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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 FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (School of Human Kinetics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRI1ISHTJOLUMBIA September 1999 ©Yuanlong Liu 1999 U B C Special Collections - Thesis Authorisation Form http://www.library.ubc.ca/spcoll/thesauth.htmi In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of 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 granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g ain s h a l l not be allowed without my w r i t t e n permission. The U n i v e r s i t y of 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 ANOVA 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 R2, and (e) high reliability of the derived ratio data. Simulation results show that the characteristics (exi, £ x 2 , V xi/V X 2, and p xi X 2 ) of the two component variables strongly affect e xi/X2 and the type I error rate. In the condition £ xi=£ X 2, the magnitude of e xi/X2 is virtually the same as that of E x i and eX2, regardless the level of p x i X 2 and V xi/V X 2. When £xi<1.0 and eX2=1.0, e xi/X2 tends to have smaller magnitudes when p x i X 2 and V xi/V X2 are high, and greater magnitudes when p x i x 2 and V xi/V x 2 are low. A£xi/X2 exhibited the greatest bias and the largest standard deviation, resulting in a serious inflation of type I error rate in the condition Vxi/Vx2=0.5, regardless of the conditions e xi, eX2, and p x i x 2 . If homogeneity of the denominator variable (small V x 2) and large sample size are present, it may reduce the likelihood of bias in Ae x ]/ x 2 and protect the type I error rate. iv TABLE OF CONTENTS Abstract ii Table of Contents iv List of Tables vii List of Figures ix Acknowledgements x CHAPTER I. Introduction 1 Introduction 1 Purpose of the Study 6 II. Literature Review 8 Statistical Characteristics of Ratios 8 The Distribution of Ratio Variables 8 Variance, Covariance, and Correlation of Ratios 12 Reliability of Ratio Variables 13 Ratio Variable Issues in the Literature 15 Using Ratio Variables for the Purpose of Deflation 15 Ratio Variables in Regression Analysis 17 Ratio Variables in Analysis of Variance 20 Ratio Variables and Unsolved Issues in Human Kinetics Research 22 Deflation Issues 23 Reliability Issues 25 Issues in RM ANOVA with Ratio Data 25 III. Methods and Procedures 28 Ratio Selection 28 Deflation Model Evaluation 29 The Models 29 Controlling for the Denominator Effect 32 Reliability of a Ratio 35 Computer Simulation Investigation on Ratio Issues 37 Overview of the Simulation Conditions and Investigations 39 Preparing Population Covariance Matrices 45 Data Generation 49 Overview of Methods Used to Answer Questions 51 IV. Results and Discussion: Characteristics of Ratio Variables 52 Ratio Variables Used for Deflation Purposes 52 Empirical Data Set 1: Vo2max/Body Mass 52 V Simple Ratio Model 53 Alternate Models 60 Empirical Data Set 2: D L C O / V A 69 Simple Ratio Model 70 Alternate Models 74 A Summary of the Evaluation of the Deflation Models 76 Ratio Variables Used Not for Deflation Purposes 76 Empirical Data Set 3: Nondominant Torque/Dominant Torque 77 Empirical Data Set 4: Waist Girth/Hip Girth 79 A Discussion on the Reliability of the Ratio Variable: Data Sets 3 and 4 80 A Summary of the Evaluation of the Ratio Variables Not for Deflation 86 V. Characteristics of Ratio Variables as A Function of the Component Variables: A Simulation Study 88 Population Characteristics of e xi / X 2 88 The Effect of £Xj,Vxi/Vx2 and p xi X 2 on the Magnitude of e x ]/X2 88 Magnitude of e xi/X2 When the Ratio Components Meets the Assumption of Circularity 90 Magnitude of £xi/X2 When the Ratio Components Violate the Assumption of Circularity 90 The Effect of e xi,V xi/V x 2 and p x i x 2 on the Magnitude of P(Xi/x2)ij 93 Sample Characteristics of A e x i / x 2 102 Condition 1 (exi=ex2=1.0) 103 Condition 2 (exi=ex2=0.7) 105 Condition 3 ((Exi=0.7, Ex2=1.0) 105 Summary of Characteristics of Ae x j / X 2 and exi/X2 111 Type I Error Rate 112 Expected effects of Characteristics of Xi and X 2 on Type I Error Rates for X 1 / X 2 112 Monte Carlo Simulation investigation on Type I Error Rate 114 Sample Estimates Ae xi/ X 2 and Type I Error Rates 119 VI. A Summary of the Investigation 128 References 136 VI Appendices 142 Appendix A: Approximating p x i i , X2j 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) 78 4-8. Means, Standard Deviations, and Reliability Coefficients of the Torques and Nondominant/Dominant Ratio (Supination) 78 4-9. 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 V l l l 5-1. 5-2. 5-3. 5-4. 5-5. 5-6. 5-7. 5-8. 5-9. 5-10. 5-11. Population Values of 8 xi/X2 for the Ratio Variable Under Different Component Variable Conditions Intertrial Correlations of Ratio Variables for Selected Conditions of the Component Variables The Correlation Matrices of the Component and Ratio Variable Populations (exi=eX2=1.0) The Correlation Matrices of Ratio Variable populations (Pxlx2=0.7) Comparison of the Correlation Matrices of the Ratio Variable under Selected Conditions Distribution Characteristics of Ae xi / X 2 (exi=ex2=1.0, u.xi=75, u.x2=100) Distribution Characteristics of A £ x i / X 2 (Exi=£x2=0.7, p,xi=75, |ax2=100) Distribution Characteristics of A £ x i / X 2 (exl=0.7, ex2=1.0, u.xl=75,1^=100) Empirical Type I Error Rates Effect of Ae xi/ X 2 and p x t x 2 on Type I Error Rates A Summary of the Effect of V xi/V x 2 on the Type I Error Rates (pxiX2=0.7, a=.05, n=15) 89 94 96 97 100 104 106 109 116 118 126 ix LIST OF FIGURES Figure 4-1. The Scatterplots of the Simple Ratio and the LRM Ratio: Data Set 1 58 4-2. The Scatterplots of the NLRM1 Ratio and NLRM2 Ratio: Data Set 1 63 4-3 Comparison of the Fit of the Three Deflation Models: Data Set 1 66 4- 4. The Scatterplots of the Simple Ratio, LRM, NLRM1 And NLRM2 Ratios: Data Set 2 73 5- 1. A E x i / x 2 in the Conditions 8xi=EX2=1.0 and exi=eX2=0.7 107 5-2. A e x i / x 2 in the Condition 8xi=0.7 and 8X2=1.0 110 5-3. Mean of epsilon estimates with +/- 2sd for exi=sX2=1.0, exi/X2=1.0 121 5-4. Mean of epsilon estimates with +/- 2sd for exi=eX2=0.7, exi/x2=0.7 122 5- 5. Mean of epsilon estimates with+/-2sd for Exi=0.7 and eX2=1.0 124 6- 1. The E x i / X 2 values and the empirical type I error rates in the condition ex)=0.7 and £x2=1.0 ((X=0.05, n=45) 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 Rat io variables have a long history o f use in different discipl ines related to human kinetics research (e.g., b iomechanics , exercise ep idemio logy , phys ica l education, sport phys io logy , and sport psychology) . A ratio variable is a composite variable that consists o f a numerator variable and a denominator variable. F o r example, the moment o f force in biomechanics is usual ly expressed as a ratio variable w h i c h consists o f the measured moment of force d iv ided by body mass, and the V o 2 m a x i n phys io logy is usual ly expressed as a ratio variable w h i c h consists o f the measured V o 2 m a x d iv ided by body mass. Per capita income and other variables i n soc io logy are ratios w i th populat ion size as the denominator to compare different countries. W a i s t and hip girth and nondominant and dominant strength ratio are popular ly used in the ep idemio logy and health f ields, and many sport statistics are expressed as a ratio of successful attempts d iv ided by the total number of attempts (e.g., basketball shooting percentage). Ra t io variables are popular ly used in different areas of human kinet ics research, and they can be class i f ied by the purpose o f use. Ra t io variables are typ ica l ly used for scal ing data, wi th the variable o f pr imary interest being the numerator variable. D i v i d i n g by the denominator is an attempt to express the numerator on a per-unit o f the denominator basis in order to remove or deflate the effects of the denominator 's variat ion from the numerator 's var iat ion, and full deflation w o u l d y i e l d a zero correlation between the ratio variable and its denominator variable. For 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 of (or at least min imiz ing 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 of similar body mass. In other situations the formation of homogeneous groups is either not possible or not desirable. For example, in schoolboy weightlifting competitions there often are not enough competitors to form different weight classes. In this situation, a ratio variable, weight lifted divided by body mass, is computed and used to declare an overall champion (Keller , 1977). In many research studies, it is desirable to examine changes in strength, or between-group differences in strength, independent of body mass changes on group differences. The ratio variable, strength/body mass, is used in such situations to "deflate" the effect of body mass. The term "sca l ing" , "e l imina t ing" , " r e m o v i n g " and "deflat ing" are equivalent terms used in the literature, but the term "def la t ing" is used through this dissertation for consistency and famil iar i ty i n the f ie ld . Ra t io variables der ived for deflation purposes (attempting to remove the effect of the denominator variable from the variat ion o f the numerator variable) serve as dependent variables in many studies i n different d isc ipl ines . Ra t io 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 special aspect not accounted for by the numerator or denominator variables, either i n d i v i d u a l l y or addi t ively. F o r example, the ratio o f waist and hip girths is treated as a combined measurement variable, ca l led the waist- to-hip ratio ( W H R ) . It is used to describe body shape, and has been shown to be very useful in ep idemio logy and obesity studies to examine morbid i ty , stroke, heart disease, breast cancer, and other diseases (Darroch & M o s i m a n n , 1985; Sonnenschein, 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 of the denominator variable increases). If the hypothesis to be tested has in i t i a l ly been formulated i n terms o f a ratio variable, some researchers (Cor rucc in i , 1995; K u h & M e y e r , 1955) have argued that the question of spurious correlation between a ratio and its component variables does not arise. A c t u a l l y , for those ratio variables (e.g., waist /hip ratio), al though a zero correlat ion is not an issue, the statistical consequences o f using such variables should be examined because the re l iabi l i ty o f the ratios is affected by variat ion o f the two component variables. Howeve r , the re l iab i l i ty issue of us ing ratio variables is usual ly neglected by researchers. A l t h o u g h in practice there may be two different purposes for us ing ratio variables, the consequences of using ratio variables in statistical inferential analyses are the same. It can be shown that the variat ion and covar ia t ion o f the component variables strongly affect the statistical characteristics of a der ived ratio variable i n descript ive and inferential analyses. Some o f the issues regarding the va l id i ty o f using ratio variables have been noted i n the literature. F o r example , the issues o f using ratio variables in correlat ion and linear regression analyses have been discussed for a long t ime. Fr iedlander (1980), and many others, pointed out that a l inear regression model wi th a ratio as the dependent variable or the independent variable w o u l d not appropriately control for the effect o f the denominator, and therefore c o u l d y i e l d mis leading results. It has been indicated that a reciprocal component o f the denominator should be inc luded i n the regression model . H o w e v e r , some other issues concerning the use o f ratio variables have not been resolved. In human kinetics research, ratio variables are c o m m o n l y used in both descriptive and inferential statistics. W h e n a ratio variable is der ived for deflation purposes, a 5 residual correlation between the ratio and its denominator variable may seriously affect the va l id i ty o f the ratio variable. However , it appears that this is rarely examined, and the impact of the denominator on related ratio variables should be investigated empi r i ca l ly in al l studies. In addit ion to the p rob lem of a non-zero correlat ion between a ratio and its denominator, the use o f a ratio variable cou ld introduce denominator-related distortions of variances. In some studies in human kinetics research, the denominator related variance distortion i n a ratio variable may not be an issue because o f the smal l range o f the denominator variable i n the data, wh i l e in others it may be a serious issue because o f the large range of the denominator variable. F o r non-deflat ion type ratio variables, g iven that the ratio variable has content va l id i ty , information about the re l iab i l i ty o f the ratio variable and variat ion effect o f the component variables i n the inferential analysis needs to be k n o w n . H o w e v e r , such information is usual ly mi s s ing i n human kinet ics research publicat ions. Researchers in human kinetics often use ratio variables i n inferential analyses, especial ly i n the analysis o f variance wi th repeated measures ( R M A N O V A ) . It is apparent that the characteristics o f a ratio variable in the repeated measures w i l l be strongly inf luenced by the characteristics o f the numerator and denominator variables. H o w the characteristics o f the component variables affect the type I error rate in R M A N O V A is s t i l l not k n o w n . A n investigation o f these issues has not yet been done for the c o m m o n l y used ratio variables in human kinet ics research. 6 Purpose of the Study A pr imary purpose o f this study was to investigate whether ratio variables are appropriately used in the human kinet ics research domain . F o r ratio variables used for deflation purposes, the va l id i ty o f the ratio variable was a ma in concern. A s imple ratio variable and adjusted ratio variables were investigated and compared us ing variables c o m m o n l y used in our f ie ld . The val id i ty o f deflation models for each ratio variable was evaluated. Because the re l iab i l i ty o f a measurement variable is one of the most important considerations in any empi r i ca l research, this study examined the re l iab i l i ty properties o f ratio variables. T h e re l iab i l i ty of a ratio variable is more complex than the re l iab i l i ty o f a raw score variable because it is affected by several factors, speci f ica l ly , the re l iab i l i ty o f the numerator and denominator variables, the correlat ion between the two component variables, and the relative variat ion o f the two variables. These effects are rarely addressed in the literature. T h e re l iabi l i ty of c o m m o n ratio measures i n human kinetics research needs to be examined for possible bias as a result o f these effects. The second purpose was to investigate how the R M A N O V A test using a ratio variable violates the assumptions o f this statistical procedure. Since a ratio variable Y* = X i i / X 2 i is a combined variable w h i c h contains numerator and denominator components, the covariance structure o f ratio variables for tr ial i and j , Y j = X n / X 2 i and Y j = X 1 / X 2 J say, is different f rom that o f the raw scores. The covariance structure o f ratio variables in R M A N O V A is affected by not on ly the covariances o f the component variables across trials, but also the covariance between the numerator and denominator variables w i th in each tr ial . These effects may cause serious violat ions 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 ANOVA 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 ANOVA 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 ANOVA 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 One o f the basic assumptions of the statistical models c o m m o n l y used in human kinet ics is that, i n the populat ion, the dependent variable is distributed normal ly . A question o f interest in the literature is , g iven the distr ibution characteristics o f the component variables that are used to fo rm a ratio dependent variable, what is the distr ibution o f the ratio variable? It is noted that the normal i ty assumption i n most statistical procedures is not cr i t ical unless sample sizes are smal l (e.g., n<15 in R M A N O V A procedures). The general statistical properties o f ratios o f random normal variables have been examined by a number o f statisticians (Fiel ler , 1932; K e n d a l l & Stuart, 1969; M a r s a g l i a , 1965; Schneeberger & Fleischer , 1993). F o r example , it can be shown that the distr ibution o f the ratio o f two independent standard normal random variables is a C a u c h y distr ibution (Kenda l l & Stuart, 1969). The density function is \ dy df(y)= 2^ - o o < y < o o (1) 71(1+ y ) where Y=Z]/Z2. In this function the ratio o f Z\ and Z 2 is treated as a new independent variable and the effect o f Z\ and Z 2 on the ratio cannot be seen. The general form of the density function o f the ratio o f two random variables can be derived. If/(Xi, X 2) is the jo in t density o f (Xi, X 2), then y/(y), the density o f the ratio Y=Xi/X2 is g iven by: Y(y) = { _ / ( V * 2 > * 2 ) K | r f X 2 - o o < y < + o o (2) Rat io distributions under bivariate normal condit ions have examined analyt ica l ly (Fiel ler , 1932; Schneeberger & Fle ischer , 1993), and also through the use o f s imula t ion procedures (Friedlander, 1980). T h e density function o f the bivariate normal distr ibution for X i and X 2 is g iven by Schneeberger and Fle ischer (1993): /(*! ,x2) = : 1 , exp 1 2 ( 1 - P 1 2 2 ) (X 2 - 2 -2p12 2- x !