 Pxlx2= • 5  117  £x2=l-0), and is affected b y p i 2 and V i / V  x 2  these effects and their r e l a t i o n s h i p to e i  and e i / , d e t a i l e d results based o n a = 0 . 0 5  x  X  x  x  i n the latter c o n d i t i o n . T o c l e a r l y s h o w A  / x 2  x  x 2  w i t h n=45 are presented i n T a b l e 5-10. T h e results i n T a b l e 5-10 indicate that the type I error rate is c o n s i s t e n t l y inflated w h e n the c o m p o n e n t variables have the same degree o f v i o l a t i o n o n the a s s u m p t i o n o f c i r c u l a r i t y (i.e., e i = e = 0 . 7 ) . F o r e x a m p l e , the mean o f £ i / A  x  x2  x  x 2  ranged between 0.65 a n d  0.70 (n=45), and the actual type I error rate ranged between 0.061 a n d 0.068 (a=0.05). T h e r e is no clear effect pattern o f p j x  and V i / V  x 2  x  o n the type I error rate, w h i c h is to be  x 2  e x p e c t e d based o n the f i n d i n g s o f the p o p u l a t i o n e i / x  s h o w n that e i / x  x 2  seems independent o f p i x  x 2  i n the p r e v i o u s section (it was  and V i / V  x 2  x  x 2  ).  T h e l o w e s t panel o f T a b l e 5-10 shows the effect o f p i x  x 2  and V i / V x  x 2  o n the  e m p i r i c a l type I rate i n the c o n d i t i o n e i = 0 . 7 and e = 1 . 0 . T h e results suggest that w h e n x  x2  o n l y the numerator v a r i a b l e violates the a s s u m p t i o n o f c i r c u l a r i t y , the e m p i r i c a l type I error rate has a p o s i t i v e r e l a t i o n s h i p w i t h p  x i x 2  and V  x l  /V  x 2  .  F o r e x a m p l e , based o n  a = 0 . 0 5 and n=45 i n the c o n d i t i o n £ i = 0 . 7 and 8 =1.0, the greatest type I error rate i n this x  x2  study was 0.073 w h e n p i = 0 . 9 a n d V i / V = 2 . 0 , and the smallest type I error rate was x  x 2  x  x 2  0.041 w h e n p i = 0 . 5 and V i / V = 0 . 5 . T h e s e results are expected, based o n the x  x 2  x  x 2  differences i n the p o p u l a t i o n value o f e i x  Pxix2-  / x 2  T h a t is, the higher the value o f V i / V x  for the different c o n d i t i o n s o f V i / V x  x 2  x2  and  and the l o w e r the v a l u e o f p i 2 , the l o w e r  the m a g n i t u d e o f 8 / and the higher the type I error rate. x)  x 2  x  X  118  vn JL ^  o  ll  o  co  8s  CO  CD  lH  o  in o o O  ro ro ro  in in m  o o ro co 00  oo oo  ON r-  ro ro ro ID ID ID  ID  00 ON ON ID ID ID  00 00 00  o ON VO ID VD  O  CM CN]  ID ID ON 00 ON ON  oo 00 ID ID ID o o o  o in CN VO in in  O ON ON ON 00 00  o l>  rH  OJ  o cn  IT) in 00 00 00  00 ID in ID ID ID  00 o ON r - 00  in in  ro  CN <H ID ID ID  o  ro  rH  t~- rH rH 00 ON ON  ro in in m  rH ID ID ID ID ID o o O  CO CN CO ID o o O o  o o rH ON ON CfN  r~ r~ ID ID ID  rm  ID ID [> OO 00 00  in in in  VO O in in VO VD  ID ID rf- r - r -  o  rH rH ID ID ID  ro rm in  rH VD  O  r-  r-  r-  ON  o  o  o  o  ON  r- in  O  ID ID ID  in in m  o  rH  O in  ON  o  f-  >  r-  rH rH  O  i—i X  r~  O  e-» r -  o o o co  W  ON  o o O  ON  3  CM X  ID ID ID ID  l>  CN  >  in oo  o o o  00 00 00  d  X  ro  rH rH  rH  o  00 ID ID  rH r-  rH  ID CO r~  o  oo  r-  o  ro  00 00  rH rH t - t-~  i—i CD  00  <n in q  ^  o  o  o  OJ  o  •c  1  o  CN II CN X  W  c o  Q.  x >  m O co CO < i-H  ID ID ID  II  CN ID VD  o  o  rH rH rH  H  CH  o  in  ON  w O  g  CD HH  in io  •*-»  •  •  ON  r- in  r~  H—»  )H  CD  c co  m u  CD  II  CN  CD  II  II  CN  ll  CN  X rH X rH X X X X Q. Q. Q. <-t  o .  •  ll I II II c* » ICN CN CN X CO HX rH X X II X X X rH Q. Q. Q. X CO  t> o  II  CN  II  CN  o r^  II  CN  rH X HX rH X o rH IrH I ICN I X X X Q. Q. CL X X CO CO  o  \  119  Sample Estimates  A  8vi/ ?, Y  and Type I Error Rates  In R M A N O V A designs, the B o x e and H u y n h - F e l d t ~e are sometimes used to A  protect against a p r o b a b l e inflation i n the type I error rates i n the within-subject F test w h e n e v e r the e o f a s a m p l e is less than 1.0 (the d e f i n i t i o n s for e and ~e are i n the A  A  methods and procedures chapter). C o l l i e r et a l . (1967), D a v i d s o n (1972), and m a n y others have s h o w n that, i n general, v i o l a t i o n o f the a s s u m p t i o n o f c i r c u l a r i t y (e<1.0) results i n the artificial inflation o f the F values for the w i t h i n - s u b j e c t m a i n effects, thus i n f l a t i n g the type I error rate. A s i n d i c a t e d i n the chapter on methods and procedures, i n practice, the p o p u l a t i o n e is u n k n o w n and the sample 8 is used to estimate the p o p u l a t i o n A  e, and an e-adjusted F test m a y be c o n d u c t e d to correct the inflation o f the type I error A  rate. H u y n h and F e l d t (1970, 1976) i n d i c a t e d that the e-adjusted F test is n e g a t i v e l y A  biased and w o u l d m a k e the F test too conservative. T h e y s h o w e d that the bias is most serious w h e n a p o p u l a t i o n 8 is above 0.75, e s p e c i a l l y w h e n the sample size is s m a l l . T h u s a less b i a s e d estimate, ~e, was suggested (see F o r m u l a (19)). H u y n h and F e l d t also i n d i c a t e d that the ~e-adjusted F test m a y result i n an i n f l a t e d type I error rate w h e n ~e is less than 0.75, and thus a m o r e conservative approach (e.g., e ) is needed i n such A  situations. In this study, w e investigated h o w the bias o f e i  and the type I error rate  A  x  are affected b y p  x ( x 2  , V i/V x  x 2  / x 2  , and e i and e . x  x2  T o investigate the s a m p l i n g characteristics o f e i A  x  / x 2  , the s i m u l a t i o n results s h o w n  i n T a b l e s 5-6, 5-7, and 5-8 are e x p a n d e d and g r a p h i c a l l y presented i n F i g u r e s 5-3, 5-4, and 5-5. F i g u r e s 5-3, 5-4, and 5-5 show the mean o f the 2,000 e i / A  x  x 2  values w i t h plus  and m i n u s t w o standard deviations i n each o f the s i m u l a t e d c o n d i t i o n s .  120  A s s h o w n i n F i g u r e 5-3, even though £ i / 2 = 1 . 0 , the values o f £ i / 2 were A  x  x  X  X  c o n s i d e r a b l y s m a l l e r than unity for a l l the levels o f p i 2 and V i / V 2 , w h i c h indicates x  x  X  X  biased estimation o f £ i / 2 - F o r e x a m p l e , the l o w e r b o u n d o f £ i / 2 was m u c h l o w e r than A  x  x  X  0.75 for n=15 i n a l l three V i / V x  x 2  X  c o n d i t i o n s . W h e n the sample size increased, the l o w e r  b o u n d increased substantially, e s p e c i a l l y i n the c o n d i t i o n s V i / V x  x 2  = 2 . 0 and V / V x l  x 2  =1.0.  G i v e n £ i / 2 = 1 . 0 , w h e n a within-subject F test is c o n d u c t e d , there s h o u l d be n o x  X  c o r r e c t i o n . H o w e v e r , because the p o p u l a t i o n e p s i l o n is not k n o w n i n practice, £ i / 2 w i l l A  x  X  result i n a "false" c o r r e c t i o n o f the F test i n a l l cases o f F i g u r e 5-3, and ~£ i/ 2 i n m a n y o f x  X  them. F o r n=30 and 45 £ i / 2 is u s u a l l y greater than 0.75 and thus ~£ adjustment w o u l d A  x  X  u s u a l l y be a p p l i e d , r e s u l t i n g i n o n l y a slight bias. B u t for n=15, because £ i / 2 is quite A  x  X  often s m a l l e r than 0.75, w e w o u l d use the B o x £ adjustment i n the F test and that w i l l A  cause a c o n s i d e r a b l e decrease i n type I error rate. In the c o n d i t i o n V ] / V 2 = 0 . 5 , because X  X  the l o w e r b o u n d is m u c h l o w e r than i n the other t w o c o n d i t i o n s , the £ y 2 adjustment A  x  X  w o u l d be used for those l o w £ i / 2 values ( £ i / 2 < 0 . 7 5 ) , and an serious o v e r adjustment A  A  x  i n the F test is expected.  x  X  X  M o r e subjects i n a s a m p l e i n the c o n d i t i o n V i / V = 0 . 5 are x  x 2  needed to reduce the bias o f the £ i / 2 , and even w i t h n=45 there is still c o n s i d e r a b l e bias A  x  X  and large s a m p l i n g v a r i a b i l i t y . F i g u r e 5-4 s h o w s the range o f £ i / 2 +/- 2 s d i n the c o n d i t i o n £ = £ = 0 . 7 .  The  A  x  X  x l  x 2  results s h o w that the v a r i a b i l i t y o f £ i / 2 increases w h e n the variation o f the d e n o m i n a t o r A  x  X  variable increases. T h e upper b o u n d o f £ i / 2 was a p p r o x i m a t e l y 0.8 for a l l the v a r i a t i o n A  x  X  c o n d i t i o n s , but the l o w e r b o u n d was l o w e r w h e n V 2 increased. X  section, i f £ i = £ = 0 . 7 , the values o f £ i / x  x 2  x  x 2  A s shown i n a previous  ranged f r o m 0.70 to 0.73 i n different V i / V x  x 2  121  aieiujisg uo|isd3  122  OTCOr^COWTtCOCM  d  d  d  d  d a)euij)S3  d u o i j s d g  d  d  123  a n d p i x 2 c o n d i t i o n s . T h e results s h o w that some values o f £ i /  were equal or very  A  x  x  c l o s e to the £ / x ]  x2  a n d some values o f s i / 2 over- or under-estimated e i A  x 2  X  x  c o n s i d e r a b l e degree.  x  A n adjusted F test, u s i n g E or ~£, w i t h the £ i / a  by a  equal or c l o s e to  A  x  / x 2  x2  the £ i/ w o u l d appropriately estimate the p o p u l a t i o n £ i/ , and appropriately adjust the x  x2  x  x2  degrees o f freedom i n the F test to protect the type I error rate. F i g u r e 5-4, there are m o r e values o f £ i/ A  x  x2  H o w e v e r , as s h o w n i n  l o w e r than e i / , thus an o v e r - c o r r e c t i o n is x  x2  m o r e l i k e l y than an under-correction i n the c o n d i t i o n e i = e = 0 . 7 , r e s u l t i n g i n an o v e r l y x  x2  c o n s e r v a t i v e F test. F i g u r e 5-5 s h o w s the range o f £ i / A  x  x 2  +/- 2sd i n the c o n d i t i o n £ i=0.7 and £ = 1 . 0 .  In this c o n d i t i o n , it has been s h o w n that the magnitudes o f £ i / x  different l e v e l s o f p i x  and V i / V  x 2  x  x 2  . The £ i / x  x 2  x 2  x l  the c o n d i t i o n s V / V x (  x 2  x 2  = 1 . 0 o r V i / V = 0 . 5 . In x  = 1 . 0 a n d V i / V = 0 . 5 , the magnitudes o f E I  substantial bias i n £ i /  x  A  x  are different o v e r the  h a d l o w values (i.e., l o w e r than 0.7) w h e n  x 2  V i / V = 2 . 0 a n d h i g h values (i.e., higher than 0.85) w h e n V / V x  x 2  X  was s h o w n . Therefore, the £ i/ A  x2  x 2  x  x  x2  / x 2  x 2  w e r e h i g h and  adjustment c o n d u c t e d i n m o s t  samples ( e s p e c i a l l y w h e n n=15 where £ i / < 0 . 5 0 s o m e t i m e s ) w o u l d overcorrect the A  x  x 2  degrees o f freedom associated w i t h the F test, resulting i n type I error rate c o n s i d e r a b l y less than the n o m i n a l l e v e l . In the c o n d i t i o n V | / V = 2 . 0 , the o v e r - c o r r e c t i o n is also x  x 2  m o r e evident than the under-correction because m o r e samples have l o w e r £ ] / A  X  x 2  values.  In general, g i v e n £ i/ =1.0, no adjustment s h o u l d be needed, h o w e v e r , i n pratice x  x2  the adjusted F test w o u l d be used i n a l l the samples and result i n F tests that were too c o n s e r v a t i v e . If £ i and/or £ x  x 2  are not equal to 1.0, u s i n g an adjusted F test i n a R M  A N O V A m a y result i n o v e r - c o r r e c t i o n i n some c o n d i t i o n s and under-correction i n other  124  125  conditions. T a b l e 5-11 s h o w s the s u m m a r y for the results o f £ i = £ = 0 . 7 , and £ i = 0 . 7 a n d x  x 2  x  £ 2 = 1 . 0 c o n d i t i o n s . B e c a u s e s i m i l a r patterns were s h o w n i n different p X  x ) X  2 conditions,  o n l y the results for p i 2 = 0 . 7 (n=15) are presented i n T a b l e 5-11. In T a b l e 5-11, an x  X  "appropriate c o r r e c t i o n " is defined as the adjusted F tests based on the magnitudes o f A  £xi/x2  i n the range £ i/ 2+/-0.05, an " u n d e r - c o r r e c t i o n " as an insufficient c o r r e c t i o n o f the x  X  F test o c c u r r i n g because £ i /  exceeds £ i /  A  x  x 2  x  x 2  b y m o r e than 0.05, and an " o v e r - c o r r e c t i o n "  as too c o n s e r v a t i v e an adjustment i n the F tests r e s u l t i n g f r o m the c o n d i t i o n s w h e r e  A  £ i/ x  x 2  is less than £ i / 2 b y m o r e than 0.05. T h e results o f the adjusted F tests w i t h an ~ £ i / 2 x  X  x  X  adjustment i n the different c o n d i t i o n s w a s c o m p a r e d to the results w i t h the £ i / 2 A  x  X  adjustment. G i v e n that the m e a n o f £ i / 2 is s m a l l e r than £ i / 2 i n a l l cases i n T a b l e 5-11, the A  x  A  £ i/x2 x  X  x  X  adjustment is l i k e l y to s h o w an o v e r - c o r r e c t i o n i n the F test. W h e n the  A  £ i/ 2 x  X  adjustment is used, the conservative F tests w o u l d result i n type I error rate s e r i o u s l y less than the n o m i n a l l e v e l i n the c o n d i t i o n £ i = 0 . 7 a n d £ = 1 . 0 . T h a t is, the percent o f x  x 2  o v e r - c o r r e c t i o n was 5 8 . 7 % , 9 5 . 1 % , and 9 8 . 6 % for the three V i / V x  respectively. T h e results also indicate that u s i n g an ~ £ i / x  x2  x 2  conditions,  c o r r e c t i o n can not substantially  increase the p r o b a b i l i t y o f a correct adjustment i n the F test i n the t w o c o n d i t i o n s 8 = E = 0 . 7 a n d £ i = 0 . 7 and £ 2 = 1 . 0 . A l t h o u g h it can not increase the percent o f x l  x ]  x  X  appropriate corrections i n the F test, as s h o w n i n T a b l e 5-11, the ~ £ i / 2 adjustment x  X  substantially reduces the percent o f the o v e r - c o r r e c t i o n and increases the percent o f the under-correction.  126  T a b l e 5-11 A S u m m a r y o f the E f f e c t o f  Vxj/Vx?  on the T y p e I E r r o r Rates ( p x i x 2 = 0 - 7 , a=.Q5, n=15)  %am Exi  £x2  V i/V x  x 2  e i/x2  A  x  Type I Error  £xi/x2  Appropriate Correction' e ~e  UnderCorrection e ~e  OverCorrection e ~e  1  A  3  A  A  0 7  0 7  2 0 1 0 0 5  0 70 0 71 0 72  0 61( 09) 0 62 ( 10) 0 58( 11)  . 061 . 065 . 067  27 0 25 6 26 4 22 2 16 4 21 9  6 0 49 6 8 1 50 0 4 2 34 8  67 0 24 65 5 27 79 4 43  8 8 3  0 7  1 0  2 0 1 0 0 5  0 64 0 91 0 96  0 57 ( 09) 0 71 ( 09) 0 67 ( 11)  . 068 . 054 . 055  32 1 26 1 4 6 4 8 1 3 17 1  9 2 50 0 0 3 47 6 0 1 15 9  58 7 23 95 1 47 98 6 67  9 6 0  A p p r o p r i a t e c o r r e c t i o n refers to percent o f times an adjustment i n the F test w o u l d be  "appropriate" defined as | £ i/x2-£xi/x2|<=0.05. U n d e r - c o r r e c t i o n is defined as insufficient A  x  adjustment w h i c h occurs w h e n £ i / x 2 > £ i / x 2 + 0 . 0 5 , and o v e r - c o r r e c t i o n is defined as too A  x  x  conservative adjustment w h i c h occurs w h e n £ i / 2 < £ x i / x 2 - 0 . 0 5 . A  x  X  127  In general, the o v e r - c o r r e c t i o n o f the F test is more p r o n o u n c e d i n the t w o c o n d i t i o n s e i = e = 1 . 0 , and e i = 0 . 7 and e i=1.0, than i n the c o n d i t i o n e i = e = 0 . 7 . T h e x2  x  x  x  x2  x  most severely over-corrected F test w o u l d be i n the c o n d i t i o n V / V = 0 . 5 and e i = 0 . 7 x )  x 2  x  a n d 8 i = 1 . 0 . T h e results indicate that large bias and extreme v a r i a b i l i t y o f £ i/ A  x  x  x2  would  be e x p e c t e d w h e n V i < V , e s p e c i a l l y w i t h s m a l l sample size (the s a m p l e size equal or x  x 2  s m a l l e r than 15 w h i c h is very c o m m o n i n s o m e fields i n h u m a n k i n e t i c s research). I f V i<V x  x 2  i n the c o m p o n e n t variable data, it is quite p o s s i b l e that a researcher c o u l d get a  very l o w e i / A  x  x 2  and the over-corrected F test w o u l d m a k e the w i t h i n - s u b j e c t F test too  conservative. In this case, the ~ e i / x  x2  adjustment m a y be an alternative c h o i c e but it m a y  result i n an under-correction. T h e results suggest that, w h e n a ratio v a r i a b l e is used, i f the d e n o m i n a t o r variable has l o w e r v a r i a t i o n than the numerator variable and s a m p l e size is large, the r i s k o f the o v e r - c o r r e c t i o n i n the R M A N O V A F test c o u l d be substantially reduced.  128  CHAPTER VI: A SUMMARY OF T H E INVESTIGATION  T h e purposes o f this dissertation were t w o f o l d . T h e first was to use a p r a c t i c a l procedure to evaluate four deflation m o d e l s for ratio scores, and to e x a m i n e the v a l i d i t y and r e l i a b i l i t y o f c o m m o n l y used ratio variables i n h u m a n k i n e t i c s research. T h e s e c o n d was to investigate h o w the statistical characteristics o f the c o m p o n e n t variables affect the assumptions and results o f a R M A N O V A test. T h i s chapter presents a s u m m a r y and i m p l i c a t i o n s o f the findings o f this study.  Empirical Ratio Data Study In this i n v e s t i g a t i o n it was s h o w n that the s i m p l e ratio m o d e l c o m m o n l y used for deflation purposes d i d not appropriately deflate the effect o f the d e n o m i n a t o r for the t w o e m p i r i c a l data sets. T h e l i n e a r regression m o d e l w i t h an intercept ( L R M ) and the n o n l i n e a r regression m o d e l w i t h o u t an intercept ( N L R M l ) seemed e q u a l l y preferable for deflation purposes i n these data. T h e s e results i m p l y that an o p t i m a l deflation m o d e l useful for a l l ratio variables m a y not exist, and different m o d e l s s h o u l d be a p p l i e d to a data set to determine the appropriate deflation m o d e l . It is r e c o m m e n d e d that the procedures d e v e l o p e d i n this study be used to determine the best deflation m o d e l for c o m m o n l y used ratio variables i n o u r f i e l d . T o obtain the best ratio v a r i a b l e for deflation purposes, one s h o u l d fit each o f the four deflation m o d e l s to the data and evaluate the v a l i d i t y o f the m o d e l u s i n g five criteria; (a) zero c o r r e l a t i o n between a d e r i v e d ratio v a r i a b l e and the d e n o m i n a t o r variable, (b) n o c u r v i l i n e a r r e l a t i o n s h i p between the d e r i v e d ratio and