- + ( >-)2 cr. cr a, (3) A s we can see f rom (2) and (3), the distr ibution o f the ratio o f continuous variables X ] and X 2 is affected by the f o l l o w i n g factors: M x i , M-x 2 : t n e populat ion means, a x p ° x 2 : the populat ion standard deviations, P i 2 : the populat ion correlation o f X i and X 2 . The distr ibution o f the ratio variable der ived f rom (2) and (3) has infinite moments when the component variables can take negative values, but this is usual ly not the case i n human kinetics because the component variables used are usual ly posi t ive and fal l w i th in a certain range. F i e l l e r (1932) pointed out that the l imi ta t ion o f the component variables to the posi t ive quadrant may change the moments o f the ratio distr ibution to finite values without having a v is ib le effect on the appearance of the dis tr ibut ion. U s i n g the aforementioned functions and a computer s imulat ion, Fr iedlander (1980) demonstrated that when the correlat ion o f the component variables is h igh and the coefficients o f variat ion are large, there c o u l d be a considerable deviat ion o f the ratio distr ibution f rom normali ty . The s imulat ion also showed that when the correlat ion 10 between the numerator and denominator increases, the mean and standard deviat ion o f the ratio variable tend to decrease. Because o f the complex relationship between a ratio and its component variables, the effect o f the characteristics o f the component variables (e.g., ( i ^ ( i x 2 , axi, ox2, and Pxix2) on the distr ibution characteristics o f a ratio variable has attracted the attention o f many researchers in different d isc ip l ines . F o r example, Schneeberger and F le i scher (1993) showed different ratio distr ibution shapes under various condi t ions on X i and X 2 . H i n k l e y (1969) discussed the distr ibution o f the ratio of two correlated normal random variables, and K o r h o n e n and N a r u l a (1989) proposed a method to compute the distr ibution o f the ratio o f the absolute values of two correlated normal random variables. Issues relating to the normal i ty of ratio variables were also discussed by Samuel and N o r m a n (1985, 1986), and A r n o l d and Brocket t (1992). S u m m a r i z i n g these studies, the distr ibution o f a ratio variable is affected by the covar ia t ion and the distr ibution characteristics of the numerator and denominator variables. W h e n the numerator variable is approximately normal and the denominator is less variable or non-stochastic, the ratio variable is s t i l l approximately normal . W h e n the variat ion of the component variables is large, and the correlat ion between the component variables is high, considerable deviat ion o f the ratio dis tr ibut ion f rom normal i ty may occur. F o r some ratio variables used in human kinet ics , the denominator may have large variat ion and the component variables may be strongly skewed. W h e n a ratio variable is der ived from the component variables wh ich have large variat ion (e.g., V=rj/u>0.2, V refers to the coefficient o f variat ion defined as the standard deviat ion d iv ided by the 11 mean) or are strongly skewed, the ratio variable m a y be less skewed but the distr ibution remains non-normal (Friedlander, 1980). A l o g transformation is a method to make the ratio data more normal ly distributed, however, this option may lack appeal because o f the diff icul ty o f interpretation in some situations. U s i n g a computer s imulat ion, A t c h l e y et a l . (1976) demonstrated how changes in the coefficients o f variat ion o f two bivariate no rma l ly distributed variables (V x i=a x i / u , i , VX2=C7X2/|J.2) affect the skewness and the kurtosis o f a ratio variable. T h e results showed that the distr ibution o f the ratio variable was strongly skewed to the right when the variat ion o f the denominator was relat ively greater than that o f the numerator. F o r example, the coefficient o f skewness reached 2.06 when V x i / V X 2 = 0 . 1 , and as V x i / V x 2 approached 1.0 the skewness o f the distr ibution o f the ratio variable decreased to 0.1. A d d i t i o n a l l y , when the variat ion of the denominator was relat ively greater than that of the numerator, the kurtosis o f the ratio variable was very h igh (e.g., the coefficient o f the kurtosis>10.0 when V x i / V x 2 < 0 . 2 ) . W h e n V x i / V x 2 approached 1.0 the kurtosis of the ratio variable decreased more q u i c k l y than skewness. The kurtosis also became slight when V x i / V X 2 > 0 . 3 . B o t h skewness and kurtosis values were unaffected (i.e., remained near zero) for condit ions i n w h i c h V x i / V x 2 > 1 . 0 . In summary, the distr ibution o f a ratio variable is strongly affected by the relative variat ion of its component variables. A l t h o u g h the component variables may be normal ly distributed, as the coefficient o f variat ion o f the denominator variable (V X 2) increases relative to that o f the numerator, the distr ibution o f the ratio variable becomes skewed and leptokurtic. O b v i o u s l y , it is not appropriate to assume that ratio variables meet the normal i ty assumption based on knowledge o f the normal i ty o f the component 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-Px,vxyx^ +V<) (4) ,2 < , 2 = V a i A « ^ ( V : + Vl - 2p,xVxVX2) (5) A 2 f*x, a x x = Cov(^,^-) « ^ L^(px XVXVX +pxxVx Vx -pxxVxVx -pxx Vx Vx ) c'l,, 1!, v V V II II 2 4 2 4 1 3 1 3 2 3 2 3 1 4 1 4 , r. A 2 A 4 i"2 ^4 (6) _ „ I ^ 1 __3 \ -"i* 3 *l X3 " ^ l - ^ •*! x4 **2X3 xl xl ^ X2X4 x2 x4 (n\ x2 ^ -2pX A7 xy x i Jvx] +vl -2pvyxyXt Here V xi, Vx2, Vx3, and V x 4 are coefficients of variation (standard deviation divided by mean) for variables Xu X2, X3, and X4, and p x ] x 2 , p xi x 3, p x , x 4 , p x 2 x3, 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 correlat ion between X 1 / X 2 and X 2 can be estimated by taking X 3 = X 2 and X 4 = l . It should be noted that the formulas are approximations, and they are accurate on ly when the coefficients o f variat ion for al l component variables are smal l enough to render cubic and higher order variat ion terms of the T a y l o r series negl ig ib le . Chayes (1971) showed that Pearson's approximate formula became increasingly inaccurate when the coefficient o f variat ion o f the component variables in the formula was greater than 0.15. Therefore, the accuracy o f us ing the formulas for human kinet ics measures wi th greater variat ion becomes an issue. Formulas (4) to (7) show that the statistical characteristics o f a ratio variable can be very different f rom that of the component variables. Firs t , Formulas (4) to (7) show that the statistical characteristics of a der ived ratio variable are complex functions o f the characteristics o f the component variables. Second, by k n o w i n g the characteristics o f the component variables, researchers can approximate the characteristics o f a ratio variable using these formulas. T h i r d , Formulas (4) to (7) are approximations and the accuracy o f these formulas depends upon the properties o f the component variables. In some situations these approximations may be acceptable, w h i l e in other situations they m a y not be acceptable due to large variation in the component variables. Reliability of Ratio Variables Wheneve r a measurement is administered to subjects in a study, researchers w o u l d l ike to have some assurance that the measurement scores c o u l d be replicated i f the same ind iv idua ls were measured again under s imi la r circumstances. One aspect o f this repl icat ion is the degree o f consistency in the in ter - individual var iabi l i ty , often referred to the re l iab i l i ty o f a measure. Based on classical 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 RM ANOVA model, assuming compound symmetry of the covariance matrix, is defined as rint=(cr7i2)/(cfn2 + Cfe2). The estimate of the intraclass correlation from data can be obtained as Arint=(MSs-MSres)/MSs, where MS S is the mean square for subjects and MSres 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): where p x i x i and p X2 X2 are the reliability coefficients of the numerator and denominator variables, and V xi and V x 2 are the coefficients of variation of the component variables. This formula is derived by substituting Xi for X3 and X 2 for X 4 in Formula (7). However, Formula (8) has never been used in the literature. The validity of Formula (8) for the approximation of reliability is suspect, because substituting Xi for X 3 and X 2 for X4 in Formula (7) and assuming that the variable Y=Xi/X2 correlates with itself would vx2l-2PxiXyxyXi+vx] (8) 15 result in pyy=1.0. Thus, this formula may not be appropriate to estimate the reliability of 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 RM 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 der ived the pattern o f correlat ion between the ratio (Y=Xi/X2) and its denominator ( X 2 ) under different combinat ions o f the coefficient o f variat ion (VX1/VX2). In general, the value o f py>X2 is a lways less than pxi,X2 and only approaches p x i , x 2 for high values of Vxi / V x 2 . W h e n V x i / V x 2 < 1.0, py,X2 is negative, and when p x ^ x 2 approaches zero, py,X2 rapidly approaches -1.0. Therefore, one should not assume that a s imple ratio w o u l d have zero correlation wi th its denominator. The more interesting question is how to interpret the correlat ion p y x2 g iven the difference between p y j X 2 and p x i j X 2 . M a c M i l l a n and Daft (1979, 1980) gave a reasonable explanation for the case py,X2=0. T h e y showed that p y j X 2 should be zero, not when X f and X 2 are unrelated, but when X) and X 2 are related in a special way (a l inear relat ionship between Xi and X 2 w i th a zero intercept). py,X2 is sometimes recommended as a measure of "nonproport ionali ty" (Firebaugh & G i b b s , 1985; Kasards & N o l a n , 1979). If p y , x 2 equals zero, it suggests that there is a l inear relationship between Xi and X 2 w i th a zero intercept, however , i f py,X2 is not zero, one cannot evaluate the degree of nonproport ional i ty by py,X2 alone. Sources o f nonproport ional i ty include a non-zero y-intercept, non-l ineari ty, or both. T o overcome the inadequacy of using a s imple ratio as a deflation method, various adjusted deflation models have been recommended (Albrecht et a l , 1993; A t c h l e y , 1978; G e l v i n & Albrech t , 1985). In order to get fu l l deflation, it is necessary to derive a new scaled variable in w h i c h the variat ion is not affected by the variat ion o f the denominator variable. T h e ma in statistical cri terion for fu l l deflation is that the adjusted ratio Yadj should have a zero correlation wi th its control variable (the denominator variable in a s imple ratio). A lb rech t et al . pointed out that deficiencies in s imple 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 2 , where a is a constant, (c) an allometrically adjusted ratio model, Y = X i / X 2 \ where k is a constant, and (d) a fully adjusted ratio model, Y= ( X i-a ) / X 2 k , 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 Yadj decreases as the value of the denominator X 2 increases). The reduction of variance in Y a aj as the denominator X 2 increases is most marked for 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 py,X2=0, but also a ratio model evaluation to examine the degree of distortion of 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 Xi and X 2 are 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/X2 as a regressor (X 2 is the 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 RM 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 RM ANOVA, although, Anderson and Lydic did look at ratio variables in the randomized groups ANOVA 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 ANOVA (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 ANOVA model ignoring the covariate: Third, for the ratios, an ANOVA model using the measured dependent variable divided by the covariate as the new dependent variable: (9) Second, an ANCOVA model: (10) xXl]lx2l]=n I- r, t (>:j (11) xnj: numerator variable; X2y: denominator variable; 21 \x: the grand mean; TJ: a constant associated wi th they'th treatment; eij: the experimental error associated wi th the ith subject under the jth treatment. The data were generated based on f o l l o w i n g condi t ions: 1. The correlations between the numerator and denominator 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 condit ions (u, x l=10 and n x 2 = 5 , p. x i=10 and n X2=10, p. xi=5 and u . x 2=10). 4. Combina t ions o f the coefficient o f variat ion between the numerator and denominator variables ( V x i / V X 2 = 0 . 5 , 1.0, 2.0). The variances of the component variables 2 2 were f ixed for a l l the condit ions ( o x i =aX2 =1). T w o groups were used and the sample size was 10 observations per group. The useful f ind ing o f this study was that the power o f the ratio A N O V A model was h igh ly dependent upon the ratio of the coefficients o f variat ion o f the component variables. H o w e v e r , comparisons among the three proposed models are not meaningful , because separate A N O V A models wi th different hypotheses were compared i n Ander son and L y d i c ' s study. T h e raw score mode l , the A N C O V A mode l , and the ratio mode l actually tested different hypotheses. In their two-group A N O V A design, the nu l l statistical hypothesis o f the raw score mode l w o u l d be that no difference exists between the raw score populat ion means i n the two groups. The n u l l hypothesis for the A N C O V A mode l is that no difference exists between the two populat ion means after these group means have been adjusted for differences between the groups on the relevant covariate variable. In general, A N C O V A is more powerful than A N O V A because o f the smaller 22 error variance in A N C O V A (this statement is true on ly when a good covariate variable is used and the assumptions are met). Therefore, it is obvious that A N C O V A has greater power than A N O V A in Ander son and L y d i c ' s study. The power o f the F-tests in these three A N O V A models i n this study are not comparable, but this study d i d show that the power wi th in the ratio A N O V A mode l is affected by the coefficients o f variat ion o f the component variables. The study indicated that the ratio mode l performed poor ly when the coefficient o f variation o f the denominator variable was greater than the coefficient o f variat ion of the numerator variable. A l t h o u g h there are intensive discussions on the issue o f v io la t ing the assumptions of R M A N O V A for raw score variables, there appears to be no publ icat ion invest igat ing ratio issues in 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 ci rcular i ty assumption o f the variance and covariance matr ix for the within-subject factor is affected not on ly by the variat ion and covar ia t ion o f the numerator variable over trials but also by that o f the denominator variable. W h e n a ratio variable is used in R M A N O V A , the variance and covariance o f the denominator ( inc luding the c o m m o n denominator condi t ion) over trials may seriously affect the populat ion characteristics o f the covariance matr ix (i.e., affect the c i rcular i ty o f the populat ion covariance matr ix in the within-subject factor). Ratio Variables and Unsolved Issues in Human Kinetics Research Rat io variables are c o m m o n l y used in human kinetics research, and are used i n almost al l o f the areas in the human kinetics domain (e.g., f rom physical education to 23 phys io logy) . Table 2-1 shows how ratio variables are used in a number o f different d isc ipl ines . In these studies, some ratio variables are used for deflation purposes and some are not, and usual ly serve as measurement variables in inferential analyses, especial ly in A N O V A . The analysis o f ratio variables wi th A N O V A procedures is c o m m o n i n human kinet ics research, for example , between 1992 and 1995, 6 2 % of the articles publ ished i n the Research Quarter ly for Exerc i se and Sport used A N O V A or M A N O V A , and 3 9 % of these articles used ratio variables. A l t h o u g h ratio variables are w i d e l y used, the va l id i ty and re l iabi l i ty issues o f using these ratio variables are rarely acknowledged and investigated in human kinet ics research. The f o l l o w i n g are the most important issues regarding the use o f ratio variables in our f ie ld . Deflation Issues In human kinet ics research, as shown above, many ratio variables are used for deflation purposes. That is , using ratio variables is an attempt to express the numerator on a per-unit o f the denominator basis i n order to remove the effects o f the denominator ' s variation f rom the numerator. W h e n the in i t ia l purpose o f using a ratio variable is deflation, it has been shown that two assumptions (homogeneous variances o f a ratio variable over the range o f the denominator variable, and zero correlation between the ratio variable and its denominator) should be examined . If the assumptions are v iola ted by the s imple ratio variable, other alternative deflation models should be considered. Recent ly , the va l id i ty o f the s imple ratio V o 2 m a x score ( V o 2 m a x / b o d y mass) has been examined and compared wi th alternative deflation ratios ( H e i l , 1997; N e v i l l , 1994; Vanderburgh & K a t c h , 1996; W e l s m a n , Arms t rong , N e v i l l , Win te r , and K i r b y , 1996). The mode l Y = X i / X 2 k was preferred for V o 2 m a x in some o f these studies and the magnitudes o f k reported varied from 0.41-0.80. H o w e v e r , a systematic invest igat ion o f 24 Table 2-1 Ratio Variables in Human Kinetics Research Discipline Variable Definition 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). Total cholesterol divided by high-density lipoprotein cholesterol (a stronger indicator of potential coronary risk). Hofstetter, C. R. et al. (1991). 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. Hi l l , D. W. etal. (1992). Education Defining intelligence quotient as IQ = 100 * mental age (MA) / chronological age (CA) and educational quotient as EQ = 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). R M M A N O V A 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 homogeneous variance assumption was not examined in these studies. A s shown in Table 2-1, s imple ratio data are s t i l l c o m m o n l y used in analyses without any val idat ion. Reliability Issues Because data are vir tual ly always measured wi th error, the considerat ion o f the re l iab i l i ty o f a measurement variable is c ruc ia l for any study. W h e n a ratio variable is used i n an analysis, the re l iab i l i ty issue becomes more important due to the complex i ty o f c o m b i n i n g two variables in one. Surpr i s ing ly , the re l iabi l i ty issue is rarely addressed in our f ie ld and needs to be investigated. Issues in RM ANOVA with Ratio Data A s mentioned in previous sections, R M A N O V A wi th ratio data is a c o m m o n l y used technique in human kinetics research. Issues o f us ing raw scores as a measurement variable in R M A N O V A have been discussed extensively in the literature. Schutz and Gessarol i (1987) have given a comprehensive review on this topic wi th empi r i ca l examples in our f ie ld . F o r the one-way R M A N O V A , the assumptions o f c i rcular i ty , a normal distr ibution o f the measurement variable, and homoscedast ici ty o f the error variances are required for the exact F statistics o f the treatment effect to be va l i d . In practice, A N O V A procedures are very robust wi th respect to violat ions o f the normal i ty assumption, but very sensitive to v io la t ion o f the assumptions o f c i rcular i ty and homoscedast ici ty o f the error variances. E o m and Schutz (1993) showed that the inflat ion in T y p e I error rates in the naive F tests when e<1.0 is not, as c o m m o n l y assumed, merely a function o f the populat ion epsi lon value. Rather, it is a function of the nature of the relative magnitudes o f the populat ion values and the sample estimate o f 26 eps i lon . That is , a large T y p e I error rate is associated wi th an overestimated epsi lon. O n the other hand, when the populat ion epsi lon is underestimated, the T y p e I error rate is considerably smaller than the corresponding nomina l leve l . Howeve r , the effects o f using ratio score variables on the assumptions i n R M A N O V A are hardly not iced and discussed in our f ie ld . A s shown in F o r m u l a (4), (5), (6) and (7), the variances and covariances of ratio variables are affected by characteristics o f the numerator and denominator variables. E v e n though there are the formulas available to approximate the variance and covariance of ratio variables, there are few publicat ions that investigate the impact o f us ing ratio variables in R M A N O V A . T h e f o l l o w i n g are some o f the issues that need to be investigated. Firs t o f a l l , we do not k n o w to what extent the ratio transformation affects the c i rcular i ty assumption o f the ratio populations in R M A N O V A . F o r example , i f the raw score populations meet the c i rcular i ty assumption, we do not k n o w how changes i n the variance and covariance of the c o m m o n denominator affect the c i rcular i ty property in R M A N O V A using ratio variables. Second, when an inference test is conducted f rom a sample o f ratio data based on the R M A N O V A , it is not clear how changes in the variances and covariances o f the component variables affects the sampl ing characteristics o f the variance and covariance matr ix o f the ratio variable. In summary, when ratio variables are used i n human kinet ics research, the f o l l o w i n g questions should be addressed: 1. W h a t is the purpose o f using ratio variables in the study? If it is for deflation purposes, has an appropriate deflation model been considered? 2. Does the ratio variable meet the re l iabi l i ty cri ter ion 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 shown in previous chapters, ratio variables are c o m m o n l y used in almost al l areas o f human kinet ics research. F o r the purposes o f this study, four o f the most frequently used ratio variables f rom several d isc ip l ines w i th in the f ie ld were selected and studied, but the pr inciples developed based on these four variables are appl icable to a l l ratio variables. Pub l i ca t ion and research databases c o m m o n to the f ie ld were searched for relevant studies that used ratio variables in repeated measurement situations, and letters were sent to researchers requesting their data. There were three cri teria used to choose appropriate studies and data sets for this research. Firs t , the study had to be i n the human kinetics research domain , and the ratio variable used in the study had to be a c o m m o n l y used variable. Second, the data sets inc luded the ratio variable and one o f its component variables, or both the numerator variable and the denominator variable, so that the ratio variable c o u l d be derived. T h i r d , the ratio variable data (both the numerator and denominator variables) were repeatedly measured. Data sets u t i l i z ing four c o m m o n l y used ratio variables meeting the above cri ter ia were obtained and analyzed. There were two data sets i n w h i c h the ratio variable was used for deflation purposes. The first data set was obtained f rom a study conducted at the Un ive r s i t y o f B r i t i s h C o l u m b i a (Taunton, 1992). The ratio variable used in this study was V o 2 m a x / k g and the sample size was 52. The second data set was from B a c o n ' s research (1997). The diffusing capacity o f the lungs for carbon monox ide ( D L c o ) d i v i d e d 29 by alveolar venti lat ion vo lume ( V A ) was the ratio variable ( D L C O / V A ) used for deflation purposes, and 13 subjects were in B a c o n ' s study. The other two sets of ratio data selected contained ratio variables that were not used for the purpose o f deflation. One o f these two data sets was f rom a project conducted by Kramer , N u s c a , B isbee , M a c D e r m i d , K e m p , and B o l e y (1994). T h e nondominant /dominant forearm strength ratio was the ratio variable used, and 43 subjects were in the project. T h e other ratio data set was the waist and hip girth ratios obtained f rom the study conducted by Sonnenschein, K i m , Pasternack, and T o n i o l o (1993), and the sample size was 1851. These four data sets served as a basis for the invest igat ion in this dissertation. Deflation Model Evaluation The Models In human kinet ics research, a s imple ratio o f two component variables is c o m m o n l y used for the purpose o f deflation. A s shown in the literature review chapter, a s imple ratio may not adequately el iminate the denominator effect, and alternative models need to be considered. T h e va l id i ty of the four deflation models used i n literature was evaluated for each o f the ratio variables chosen for the purpose o f deflation in this study. Simple ratio model. Based on classic measurement theory, a variable can be expressed as a populat ion mean wi th a error term. A s imple ratio Y = X i / X 2 can be presented as Y j = X n / X 2 i = H + e* (where p. is the populat ion mean for the ratio variable Y , and e; is the error term for subject i ; i = l , 2 , . . . , N ) . In the literature, a l inear regression function omit t ing the intercept is c o m m o n l y u t i l i zed to represent the relationship between the expected mean o f the s imple ratio and its component variables (Albrecht , et a l , 1993; 30 A l l i s o n , Paultre, Goran , Poeh lman , & H e y m s f i e l d , 1995; N e v i l l , Ramsbot tom, & W i l l i a m , 1992), i.e.; X „ t = p t X 2 l t + e i t ( i = l , 2 , . . . , n ; t = l , 2 , . . . , k ) (12) Where X i i t is the numerator variable, X 2 i t is the denominator variable. The s imple ratio score i n trial t for a subject i based on this mode l can thus be presented as Y i t =Xi i t /X2it=p\+e i t , where the slope p\ i n F o r m u l a (12) serves as the expected populat ion mean p.t, and elt is the error term (e;t =e i t /X 2 i t ) . Occas iona l ly , ordinary least-squares l inear regression based on Equa t ion (12) has been used to fit the component variable data (separately for each tr ial t) and obtain the estimated slope (A(3 t) o f the mode l , and A p \ used as the estimate o f the expected value o f the s imple ratio (e.g., N e v i l l et a l . , 1992). T h e weakness o f using this mode l is that the error term en is related to the denominator variable e^ =ei t /X2i t . T h i s impl ies that the expected value o f e^ given X 2 i t is s t i l l zero but its variance varies as a function o f X 2 j t ( A l l i s o n , et a l . , 1995). A l t h o u g h us ing this mode l cou ld alter the error dis t r ibut ion, it has been used by researchers to estimate the expected mean o f the s imple ratio variable. A l l i s o n stated: "hav ing been alerted to the fact that taking ratios alters the error distr ibution, the case for careful ly check ing the residuals for homoscedastici ty becomes even more compe l l i ng" . Therefore, a graphical cr i ter ion was inc luded i n this study to identify the v is ib le distort ion o f the variat ion o f the ratio variable. Linear regression model (LRM). T h i s mode l is an intercept adjusted ratio mode l : Xi i t =P ( a d j ) tX 2 i t+at+eit (13) 31 Where at is the intercept and p\adj)t is the slope of the regression equation for trial t, the 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(adj)it=(Xiit-at)/X2it=P(adj)t+eit, where eu is the error term (eit =eit/X2it). This linear regression model was used to fit the component variable data (separately for each trial t) to obtain the estimates of the slope, AP(adj)t and Aa t in Equation (13). Ap\adj)t serves as the estimate of the expected value of the adjusted ratio variable. The central issue hinges on the desire for a simple ratio (or an adjusted ratio variable based on the LRM 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 (at) equal to zero is a necessary and sufficient condition for the population correlation (py,X2) to be zero. Therefore, in the model LRM, subtracting the regression intercept at was necessary in order to meet the zero correlation criterion. As with the simple ratio model, the weakness of the LRM model is that the error distribution is related to the denominator variable due to ejt =ei t/X2i t. Nonlinear regression model 1 (NLRM1). This model has the form of an exponent adjusted ratio: Xlit=P(adj)tX2itkeit (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 X2it values when plotted against each other, provided these scores diverge at a 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 Ap\adj)t serves as the estimate of the expected value 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=Xlit/X2itk=P(adj)teit. Nonlinear regression model 2 (NLRM2). This model is a fully adjusted ratio model of the form: Xiit=P(adj)tX2itkeit+at (15) This model, which was suggested by Huxley (1972) and used by Albrecht et al. (1993), has both the intercept at and constant k. 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\aaj)t) in the model, and p\aaj)t serves as the estimate of the expected value of the adjusted ratio variable. The ratio (3(adj)it for subject i in trial t is presented by the equation Y(adj)it=(Xiit-a)/X2itk=P(adj)tejt. 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 cr i ter ion. The linear relationship between the der ived ratio variable (the s imple ratio variable Y or the adjusted ratio variable Y a d j ) and the denominator variable ( X 2 ) is zero or nearly so. T h e Pearson correlation coefficient between the der ived ratio variable and X 2 was calculated to ascertain the strength o f the l inear relat ionship. 2. Graph ica l cr i ter ion. The relationship between Y t and X 2 t can be plotted as a horizontal l ine (i.e., the least-squares regression l ine has a slope o f zero). 3. A lgeb ra i c cri ter ion. T h e estimate of the expected value A (3 t is equal to the empi r i ca l mean o f the ratio variable. F o r example , i n a s imple ratio mode l (XiJt=p\X 2it+ej t), the expected mean i n the ratio variable data Y t = E ( X i j t / X 2 j t ) / n should be equal to the estimated parameter A P t . It is noted that i n the s imple ratio mode l the expected value w o u l d be E(Yj t)=E[Pt+ejt/X2jt]= p\+E[eit /X 2 i t] . G i v e n that the error term e^ has mean zero (E(ei t)=0), the expected value E[e j t /X 2 i t ]=0 p rov ided e;t and X 2 j t are uncor rec t ed (in this case, E ( e i t / X 2 i t ) = E ( e j t ) E ( l / X 2 i t ) = 0 ) . Therefore, the expected value o f Y;t is equal to p\- In practice, the estimate o f the expected value A P t should be equal to the empi r ica l mean Y t . 