the d e n o m i n a t o r i n  129  the scatterplots, (c) e q u a l i t y o f the estimated expected value o f the m o d e l and c a l c u l a t e d e m p i r i c a l m e a n o f the d e r i v e d ratio data, (d) h i g h R , and (e) h i g h r e l i a b i l i t y o f the d e r i v e d ratio data. It was s h o w n that the r e l i a b i l i t y o f a ratio variable is strongly affected b y not o n l y w i t h i n and between trial correlations, but also b y the relative v a r i a t i o n o f the c o m p o n e n t variables ( V i / V 2 ) . x  X  I f the coefficients o f v a r i a t i o n o f the c o m p o n e n t variables i n t w o  trials are the same, the results s h o w that the r e l i a b i l i t y o f the ratio v a r i a b l e is a function o f o n l y the w i t h i n and between trial c o r r e l a t i o n o f the c o m p o n e n t variables, and is not affected b y the coefficients o f v a r i a t i o n . H o w e v e r , g i v e n that the w i t h i n and between trial correlations o f the c o m p o n e n t variables do not change, unequal coefficients o f v a r i a t i o n o f the numerator and d e n o m i n a t o r variables m a y result i n the h i g h r e l i a b i l i t y for the ratio variable, and the effect is most p r o n o u n c e d w h e n the d e n o m i n a t o r v a r i a b l e has the s m a l l e r coefficient o f v a r i a t i o n ( V i > V 2 ) . It i m p l i e s that researchers s h o u l d c o m p u t e x  X  the r e l i a b i l i t y o f the d e r i v e d ratio scores, and not assume that strong r e l i a b i l i t i e s i n the c o m p o n e n t measures a u t o m a t i c a l l y l e a d to strong r e l i a b i l i t y i n the ratio measures.  Simulation Investigation Characteristics of  and e i / 2  Evi/v2, p \ ^  A  x  v v  X  T h e m a g n i t u d e o f e p s i l o n o f a ratio variable, e i/ 2, and its sample estimate, x  A  X  £ x i / x 2 , were s h o w n to be affected not o n l y b y the magnitudes o f e p s i l o n o f the  c o m p o n e n t variables e i and e 2, but also b y the relative v a r i a t i o n V i / V 2 and the x  X  x  X  c o r r e l a t i o n ( p i 2 ) between the c o m p o n e n t variables. T h e nature o f these relationships x  X  can be s u m m a r i z e d as f o l l o w s .  130  Characteristics of epsilon (E I= £ ) , X  I f the c o m p o n e n t variables have the same l e v e l o f  e^^i.  ratio v a r i a b l e has v i r t u a l l y the same e p s i l o n value as the c o m p o n e n t  a  x 2  variables. H o w e v e r , the m a g n i t u d e o f e p s i l o n o f a ratio v a r i a b l e £ i/ 2 is strongly affected x  X  b y the relative v a r i a t i o n and the c o r r e l a t i o n o f the c o m p o n e n t variables i f the c o v a r i a n c e m a t r i x o f the numerator v a r i a b l e or o f the d e n o m i n a t o r v a r i a b l e has a heterogeneous structure ( £ i o r £ x  is s m a l l e r than u n i t y ) .  x 2  If the numerator v a r i a b l e violates the a s s u m p t i o n o f c i r c u l a r i t y and the d e n o m i n a t o r does not ( E I = 0 . 7 a n d £ 2 = 1 . 0 ) , the results s h o w that relationship with V i / V x  x 2  £ i/ 2  X  X  x  X  has a negative  a n d p i 2- W h e n V i / V 2 a n d p i 2 decrease the p o p u l a t i o n x  X  x  X  x  X  e p s i l o n o f the ratio v a r i a b l e £ i/ 2 increases. F i g u r e 6-1 s h o w s these f i n d i n g s . W h e n o n l y x  X  the d e n o m i n a t o r v a r i a b l e violates the assumption o f c i r c u l a r i t y ( £ i = 1 . 0 a n d £ 2 = 0 . 7 ) , the x  X  pattern o f £ i/ 2 o v e r the three v a r i a t i o n c o n d i t i o n s is the opposite f r o m the c o n d i t i o n x  X  £ i = 0 . 7 a n d £ = 1 . 0 , but the effect o f p i x  x 2  x  x 2  on  £ i x  is the same i n the t w o e p s i l o n  / x 2  c o n d i t i o n s . T h a t i s , i n the c o n d i t i o n £ i = 1 . 0 and £ = 0 . 7 , w h e n V i / V x  Pxix2  x  x 2  x 2  increases a n d  decreases the p o p u l a t i o n e p s i l o n o f the ratio v a r i a b l e £ i / 2 increases. x  X  Characteristics of p i i. T h e results s h o w that the magnitudes i n the c o r r e l a t i o n v  v  matrices o f the ratio v a r i a b l e are m a i n l y affected b y the between trial c o r r e l a t i o n o f the numerator a n d o f the d e n o m i n a t o r variables, a n d the relative v a r i a t i o n o f the t w o c o m p o n e n t variables ( V / V x ]  x 2  ) . Varying  p  x l x 2  has n o substantial effect on the m a g n i t u d e  o f the correlations i n the p i j matrices. y  y  W h e n £ i = £ = 1 . 0 , the c o r r e l a t i o n m a t r i x o f the ratio v a r i a b l e has a constant x  x 2  pattern a n d the magnitudes o f p i j decrease w h e n V y  y  x )  /V  x 2  decreases f r o m 2.0 to 0.5.  131  F i g u r e 6 - 1 . T h e £ i/ 2 values and the e m p i r i c a l type I error rates i n the c o n d i t i o n 8 i = 0 . 7 x  x  X  and 6x2=1.0 ( 0 = 0 . 0 5 , n=45).  Vx1 / V x 2 = 0.5  Vx1/Vx2=0.5  132  W h e n 8 i=e 2=0.7 and the correlations o f the c o m p o n e n t variables f o l l o w a s i m p l e x X  x  pattern, the c o r r e l a t i o n matrices o f the ratio v a r i a b l e e x h i b i t a s l o w e r d e c r e a s i n g pattern than the s i m p l e x pattern i n a l l V i / V 2 c o n d i t i o n s . In the t w o c o n d i t i o n s e i=0.7 and X  x  x  e 2=1.0, a n d 8xi=1.0 a n d e 2=0.7, the results i n d i c a t e that the c o r r e l a t i o n pattern o f the X  X  ratio v a r i a b l e is m o r e s i m i l a r to that o f the c o m p o n e n t v a r i a b l e (numerator or denominator) w h i c h has a larger coefficient o f v a r i a t i o n than the other c o m p o n e n t .  Characteristics of  A  e i / ? , . T h e value o f e p s i l o n estimated f r o m a s a m p l e , e i / 2 , A  v  V  x  X  is a b i a s e d estimator o f the p o p u l a t i o n e p s i l o n o f the ratio v a r i a b l e e i/ 2, a n d the bias X  x  varies a c c o r d i n g to the c o n d i t i o n s o f e i , e 2, V i / V 2 , and p i 2 X  x  X  x  x  X  W h e n exi=8 2=1.0, the bias o f e i / 2 is affected b y the coefficient o f v a r i a t i o n o f A  X  x  X  the c o m p o n e n t v a r i a b l e s , but the c o r r e l a t i o n p i x  x 2  does not have substantial effect o n the  bias i n this c o n d i t i o n . T h a t i s , i f V i / V 2 > 1 . 0 , the m e a n values o f e i / 2 for n=15 are A  X  x  X  x  between 0.74 a n d 0.77 (Sd= 0.08-0.09), and the m e a n bias is between 0.23 to 0.26. W h e n the v a r i a t i o n o f the d e n o m i n a t o r increases ( V ] / V 2 = 0 . 5 ) , the bias o f e i / 2 is m o r e A  X  X  X  x  serious, a n d the m e a n v a l u e o f e i / 2 is 0.68 (Sd=0.12), and the mean bias is 0.32. A  x  X  W h e n 8x1=8x2=0.7, the bias o f e i / x 2 is also m o r e serious w h e n V i / V 2 has a l o w A  x  value ( V i / V 2 = 0 . 5 ) , but the effect o f V i / V x  X  x  The correlation p  x t x 2  mean bias o f e i /  x 2  A  x  X  x  x 2  is less than i n the c o n d i t i o n e i=e 2=1.0. X  x  does not have a substantial effect i n this c o n d i t i o n . T h e largest  is 0.14 and the standard d e v i a t i o n o f e i A  x  / x 2  is 0.11 ( V i / V = 0 . 5 , x  x 2  n=15). If the numerator v a r i a b l e violates the a s s u m p t i o n o f c i r c u l a r i t y a n d the d e n o m i n a t o r does not (e.g., e i=0.7 and e 2=1.0), both V i / V x  X  x  x 2  and p i x  x 2  affect the bias o f  133  A  A  Exi/x2-  T h a t i s , the bias o f E I / a  X  x 2  increases w h e n V i / V 2 and p x  X  x ! x  2 decrease.  T h e bias o f  is m o r e p r o n o u n c e d i n the c o n d i t i o n £ i = 0 . 7 a n d e = 1 . 0 than the other t w o  Exi/x2  x2  x  c o n d i t i o n s ( E = £ = 1 . 0 and £ I = E = 0 . 7 ) . A m o n g the cases c o n s i d e r e d , the largest bias o f x 1  A  x 2  X  x2  was 0.3 a n d the standard d e v i a t i o n o f £ i  £ i/x2  A  x  x  /  x  2  is 0.11 ( V i / V = 0 . 5 , n=15). x  x 2  If a ratio v a r i a b l e is used, g i v e n certain m e a n differences, and m a g n i t u d e s o f £ i x  a n d £ 2 o f the t w o c o m p o n e n t variables, the results indicate that the m a g n i t u d e s o f £ i/ 2 X  x  a n d £ i / 2 are substantially affected b y V i / V A  x  x  X  x 2  and p i x  x 2  X  . W h e n the magnitudes o f E I X  / X 2  a n d E i / 2 v a r y , the type I error rates o f the R M A N O V A tests is affected. T h e i m p a c t o f A  x  X  u s i n g ratio v a r i a b l e s o n the type I error rate i n a R M A N O V A test is s u m m a r i z e d i n the f o l l o w i n g sections.  Type I Error When Using Ratio Variables in R M ANOVA Type I Error Rate.  