4. R 2 . H i g h R 2 in the deflation mode l . 5. Re l i ab i l i t y . T h e re l iab i l i ty o f the der ived ratio data was assessed by using an intraclass correlation approach, the details o f wh ich are shown in next section. W h e n evaluating the second cri ter ion, by graphing Y t or Y ( a (jj)t and X 2 t in a scatterplot, we can examine the impact of denominator related distortions o f variances o f a ratio variable. A distortion o f the variances as the value o f the denominator increases may affect the val id i ty o f the deflation models . Bivar ia te plots and the algebraic cri terion 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 2 t and Yt or Y(adj)t. A valid deflation model for the ratio variable should meet all of these criteria. The traditional ANCOVA 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 ANCOVA model may not be a good model for deflation purposes. Vanderburgh (1998) has shown that the traditional ANCOVA model is not a most parsimonious and plausible model for scaling (deflation) purposes and nonlinear regression is better than the ANCOVA model. That is, in traditional ANCOVA 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 ANCOVA 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 ANCOVA 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 model should be developed was discussed. Reliability of a Ratio W h e n a ratio variable is used, the re l iab i l i ty o f the ratio variable is c ruc ia l for the va l id i ty o f the study. Unre l i ab le repeated measurements o f a ratio variable in an inferential analysis c o u l d seriously affect the va l id i ty o f the study. Therefore, the re l iabi l i ty o f the four ratio variables is investigated in this research. A s argued in previous sections, the re l iab i l i ty issues are more complex for ratio variables because a ratio variable is a combined variable w h i c h consists o f two component variables. It was shown i n F o r m u l a (8) that the re l iabi l i ty o f a s imple ratio variable is affected not on ly by the re l iabi l i ty o f the component variables but also by the variances, covariances, and the correlation o f the component variables. E v e n when the component variables are rel iable, it does not automatical ly guarantee that the der ived ratio variable has appropriate re l iabi l i ty . It is expected that a ratio variable is less rel iable than its component variables in repeated measurements because both the numerator and denominator variables are sources o f lack o f re l iabi l i ty . A researcher m a y ask how the re l iabi l i ty o f ratio data can be approximated f rom re l iabi l i ty o f the component variable data. If the assumptions o f the formula are met, the re l iabi l i ty o f s imple ratio variables wi th test-retest measurement data can be evaluated by Pearson's approach (Formula (7)) based on the correlation o f the component variables. The most important assumption for using this formula is that the coefficients o f variat ion for a l l the component variables are smal l enough (V<0.15) so that the cubic and higher 36 order variat ion terms i n the T a y l o r series are negl ig ib le . The variat ion and the correlat ion of the component variables may also strongly affect the va l id i ty o f using F o r m u l a (7). W i t h repeated measurement data i n v o l v i n g more than two repeated observations (e.g., trials), the intraclass correlation coefficient can be used to compute the re l iab i l i ty of the observed scores. A c c o r d i n g to classic measurement theory, re l iabi l i ty is defined as 2 2 2 the ratio o f true score variance (an ) to observed score variance (o n + o~e ). U s i n g the addit ive mode l w i th the assumption that subjects are a random variable and the t ime (trial) factor is a f ixed variable (the most c o m m o n situation in human kinet ics research), the intraclass correlat ion coefficient can be estimated f rom the mean squares i n a one-way R M A N O V A test (Winer , 1991): = MS, - MSe mt MSs+(k-l)MSe' K ) where M S s is the mean square due to subjects, and M S e is the mean square due to error, k is the number o f trials. In our f ie ld , the measured scores for several trials are usual ly averaged to increase the re l iabi l i ty . The intraclass correlat ion for the mean o f measured scores in several trials is computed as: _ MS. - MS, MS, ( ' These formulas have not inc luded the variance due to the treatment, because it is not a part o f the true variance or the total variance in the re l iab i l i ty evaluations. B o t h the two formulas appear i n publicat ions to report the re l iabi l i ty . In summary, using the above procedures (i.e., ratio variable selection, deflation mode l evaluation, and re l iab i l i ty evaluation), we examined the empi r ica l ratio variable data and investigated issues related to the va l id i ty o f ratio variables used i n our f ie ld . F o r 37 the selected ratio variables, this study evaluated the va l id i ty o f the four deflation models . A most rel iable and v a l i d deflation mode l was suggested for each o f the selected empi r ica l data sets. Computer Simulation Investigation of Ratio Issues The va l id i ty issues o f some ratio variables used in our f ie ld were addressed by examin ing four empi r ica l data sets according to the criteria shown in the above section. H o w e v e r , the general izabi l i ty of those f indings is l imi t ed because the procedures were based on specific empi r ica l samples. F o r example , how the variat ion and correlat ion o f the component variables affect the va l id i ty o f using ratio variables i n R M A N O V A was not examined. In fact, we cannot k n o w how different combinat ions o f variat ion o f the component variables affect the va l id i ty o f a ratio variable i n R M A N O V A by invest igat ing l imi t ed empir ica l studies. Because R M A N O V A is a c o m m o n l y used technique in our f ie ld , the va l id i ty of us ing ratio variables i n R M A N O V A requires further invest igat ion. S imula ted populat ion 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 may lead to a v io la t ion o f assumptions i n the one-way R M A N O V A . The effects of the relative variat ion, covar ia t ion, and correlation between the numerator and denominator variables were examined under different condit ions. The parameters of the ratio variables for generating the populat ion data were based on the statistical characteristics o f the selected component variables. B y vary ing these control led condit ions for a broad range o f the parameters, we 38 investigated how certain condit ions o f the component variable distributions lead to v io la t ion o f the assumptions of R M A N O V A when a ratio variable was used. The effects of using ratio variables on the type I error rate in the one-way R M A N O V A model (treatment f ixed and subjects random) were investigated us ing M o n t e C a r l o s imula t ion procedures. It is w e l l k n o w n that v io la t ion o f the c i rcular i ty assumption results in inf lat ion in the type I error rate, and that the inflat ion can be al leviated by using the eps i lon correction procedures (Greenhouse & Geisser , 1959; H u y n h & Feldt , 1976). In this study, the effect o f the ratio transformation on the c i rcular i ty property o f the populat ion covariance matrix and the consequences o f using sample data o f ratio variables in a R M A N O V A test were examined. Furthermore, the effect o f us ing ratio variables on the type I error rate i n one-way R M A N O V A test was investigated. In general, by us ing s imulat ion procedures, it was expected that the f o l l o w i n g questions c o u l d be answered: 1. I f the component variables produce a covariance matrix that meets the c i rcular i ty assumption in R M A N O V A , does the covariance matr ix o f the ratio variable compr i sed o f these components also exhibi t c i rcular i ty? 2. If the component variables produce a covariance matr ix that violates the c i rcular i ty assumption, under what condit ions do the ratio variables produce a covariance matr ix w h i c h leads to less or greater v io la t ion o f the assumption? 3. W h a t is the sampl ing distr ibution o f the 8 estimate (a measure o f the degree to w h i c h the c i rcular i ty assumption is v iola ted; see below for a complete defini t ion) when a ratio variable is used in R M A N O V A , and how 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 and five trials ( g= l , k=5) R M A N O V A in different control led condit ions was designed for the invest igat ion. R a w score populations o f the component variables were first generated based upon control led condit ions. Because the raw score variables i n our f i e ld have posi t ive values and l ie w i th in a certain finite range (e.g., V o 2 , Height , Weigh t , and waist and hip girths), a generated raw score populat ion was a large posi t ive populat ion. Table 3-1 shows the design and layout of the populat ion data o f the component variables. X H J is defined as the numerator variable and X 2 i j as the denominator variable, i is the subject number and j is the trial number ( X j and X 2 are used through this dissertation to represent the numerator and denominator variables). The ratio score variables ( Y = X i / X 2 ) were der ived by transformations f rom the raw score populations. The f o l l o w i n g sections describe the condi t ions under w h i c h the simulat ions were conducted. Box's Epsilon (e). B o x ' s e, a measure o f the degree o f departure f rom ci rcular i ty i n a populat ion covariance matr ix, is a function o f the variances and covariances in the populat ion matrix. B o x (1954) defined 8 as: (k-l)(JJyZojJ,-2kJjal + k2o2) (18) Where k: the order o f the covariance matrix (number of trials), Gjj: the mean o f the variances (diagonal elements), 40 Table 3-1 Da ta M a t r i x o f the Componen t Var iab les Numerator Denominator Trials T2 T3 T4 T5 T2 T3 T4 Ts Subject X i i i X112 Xu3 X114 X115 X211 X212 X213 X214 X215 s2 X121 X122 X]23 X124 X125 X221 X222 X223 X224 X225 St X n i Xii 2 Xii3 X)i4 Xii5 X 2 H X2i2 X2i3 X2i4 X2i5 5/v XiNl XiN2 XiN3 X]N4 X i N 5 X2N1 X 2 N 2 X2N3 X 2 N 4 X 2 N 5 41 a : the grand mean o f a l l elements in the covariance matr ix , Oj.: the mean o f j th row of the covariance matr ix , Ojj': an ind iv idua l entry o f the covariance matr ix (j, j ' = l , 2 , . . . , k ) . W h e n the populat ion data matrix meets the c i rcular i ty assumption, 8 = 1.0; otherwise, 8 < 1.0, wi th a m i n i m u m of l / ( k - l ) , where k is the number o f repeated trials in this study (the order of the covar iance matr ix) . E p s i l o n is used to correct a posi t ive bias i n the naive F test by adjusting the degrees o f freedom by an amount proport ional to e. That is , the computed F statistic for the repeated measures factor in a repeated measures design is distributed as F [ ( k - l ) e , ( n - l ) ( k - l ) e ] . In general, the characteristics o f a populat ion covariance matr ix are unknown and are estimated by a sample covariance matr ix . C o l l i e r , Baker , M a n d e v i l l e , and Hays (1967) showed that replacing the components i n F o r m u l a (18) wi th those estimated f rom a sample covariance matr ix and comput ing an estimate o f eps i lon ( a E ) to adjust the degree o f freedom results in a relat ively robust F-test for reasonable sample sizes o f 15 or larger. That is , the naive F statistic is approximately distributed as F [ ( k - 1 ) A 8 , ( n - l ) ( k - l ) A e ] . W e w i l l refer to this procedure for testing as the A e-adjusted F test. H u y n h and Fe ld t (1976, 1978) subsequently showed that the A s-adjusted test is negatively biased and w o u l d make the F test too conservative. The bias is most serious when a populat ion 8 is greater than 0.75, especial ly when the sample size is smal l . T h e y recommended 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 equali ty holds when A £ = l / ( k - l ) . The upper bound o f ~£ was set to unity, though it c o u l d be greater that unity. In this study, there were two £ condit ions for the two component variables generated. T h e values o f E were set to provide v io la t ion ( £ = 0 . 7 0 ) , and no v io la t ion (E=1.0) of the c i rcular i ty assumption (e.g., H e r t z o g & R o v i n e , 1985; H u y n h , 1978). The two £ condit ions for the component variables resulted in four combinat ions for the ratio populations (i.e., £ x i = 1 . 0 , £ x 2 = 1 . 0 ; £ x i = 1 . 0 , £ x 2 = 0 . 7 ; £ x i = 0 . 7 , £ x 2 = 1 . 0 ; £ x i = 0 . 7 , £ X 2=0.7) . K n o w i n g the £ value in a raw score populat ion and systematical ly va ry ing the correlation and relative variances o f the component variables a l l owed us to investigate the impact o f us ing the ratio variable on the c i rcular i ty assumption o f the covar iance matrices. F o r £ = 1 . 0 o f the component variable populat ion, the populat ion variances and the correlation between the repeated trials were constant, and the values were set based upon the waist and hip girth variables in Sommensche in , K i m , Pasternack, and T o n i o l o ' s study (1993). F o r the £ = 0 . 7 0 condi t ion , i n each populat ion data o f the component variables, the correlat ion between the repeated trials fo l lowed a s implex pattern in w h i c h the adjacent trials were more h ighly correlated than the trials w h i c h were farther apart (the details o f correlat ion condit ions are discussed later). G i v e n the characteristics o f the component variables, the effect of us ing ratio variables on the populat ion £ was evaluated by compar ing the £ values o f the ratio score populat ions ( £ x i / x 2 ) . The effect o f us ing ratio variables on the sampl ing characteristics o f A £ x i / X 2 was evaluated by compar ing the means and standard deviat ions o f the A £ x i / x 2 in different s imulat ion condit ions. It can be 43 speculated that a sample covariance matr ix is always expected to violate the c i rcular i ty assumption to some degree, even when the populat ion matrix does not. B y invest igat ing the e estimates o f samples drawn f rom the ratio populat ion under different condi t ions, we c o u l d examine the effect o f us ing ratio variables on the sampl ing behavior o f the estimated e coefficient ( A e x i/x2)-Population means of the component variables. The populat ion means o f the component variables were set to u, xij=75.0 and p. x 2j=100.0 for a l l treatment trials based on the waist and hip girths in Sonnenschein, K i m , Pasternack, and T o n i o l o ' s study (1993). Here j denotes the j th trial of the repeated measurement experiment. B y setting the means to be constant over trials, we were able to investigate how 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 combina t ion condi t ions of the variances and covariances o f the component variables. PY1Y2, V Y I , and V ^ . A s shown i n Formulas (4), (5), (6), and (7), both the correlations o f the component variables wi th in trials and over trials, and the coefficients o f variat ion o f the component variables p lay an important role in determining the ratio variable means, variances, covariances, and correlations. A m o n g the four types o f correlations between the numerator and denominator variables (i.e., p x n ; X 2 i , Pxii,x2j, P x i i . x i j , Px2i,x2j, i , j=l ,2 , . . . ,5) , the most meaningful correlations i n practical research are p x n , X 2i , P x i i . x i j , and pX2i,x2j- Here p x n , X 2i is the correlation between the numerator and denominator variables in the same trial (i), p x i i , x i j is the correlat ion between the numerator variables in different trials ( i , j ) , and pX2i,x2j is the correlat ion between the denominator variables in different trials. The correlation p x n i X 2j (i * j ) is the correlation between the numerator variable in trial i and the denominator variable in trial j . Its magnitude was based on the 44 other correlation components i n the correlat ion matrix (the details on comput ing pxn,X2j are shown in a subsequent section). The in i t ia l correlation among the f ive trials was also based on the empi r ica l data (Sonnenschein et a l , 1993). The main objective of the invest igat ion was to examine how the relative variat ion and the correlat ion pattern over trials between the numerator and denominator variables affect the characteristics o f the covariance matr ix o f a ratio populat ion. Therefore, to c lear ly show the main effect i n the invest igat ion, the magnitudes o f p x i i ,xij and pX2i,x2j were set to f ixed values for the condi t ion o f EXI=EX2=1.0. That is , for the £ xi=e X2=1.0 condi t ion , the correlation between the two adjacent trials i n Sonnenschein 's data set (pxn,xij=0.87 and pX2i,x2j=0.82, i?*j) was employed to fo rm the covariance matrix for the raw score populat ion data w i th equal variances and equal covariances over the f ive trials. These two values also served as a baseline for other eps i lon condit ions (i.e., p xii, xi2=0.87 and pX2i,X22=0.82 in the condit ions e xi=0.7 and ex 2=0.7). F o r the E=0.70 condi t ion , the pattern o f the correlations p x n, x i j and pX2i,x2j fo l l owed a s implex pattern showing the correlation between the first trial and subsequent trials decreasing exponent ia l ly (i.e., p, p 2, p 3, p4...). T o investigate how the correlation between the numerator and denominator affect the result o f us ing ratio variables in R M A N O V A , the three levels o f the correlat ion between the numerator and denominator in each trial (p xnX2i) were chosen to represent a range of low (.50), m e d i u m (.70), and high (.90) relationships (e.g., A t c h l e y , 1976). The value o f p xu X2i was f ixed i n a l l f ive trials i n each populat ion to reflect a fact that many ratios in human kinet ics have a near constant correlat ion between the numerator and denominator variables in a repeated measurement experiment (waist/hip, weight/height, and nondominant /dominant strength). Therefore, this invest igation was l imi t ed to a f ixed p x n X 2 i over trials. V x ! and V X 2 are the coefficients of variat ion of the numerator X i and denominator X 2 , respectively, where the coefficient of variat ion is Vx = a x I / i t . Three relative variat ion situations were s imulated ( V x i / V x 2 = 2 . 0 , 1.0, 0.5). The three combinat ions o f the coefficient o f variat ion between the numerator and denominator variables were designed to investigate the ratio effect and to cover different relative variations between the numerator and denominator variables. Preparing Population Covariance Matrices. The first step i n generating the populat ion data was to prepare the populat ion covariance matrices o f interest. Table 3-2 shows the format of the populat ion covariance matr ix wi th f ive repeated measurement trials. Because this covariance matrix is symmetr ica l , on ly the upper right cel ls are shown i n Tab le 3-2. The two shadowed parts are the sub-covariance matrices for the numerator and denominator variables, respectively. The elements o f the two shadowed sub-covariance matrices were der ived according to the defined populat ion epsi lon (eXj, i = l , 2), correlations of the two component variables over the repeated trials, and coefficient o f variat ion ( V x i , and V x 2 ) for each o f the s imulat ion condi t ions. The unshadowed part o f the matrix contains the covariances between the two component variables i n the f ive measurement trials. The elements i n the diagonal o f the unshadowed sub-matrix are the covariances between the component variables in the same tr ial . These covariances were calculated by using the f o l l o w i n g equation: (20) 46 a) c o 13 a c o I o o D o o GO <+-( o X •a 3 CO o s S o c a s s a s IS X x. X ~> o U x. X X. X o U X. x x. X o u X. X X o u x X o o x. X o CJ X. X o X. x X X o u o CJ o O X. IN X o O X. ro X cj x. X o u x X o CJ X. X x X cj u x X ~> o CJ x. X o CJ X. 6 X cB X x X > 5 - CJ 6 x. X c3 X > x > g 3 X ¥ > x ¥ > "a a "a 5 ro X — X ^> CJ 3 •k | : 1 -* 3 > 8 X > d x o ftSiCJ: x vm X -..iiiiM.:: IXS -c !!!©;! •X X > x I 47 where p x i i , X 2i denotes the correlation between the numerator and denominator variables in the same tr ial . S ince the variances and the correlat ion between the numerator and denominator variables i n the same trial were defined f rom the control led condi t ions , the Cov(Xu X2i) in the f ive trials was easily der ived. T h e off-diagonal elements o f the unshadowed sub-matrix are the covariances between the numerator and denominator variables in different trials (trial i and j ) : Oxn,x2r=Cov(XuX2J) = PXu<Xll^oxUax2. i , j = 1, 2 , . . . , 5 i * j (21) where Px.ii.x2j is the correlation between the numerator variable in the ith trial and the denominator variable i n the j th tr ial . T h i s correlat ion can not be obtained d i rec t ly f rom the defined control condit ions, and thus must be estimated. It is expected that the correlation between X i and X 2 in different trials (p xiu2j, i^ j ) w i l l be lower than that between X i and X 2 i n the same tr ial . The rationale for this is twofo ld . Firs t , i f X i and X 2 were perfectly rel iable, it might be expected that the correlation between X i and X 2 in different trials w o u l d be identical to that between X i and X 2 in the same tr ial . However , the rel iabi l i t ies o f X i and X 2 are considerably less than unity, and this trial-to-trial unrel iabi l i ty o f both X i and X 2 w i l l affect (reduce) the correlations between X i and X 2 in different trials. It cou ld be hypothesized that this reduction w o u l d be a function o f the re l iab i l i ty o f X i and X 2 . Second, the correlat ion matrices o f the empi r ica l data sets (e.g., K r a m e r ' s data, 1994, and Sonnenschein 's data, 1993) do show that p x n, x 2 j is less under the condi t ion i^j than i n the i=j condi t ion . The relationships between p xn,X2i and p xii,x2j ( i^j) were examined using two empi r i ca l data sets (Kramer et a l , 1994, Sonnenschein et a l , 1993), and the examinat ion suggested that an appropriate approximat ion for the correlat ion between the numerator in trial i and the denominator i n trial j cou ld be presented as: p x i l , x 2 1 = p J t „ . x 2 , v ^ " v ^ r ( 2 2 ) The estimated values o f the correlat ion p xn , X2j ( i^j) using F o r m u l a (22) were very close to the empi r i ca l correlat ion values in the correlat ion matrices i n these two data sets (the largest deviat ion was only 0.02). A n analytic approach based on the attenuation correlat ion formula (Nunna l ly , 1978, pp215-221) also supported the va l id i ty o f F o r m u l a (22). The details o f the support based on the attenuation correlat ion fo rmula are presented i n A p p e n d i x A . T o derive the covariances between the numerator and denominator i n different trials, we replaced p x n X 2j i n F o r m u l a (21) w i th F o r m u l a (22): ax,i,x2j = Cov(XuX2j) = pXiiXii7P^Vp*2*2 ^axuax2j i , j = l , 2 , . . . , 5 i * j (23) Based upon the defined condit ions, F o r m u l a (23) was used to calculate the off-diagonal elements in the unshadowed part of the covariance matr ix . In order to generate the raw score populations for this invest igat ion, each o f the elements in the covariance matrix in Table 3-2 was defined or calculated based upon the contro l led condit ions and the procedures shown above. A Turbo P A S C A L computer program developed by E o m (1993) was employed to construct the covariance matrices for X i and X 2 (i.e., shadowed parts o f the populat ion matrix in Tab le 3-2). W i t h the specif ied populat ion parameters (i.e., variances, correlations, and means), the computer program first constructed a populat ion covariance matrix wi th an epsi lon equal to unity (e=l) for the component variables. Second, each o f the matrices was further manipulated 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 2 (i.e., unshadowed part of the 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 2 ) over five 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; Pxix2 v 6x1 = 6x2=1.0 0.9 Vxi=0.2Vx2=0.1 6xi = 6x2 = 0.7 Ox, = 75 ( i x 2 = 100 0.7 Vxi=V x 2=1.55 8x1 = 0.7 < e x 2= 1.0 0.5 Vxi=0.1 V x 2=0 .2 8x1= 1.0 > 8 x 2 =0.7 51 Overview of Methods Used to Answer Questions I first investigated the characteristics o f the ratio variable populations under different variat ion and correlation condit ions of the component variables. T h i s process provided evidence to examine how characteristics of the component variables affected the c i rcular i ty assumption o f the covariance matrices for the der ived ratio variable populat ion. B o x ' s E values o f the ratio score populations in different condit ions were computed to provide information regarding the degree o f non-ci rcular i ty o f the covariance matrices o f the ratio variables. Second, the behavior o f the sample estimates o f £ (e ) for ratio data across the simulated condit ions was examined. The F O R T R A N M o n t e C a r l o s imulat ion program was used for the purpose o f investigating the effect o f us ing ratio variables on the sampl ing characteristics o f e . Af ter d rawing samples o f n subjects f rom the ratio score populations (n=15, 30, 45), the F O R T R A N program conducted 2000 one-way R M A N O V A tests in each condi t ion for the ratio score data. T h e f o l l o w i n g sample estimates were computed: mean squares in the F test, standard deviat ion o f the mean squares, mean of £ , standard deviat ion o f £ , and type I error rate. B y compar ing these sample estimates i n different condit ions, we were able to investigate how the relative variance and correlation changes between the numerator and denominator variables affected the type I error rate o f the F test wi th ratio variable data. 52 CHAPTER IV. RESULTS AND DISCUSSION: CHARACTERISTICS OF RATIO VARIABLES In this chapter, the results emanating f rom each o f the four empi r ica l data sets are presented. Firs t , I present detailed evaluation results for each o f the two data sets in w h i c h the ratio variables were used for the purpose o f deflat ion. The four ratio deflation models are compared and evaluated based on the cri ter ia l is ted in the methods and procedures chapter. T h e f ive cri teria; statistical, graphical , algebraic, R2, and re l iab i l i ty , are examined for each o f the deflation models . Then , the best deflation model is identif ied for each o f the two data sets, and general issues related to the development o f an appropriate deflation mode l for ratio variables i n human kinet ics research are discussed. Second, the statistical characteristics o f the other two 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 re l iab i l i ty o f us ing ratio variables is discussed. Ratio Variables Used for Deflation Purposes Empirical Data Set 1: Vo2max/Body Mass These data were obtained f rom a study conducted at the V a n c o u v e r General Hosp i ta 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 women aged 65 to 75 years. Seventy-six subjects were randomly assigned into a water-based exercise group, a land-based aerobic group or a c lassroom health promot ion group (control group). E a c h group met for one hour, three times a week, over a twelve weeks period. The variables o f interest were measured at three t ime points (trials) dur ing the twelve weeks (pre-test, mid-program test, and post-test). The measured variables inc luded m a x i m a l fatigue, strength, body composi t ion , f l ex ib i l i ty , balance, and V o 2 m a x , wi th the latter measure being one o f the most important variables measured in the study. T h e measured V o 2 m a x was d iv ided by the body mass in order to express the V o 2 m a x on a per-unit o f body mass basis, thus deflating the effects o f the body mass variation f rom the measured V o 2 m a x variable. T h e der ived ratio V o 2 m a x variable, V o 2 m a x / k g , then served as a measurement variable i n a 3 x 3 (group x time) R M A N O V A to answer the research questions. The researchers' p r imary interest was to investigate whether cardiovascular endurance, as measured by V o 2 m a x per-unit o f body mass, was affected by the different t raining techniques. T o evaluate the va l id i ty o f the V o 2 m a x / k g ratio variable and the alternative deflation models i n these data, the in i t ia l plan was to evaluate the three groups separately (Gi=water, n i=27; G2=land, n2=25, G3=control, n 3 =23; k=3). Howeve r , as shown i n Table 4-1 , the land and control groups had a considerable amount of mis s ing data, especial ly in the second and thi rd measurement points, due to subject drop out. Because the F tests in the 3 x 3 (group x t ime) m i x e d mode l A N O V A showed no significant differences in V o 2 m a x / k g among the groups and no significant groups by t ime interaction, the complete cases i n the three groups were combined and analyzed together (n=52). Simple Ratio Model The results reported in Tab le 4-2 are based on the s imple ratio data. A s shown in Table 4-2, the correlation between V o 2 m a x ( X i , raw score) and body mass ( X 2 ) ranged f rom 0.46 to 0.54 i n the three repeated trials. T h i s result indicates that the variat ion o f 54 CO co ts Q bi x s CN O B CN O > 8 o o Pi GO GO C3 > -a o PQ o GO C o 1 ' > CO Q Ii on GO co co H CN H co H 3 S S3 > CN T3 O PQ C O H CN H CN 53 CN S3 ? oo r -2 G • 2 R £ oo S O ^ ^ ^ g CN so Q q © ^ O r- CN co in o ^ so *™H m CN S O c o oo oo ~> S O CN 2 ^ 3 CN S " co CN © ^ oo Os CN O CN C O o CN ^ ^ 8 OO  CN ^ ^ ^ 2 Fi ©? o-i _• O 2 a ^ S O CN 2 Os Os i n oo g, l> - 1 SO i—l «n CN r - W CN s o <—i Os CO S O S O so CN r - W M so C CN C U V O 3 CN 5 ll 6 c oo CN co CN ! £ oo °) 3 JO 2 K H ^ S c o CN oo Os IT) co co r—( • CN r~- ,—i oo CN ^ ^ SO o S? '-' 5 ^ CN CN CO CN SO ^  ^ so O in ~ V O S O 00 ^o CO 1^ CN CN r--S O © ii CO CU CO 3 CN 5 'I 6 c g s o r -S © ^ o CN C O O 2 d , © ' co ^ H H m £ >n ~! ?o © © H h 5 6 1 £ 2 ° 2 . so OS CO CN CO O 00 CN | ^ *n £ so —•< " ^ F - oo n 2 S O -H Q SO <-» O N l> CN Os SO - H so i—i oo > - | U C N o 13 GO CO 3 > C l l > <+-< o G CO co o o § GO e o "fj I-8 ' i Table 4-2 Eva lua t ion o f the S i m p l e Ra t io M o d e l : Da ta Set 1 Ra t io r r Es t imated E m p i r i c a l M o d e l X i VS X 2 Y vs X 2 Expec ted M e a n Y V a l u e A |3 Y = X , / X 2 T r i a l 1 0.48 -0.45 18.28 18.85 T r i a l 2 0.54 -0.61 19.57 20.22 T r i a l 3 0.46 -0.58 20.30 21.02 R e l i a b i l i t y Coeff ic ient X I x 2 Y Intraclass r Average 0.88 0.99 0.89 Trials Single Trial 0.72 0.99 0.72 56 body mass does affect the variat ion o f the raw score V o 2 m a x variable. The 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 fu l ly remove the effect o f body mass f rom the V o 2 m a x variable. The question o f interest w i th these data is whether the s imple ratio ( V o 2 m a x / k g ) can appropriately deflate the effect o f the body mass f rom the V o 2 m a x . A s shown i n the methods and procedures chapter, one important cri ter ion is that the correlat ion between the der ived V o 2 m a x / k g and body mass should be zero or nearly so i f the s imple ratio can appropriately deflate the body mass f rom the V o 2 m a x . Th i s cri ter ion is not met, as the results (ryix2i in Tab le 4-2) show substantial negative correlations (0.45-0.61) between the V o 2 m a x / k g ( Y ) and body mass ( X 2 ) in a l l three trials. G i v e n a posi t ive correlat ion between the two component variables, us ing Pearson's approximat ion function (Formula (7)), A lb rech t (1978) showed that the correlation between a ratio and its denominator variable is negative when the coefficient o f variat ion o f the denominator variable is equal to or greater than that of the numerator variable. A s can be seen in Tab le 4 -1 , the coefficients o f variat ion o f the body mass are almost equal to or greater than that o f the raw score V o 2 m a x in a l l three