T h e s i m u l a t i o n i n v e s t i g a t i o n was d e s i g n e d to e x a m i n e h o w  the characteristics o f the c o m p o n e n t v a r i a b l e s affect the type I error rate o f the F test i n a o n e - w a y R M A N O V A based o n the t r a n s f o r m e d ratio variable. T h e advantage o f this study is that w e c o u l d investigate the i m p a c t o f u s i n g the ratio v a r i a b l e i n a o n e - w a y R M A N O V A , g i v e n that the characteristics o f the ratio v a r i a b l e p o p u l a t i o n a n d the related characteristics o f the c o m p o n e n t variables w e r e k n o w n . In the M o n t e C a r l o s i m u l a t i o n i n v e s t i g a t i o n , results s h o w that the n a i v e F tests i n a f i v e - t r i a l o n e - w a y R M A N O V A o n a ratio v a r i a b l e result i n a type I error rate that is c l o s e to the n o m i n a l l e v e l i n the c o n d i t i o n E i = £ x 2 = l - 0 ( £ x i / x 2 = l - 0 ) , regardless o n the c o n d i t i o n o f V x l / V x 2 and p i 2 - T h e type I x  x  X  error rate o f the F test e x h i b i t s inflation i n the c o n d i t i o n E I = £ 2 = 0 . 7 ( £ i / 2 = 0 . 7 ) , but there X  is n o clear effect o f V x l / V x 2 and p i . x  x 2  X  x  X  If o n l y the numerator v a r i a b l e violates the  a s s u m p t i o n o f c i r c u l a r i t y (e i=0.7 and £ = 1 . 0 ) , the type I error rate o f the F test increases x  x 2  134  w h e n V x l / V x 2 and p  x i x  2 increase. B e c a u s e the value o f e i x  / x 2  indicates the l e v e l o f  v i o l a t i o n o f c i r c u l a r i t y i n the p o p u l a t i o n c o v a r i a n c e m a t r i x o f the ratio v a r i a b l e , the h i g h l e v e l o f the inflation o f the type I error rate i n the F test s h o u l d c o r r e s p o n d to the l o w value o f E i/x2- A s s h o w n i n the first graph i n F i g u r e 6-1, E i/ 2 decreases w h e n V x l / V x 2 x  x  X  and p i x 2 increase. Therefore, it is expected that the inflation o f the type I error rate is x  m o r e severe w h e n V x l / V x 2 and p i x  x 2  increase (see F i g u r e 6-1).  Sample Estimates e i/v7 and Type I Error Rates. B e c a u s e the v a l u e o f e i / A  v  x  x2  is not k n o w n i n practice, i n R M A N O V A designs, the B o x e p s i l o n e a n d H - F ~e are used A  to protect against a probable i n f l a t i o n i n the type I error rates i n the w i t h i n - s u b j e c t F test w h e n e v e r e o f a sample is less than 1.0. U s i n g M o n t e C a r l o procedures, the s a m p l i n g A  characteristics o f e i / A  x  was investigated a n d the effects o f e i 2 and H - F ~£ i/ 2 i n the A  x 2  x  /X  x  X  R M A N O V A were investigated. T h e results indicate that over-adjustment i n the e-adjusted F test is m o r e A  p r o n o u n c e d i n the t w o c o n d i t i o n s e i= e =1.0, and e i=0.7 and e =1.0. T h e large bias x  x2  x  x2  and extreme v a r i a b i l i t y o f 8-adjusted F tests w o u l d be expected w h e n V / V = 0 . 5 A  x l  x 2  e s p e c i a l l y w i t h s m a l l e r sample size (e.g., n=15). I f the H - F ~e c o r r e c t i o n is used, it reduces the r i s k o f over-adjustment but increases the r i s k o f under-adjustment.  In s u m m a r y , this dissertation investigated the v a l i d i t y and r e l i a b i l i t y o f some c o m m o n l y used ratio variables i n h u m a n k i n e t i c s research, and the effect o f u s i n g ratio variables o n the c i r c u l a r i t y a s s u m p t i o n o f the c o v a r i a n c e m a t r i x , type I error rates i n R M A N O V A tests. It shows that different m o d e l s s h o u l d be used to derive an appropriate deflation m o d e l i n e m p i r i c a l research. A c o m m o n l y used deflation m o d e l for a l l ratio  135  variables m a y not exist. T h e results indicate that high r e l i a b i l i t y o f the c o m p o n e n t variables does not necessarily result i n high r e l i a b i l i t y o f the transformed ratio variable. T h u s , w h e n a ratio variable is used the r e l i a b i l i t y s h o u l d be e x a m i n e d based on the ratio variable data. S i m u l a t i o n results s h o w that the characteristics o f the t w o c o m p o n e n t variables (e i, e , V i/V , and p i ) strongly affect the c i r c u l a r i t y o f the c o v a r i a n c e x  x2  x  x2  x  x 2  matrix o f the ratio variable, and the type I error rate i n R M A N O V A tests. In general, the mean E i A  x  / x 2  e x h i b i t e d the greatest bias and the largest standard d e v i a t i o n , resulting i n a  serious inflation o f type I error rate i n the c o n d i t i o n V i/V =0.5, regardless the x  x2  c o n d i t i o n s e , e , and p i . I f h o m o g e n e i t y o f the d e n o m i n a t o r v a r i a b l e ( s m a l l V ) and x)  x2  x  x 2  x2  large sample size are present, it m a y reduce l i k e l i h o o d o f bias i n £ i / A  x  type I error rate.  x 2  and protect the  136  References  A l b r e c h t , G . 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T h e c o r r e l a t i o n between the 2  2  2  t w o o b s e r v e d variables c a n be s h o w n as f o l l o w s :  n  _ <?„,2  Pxlx2  _l/JVX(r +e )(T +e ) i  1  2  2  ~ GxPx2  xl°x2  a  °xl°x2  U n d e r the postulated m o d e l , a l l terms but the first i n the numerator are zero, so the f o r m u l a can be written as  ®T\T2  Px\x2  °x\ x2 G  T h i s shows that the c o v a r i a n c e o f o b s e r v e d scores i s equal to the c o v a r i a n c e o f true scores a 2=pxix2(7xiC?x2=crxix2- B e c a u s e the r e l i a b i l i t y is defined as true variance d i v i d e d TIT  b y o b s e r v e d variance p = G / o 2  xx  T  2 x  (p  x x  refers to the r e l i a b i l i t y for a v a r i a b l e X ) , i f there  were n o errors present, the true c o r r e l a t i o n between t w o variables X ] and X ( p m ) w o u l d 2  be as f o l l o w s :  T  143 _ ®T\TI PT\TI ~  0 0 Ty  ^°x\x2  ~  ®xix2  I  /  Jp O Jp O  T2  xlxl  1^x^x2)  VPx\x\  ^Px2x2  _  xl  x2x2  x2  Px\x2  V'Px\x\  -\Px2x2  T h i s f o r m u l a is s p o k e n o f as the " c o r r e c t i o n " for attenuation, but it is r e a l l y an estimate o f the magnitude o f a correlation i f t w o variables were made perfectly r e l i a b l e ( N u n n a l l y , 1978,p215-221). In this study, the attenuation f o r m u l a was e m p l o y e d to estimate the c o r r e l a t i o n between the n u m e r a t o r and denominator variables i n different trials p ii, 2j: x  Px  X  X  PTUTU ~J=^'r  ( )  =  A-1  V Px^ -\J Px x 2  where p  Tli>T2j  2  is true correlation (measured w i t h o u t error) between X j i n the i t h trial and X  2  i n the j t h trial, p ii,x2j is the observed p o p u l a t i o n correlation between X i i n the ith trial and x  X  2  i n the j t h trial, and p i i and p x  x  x 2 x 2  are the r e l i a b i l i t y coefficients o f X i and X , 2  respectively. T h e r e l i a b i l i t y coefficients o f X i and X were d e r i v e d f r o m the defined 2  c o v a r i a n c e m a t r i x o f the c o m p o n e n t variables i n each o f the c o n t r o l l e d c o n d i t i o n s . A s s h o w n above, this investigation assumed that the w i t h i n trial correlation between the numerator and denominator variables was the same, and the variance o f the c o m p o n e n t variables was constant, over all five trials. Therefore, the best a p p r o x i m a t i o n for the true correlation between the numerator i n trial i and the d e n o m i n a t o r i n trial j w o u l d be the correlation between the t w o variables i n the same trial. T h a t i s , PTii,T2j=Pxii,x2i- Therefore, F o r m u l a ( A - l ) can be rewritten to show the o b s e r v e d correlation between the numerator and denominator variables i n different trials:  (A-2)  145  Appendix B Covariance Matrices Used to Generate the Population Data  Exi=Ex2=1.0, V x l / V x 2 = 2 . 0 , p 225 195 195 195 195 131 110 110 110 110  000 000 000 000 000 582 996 996 996 996  195 225 195 195 195 110 131 110 110 110  000 000 000 000 000 996 582 996 996 996  195 195 225 195 195 110 110 131 110 110  000 000 000 000 000 996 996 582 996 996  195 195 195 225 195 110 110 110 131 110  6x1=6x2=1.0, V x l / V x 2 = 1 . 0 , p 135 117 117 117 117 162 135 135 135 135  000 000 000 000 000 000 942 942 942 942  117 135 117 117 117 135 162 135 135 135  000 000 000 000 000 942 000 942 942 942  117 117 135 117 117 135 135 162 135 135  000 000 000 000 000 942 942 000 942 942  x2  53 45 45 45 45 131 109 109 109 109  000 000 000 000 000 042 675 675 675 675  45 53 45 45 45 109 131 109 109 109  000 000 000 000 000 675 042 675 675 675  45 45 53 45 45 109 109 131 109 109  000 000 000 000 000 675 675 042 675 675  195 195 195 195 225 110 110 110 110 131  000 000 000 000 000 996 996 996 996 582  131 110 110 110 110 95 78 78 78 78  582 996 996 996 996 000 000 000 000 000  110 131 110 110 110 78 95 78 78 78  996 582 996 996 996 000 000 000 000 000  110 110 131 110 110 78 78 95 78 78  996 996 582 996 996 000 000 000 000 000  110 110 110 131 110 78 78 78 95 78  996 996 996 582 996 000 000 000 000 000  110 110 110 110 131 78 78 78 78 95  996 996 996 996 582 000 000 000 000 000  000 000 000 000 000 942 942 942 942 000  162 135 135 135 135 240 195 195 195 195  000 942 942 942 942 000 000 000 000 000  135 162 135 135 135 195 240 195 195 195  942 000 942 942 942 000 000 000 000 000  135 135 162 135 135 195 195 240 195 195  942 942 000 942 942 000 000 000 000 000  135 135 135 162 135 195 195 195 240 195  942 942 942 000 942 000 000 000 000 000  135 135 135 135 162 195 195 195 195 240  942 942 942 942 000 000 000 000 000 000  000 000 000 000 000 675 675 675 675 042  131 109 109 109 109 400 330 330 330 330  042 675 675 675 675 000 000 000 000 000  109 131 109 109 109 330 400 330 330 330  675 042 675 675 675 000 000 000 000 000  109 109 131 109 109 330 330 400 330 330  675 675 042 675 675 000 000 000 000 000  109 109 109 131 109 330 330 330 400 330  675 675 675 042 675 000 000 000 000 000  109 109 109 109 131 330 330 330 330 400  675 675 675 675 042 000 000 000 000 000  =0.9  000 000 000 000 000 942 942 942 000 942  x l x 2  45 45 45 53 45 109 109 109 131 109  =0.9  000 000 000 000 000 996 996 996 582 996  x l x 2  117 117 117 135 117 135 135 135 162 135  e i=e =1.0, V x l / V x 2 = 0 . 