trials, and the raw score V o 2 m a x and body mass have posi t ive correlat ion. Thus , negative correlations between V o 2 m a x / k g and body mass i n these data are to be expected. The graphical cri terion expla ined in the previous chapter was designed to identify any v is ib le pattern o f a relationship between a ratio variable and its denominator variable, and any denominator related distortion o f the variances for the ratio variables. If the s imple deflation mode l is va l id , V o 2 m a x / k g should have not on ly a zero l inear correlat ion but also zero curvi l inear relation wi th the body mass. A s shown in the three scatterplots 57 i n the first panel o f F igure 4-1 , a negative relationship pattern is c lear ly shown in a l l three trials. It confirms that the s imple ratio variable favors the lighter subjects and disadvantages the heavier subjects. That is , the l ighter subjects show larger V o 2 m a x / k g than the heavier subjects. There is no suggestion o f a curvi l inear relat ionship between the V o 2 m a x / k g and the body mass in any o f the three trials. If there were curvi l inear relat ionship between V o 2 m a x / k g and body 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 in these data. T h e scatterplots, and the negative Pearson correlat ion 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 denominator effect and may not be v a l i d for deflation purposes based on these two cri teria. A s stated in the previous chapter, a correct ly formulated ratio variable should result i n vir tual equali ty o f the estimate of the expected value o f the ratio mode l and the calculated mean average o f the empi r i ca l data. In the s imple ratio mode l , the estimate o f the expected value ( A H y ) o f the V o 2 m a x / k g was computed f rom the (3 coefficient in the l inear regression equation without an intercept (i.e., Xii t =p\X2it+ej t ) . T h e empi r ica l means in the three trials were computed f rom the ratio V o 2 m a x / k g data. T h e calculat ions show that the empi r ica l means are consistently greater than the estimate o f the expected values i n the s imple ratio mode l (see Table 4-2). T h e over-estimation in the three trials is between 0.57 to 0.72, about 4 % error relative to the actual mean o f the empi r i ca l data. The last cri terion for evaluat ing the va l id i ty o f a ratio data is its re l iabi l i ty . The re l iab i l i ty o f these data was calculated based on Formulas (16) and (17) shown in the methods and procedures chapter. The mean squares used in Formulas (16) and (17) were der ived f rom a three trials R M A N O V A test performed on each o f the three dependent •c H o o in in ra 5 >. •o o CO 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 13 •c H >. •o o m o co o O C O C O - ^ C M O O O C D M - C O O 6>(/XBUJ20A >» T J O m o o c D T f o a o o o c o - ^ C A i o (wan) Z A 0 3 •c O l O J I t N O O O O ^ N O n ( M ( M « « « ' - ' - < - r - r -o o T J O m o C O C O - t f O U O C O C D ' t f C M O 6>|/xeuizoA (lAIHl) LA SI variables, the raw score V o 2 m a x , body mass, and s imple ratio V o 2 m a x / k g . Then , the intraclass correlation coefficients ( r x i x i , rX2X2, % ) were calculated based on the der ived mean squares o f the R M A N O V A tests. The intraclass correlations are inc luded in reporting of results. The discussion of re l iabi l i ty is based on the intraclass correlat ion for the mean o f a l l trials (Fo rmula (17), but the single trial re l iab i l i ty (Formula (16) is also reported as both forms are often presented in the literature. Based on the def ini t ion, the single trial re l iab i l i ty reported is the same as the average pairwise correlation o f the trials i n the data. The results (see Table 4-2) show that the body mass data are very rel iable (rX2x2=0.99), and the re l iab i l i ty o f the raw score V o 2 m a x data is also high ( r x l x ) = 0 . 8 8 ) . The high re l iab i l i ty o f body mass i n these data is expected because body mass should not be substantially affected in the twelve week experiment period. The s imple ratio d i d not reduce the re l iab i l i ty o f the ratio data, wi th the re l iab i l i ty o f the s imple ratio data be ing close to that o f the measured V o 2 m a x variables ( r y i y i=0 .89) . Because the re l iab i l i ty o f the denominator variable is so h igh, the re l iab i l i ty o f the der ived s imple ratio variable is very close to the re l iab i l i ty o f the numerator variable. T h i s confirms Cronbach ' s deduction (1941). That is , when the denominator variable is h igh ly reliable (e.g., r x 2 x 2 close to unity), it can be proved that the re l iab i l i ty o f the ratio variable is close to the re l iabi l i ty o f the numerator variable. In summary, for the evaluation of the s imple ratio i n Data Set 1, the ratio V o 2 m a x / k g does not appropriately deflate the effect o f body mass from the V o 2 m a x variable. A l t h o u g h the s imple ratio has acceptable re l iabi l i ty , the s imple ratio does not fu l ly deflate the effect o f body mass. It on ly reverses the relationship between the 60 measured V o 2 m a x and the body mass f rom posi t ive to negative. Based on the results o f examin ing the cri teria for deflation purposes i n these data, the s imple ratio does not meet the important cr i ter ia o f fu l l deflation, especia l ly zero correlat ion between the ratio variable and the denominator variable, and equali ty o f calculated empi r ica l mean and the estimate of the expected value. Therefore, it is conc luded that the s imple ratio does not appropriately deflate the effect o f the body mass for the V o 2 m a x variable i n these data. A n appropriate alternate deflation mode l may be needed for this c o m m o n l y used measure. Alternative Ratio Models These data were fitted to the three alternative deflation models ( L R M , N L R M 1 , and N L R M 2 , see p31 for definitions). The results are shown i n Table 4-3. A s shown i n Tab le 4-3, the adjusted ratio V o 2 m a x variables have a near-zero correlation wi th the denominator variable (body mass) i n a l l three alternative models . F o r the models L R M and N L R M 2 , the correlations between the adjusted ratio V o 2 m a x and the body 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 smal l as 0.01 i n a l l three trials. Thus , i n this data it is conc luded that a l l three models meet the cri terion o f zero correlat ion. Some researchers (e.g., Albrech t , 1993) c l a i m e d that adjusted ratios especia l ly wi th the nonl inear models ( N L R M 1 and N L M R 2 ) b r ing about a serious denominator-related distort ion of variances. T h i s distort ion o f variance w o u l d be indicated i n scatterplots of the adjusted ratio variable versus the denominator variable. That is , the adjusted ratio variable has less variance for greater values o f the denominator and higher variance for smaller values o f the denominator. A s shown in Figures 4-1 61 Table 4-3 Eva lua t ion o f the Alternate M o d e l s : Da ta Set 1 r Y vs X 2 R e l i a b i l i t y Intraclass r R 2 E m p i r i c a l M e a n Y Es t imated Expec t ed V a l u e A |3 Parameters Y = ( X i - a ) / X 2 (LRM) T r i a l 1 T r i a l 2 T r i a l 3 0.02 -0.01 -0.01 all Trials 0.83 Single Trial 0.62 0.229 0.293 0.214 9.22 8.84 8.24 9.23 8.84 8.23 a 619.53 732.98 826.05 k Y = X ! / X 2 k ( N L R M l ) T r i a l 1 T r i a l 2 T r i a l 3 0.01 -0.01 -0.01 0.83 0.62 0.233 0.289 0.210 148.63 198.41 251.51 148.67 198.38 251.47 • 0.50 0.45 0.41 Y = ( X ! - a ) / X 2 k (NLRM2) T r i a l 1 T r i a l 2 T r i a l 3 0.01 -0.02 -0.02 0.82 0.60 0.233 0.291 0.212 312.64 56.20 44.88 309.47 55.42 45.73 -367.80 426.96 563.89 0.39 0.66 0.69 62 (panel 2) and 4-2, the scatterplots of the three ratios in the three trials do not indicate v i s ib le variance decreases when the value o f the denominator variable (body mass) increases (although the sample size might be too smal l to v i sua l ly reveal any such effects). Therefore, body mass related distort ion o f variance does not seem to be an issue for these three models i n this data set. T h e th i rd cri terion for the evaluation o f a deflation mode l is the equal i ty o f the estimated expected value and the empi r i ca l means. The calculations (see Tab le 4-3) indicate that the expected means are accurately estimated (wi th in +/- 1% difference) by al l three models . T h e re l iab i l i ty o f the adjusted ratio variables is examined and shown i n Tab le 4-3. The re l iab i l i ty o f the adjusted ratio V o 2 m a x variables was lower than the re l iab i l i ty o f the two component variables (e.g., 0.83 compared to 0.88 and 0.99). There was no substantial difference in the re l iabi l i ty among these three models for these data. T h e relative lower re l iab i l i ty for the adjusted ratio variables was expected, and for the f o l l o w i n g reasons. The L R M ratio is based on a s imple l inear regression relationship X n t = a t + p \ X 2 i t + e i t ; where ej t is the error term and p t serves as the expected value o f the adjusted ratio data. The adjusted ratio variable ( L R M ) can be expressed as Yj t=(Xij t-a t)/X2it=Pt+eit/X2it. O f interest in the L R M model is the error term e j t / X 2 j t . Heteroscedastici ty occurs when the variance o f Y ; t is not constant for a l l values o f X 2 j t , and the level of heteroscedasticity w o u l d direct ly affect the re l iab i l i ty of the data. The larger the range o f the variable X 2 H , the greater the heteroscedasticity o f the data, and the lower the re l iab i l i ty o f Y i t . The heteroscedasticity o f the L R M mode l is very sensitive to the range o f the denominator variable (Pedhazur, 1982). Thus , the heteroscedasticity that is introduced by 63 •c H o o o o o o o o i n o m o m o m c r > C O C M C M i - i -> T J O m (Hflld1N)£A o o o r - c o i n o o o CO OJ i -(ewtnN) EA 13 •a H •o o m o o o o o o o o o i n o i n o m c o CM CM ••- r-o o o o o o o o o o o o c t e o s i D u i * n w < -(iWHnN)ZA tewunN) Z A •c H V) CO s T J o o o o o o o m o i n o m CM CM i - * -co 10 (0 > T J O CO UwanN) LA o o o o o o o o o o i n o m o i n o m o i n (ZIAIUIN) LA PH the ratio transformation is the main factor to reduce the re l iab i l i ty o f the adjusted ratio variable in the L R M mode l . Because the range of the body mass in these data is relat ively smal l , the re l iab i l i ty o f the adjusted ratio variable is reduced, but on ly s l ight ly , and the re l iabi l i ty o f Y under both the L R M and N L R M 1 models is s t i l l reasonably strong (0.83). The N L R M 1 ratio is based on the relationship X i J t = Y t X 2 i t k - e i t . A l l i s o n (1995) c l a imed that this approach might be quite desirable and appropriate. Howeve r , when the adjusted ratio variable is used i n group comparisons or R M A N O V A , it is v a l i d on ly when the exponent parameters o f the adjusted ratio variable in the different trials are ident ical . Because the exponents are different in the different trials, the units of the adjusted ratio variable are different. F o r example, when calcula t ing the re l iab i l i ty o f the adjusted ratio variable, one c o u l d not calculate the re l iab i l i ty coefficient by us ing the N L R M 1 mode l wi th different parameters such as k i=0 .50 , k 2 =0.45, and k 3 =0.41 in the three trials. Therefore, a c o m m o n exponent (k=0.45) was der ived by averaging the three exponent parameters (i.e., k=(ki+k 2 +k3)/3). The c o m m o n k=0.45 was used to derive the N L R M 1 ratio for each tr ia l , and the re l iabi l i ty calculat ions were based on those data. W h e n the c o m m o n exponent k is used in the different trials, it increases the variances o f the adjusted ratio variable. Because a c o m m o n k is used, the inf lat ion o f the variances is more serious in the first and third trials. Therefore, it reduces the re l iabi l i ty o f the adjusted ratio variable in the N L R M 1 model . F o r the same reason, a c o m m o n k has to be used in the N L R M 2 mode l , thus reducing the re l iab i l i ty o f the adjusted ratio data. A s expla ined i n the methods and procedures chapter, the ordinary least squares linear regression option i n S P S S was used to fit the L R M mode l . The nonlinear 65 regression option wi th a Levenberg-Marquard t a lgor i thm in S P S S was used to fit the N L R M 1 and N L R M 2 models . It has been shown that the s imple ratio over-adjusts the effect o f the denominator and is not an appropriate deflation model for these data. T o compare the relative merits o f the three alternative models , R , the proport ion o f variance accounted for by the mode l , was examined. It indicates that the differences o f R among the three adjusted ratios are t r iv ia l (see Table 4-3). The magnitudes o f the R 2 obtained using the N L R M 2 mode l are s l ight ly higher than those for the N L R M 1 mode l , and the L R M model has s l ight ly higher R than the other two nonlinear models in these data but difference is not substantial. Results also show that the re l iab i l i ty o f the three adjusted ratio data was v i r tua l ly the same. These results suggest that there is no substantial difference among the three models in their ab i l i ty to provide a v a l i d ratio score for these data. It may seem contradictory that the three deflation models wi th different parameters fit these data about equal ly w e l l . The explanat ion for this phenomenon is that the regression lines diverge outside the range o f the observed data i n accordance wi th the very different regression models . F o r example , as shown in F igure 4-3, the fitted curvi l inear regression lines of the second tr ial i n Da ta Set 1 are very close to l inear in the relat ively narrow range o f these observed data. H o w e v e r , the nonlinear models do show the better fit than the l inear mode l in some other situations (Albrecht et a l . , 1993). G i v e n the s imi la r goodness of fit in the models , the L R M and N L R M 1 models may be more preferable than the N L R M 2 mode l for these data because the L R M and N L R M 1 models are easier to use. If the L R M mode l is used, the l inear regression approach (which is avai lable in most commerc ia l statistical software) can be used to fit 67 the data. A s shown above, L R M and N L R M 1 have successfully met the cri ter ia and are s impler than N L R M 2 for deflation purposes i n Da ta Set 1, although wi th some reduction in the re l iabi l i ty . The s imple ratio model fa i led to meet these criteria. Therefore, i f the re l iab i l i ty o f the two models is acceptable, it may be suggested that L R M and N L R M 1 are statistically v a l i d and appropriate for the purpose o f deflation i n Da ta Set 1. E v e n though the L R M and N L R M 1 ratios have s imi la r va l id i ty based on statistical cri teria, physiologis ts may have reason to choose one over the other. In recent years, N L R M 1 has been used in some human kinet ics research (e.g., H e i l , 1997; N e v i l l , Ramsbo t tm, & W i l l i a m s , 1992; Vanderburgh & K a t c h , 1996). Based on the theory o f geometric s imi lar i ty , a c o m m o n value of 0.67 for the k parameter was developed for V o 2 m a x (Ast rand, & R o d a h l , 1986; H e i l , 1997). H e i l (1997) pointed out that the 0.67 exponent appears to be more appropriate for samples that are very s imi la r w i th respect to age, t raining background, body height, etc. H o w e v e r , a sample that is homogeneous on so many measurement variables is hard to f ind in an empi r ica l study. T h i s study supports that o f other researchers who have shown that a c o m m o n magnitude o f the k parameter in the N L R M 1 mode l for V o 2 m a x data is questionable (e.g., B e r g h , S jod in , Forsberg , & Svedenhag, 1991; N e v i l l , 1994; W e l s m a n , Arms t rong , N e v i l l , Win te r , & K i r b y , 1996). The der ived magnitudes of k for V o 2 m a x reported in the literature range f rom 0.41 to 0.80, indicat ing that it is a data dependent parameter, and therefore it is not appropriate to use a c o m m o n parameter k in the N L R M 1 