5 , p x  x l x 2  117 117 117 117 135 135 135 135 135 162  =0.9  000 000 000 000 000 675 675 675 042 675  45 45 45 45 53 109 109 109 109 131  146  Exl=E 2=1.0, V x l / V x 2 = 2 . 0 , p ix2=0.7 X  225 195 195 195 195 102 86 86 86 86  000 000 000 000 000 341 330 330 330 330  x  195 225 195 195 195 86 102 86 86 86  000 000 000 000 000 330 341 330 330 330  195 195 225 195 195 86 86 102 86 86  000 000 000 000 000 330 330 341 330 330  195 195 195 225 195 86 86 86 102 86  000 000 000 000 000 330 330 330 341 330  195 195 195 195 225 86 86 86 86 102  000 000 000 000 000 330 330 330 330 341  102 86 86 86 86 95 78 78 78 78  341 330 330 330 330 000 000 000 000 000  86 102 86 86 86 78 95 78 78 78  330 341 330 330 330 000 000 000 000 000  86 86 102 86 86 78 78 95 78 78  330 330 341 330 330 000 000 000 000 000  86 86 86 102 86 78 78 78 95 78  330 330 330 341 330 000 000 000 000 000  86 86 86 86 102 78 78 78 78 95  330 330 330 330 341 000 000 000 000 000  000 000 000 000 000 732 732 732 732 000  126 105 105 105 105 240 195 195 195 195  000 732 732 732 732 000 000 000 000 000  105 126 105 105 105 195 240 195 195 195  732 000 732 732 732 000 000 000 000 000  105 105 126 105 105 195 195 240 195 195  732 732 000 732 732 000 000 000 000 000  105 105 105 126 105 195 195 195 240 195  732 732 732 000 732 000 000 000 000 000  105 105 105 105 126 195 195 195 195 240  732 732 732 732 000 000 000 000 000 000  000 000 000 000 000 302 302 302 302 922  101 85 85 85 85 400 330 330 330 330  922 302 302 302 302 000 000 000 000 000  85 101 85 85 85 330 400 330 330 330  302 922 302 302 302 000 000 000 000 000  85 85 101 85 85 330 330 400 330 330  302 302 922 302 302 000 000 000 000 000  85 85 85 101 85 330 330 330 400 330  302 302 302 922 302 000 000 000 000 000  85 85 85 85 101 330 330 330 330 400  302 302 302 302 922 000 000 000 000 000  e ,=e =1.0, V x l / V x 2 = 1 . 0 , P x , = 0 . 7 x  x2  135 117 117 117 117 126 105 105 105 105  000 000 000 000 000 000 732 732 732 732  x 2  117 135 117 117 117 105 126 105 105 105  000 000 000 000 000 732 000 732 732 732  117 117 135 117 117 105 105 126 105 105  000 000 000 000 000 732 732 000 732 732  117 117 117 135 117 105 105 105 126 105  000 000 000 000 000 732 732 732 000 732  117 117 117 117 135 105 105 105 105 126  e ,=e =1.0, V x l / V x 2 = 0 . 5 , p i = 0 . 7 x  x2  53 45 45 45 45 101 85 85 85 85  000 000 000 000 000 922 302 302 302 302  x  45 53 45 45 45 85 101 85 85 85  000 45 000 000 45 000 000 53 000 000 45 000 000 45 000 302 85 302 922 85 302 302. 101 922 302 '• 85 302 302 '• 85302  45 45 45 53 45 85 85 85 101 85  x 2  000 000 000 000 000 302 302 302 922 302  45 45 45 45 53 85 85 85 85 101  147  6x1=6x2=1.0, V x l / V x 2 = 2 . 0 , p  225 195 195 195 195 73 61 61 61 61  000 000 000 000 000 101 664 664 664' 664  195 225 195 195 195 61 73 61 61 61  000 000 000 000 000 664 101 664 664 664  195 195 225 195 195 61 61 73 61 61  000 000 000 000 000 664 664 101 664 664  x ] x 2  195 195 195 225 195 61 61 61 73 61  =0.5  000 000 000 000 000 664 664 664 101 664  195 195 195 195 225 61 61 61 61 73  000 000 000 000 000 664 664 664 664 101  73 61 61 61 61 95 78 78 78 78  000 000 000 000 000 523 523 523 523 000  000 000 000 000 000 930 930 930 930 801  101 664 664 664 664 000 000 000 000 000  664 101 664 664 664 000 000 000 000 000  61 61 73 61 61 78 78 95 78 78  664 664 101 664 664 000 000 000 000 000  61 61 61 73 61 78 78 78 95 78  664 664 664 101 664 000 000 000 000 000  61 61 61 61 73 78 78 78 78 95  664 664 664 664 101 000 000 000 000 000  90 75 75 75 75 240 195 195 195 195  000 75 523 523 90 000 523 75 523 523 75 523 523 . 75523 000 195 000 000 240 000 000 195 000 000 195 000 000 195 000  75 75 90 75 75 195 195 240 195 195  523 523 000 523 523 000 000 000 000 000  75 75 75 90 75 195 195 195 240 195  523 523 523 000 523 000 000 000 000 000  75 75 75 75 90 195 195 195 195 240  523 523 523 523 000 000 000 000 000 000  72 60 60 60 60 400 330 330 330 330  801 930 930 930 930 000 000 000 000 000  60 60 72 60 60 330 330 400 330 330  930 930 801 930 930 000 000 000 000 000  60 60 60 72 60 330 330 330 400 330  930 930 930 801 930 000 000 000 000 000  60 60 60 60 72 330 330 330 330 400  930 930 930 930 801 000 000 000 000 000  61 73 61 61 61 78 95 78 78 78  6*1=6*2=1.0, V x l / V x 2 = 1 . 0 , p i = 0 . 5 x  135 117 117 117 117 90 75 75 75 75  000 000 000 000 000 000 523 523 523 523  117 135 117 117 117 75 90 75 75 75  000 000 000 000 000 523 000 523 523 523  117 117 135 117 117 75 75 90 75 75  000 000 000 000 000 523 523 000 523 523  117 117 117 135 117 75 75 75 90 75  x 2  000 000 000 000 000 523 523 523 000 523  117 117 117 117 135 75 75 75 75 90  e i=8 =1.0, V x l / V x 2 = 0 . 5 , p i = 0 . 5 x  53 45 45 45 45 72 60 60 60 60  x2  000 000 000 000 000 801 930 930 930 930  x  45 53 45 45 45 60 72 60 60 60  000 000 000 000 000 930 801: 930 930 930  45 45 53 45 45 60 60 72 60 60  000 000 000 000 000 930 930 801 930 930  45 45 45 53 45 60 60 60 72 60  x 2  000 000 000 000 000 930 930 930 801 930  45 45 45 45 53 60 60 60 60 72  60 72 60 60 60 330 400 330 330 330  930 801 930 930 930 000 000 000 000 000  148  e =e =0.7, V x l / V x 2 = 2 . 0 , p xl  225 195 170 160 150 131 110 92 83 75  x2  000 000 000 000 000 582 996 398 656 568  195 225 196 170 160 110 131 109 90 83  000 000 000 000 000 996 582 844 896 656  170 196 225 195 170 92 109 131 109 92  000 000 000 000 000 398 844 582 564 398  160 170 195 225 192 83 90 109 131 105  e =e =0.7, V x l / V x 2 = 1 . 0 , p xl  135 117 104 95 81 162 134 116 101 84  x2  000 000 000 000 000 000 188 097 923 954  117 135 116 104 95 134 162 133 116 101  000 000 000 000 000 188 000 613 097 923  104 116 135 116 102 116 133 162 133 114  000 000 000 000 000 097 613 000 613 975  53 45 40 35 32 131 109 91 78 70  x2  000 000 000 000 000 042 675 782 975 177  45 53 45 40 36 109 131 106 91 80  000 000 000 000 000 675 042 299 782 095  40 45 53 45 40 91 106 131 108 90  000 000 000 000 000 782 299 042 000 000  150 160 170 192 225 75 83 92 105 131  000 000 000 000 000 568 656 398 818 582  131 110 92 83 75 95 78 62 54 47  582 996 398 656 568 000 000 000 000 000  110 131 109 90 83 78 95 76 60 54  996 582 844 896 656 000 000 000 000 000  92 109 131 109 92 62 76 95 76 62  398 844 582 564 398 000 000 000 000 000  83 90 109 131 105 54 60 76 95 72  656 896 564 582 818 000 000 000 000 000  75 83 92 105 131 47 54 62 72 95  568 656 398 818 582 000 000 000 000 000  000 000 000 000 000 954 923 975 012 000  162 134 116 101 84 240 190 160 135 110  000 188 097 923 954 000 000 000 000 000  134 162 133 116 101 190 240 190 160 135  188 000 613 097 923 000 000 000 000 000  116 133 162 133 114 160 190 240 190 160  097 613 000 613 '975 000 000 000 000 000  101 116 133 162 130 135 160 190 240 188  923 097 613 000 012 000 000 000 000 000  84 101 114 130 162 110 135 160 188 240  954 923 975 012 000 000 000 000 000 000  000 000 000 000 000 177 095 000 402 042  131 109 91 78 70 400 330 260 220 190  042 675 782 975 177 000 000 000 000 000  109 131 106 91 80 330 400 310 260 220  675 042 299 782 095 000 000 000 000 000  91 106 131 108 90 260 310 400 320 250  782 299 042 000 000 000 000 000 000 000  78 91 108 131 103 220 260 320 400 300  975 782 000 042 402 000 000 000 000 000  70 80 90 103 131 190 220 250 300 400  177 095 000 402 042 000 000 000 000 000  =0.9  000 000 000 000 000 923 097 613 000 012  x [ x 2  35 40 45 53 44 78 91 108 131 103  =0.9  000 000 000 000 000 656 896 564 582 818  x l x 2  95 104 116 135 111 101 116 133 162 130  e =e =0.7, V x l / V x 2 = 0 . 5 , p xl  x l x 2  81 95 102 111 135 84 101 114 130 162  =0.9  000 000 000 000 000 975 782 000 042 402  32 36 40 44 53 70 80 90 103 131  149  e i=£ =0.7, Vxl/Vx2=2.0, p x  x 2  225 195 170 160 150 102 86 71 65 58  000 000 000 000 000 341 330 865 066 775  195 225 196 170 160 86 102 85 70 65  000 000 000 000 000 330 341 434 697 066  170 196 225 195 170 71 85 102 85 71  000 000 000 000 000 865 434 341 216 865  160 170 195 225 192 65 70 85 102 82  £xi=e 2=0.7, V x l / V x 2 = 1 . 