mode l for V o 2 m a x . It is suggested here that researchers should fit a N L R M 1 mode l to their data and derive a sample-dependent k value for the adjusted ratio variable V o 2 m a x i n each study. A l t h o u g h the N L R M 2 model 6 8 is not suggested for this specific data set, it should be considered by researchers as a possible deflation model . W h e n alternative deflation models are used, special caution must be taken when compar ing the means o f the adjusted ratio variables among repeated trials or different groups, and when interpreting the meaning o f the actual values o f the adjusted ratio variables. Firs t , when the nonl inear regression models are used, group or trial comparisons are appropriate on ly when the exponents (k) for different groups or trials are ident ical . F o r example, i n Data Set 1, because the magnitudes o f the parameter k o f the N L R M l model i n the three trials are s l ight ly different (i.e., k i=0.50, k 2 =0.45 , and k 3 =0.41) , 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. That is , the scales o f the adjusted V o 2 m a x i n the three trials are ml-kg~ 0 ' 5 0 -min~ 1 , ml -kg" 0 4 5 - m i n " ' , and m l - k g ^ ' - m i n " 1 , respectively. Thus , one c o u l d not compare the three trial means of the adjusted V o 2 m a x der ived from the N L R M l mode l under the different scales. T o compare the three group means, the scale o f the adjusted ratio variable in a l l trials or groups should be ident ical . In these data, a c o m m o n k parameter o f N L R M l can be obtained by the average o f the three k parameters over the three trials (i.e., k=(k,+k 2 +k 3 ) /3) . A v e r a g i n g different values of the k parameter is not the on ly method for getting the same unit scale in different trials or groups. If different groups are compared (not the same group in different trials), the data cou ld also be pooled and a c o m m o n k can be der ived by fi t t ing the N L R M l model to the pooled data. H o w e v e r , when the N L R M l mode l is used to fit the data in repeated measurement data l ike Data Set 1, the pooled data may 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 over subjects. That is , i f the repeatedly measured data were pooled, there w o u l d be a between-subject error correlat ion because the repeated measurements for the same subject were treated as different subjects in the pooled data. A second caution is that nonl inear regression mode l ing yields adjusted ratio variables that may not be as easy to interpret as the raw score component variables due to the complex scale (e.g., m l - k g " 0 4 5 - m i n _ 1 in Data Set 1). H o w e v e r , the relative differences among groups or trials are comparable. T h e actual group or trial differences can be described dimensionless ly and expressed as a percentage difference. F o r example, the sample means of the adjusted V o 2 m a x using N L R M 1 wi th c o m m o n k=0.45 in the first and second trials are 184.72 and 198.41, respectively. T h i s difference can be expressed by saying that based on the N L R M 1 model the first trial had 9 3 % of the body mass adjusted V o 2 m a x of the second tr ial . It should also be noted that the percentage difference compar ison depends on w h i c h mode l is be ing employed . Empirical Data Set 2: D L C D / V A T h i s data set was f rom a research project conducted i n the Sports M e d i c i n e Center at the Un ive r s i t y o f B r i t i s h C o l u m b i a (Bacon , 1997). One o f the purposes o f the research was to investigate whether there was a change in "dif fusing capaci ty o f the lungs for carbon m o n o x i d e " (DLCO) w i th in an ovulatory menstrual cyc le in regular ly menstruating women . In order to deflate the variat ion o f the inspired volumes , w h i c h was measured as alveolar venti lat ion vo lume ( V A ) , the ratio variable D L C O / V A was used in 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 variable in s imi la r studies in the related literature (e.g., Crapo & Forster, 1989). Howeve r , un l ike the 70 V o 2 m a x / k g ratio measure, the statistical and va l id i ty issues o f this ratio have never been addressed i n the literature. There were 13 subjects i n the study, and D L C O and V A were measured at f ive points dur ing the menstrual cyc le . The f ive test points were early menstrual, late menstrual, early fo l l icular , m i d c y c l e , and m i d luteal. A one-way R M A N O V A (n=13, k=5) was used to analyze these data. Tab le 4-4 shows the descript ive characteristics o f the data. Simple Ratio Model The cr i ter ia to evaluate the va l id i ty o f the mode l w i t h these data are the same as i n D a t a Set 1. A s shown i n Tab le 4-5, the correlations between the measured D L C O and V A vary from 0.29 to 0.59 i n the f ive measurement trials. T h i s indicates the posi t ive relationship between V A and D L C O , and thus an appropriate deflation mode l c o u l d be used to deflate the effect o f V A f rom the D L C O data. The results show negative correlations between the s imple ratio D L C O / V A and V A i n the f ive trials, indica t ing that the s imple ratio mode 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 igure 4-4 also show that the subjects w h o have greater V A tend to have smaller D L C O / V A (this trend was not c lear ly shown in the first trial), but there is no suggestion o f a possible curv i l inear relationship between the s imple ratio D L C O / V A and V A . T h e estimated expected value and the empi r ica l means are not the same over the f ive trials. The intraclass correlat ion shows that the s imple ratio D L C O / V A has a re l iab i l i ty (r y y =0.90) w h i c h is approximately the same as its components. In summary, although the s imple ratio mode l meets some of the criteria, the ratio D L C O / V A may not be an appropriate mode l for deflation i n these data because it over-deflates the effect o f V A f rom D L C O A n alternative deflation model should be considered. 71 > co oo P 0 3 O Q u 6 Q o o oo £ ca OS > 0 0 C O 'I ' > co Q 1 -t—> 00 T3 ^ 0 0 CO c o oo co i-H CO 1—1 H CO d d VO vq vo •n m i—t CO 1—I H co d d c3 T(- os CO CO 1-H O H co d d 3 ON CN oo »—1 CO .—1 H CO d d (N 00 oo CO co 1—1 H CO d d i n H 21.69 2.23 0.10 CO <N CO CN VO co H 1—1 CN CO d o o m co T—4 H in co Q H CN CN CO d r- co m CO 1—1 H CN CO d o (N oo CN m co i—1 H CN CN co d IT) CO m oo i—1 CO i-H H in d d OS 00 o os m i — i CO 1—1 H in d d Va m H 5.95 0.85 0.14 CO 1—1 CM . VO m 00 >n i-H CO 1—1 H in d d CN oo m m Os o co H in d d Mean PS > c Table 4-5 Eva lua t ion o f the S i m p l e Ra t io M o d e l : Da ta Set 2 Ra t io r r Es t imated E m p i r i c a l M o d e l X , vs X 2 Y vs X 2 Expec ted M e a n Y V a l u e A p Y = X i / X 2 T r i a l 1 0.58 -0.08 3.81 3.82 T r i a l 2 0.29 -0.59 3.80 3.91 T r i a l 3 0.59 -0.35 3.69 3.74 T r i a l 4 0.49 -0.49 3.58 3.66 T r i a l 5 0.50 -0.68 3.61 3.70 R e l i a b i l i t y Coeff ic ient X I x 2 Y Intraclass r A l l Trials 0.90 0.93 0.90 Single Trial 0.62 0.71 0.64 CO I-^ co • I /* I—I—I—h ID in t n 01 CM 10 T- in o • • • W 4 co co • • • • • • l - t - ) - l -< > co CM ••- o a> CO • H — I — h (O U) f CO CM H h • 4 i I I I I c\jT-oo>oor--io H — I — t -+ to < (O Ul ^ CO CM CO CM •>- O H—I—t-< > to CO CO CM CO CM ••- o O) GO I CO CO O) CO N (O U) CM CO • • •4. +4 _ l — l _ CO LO ^ CO CM CO CM i - O O CO CM CO CM ••- i -< • • • • • < > L__ H < > O CO CM 00 CM t -74 Alternative Ratio Models Three alternative deflation models were evaluated us ing the same procedures used wi th Da ta Set 1. A s shown in Tab le 4-6, when the alternative deflation models ( 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 correlat ion wi th V A . The scatterplots in F igure 4-4 do not show a curvi l inear relat ionship between the adjusted ratio variable and its denominator variable, and the estimated expected value and the empi r ica l mean o f the data are v i r tua l ly the same. W h e n the N L R M 2 mode l was fitted to the data for the second, third, and fifth trials, the estimate o f the k parameters were s imi la r to those for the N L R M 1 mode l i n these three trials. H o w e v e r , when the R was used to evaluate goodness o f fit for the models w i th these data, as shown i n Table 4-6, the N L R M 2 does not have a better fit than the N L R M 1 mode l in the three trials. W h e n the N L R M 2 model was fitted to the data for the first trial and fourth t r ia l , the estimate o f the k parameter was unity, and the N L R M 2 mode l is the same as the L R M model in the two trials. It suggests that the N L R M 2 model does not have a better fit than the L R M model in the two trials. T h e R 2 for the N L R M 1 and N L R M 2 models in the other three trials d i d not substantially differ. T h e L R M and N L R M 1 models cou ld be suggested for these data because they are more parsimonious than the N L R M 2 mode l . It is noted that the sample size is smal l in Data Set 2 (n=13). If the sample size is smal l , it may result i n very different magnitudes of the parameters in the fi t t ing model (especially for the nonl inear regression models) for s imi la r data in different trials. F o r example , the magnitude o f the estimated parameter k in the N L R M 1 model ranged f rom 0.30 to 0.93 over the f ive trials in D a t a Set 2. The serious inconsis tency o f the estimated 75 Table 4-6 Evaluation of the Alternate Models: Data Set 2 r Reliability R Estimated Empirical Y vs X 2 Intraclass r Expected Mean Y a k Value A(3 Y=(X,-a)/X2 All Single (LRM) Trials Trial Trial 1 -0.010 0.34 3.51 3.51 1.76 Trial 2 0.003 0.10 1.16 1.16 15.52 Trial 3 0.040 0.90 0.62 0.35 2.37 2.37 8.00 Trial 4 -0.010 0.24 1.88 1.89 10.22 Trial 5 0.005 0.25 1.30 1.30 13.97 Y=X,/X:k (NLRM1) Trial 1 -0.0200 0.34 4.34 4.34 0.93 Trial 2 -0.0003 0.10 13.23 13.23 0.30 Trial 3 0.0300 0.89 0.62 0.35 7.13 7.12 0.64 Trial 4 -0.0100 0.24 8.46 8.46 0.52 Trial 5 0.0003 0.25 11.39 11.39 0.36 Y=(X1-a)/X2k (NLRM2) Trial 1 -0.010 0.35 3.51 3.51 1.76 1.00 Trial 2 -0.001 0.09 13.03 13.02 0.22 0.30 Trial 3 0.050 0.88 0.60 0.36 7.81 8.01 -1.00 0.61 Trial 4 -0.010 0.24 1.88 1.88 10.22 1.00 Trial 5 0.040 0.25 12.60 13.02 -1.00 0.35 76 parameters c o u l d reduce the re l iabi l i ty 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 in different trials, and the l o w re l iabi l i ty w i l l affect the va l id i ty o f us ing the deflation mode l . A larger sample size may be needed to reduce the inconsistency o f the parameters in these data. A Summary of the Evaluation of the Deflation Models Firs t , one should not assume that a s imple ratio mode l is va l id for deflation purposes, but rather that the procedures developed in this study c o u l d be used to determine a best deflation model . Second, the va l id i ty o f a deflation mode l depends on the sample size and the statistical characteristics o f the particular component variables used, and an opt imal deflation model for a l l variables may not exist. Different deflation models should be used to fit each empir ica l data set for determining the best deflation model . T h i r d , to derive a best deflation mode l , the f o l l o w i n g cri teria cou ld be considered: (a) zero correlat ion between the ratio variable and the denominator variable, (b) no curv i l inear relat ionship between the ratio variable and the denominator variable, (c) equali ty o f the estimated expected value o f the mode l and the empi r ica l mean of the ratio variable data, (d) h igh R 2 i n the deflation mode l , and (e) high re l iab i l i ty o f the ratio variable data. Ratio Variables Used Not for Deflation Purposes A s previous ly mentioned, i n human kinet ics research, ratio variables are also used as a measurement variable o f direct interest, without any intent to derive a denominator-free variable. In this case, a ratio variable has some special aspect not accounted for by the numerator or denominator variables ind iv idua l ly . The m a i n concern 77 when using this k i n d o f ratio variables as a measurement variable is the re l iab i l i ty o f the ratio variable and the effect o f covariat ion o f the component variables in a statistical analysis. In this section, two empir ica l studies were examined and the re l iab i l i ty issue o f us ing this type o f ratio variable was addressed. The effect o f covariat ion o f the component variables in a statistical analysis was investigated, using s imula t ion procedures, i n Chapter V . Empirical Data Set 3: Nondominant Torque/Dominant Torque These data are f rom a research project conducted by Kramer , N u s c a , B i sbee , M a c D e r m i d , K e m p , and B o l e y (1994). The study examined the re l iabi l i ty o f forearm pronation and supination isometric strength (torques) i n absolute units and nondominant /dominant ratios. Twenty-one men and 22 women were tested wi th two repetitions i n two occasions on each of two machines, a B T E w o r k s imulator and a C y b e x dynamometer. T h e data we received f rom the authors were col lected f rom the B T E w o r k simulator. T o reduce the var iabi l i ty o f the measurement, the measured strengths o f the two repetitions in each occasion were averaged. T h e n , the absolute torques o f the dominant arm and nondominant arm, and the nondominant /dominant ratio on the two occasions were calculated. The re l iab i l i ty o f the component variables and the nondominant /dominant ratio was evaluated. Tables 4-7 and 4-8 show the characteristics o f the ratio variable and the component variables for the forearm pronation and supination, respectively. T h e results show that the absolute torques exhibi ted large between subject var iabi l i ty (V>0.40) , and high rel iabi l i t ies (0.94-0.98). Howeve r , w i th the nondominant /dominant ratio variable 78 Table 4-7 Means , Standard Devia t ions , and R e l i a b i l i t y Coeff icients of the Torques and Nondominan t /Dominan t Ra t io (Pronation) Pronat ion Torques N D 1 a N D 2 D I D 2 R A T I O - 1 R A T I O - 2 M e a n 8.01 8.24 8.95 8.98 0.91 0.94 S d 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 R e l i a b i l i t y A l l Tr ia l s 0.97 0.94 0.68 Single T r i a l 0.94 0.89 0.51 a 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. Table 4-8 Means , Standard Devia t ions , and R e l i a b i l i t y Coeff icients o f the Torques and Nondominan t /Dominan t Ra t io (Supination) Supinat ion Torques N D 1 N D 2 D I D 2 R A T I O - 1 R A T I O - 2 M e a n 7.49 7.67 8.17 8.19 0.93 0.97 S d 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 R e l i a b i l i t y A l l Tr ia ls 0.95 0.98 0.75 Single T r i a l 0.90 0.96 0.61 79 the coefficients of variat ion were reduced considerably (0.26 > V > 0.17), as were the rel iabi l i t ies (0.68 and 0.75 for the average o f the two trials, and 0.51 and 0.61 for a single trial) . The results show that the ratio transformation substantially reduces the re l iabi l i ty i n these data even though the re l iab i l i ty o f the component variables is h igh . A s shown in F o r m u l a (7) i n the literature review chapter, the re l iab i l i ty o f ratio variables i n a single t r ia l is affected by not on ly the coefficient o f variat ion but also by the w i t h i n and between tr ial correlations of the component variables. Therefore, although the component variables are very rel iable i n these data, the re l iab i l i ty o f the ratio variable reduces substantially because o f the effect o f the w i t h i n and between trial correlations of the component variables. The details o f the discuss ion on how these factors affect the re l iab i l i ty o f the ratio variable are presented i n a f o l l o w i n g section. T h e results in the Da ta Set 3 suggest that one should not assume that ratio data are rel iable merely because the component variables are rel iable. The re l iab i l i ty o f the ratio data should be computed whenever the ratio transformation is conducted. Empirical Data Set 4; Waist Girth/Hip Girth These data were obtained f rom a study conducted i n the N e w Y o r k Un ive r s i t y M e d i c a l Center (Sonnenschein, K i m , Pasternack, & T o n i o l o , 1993). The subjects