0 , p X  135 117 104 95 81 126 104 90 79 66  000 000 000 000 000 000 368 297 273 075  117 135 116 104 95 104 126 103 90 79  000 000 000 000 000 368 000 921 297 273  104 116 135 116 102 90 103 126 103 89  000 000 000 000 000 297 921 000 921 425  53 45 40 35 32 101 85 71 61 54  000 000 000 000 000 922 302 386 425 582  45 53 45 40 36 85 101 82 71 62  000 000 000 000 000 302 922 677 386 296  40 45 53 45 40 71 82 101 84 70  000 000 000 000 000 386 677 922 000 000  150 160 170 192 225 58 65 71 82 102  000 000 000 000 000 775 066 865 303 341  102 86 71 65 58 95 78 62 54 47  341 330 865 066 775 000 000 000 000 000  86 102 85 70 65 78 95 76 60 54  330 341 434 697 066 000 000 000 000 000  71 85 102 85 71 62 76 95 76 62  865 434 341 216 865 000 000 000 000 000  65 70 85 102 82 54 60 76 95 72  066 697 216 341 303 000 000 000 000 000  58 65 71 82 102 47 54 62 72 95  775 066 865 303 341 000 000 000 000 000  000 000 000 000 000 075 273 425 120 000  126 104 90 79 66 240 190 160 135 110  000 368 297 273 075 000 000 000 000 000  104 126 103 90 79 190 240 190 160 135  368 000 921 297 273 000 000 000 000 000  90 103 126 103 89 160 190 240 190 160  297 921 000 921 425 000 000 000 000 000  79 90 103 126 101 135 160 190 240 188  273 297 921 000 120 000 000 000 000 000  66 79 89 101 126 110 135 160 188 240  075 273 425 120 000 000 000 000 000 000  000 000 000 000 000 582 296 000 424 922  101 85 71 61 54 400 330 260 220 190  922 302 386 425 582 000 000 000 000 000  85 101 82 71 62 330 400 310 260 220  302 922 677 386 296 000 000 000 000 000  71 82 101 84 70 260 310 400 320 250  386 677 922 000 000 000 000 000 000 000  61 71 84 101 80 220 260 320 400 300  425 386 000 922 424 000 000 000 000 000  54 62 70 80 101 190 220 250 300 400  582 296 000 424 922 000 000 000 000 000  =0.7  000 000 000 000 000 273 297 921 000 120  x l x 2  35 40 45 53 44 61 71 84 101 80  =0.7  000 000 000 000 000 066 697 216 341 303  x l x 2  95 104 116 135 111 79 90 103 126 101  £xi=e 2=0.7, V x l / V x 2 = 0 . 5 , p X  x l x 2  81 95 102 111 135 66 79 89 101 126  =0.7  000 000 000 000 000 425 386 000 922 424  32 36 40 44 53 54 62 70 80 101  150  E x l  =e =0.7, Vxl/Vx2=2.0, p i =0.5  225 195 170 160 150 73 61 51 46 41  x 2  000 000 000 000 000 101 664 332 476 982  x  195 225 196 170 160 61 73 61 50 46  000 000 000 000 000 664 101 025 498 476  170 196 225 195 170 51 61 73 60 51  000 000 000 000 000 332 025 101 869 332  160 170 195 225 192 46 50 60 73 58  x 2  000 000 000 000 000 476 498 869 101 788  150 160 170 192 225 41 46 51 58 73  000 000 000 000 000 982 476 332 788 101  73 61 51 46 41 95 78 62 54 47  101 664 332 476 982 000 000 000 000 000  61 73 61 50 46 78 95 76 60 54  664 101 025 498 476 000 000 000 000 000  51 61 73 60 51 62 76 95 76 62  332 025 101 869 332 000 000 000 000 000  46 50 60 73 58 54 60 76 95 72  476 498 869 101 788 000 000 000 000 000  41 46 51 58 73 47 54 62 72 95  982 476 332 788 101 000 000 000 000 000  000 000 000 000 000 196 624 875 229 000  90 74 64 56 47 240 190 160 135 110  000 549 498 624 196 000 000 000 000 000  74 90 74 64 56 190 240 190 160 135  549 000 229 498 624 000 000 000 000 000  64 74 90 74 63 160 190 240 190 160  498 229 000 229 875 000 000 000 000 000  56 64 74 90 72 135 160 190 240 188  624 498 229 000 229 000 000 000 000 000  47 56 63 72 90 110 135 160 188 240  196 624 875 229 000 000 000 000 000 000  000 000 000 000 000 987 497 000 446 801  72 60 50 43 38 400 330 260 220 190  801 930 990 875 987 000 000 000 000 000  60 72 59 50 44 330 400 310 260 220  930 801 055 990 497 000 000 000 000 000  50 59 72 60 50 260 310 400 320 250  990 055 801 000 000 000 000 000 000 000  43 50 60 72 57 220 260 320 400 300  875 990 000 801 446 000 000 000 000 000  38 44 50 57 72 190 220 250 300 400  987 497 000 446 801 000 000 000 000 000  £x,=ex2=0.7, V x l / V x 2 = 1 . 0 , p ix =0.5 x  135 117 104 95 81 90 74 64 56 47  000 000 000 000 000 000 549 498 624 196  117 135 116 104 95 74 90 74 64 56  000 000 000 000 000 549 000 229 498 624  104 116 135 116 102 64 74 90 74 63  000 000 000 000 000 498 229 000 229 875  95 104 116 135 111 56 64 74 90 72  £xi=£x2=0.7, Vxl/Vx2=0.5, p  53 45 40 35 32 72 60 50 43 38  000 000 000 000 000 801 930 990 875 987  45 53 45 40 36 60 72 59 50 44  000 000 000 000 000 930 801 055 990 497  40 45 53 45 40 50 59 72 60 50  000 000 000 000 000 990 055 801 000 000  2  000 000 000 000 000 624 498 229 000 229  x l x 2  35 40 45 53 44 43 50 60 72 57  81 95 102 111 135 47 56 63 72 90  =0.5  000 000 000 000 000 875 990 000 801 446  32 36 40 44 53 38 44 50 57 72  151  Exi=0.7, 6x2=1.0, V x l / V x 2 = 2 . 0 , p , x 2 = 0 . 9 x  225 195 170 160 150 131 110 103 100 97  000 000 000 000 000 582 996 637 543 350  195 225 196 170 160 110 131 111 103 100  000 000 000 000 000 996 582 280 637 543  170 196 225 195 170 103 111 131 110 103  000 000 000 000 000 637 280 582 996 637  160 170 195 225 192 100 103 110 131 110  000 000 000 000 000 543 637 996 582 139  150 160 170 192 225 97 100 103 110 131  000 000 000 000 000 350 543 637 139 582  131 110 103 100 97 95 78 78 78 78  582 996 637 543 350 000 000 000 000 000  110 131 111 103 100 78 95 78 78 78  996 582 280 637 543 000 000 000 000 000  103 111 131 110 103 78 78 95 78 78  637 280 582 996 637 000 000 000 000 000  100 103 110 131 110 78 78 78 95 78  543 637 996 582 139 000 000 000 000 000  97 100 103 110 131 78 78 78 78 95  350 543 637 139 582 000 000 000 000 000  162 135 128 122 113 240 195 195 195 195  000 942 167 496 110 000 000 000 000 000  135 162 135 128 122 195 240 195 195 195  942 000 360 167 496 000 000000 000 000  128 135 162 135 126 195 195 240 195 195  167 360 000 360 929 000 000 000 000 000  122 128 135 162 132 195 195 195 240 195  496 167 360 '000 410 000 000 000 000 000  113 122 126 132 162 195 195 195 195 240  110 496 929 410 000 000 000 000 000 000  123 103 97 91 87 400 330 330 330 330  762 581 658 350 348 000 000 000 000 000  103 123 103 97 92 330 400 330 330 330  581 762 581 658 646 000 000 000 000 000  97 103 123 103 97 330 330 400 330 330  658 581 762 581 658 000 000 000 000 000  91 97 103 123 102 330 330 330 400 330  350 658 581 762 424 000 000 000 000 000  87 92 97 102 123 330 330 330 330 400  348 646 658 424 762 000 000 000 000 000  e i = 0 . 7 , e 2=1.0, V x l / V x 2 = 1 . 0 , p x i x 2 = 0 . 9 X  x  135 117 104 95 81 162 135 128 122 113  000 000 000 000 000 000 942 167 496 110  117 135 116 104 95 135 162 135 128 122  000 000 000 000 000 942 000 360 167 496  104 116 135 116 102 128 135 162 135 126  000 000 000 000 000 167 360 000 360 929  95 104 116 135 111 122 128 135 162 132  000 000 000 000 000 496 167 360 000 410  81 95 102 111 135 113 122 126 132 162  000 000 000 000 000 110 496 929 410 000  6x1=0.7, 6x2=1.0, V x l / V x 2 = 0 . 5 , p x 2 = 0 . 9 x l  53 45 40 35 32 123 103 97 91 87  000 000 000 000 000 762 581 658 350 348  45 53 45 40 36 103 123 103 97 92  000 000 000 000 000 581 762 581 658 646  40 45 53 45 40 97 103 123 103 97  000 000 000 000 000 658 581 762 581 658  35 40 45 53 44 91 97 103 123 102  000 000 000 000 000 350 658 581 762 424  32 36 40 44 53 87 92 97 102 123  000 000 000 000 000 348 646 658 424 762  152 exi=0.7, e =1.0, V x l / V x 2 = 2 . 0 , p x2  225 195 170 160 150 102 86 80 78 75  000 000 000 000 000 341 330 606 200 717  195 225 196 170 160 86 102 86 80 78  000 000 000 000 000 330 341 551 606 200  170 196 225 195 170 80 86 102 86 80  000 000 000 000 000 606 551 341 330 606  160 170 195 225 192 78 80 86 102 85  x l x 2  000 000 000 000 000 200 606 330 341 664  =0.7  150 160 170 192 225 75 78 80 85 102  000 000 000 000 000 717 200 606 664 341  102 86 80 78 75 95 78 78 78 78  341 330 606 200 717 000 000 000 000 000  86 102 86 80 78 78 95 78 78 78  330 341 551 606 200 000 000 000 000 000  80 86 102 86 80 78 78 95 78 78  606 551 341 330 606 000 000 000 000 000  78 80 86 102 85 78 78 78 95 78  200 606 330 341 664 000 000 000 000 000  75 78 80 85 102 78 78 78 78 95  717 200 606 664 341 000 000 000 000 000  126 105 99 95 87 240 195 195 195 195  000 732 686 275 975 000 000 000 000 000  105 126 105 99 95 195 240 195 195 195  732 000 280 686 275 000 000 000 000 000  99 105 126 105 98 195 195 240 195 195  686 280 000 280 722 000 000 000 000 000  95 99 105 126 102 195 195 195 240 195  275 686 280 000 986 000 000 000 000 000  87 95 98 102 126 195 195 195 195 240  975 275 722 986 000 000 000 000 000 000  101 85 80 75 71 400 330 330 330 330  922 302 424 230 933 000 000 000 000 000  85 101 85 80 76 330 400 330 330 330  302 922 302 424 297 000 000 000 000 000  80 85 101 85 80 330 330 400 330 330  424 302 922 302 424 000 000 000 000 000  75 80 85 101 84 330 330 330 400 330  230 424 302 922 349 000 000 000 000 000  71 76 80 84 101 330 330 330 330 400  933 297 424 349 922 000 000 000 000 000  exi=0.7, 6x2=1.0, V x l / V x 2 = 1 . 0 , p i = 0 . 7 x  135 117 104 95 81 126 105 99 95 87  000 000 000 000 000 000 732 686 275 975  117 135 116 104 95 105 126 105 99 95  000 000 000 000 000 732 000 280 686 275  104 116 135 116 102 99 105 126 105 98  000 000 000 000 000 686 280 000 280 722  95 104 116 135 111 95 99 105 126 102  000 000 000 000 000 275 686 280 000 986  exi=0.7, 6x2=1.0, V x l / V x 2 = 0 . 5 , p  53 45 40 35 32 101 85 80 75 71  000 000 000 000 000 922 302 424 230 933  45 53 45 40 36 85 101 85 80 76  000 000 000 000 000 302 922 302 424 297  40 45 53 45 40 80 85 101 85 80  000 000 000 000 000 424 302 922 302 424  35 40 45 53 44 75 80 85 101 84  000 000 000 000 000 230 424 302 922 349  x 2  81 95 102 111 135 87 95 98 102 126  x l x 2  000 000 000 000 000 975 275 722 986 000  =0.7  32 36 40 44 53 71 76 80 84 101  000 000 000 000 000 933 297 424 349 922  153  exi=0.7, 6x2=1.0, V x l / V x 2 = 2 . 0 , p , x 2 = 0 . 5 x  225 195 170 160 150 73 61 57 55 54  000 000 000 000 000 101 664 576 857 083  195 225 196 170 160 61 73 61 57 55  000 000 000 000 000 664 101 822 576 857  170 196 225 195 170 57 61 73 61 57  000 000 000 000 000 576 822 101 664 576  160 170 195 225 192 55 57 61 73 61  000 000 000 000 000 857 576 664 101 188  150 160 170 192 225 54 55 57 61 73  000 000 000 000 000 083 857 576 188 101  73 61 57 55 54 95 78 78 78 78  101 664 576 857 083 000 000 000 000 000  61 73 61 57 55 78 95 78 78 78  664 101 822 576 857 000 000 000 000 000  57 61 73 61 57 78 78 95 78 78  576 822 101 664 576 000 000 000 000 000  55 57 61 73 61 78 78 78 95 78'  857 576 664 101 188 000 000 000 000 000  54 55 57 61 73 78 78 78 78 95  083 857 576 188 101 000 000 000 000 000  90 75 71 68 62 240 195 195 195 195  000 523 204 053 839 000 000 000 000 000  75 90 75 71 68 195 240 195 195 195  523 000 200 204 053 000 000 000 000 000  71 75 90 75 70 195 195 240 195 195  204 200 000 200 516 000 000 000 000 000  68 71 75 90 73 195 195 195 240 195  053 204 200 000 561 000 000 000 000 000  62 68 70 73 90 195 195 195 195 240  839 053 516 561 000 000 000 000 000 000  72 60 57 53 51 400 330 330 330 330  801 930 446 735 381 000 000 000 000 000  60 72 60 57 54 330 400 330 330 330  930 801 930 446 498 000 000 000 000 000  57 60 72 60 57 330 330 400 330 330  446 930 801 930 446 000 000 000 000 000  53 57 60 72 60 330 330 330 400 330  735 446 930 801 249 000 000 000 000 000  51 54 57 60 72 330 330 330 330 400  381 498 446 249 801 000 000 000 000 000  6x1=0.7, 8x2=1.0, V x l / V x 2 = 1 . 0 , Pxlx2=0.5  135 117 104 95 81 90 75 71 68 62  000 000 000 000 000 000 523 204 053 839  117 135 116 104 95 75 90 75 71 68  000 000 000 000 000 523 000 200 204 053  104 116 135 116 102 71 75 90 75 70  000 000 000 000 000 204 200 000 200 516  95 104 116 135 111 68 71 75 90 73  000 000 000 000 000 053 204 200 000 561  8xi=0.7, 6x2=1.0, V x l / V x 2 = 0 . 5 , p  53 45 40 35 32 72 60 57 53 51  000 000 000 000 000 801 930 446 735 381  45 53 45 40 36 '60 72 60 57 54  000 000 000 000 000 930 801 930 446 498  40 45 53 45 40 57 60 72 60 57  000 000 000 000 000 446 930 801 930 446  35 40 45 53 44 53 57 60 72 60  000 000 000 000 000 735 446 930 801 249  81 95 102 111 135 62 68 70 73 90  x l x 2  000 000 000 000 000 839 053 516 561 000  =0.5  32 36 40 44 53 51 54 57 60 72  000 000 000 000 000 381 498 446 249 801  154  £xi=1.0, 8 =0.7, V x l / V x 2 = 2 . 0 , p x2  225 195 195 195 195 100 85 75 70 66  000 000 000 000 000 879 097 869 805 056  195 225 195 195 195 85 100 83 74 70  000 000 000 000 000 097 879 999 635 805  195 195 225 195 195 75 83 100 83 75  000 000 000 000 000 869 999 879 999 869  195 195 195 225 195 70 74 83 100 81  x l x 2  000 000 000 000 000 805 635 999 879 758  =0.7  195 195 195 195 225 66 70 75 81 100  000 000 000 000 000 056 805 869 758 879  100 85 75 70 66 95 78 62 54 47  879 097 869 805 056 000 000 000 000 000  85 100 83 74 70 78 95 76 60 54  097 75 879 83 999 100 635 83 805 75 000 62 000 76 000 95 000. . 76 000 62  869 999 879 999 869 000 000 000 000 000  70 74 83 100 81 54 60 76 95 72  805 635 999 879 758 000 000 000 000 000  66 70 75 81 100 47 54 62 72 95  056 805 869 758 879 000 000 000 000 000  124 102 94 86 78 240 190 160 135 110  200 877 407 718 278 000 000 000 000 000  102 124 102 94 86 190 240 190 160 135  877 200 877 407 718 000 000 000 000 000  94 102 124 102 94 160 190 240 190 160  407 877 200 877 407 000 000 000 000 000  86 94 102 124 102 135 160 190 240 188  718 407 877 200 334 000 000 000 000 000  78 86 94 102 124 110 135 160 188 240  278 718 407 334 200 000 000 000 000 000  101 85 75 69 64 400 330 260 220 190  922 302 717 649 726 000 000 000 000 000  85 101 82 75 69 330 400 310 260 220  302 922 677 717 649 000 000 000 000 000  75 82 101 84 74 260 310 400 320 250  717 677 922 000 246 000 000 000 000 000  69 75 84 101 81 220 260 320 400 300  649 717 000 922 333 000 000 000 000 000  64 69 74 81 101 190 220 250 300 400  726 649 246 333 922 000 000 000 000 000  e =1.0, £ = 0 . 7 , V x l / V x 2 = 1 . 0 , p i =0.7 xl  135 117 117 117 117 124 102 94 86 78  x 2  000 000 000 000 000 200 877 407 718 278  117 135 117 117 117 102 124 102 94 86  x  000 000 000 000 000 877 200 877 407 718  117 117 135 117 117 94 102 124 102 94  000 000 000 000 000 407 877 200 877 407  117 117 117 135 117 86 94 102 124 102  000 000 000 000 000 718 407 877 200 334  6x1=1.0, e = 0 . 7 , V x l / V x 2 = 0 . 5 , p x2  53 45 45 45 45 101 85 75 69 64  000 000 000 000 000 922 302 717 649 726  45 53 45 45 45 85 101 82 75 69  000 000 000 000 000 302 922 677 717 649  45 45 53 45 45 75 82 101 84 74  000 000 000 000 000 717 677 922 000 246  45 45 45 53 45 69 75 84 101 81  000 000 000 000 000 649 717 000 922 333  x 2  117 117 117 117 135 78 86 94 102 124  x l x 2  000 000 000 000 000 278 718 407 334 200  =0.7  45 45 45 45 53 64 69 74 81 101  000 000 000 000 000 726 649 246 333 922  155  £ x i - =1.0, Ex2= 0.7,  225 195 195 195 195 73 61 54 51 47  000 000 000 000 000 101 664 977 308 867  E x l ==1.0,  135 117 117 117 117 90 74 68 62 56  000 000 000 000 000 000 549 411 839 723  195 225 195 195 195 61 73 60 54 51  000 000 000 000 000 .664 101 .869 .083 308  Vxl/Vx2=2.0, pxi =0.5 x 2  195 195 225 195 195 54 60 73 60 54  000 000 000 000 000 977 869 101 869 977  195 195 195 225 195 51 54 60 73 59  000 000 000 000 000 308 083 869 101 245  53 45 45 45 45 72 60 54 49 46  117 135 117 117 117 74 90 74 68 62  .000 .000 .000 .000 .000 .549 .000 .549 .411 .839  117 117 135 117 117 68 74 90 74 68  000 000 000 000 000 411 549 000 549 411  117 117 117 135 117 62 68 74 90 74  000 000 000 000 000 839 411 549 000 155  x2  000 000 000 000 000 801 930 083 749 233  000 000 000 000 000 867 308 977 245 101  73 61 54 51 47 95 78 62 54 47  101 664 977 308 867 000 000 000 000 000  61 73 60 54 51 78 95 76 60 54  664 101 869 083 308 000 000 000 000 000  54 60 73 60 54 62 76 95 76 62  977 869 101 869 977 000 000 000 000 000  51 54 60 73 59 54 60 76 95 72  308 083 869 101 245 000 000 000 000 000  47 51 54 59 73 47 54 62 72 95  867 308 977 245 101 000 000 000 000 000  90 74 68 62 56 240 190 160 135 110  000 549 411 839 723 000 000 000 000 000  74 90 74 68 62 190 240 190 160 135  549 000 549 411 839 000 000 000 000 000  68 74 90 74 68 160 190 240 190 160  411 549 000 549 411 000 000 000 000 000  62 68 74 90 74 135 160 190 240 188  839 411 549 000 155 000 000 000 000 000  56 62 68 74 90 110 135 160 188 240  723 839 411 155 000 000 000 000 000 000  72 60 54 49 46 400 330 260 220 190  801 930 083 749 233 000 000 000 000 000  60 72 59 54 49 330 400 310 260 220  930 801 055 083 749 000 000 000 000 000  54 59 72 60 53 260 310 400 320 250  083 055 801 000 033 000 000 000 000 000  49 54 60 72 58 220 260 320 400 300  749 083 000 801 095 000 000 000 000 000  46 49 53 58 72 190 220 250 300 400  233 749 033 095 801 000 000 000 000 000  £x2=0 . 7 , V x l / V x 2 = 1.0, Pxlx2=0.5  E i=1.0, e =0.7, V x l / V x 2 = 0 . 5 , p x  195 195 195 195 225 47 51 54 59 73  45 53 45 45 45 60 72 59 54 49  000 000 000 000 000 930 801 055 083 749  45 45 53 45 45 54 59 72 60 53  000 000 000 000 000 083 055 801 000 033  45 45 45 53 45 49 54 60 72 58  000 000 000 000 000 749 083 000 801 095  117 117 117 117 135 56 62 68 74 90  x l x 2  000 000 000 000 000 723 839 411 155 000  =0.5  45 45 45 45 53 46 49 53 58 72  000 000 000 000 000 233 749 033 095 801  156  Appendix C Accuracy of the Estimation Formulas (Ratio Mean, Sd., and Correlation) T o e x a m i n e the a c c u r a c y o f the P e a r s o n a p p r o x i m a t i o n s for ratio variables ( F o r m u l a s (4), (5), a n d (7)), the c o m p u t e d statistics o f the n i n e s i m u l a t e d ratio variable p o p u l a t i o n s s h o w n i n the first panel o f T a b l e 5-1 ( c o r r e s p o n d i n g to e i = e x  x2  = 1.0) are  c o m p a r e d w i t h the a p p r o x i m a t i o n s o f the mean, standard d e v i a t i o n , and c o r r e l a t i o n b a s e d o n F o r m u l a s (4), (5), a n d (7). B e c a u s e the c o v a r i a n c e can be expressed as a f u n c t i o n o f the magnitudes o f the standard d e v i a t i o n and c o r r e l a t i o n o f the ratio v a r i a b l e s , the a c c u r a c y o f F o r m u l a (6) w a s not separately e x a m i n e d . T h e means a n d standard d e v i a t i o n s , for both the numerator and d e n o m i n a t o r variables used to create the ratio v a r i a b l e , are s h o w n i n T a b l e D - l for each o f the nine c o n d i t i o n s o f V i / V x  T a b l e D - 2 presents p -  x ) / x 2  and a i / x  x 2  x 2  and p i . x  x 2  c a l c u l a t e d f r o m the first trial o f the p o p u l a t i o n data o f  the ratio v a r i a b l e , and the a p p r o x i m a t i o n s ~ | L I I / , a n d ~rj i/ X  x2  x  x2  based o n F o r m u l a s (4) a n d  (5). T h e a p p r o x i m a t i o n s are s h o w n i n the brackets. T h e values o f the p i y  y 2  were  c a l c u l a t e d between the first trial a n d s e c o n d trial o f the ratio p o p u l a t i o n data, and the values o f the ~ p i y  y 2  were a p p r o x i m a t e d f r o m F o r m u l a (7) based o n the characteristics o f  the c o m p o n e n t variables. T h e subscript y i refers the ratio v a r i a b l e o f the first trial, and y refers the s e c o n d trial i n the p o p u l a t i o n o f the ratio v a r i a b l e data. A s can be seen i n T a b l e D - 2 , although a slight u n d e r - a p p r o x i m a t i o n exists i n the c o n d i t i o n s V i / V = 2 . 0 a n d V / V = 1 . 0 , the ratio variable means are accurately x  x 2  x ]  x 2  a p p r o x i m a t e d by F o r m u l a (4). W h e n the v a r i a t i o n o f the d e n o m i n a t o r v a r i a b l e increased f r o m V / V = 2 . 0 to V / V = 0 . 5 , the p o p u l a t i o n mean was s l i g h t l y u n d e r - a p p r o x i m a t e d ; x )  x 2  x )  x 2  2  157 Table D - l T h e P o p u l a t i o n C h a r a c t e r i s t i c s o f the C o m p o n e n t V a r i a b l e s (Xu X ? )  v /v = =2.0 xl  v /v = = 1.0  x2  xl  Hxi  V  x2  Hxl  1^x2  x l  /V  x 2  : =0.5 1^x2  Hxl  (CTxl)  (0-x2)  (Oxl)  (Ox2)  (CTxl)  (<7x2)  75.04 (14.98) 75.04 (14.98) 75.04 (14.98)  100.03 (9.74) 100.04 (9.75) 100.04 (9.75)  75.03 (11.60) 75.03 (11.60) 75.03 (11.60)  100.05 (15.49) 100.06 (15.49) 100.06 (15.49)  75.02 (7.27) 75.02 (7.27) 75.02 (7.27)  100.07 (20.00) 100.08 (20.01) 100.07 (20.00)  75.01 (15.01) 75.01 (51.01) 75.01 (15.01)  100.02 (9.77) 100.02 (9.78) 100.03 (9.78)  75.00 (11.63) 75.01 (11.63) 75.01 (11.63)  100.02 (15.52) 100.04 (15.54) 100.04 (15.54)  75.00 (7.29) 75.00 (7.29) 75.00 (7.29)  100.04 (20.04) 100.05 (20.06) 100.06 (20.07)  Trial 1 Pxlx2=0.9 Pxlx2=0.7 Pxlx2=0.5 Trial 2 Pxlx2=0.9 P x 1x2=0.7 Pxlx2=0.5  T a b l e D-2 Comparison Between Approximations r^xi/*?, C h a r a c t e r i s t i c s (iuu^.  y  Y  V *i/V =1.0  x 2  ( Mnl/x2)  ~Pyiy2> a n d the P o p u l a t i o n  o,i<v9. p i 7 )  V *i/V =2.0 U-xl/x2  "Qxu*?,  C» xl/x2 ( O i/x2) x  V xi/V =0.5 x 2  x2  Pyly2  M^l/x2  O xl/x2  (~Pyly2)  ( M*l/x2)  ( O l/x2) x  Pyly2  Hxl/x2  (~Pvly2)  ( M*l/x2)  C* xl/x2 ( 0"xl/x2)  Pyly2 (~Pyl 2) V  Pxlx2 =0.9  .744 (.744)  .092 (.090)  .891 (.900)  .752 (.752)  .054 (.052)  .834 (.840)  .769 (.767)  .110 (.090)  .792 (.814)  Pxlx2==0.7  .747 (.747)  .113 (.111)  .875 (.876)  .756 (.755)  .094 (.090)  .831 (.838)  .772 (.769)  .133 (.112)  .797 (.820)  Pxlx2 =0.5  .750 (.750)  .131 (.130)  .866 (.869)  .760 (.759)  .123 (.116)  .829 (.838)  .776 (.772)  .154 (.130)  .801 (.823)  :  :  158 the greatest difference between the a p p r o x i m a t e d value and true value was 0.004 ( 0 . 5 % error). T h e standard d e v i a t i o n also s h o w s an u n d e r - a p p r o x i m a t e d pattern. H o w e v e r , the difference between the a p p r o x i m a t e value and the true value is m u c h h i g h e r i n the c o n d i t i o n V i / V = 0 . 5 than i n the other t w o c o n d i t i o n s . T h e a p p r o x i m a t e standard x  x 2  deviations i n the c o n d i t i o n V / V = 0 . 5 underestimated the true values b y 0.020, 0 . 0 2 1 , x )  x 2  and 0.024, and the percent error were 1 8 . 2 % , 1 5 . 8 % , and 1 5 . 6 % , respectively. T h e approximate c o r r e l a t i o n ( F o r m u l a (7)) between the first trial and the s e c o n d trial o f the ratio variable result i n overestimate w h e n the v a r i a t i o n o f the d e n o m i n a t o r v a r i a b l e increased f r o m V / V = 2 . 0 to V / V = 0 . 5 . In the c o n d i t i o n V i / V = 0 . 5 , the x ]  x 2  x ]  x 2  x  x 2  overestimations were 0.022, 0.023, and 0.022, and the percent errors were 2 . 7 % , 2 . 9 % , and 2 . 7 % , r e s p e c t i v e l y . T h e results i n d i c a t e that the three parameters (the mean and standard d e v i a t i o n i n the first trial and the c o r r e l a t i o n between the t w o trials) are not w e l l a p p r o x i m a t e d b y F o r m u l a s (4), (5), and (7) w h e n V i / V = 0 . 5 . It can be c o n c l u d e d that i f the v a r i a t i o n o f x  x 2  the denominator v a r i a b l e is r e l a t i v e l y higher than o f the numerator variable ( V ] / V = 0 . 5 ) , the a p p r o x i m a t i o n formulas ( F o r m u l a s (4), (5), and (7)) m a y not X  x 2  accurately a p p r o x i m a t e the p o p u l a t i o n mean, standard d e v i a t i o n , and c o r r e l a t i o n o f ratio variables.  159  r~ H  CN CN O O O  LD Lfl O  vo r> r- co o O O O O rH  'J (N O CI O  in vo r- co o  ^ in vo r-> o  O O O O rH  O O O O rH  rH C N C O O i n u) h o O O O rH  H  ^  O O O rH  O O O rH  H H  h O  ?i r> co  in o  >1 VO 0 O  o  C O O O rH  in  II  •3 o  M  m vo r> co o  X <3* in vo r> o » O O O O rH  Q. O O O rH  OOOOrH  ^  Q. O O O O rH  O O O O rH  vo h oi o in vo r- o  C N rH C N O  O O O rH  O O O rH  r» co o o  h cn o v D h O  >i r> co  vo r- co o  ' O O O O rH  « m in o o in vo co o  in co o vo r> o  rH C N O s  O H h O  n  Q. O O O rH  Kl I  O O O O rH  CO O  Q. O O rH  in vo o ^  o  O O O rH  o  O  >i vo r- o  Q. O O rH  II o| ll  Q. rH  $ll  rH > < f  w  c o id c o  co vo ro co o ro ^ in vo o O O O O rH  O O O O rH  H o r- vo o K r- r> co o  o  <U  H _c CO CU X  T"> O O O rH  ^  O O O rH  rH  CU  Q H  O O O O < H  in CN ro o vo r-- oo o  vo m m o vo r- co o  ^OOOrH  • n O O O rH  >» r> co o  M O O  rH rH O PICOOIO  coo  Q. O O rH  O O rH  (Tl O 0 0 O  CN rH  3  m r> o ^  O O O rH  C N O O O Pi  r- oo o  Q. O O rH  Q. O O rH  CO O C O o  in  II cs  O O O O rH  n  *  o o U  Q-OOOrH  O O O O rH  Tt  •c  cu fci  ' O O O O rH  o O  n  CN '  • o II  3  c  3 c  x  160  io H in in o vo r- r> oo o  rH VO 00 rH  o  O  o  o  o  rH  H  O  O  O  in [— o r> r- co o  vo cn ro o r> 00 o  O  o  O  O  rH  o  o  in r> o >, r> oo o  oo ro o >, t> co o  c- o  ro o oo o  00 o O  CN rH rH O  O  O  rH  O  00 CO CO 00 o  c- r> r- oo o rH  O  O  O  O rH  rH rH rH O 00 00 00 O o o  ^  rH  O  ^ o o >, oo oo  o  o  O  rH  rH  o o  00 o  rH  i n  i n  in  Ol CM (M CM CO 00 00 00 TI  O  O  O  3 CN CN CN !«j 00 00 GO Q.  O  O  CN CN  O  O  O o rH  O  O O  O  O  O  O rH  TT  O  O  O  O  O  Q.  ^ rji CO >, t- CO  0. O  O  O  rH  ro VO 00 rH  O  O  O  O  rH  rH  rH rH O CO CO O  o  ro ro ro ro o oo co co oo o  O  r> r> r> oo o  O  X  J H H ri O % CO CO 00 o  O  O  00 00 CO 00 o  oi -a r> o vo r> oo o 1  rH  00 00 o  rH rH rH rH  rjoirjmo  vo vo r~ oo o  O  O  CD Ol CO  h MB  _  O  O  O  j C d PI  O  rH  O  O rH  TT  O  O  O  O  CN O 00 o  r^ ro ro ro o ) j CO CO CO o Q. O  O  O  CN O  rH  ^ O O O r H  ro ro o 00 CO o  O  o  O  00 00 00 o  CN O  O  >, 00 oo o Q.  ro o co o  rH O CO  o o 00 00 00 00 o o o o o  rH  >, r> oo o  rH  lO r) n  O  O  rH  00 o O  rH  II  co  " o  o o  I-l  £rH  II col  1  e o  VD rH VD in o VD t> 00 o  •a c o  U a  O o O o rH  rH CN o 00 in VD r- CO o o o o o rH  rH CD r> o o  rH r> o VD r> CO o  00 ,—> • rTT  O o  o  o  rH  X  a.  -C H  o o rH o 00 o o rH  c o  co  rH  TT  o o ON o  00 o  •a  fc o O  o  CN CN CN in o  o r~ vo O r> r> 00 O  r> OS ro o r> CO o  o  o  o  rH  r> r- o o o o O rH  _  o o  o  rH  o CO in ro o VD VD r> 00 o  O in rH OS OS 00 CO r~  o  O o o o  o  o  o  rH  VD m m o r> 00 o  ,—. vo • TT  O o  o  CO ro as o CO 00 r> o o  rH  o  o  TT  3 VD r~ o CO  o  ,—,  o o VO CN O vo r> r> 00 O o o o O rH  o • CM rH  <N II  Q.  O o rH  r~ vo o r> CO o a.  OS o CO o OS  rH  o 00 o o rH  o rH  •is SC • Q . rH  O o rH  o  •  ii II  Q.  O o rH ro o 00 o o rH  •I , o  o •  CN rH  rH  ro o 00 o  3 in in o K c- CO o a.  in • «  O II  o o rH in o 00 o o rH CD o  • CN rH  >l  J5ro OS o  £<» r- o Q.O  o rH  OS o o  o rH  X o drH  rH  161 co co o r- o co co co co o  r- rH VD CN o vo r- r- co o  i n CN o  O O O O rH  O O O O rH  O O O O rH  co r - r>  t—i VD CN O r- [-- co o  rH CN C O O m VD r - o  O O O rH  O O O rH  o  C O C O C O o O O O rH  •j l>  I>  o  Lfl CN O  ^ CO CO o  ^ r- co o  *— O O rH  w O O rH Q.  >l • Q.  K  3 °  Oi r> m V D o «3< m vo r- o O O O O rH  ^  O O O O rH  O O O O rH  ^  VD  LT) V D  OJ r-  O O O rH  x  • • * •  in o r-  o  ' O O rH  Q.  o  vo co rH i n o m m V D vo o  in i n ro m ^ in vo r-  O O O O rH  O O O O rH  O O O rH  w O O O rH a  rH C O O ^ (Tl C O O  "V VO CN O . r- co o  o o  • O O rH  w O O tH Q.  w O O O rH Q.  O O O O rH  O O O rH  co ro .j m t  H  s •  o.  3 °  Q. tH  O O O O rH  O O O O rH  O O O rH  O O O rH  O O O O rH  j m in o % •H H  Q.  r> CO  CN II <N  0  o o  j!  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