were 35-65 year o l d w o m e n who vis i ted a breast screening c l i n i c and enrol led i n the N e w Y o r k Un ive r s i t y W o m e n ' s Hea l th Study between 1985 and 1991 (n=14,290). The purpose o f the or ig inal study was to investigate the relationship between endogenous hormones and cancer. The longi tudinal data inc luded the f o l l o w i n g information: A self-administered questionnaire w h i c h gathered demographic , medica l , l ifestyle, and reproductive informat ion, inc lud ing current menopausal status and self-reported height and weight; a self-administered semi-quantitative food frequency questionnaire; and waist and hip girths measured for 1,851 cohort members at each c l i n i c visi t . The waist /hip ratio has recently become a c o m m o n measure in health science research, as it purportedly reflects fat distr ibution i n the body. H i g h ratio scores are indicat ive of health r isks associated wi th obesity. Therefore, the data used in our study were the waist and hip girth measures. A l l 1,851 subjects had the first three longi tudinal measurements of waist and hip girths, 693 subjects had the first four measurements, and on ly 109 subjects had a l l f ive measurements. T o m a x i m i z e the number o f subjects i n the examinat ion o f re l iabi l i ty , on ly the measurement data i n the first three trials were inc luded i n this invest igat ion. Table 4-9 shows the descriptive statistics for the h ip , waist and ratio, for a l l three trials. The coefficients of variat ion for the waist and hip girths exhibi ted very little change over the three trials, ranging from 0.15-0.16 for the waist girth and 0.10-0.11 for the h ip girth. It indicates that waist girth is more variable than hip girth in these data ( V w a i s t > V h i P ) . The coefficients o f variat ion o f the ratio variable (waist/hip) were somewhat lower , ranging f rom 0.09-0.10 over the three trials. T h e re l iabi l i ty o f the ratio variable (0.89 for a l l trials and 0.73 for a single trial) was s t i l l quite high although it was lower than that o f the component variables. A Discussion on the Reliability of the Ratio Variable: Data Sets 3 and 4 The results o f this study show that the waist /hip ratio variable in Da ta Set 4 has much higher re l iab i l i ty than the nondominant/dominant ratio i n Da ta Set 3, al though the re l iab i l i ty o f the component variables are h igh in both data sets. A s shown in Tables 4-7, 81 Table 4-9 Means , Standard Devia t ions , and R e l i a b i l i t y Coeff icients o f the W a i s t and H i p G i r t h , and Wai s t / H i p Ra t io W a i s t H i p W / H Ra t io T l T 2 T 3 • T l T 2 T 3 T l T 2 T 3 M e a n 74.93 76.27 76.74 97.71 100.83 100.96 0.77 0.76 0.76 S d 11.64 11.97 11.80 10.77 10.00 9.86 0.07 0.07 0.07 V 0.16 0.16 0.15 0.11 0.10 0.10 0.09 0.10 0.10 R e l i a b i l i t y A l l Tr ia l s 0.96 0.94 0.89 Single T r i a l 0.89 0.84 0.73 82 4-8, and 4-9, the coefficients of variat ion o f the component variables i n Da ta Set 4 were much smaller than i n Da ta Set 3. H o w e v e r , as shown in F o r m u l a (7), the re l iab i l i ty o f a ratio variable has a complex relationship wi th the coefficients o f variat ion and the w i th in and between trial correlat ion o f the component variables. It is not immedia te ly obvious w h y the re l iab i l i ty o f the ratio in Da ta Set 4 is so much higher than that o f the ratio (pronation) in Da ta Set 3. Tables 4-10 and 4-11 presents the comparat ive 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 Da ta Set 3, and yet the re l iabi l i ty o f the ratio is lower . It was speculated that the higher re l iab i l i ty i n D a t a Set 4 cou ld be due to either the lower coefficients o f var ia t ion, or the much larger difference o f the coefficients o f variat ion between the numerator and denominator variables. T o explore these possibi l i t ies , Da ta Set 3 (the pronation variable on ly) was used to create three addit ional data condit ions by vary ing the magnitudes of the coefficients o f variat ion. T h e re l iab i l i ty o f the ratio variable was then approximated for each condi t ion by apply ing F o r m u l a (7) (a val idat ion o f the accuracy o f Formulas (4), (5), and (7) is given in A p p e n d i x C ) . The wi th in and between tr ial correlations were set to the o r ig ina l values in Da ta Set 3 (as in Table 4-10) for a l l the three addit ional condi t ions. T h e three sets of the coefficients o f variat ion were set based on the values in Da ta 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 denominator value, and (c) numerator value smaller than denominator value. Tab le 4-12 shows these values. In C o n d i t i o n 1, although the coefficient of variat ion for both the numerator and the denominator variables was reduced f rom approximately 0.45 to 0.16, the results show Table 4-10 Correlat ions i n Da ta Set 3 and D a t a Set 4 Corre la t ion Components Corre la t ion D a t a 3 Da ta 4 Difference X i a n d X 2 t r ial 1 rxi1x21 0.85 0.83 0.02 X i a n d X 2 t r ial 2 Tx12x22 0.92 0.80 0.12 X i t r ial 1 and trial 2 1x11x12 0.94 0.89 0.05 x 2 t r ial 1 and tr ial 2 Tx21x22 0.89 0.84 0.05 X i trial 1 and X 2 t r ial 2 Tx11x22 0.88 0.71 0.17 X i t r ial 2 and X 2 tr ial 1 Tx12x21 0.84 0.79 0.05 Table 4-11 Coefficients o f V a r i a t i o n i n Da ta Set 3 and Da ta Set 4 Coeff ic ient o f Var i a t i on D a t a 3 Data 4 Difference Numerator tr ial 1 v x n 0.46 0.16 0.30 tr ial 2 v x l 2 0.43 0.16 0.27 Denominator trial 1 v x 2 , 0.46 0.11 0.35 trial 2 v x 2 2 0.44 0.10 0.34 Ra t io tr ial 1 V y l 0.24 0.09 0.15 trial 2 v y ? 0.17 0.10 0.07 Ra t io Reliability ^single 0.51 0.73 -0.22 84 Table 4-12 R e l i a b i l i t y o f the Ra t io Var i ab le i n Da ta Set 3 under Different Coeff ic ient o f Var i a t ion Condi t ions Data Set 3 C o n d i t i o n 1 v x l =v x 2 Cond i t i on 2 V x i > V x 2 C o n d i t i o n 3 v x,<v x 2 Numera tor trial 1 v x „ 0.46 0.16 0.16 0.11 tr ial 2 v x l 2 0.43 0.16 0.16 0.11 Denomina tor trial 1 v x 2 , 0.46 0.16 0.11 0.16 tr ial 2 v x 2 2 0.44 0.16 0.10 0.16 Rat io R e l i a b i l i t y Tsingle 0.51 0.50 0.72 0.60 Note . A l l correlat ion values are identical to those l is ted under Da ta Set 3 in Tab le 4-10. 85 that the re l iabi l i ty o f the der ived ratio variable d id not increase (r s j n g i e =0.50) . T h i s result is expected i f we set V x n = V X2i= V xi2= V X22, F o r m u l a (7) is s impl i f i ed to: (PJC x +Px x ) " ( P , , +PX x ) V F -M1-M2 x2\x22 ' x\\x22 x2\x\2 ' 2P~P^-P,^2) (24) In F o r m u l a (24), the coefficients of variat ion are cancel led out, and the re l iabi l i ty o f the ratio variable becomes on ly a function o f the wi th in and between trial correlations o f the numerator and denominator variables. The four coefficients o f variat ion in Da ta Set 3 were approximately equal (0.43<V<0.46), and thus the s imi la r value (r S i n g i e =0.51) for the re l iab i l i ty o f the ratio variable w i th those data. Th i s indicates that the re l iabi l i ty o f the ratio variable is not affected by the coefficient of variat ion i f the coefficients o f variat ion of the component variables are the same. In Cond i t i on 2, the coefficient o f variat ion of the numerator variable was higher than that o f the denominator variable, the values being set equal to those o f trials 1 and 2 o f Da ta Set 4. A s shown i n Tab le 4-12, the re l iabi l i ty o f the ratio variable increased f rom 0.51 to 0.72, a value very close to the observed ratio re l iab i l i ty o f Da ta Set 4 (0.73). In C o n d i t i o n 3, the coefficient o f variat ion o f the numerator variable was lower than that o f the denominator variable, and the re l iab i l i ty of the ratio variable s t i l l increased beyond the 0.51 value of Data Set 3, but on ly to 0.60. G i v e n that the wi th in and between trial correlations o f the component variables d i d not change, the results o f these two condit ions suggest that the re l iab i l i ty o f a ratio variable is h igh i f the coefficients o f variat ion o f the component variables are different, and the effect is most pronounced when the numerator variable has the higher coefficient o f variat ion. A comparison of D a t a Set 4 and C2 is o f interest. C2 and Data Set 4 have identical 86 coefficients o f variat ion, but differ in the values o f the correlations ( C 2 has correlations equal to those o f Da ta Set 3). A s can be seen in Table 4-10, a l l of the correlations o f D a t a Set 3 are greater than those o f Da ta Set 4, some by as much as 0.17. H o w e v e r , this has very little effect on the re l iab i l i ty o f the ratios, as they are almost identical (0.72 i n Cond i t i on 2 and 0.73 in Da ta Set 4). The results indicate that the unequal coefficients o f variat ion o f the numerator and denominator variables result in the higher re l iabi l i ty o f the ratio variable i n Da ta Set 4 than in Da ta Set 3. A Summary of the Evaluation of the Ratio Variables not for Deflation In human kinet ics research, ratio variables s imi la r to those in Data Set 3 and Da ta Set 4 are c o m m o n l y used, however little or no considerat ion is g iven to the re l iab i l i ty o f these der ived measures. The results o f this study have c lear ly shown that even i f the component variables are h igh ly rel iable the der ived ratio variable data may have l o w re l iabi l i ty . The consequences o f such reduction i n re l iab i l i ty c o u l d seriously affect the va l id i ty o f empi r i ca l research, because unless the repeated measurements are rel iable research decisions based on these data may be questionable. F o r example, l o w re l iab i l i ty results in an increased error term (for the R M effect) in a R M A N O V A , thus reducing statistical power. A s was shown wi th Da ta Sets 3 and 4, the re l iab i l i ty o f the ratio variables der ived from component measures are affected by the coefficient o f var ia t ion, wi th in and between trial correlations between the numerator and denominator variables (see Tables 4-10 and 4-11). Therefore, whenever a ratio variable is used, researchers should compute the re l iabi l i ty of the derived ratio scores, and not assume that strong rel iabil i t ies in the component measures automatical ly lead to h igh re l iab i l i ty in the ratio 87 measures.. I f the re l iab i l i ty o f the ratio variable is too l o w , the va l id i ty of the research may be compromised and an alternative approach should be considered. F o r example , i n the case o f nondominant /dominant ratio, the strength o f the nondominant and dominant arms may have to be analyzed separately. 88 CHAPTER V. CHARACTERISTICS OF RATIO VARIABLES AS A FUNCTION OF THE COMPONENT VARIABLES: A SIMULATION STUDY T h i s chapter reports on how the characteristics o f the component variables affect the ci rcular i ty condi t ion o f the covariance matrix and type I error rates when a ratio variable is used in one-way R M A N O V A . A s mentioned in the methods and procedures chapter, I first generated the populations o f the numerator and denominator variables based on the control led characteristics o f the two component variables (see Table 3-3). T h e data for each o f the 90,000 cases w i th in a populat ion consisted o f a numerator and denominator variable for each o f the f ive repeated observations (hereafter referred to as trials). The ratio value, X i / X 2 i ( i = l , 90000, i = l , . . . , 5), was then computed for each o f the f ive paired X n and X 2 i values for each case. The effects o f the characteristics o f the component variables on the magnitude o f eps i lon (e), the measure o f the extent to w h i c h the covariance matrix o f the ratio variable departs from circular i ty , were examined for each o f the transformed ratio score populations. Then , a M o n t e C a r l o s imula t ion procedure was used to investigate the sampl ing characteristics o f A e x i / X 2 under different sample sizes. F i n a l l y , the effect o f using ratio variables on the type I error rate was investigated. Population Characteristics of £ x i / X 2 The Effect of E y j , V*i /V V 2 and p Y i V 2 on the Magnitude of e^^i Table 5-1 presents the values o f £ xi/X2 for each of the 33 s imulated ratio variable populations. The values o f £ xi/X2 for the covariance matrix o f the ratio populat ion data are 89 Tab le 5-1 Popula t ion Va lues o f e^mj for the Ra t io Var i ab le Xj /X? under Different Componen t Var i ab l e Condi t ions V x l / V x 2 = 2 . 0 V x l / V x 2 = 1 . 0 V x l / V x 2 = 0 . 5 Exi=Ex2=l-0 P x i x 2 = 0 . 9 1.00 1.00 1.00 P x i x 2 = 0 . 7 1.00 1.00 1.00 px 1x2=0.5 1.00 1.00 1.00 £xi=£ X 2=0.7 P x i x 2 = 0 . 9 0.70 0.73 0.72 P x i x 2 = 0 . 7 0.70 0.71 0.72 Pxix2=0.5 0.70 0.71 0.72 e X i=0.7 Ex2=1.0 P x i x 2 = 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 £ x i = 1 . 0 £ X 2 = 0 . 7 Px 1X2=0.7 0.78 0.81 0.71 P x i x 2 = 0 . 5 0.99 0.84 0.72 90 presented for each o f the three condit ions o f equali ty/ inequali ty o f the coefficients o f variat ion o f the component variables, four epsi lon ( e x i , e X2) condi t ions , and three levels o f correlat ion between X i and X 2 ( p x i X2). Magnitude of eYi/^  when the Ratio Components Meet the Assumption of Circularity The results in Tab le 5-1 show that the data in the transformed ratio score populat ion meet the assumption o f c i rcular i ty when the populations o f the numerator and denominator variables meet the assumption (i.e., the populat ion e o f the ratio variable equals unity e x i / x 2 = 1.0, when e x i = e x 2 = 1.0). T h i s indicates that when both the populat ion e values o f the component variables equal unity (i.e., perfect c i rcular i ty) , the ratio transformation does not affect the c i rcular i ty o f the covariance matr ix , and the homogeneity of covariance condi t ion s t i l l holds i n the populat ion o f a ratio variable. T h i s homogenei ty holds regardless o f the levels o f correlation and the relative variat ion between the numerator and denominator variables (i.e., p x i X2 and V x i / V X 2 ) . Magnitude of e v i / Y 7. when the Ratio Components Violate the Assumption of Circularity A s shown in Table 5-1, when the populat ion covariance matrices o f the numerator and denominator variables have the same degree o f v io la t ion o f c i rcular i ty (e x i = £ x 2 = 0.7), the e x i / x 2 o f the ratio populat ion covariance matrix does not deviate substantially f rom the 8 o f the component variable populat ion covariance matrices. The value o f £ x i / x 2 increased s l ight ly (from 0.70 to 0.72) when the coefficient o f variat ion of the numerator variable decreased relative to the denominator variable (i.e., V x i / V x 2 changed from 2.0 to 91 0.5), regardless o f the levels o f the correlation (p xi X 2)- The correlat ion between the numerator and denominator variables, p x i X 2, has no substantial effect on e x i / x 2 (except £xi/x2 was s l ight ly higher when V x i / V x 2 = 1 . 0 , p xi X2=0.9) i n the condi t ion £xi = £ X 2 = 0.7. In the £ x i=0.7 and £ X 2=1.0 condi t ion , the value o f £xi/X2 increased considerably as the coefficient o f variat ion o f the numerator variable decreased relative to the denominator variable. F o r example, when p xi X2=0.9 and V x i / V X 2 decreased f rom 2.0 to 0.5, the value o f £ xi / x 2 increased f rom 0.59 to 0.86. In the other two p x i x 2 condi t ions, £ X ] / X 2 exhibi ted a s imi la r pattern o f change. Add i t i ona l l y , the populat ion £ x , / x 2 is affected by the correlation between the numerator and denominator variables. A s shown i n Tab le 5-1, when the p x i x 2 decreased f rom 0.9 to 0.5, the value o f £ x i / X 2 increased, especial ly when the coefficients o f variat ion o f the component variables were not equal . T o systematically examine the characteristics o f e x i / x 2 , and to understand how the two factors, p x i x 2 and V x i / V x 2 , affect £xi/X2, the condi t ion £xi=1.0 and £ X 2=0.7 was also inc luded i n the £xi/X2 invest igation. W h e n this condi t ion was inc luded in this study, it was acknowledged that the denominator variable usual ly does not show a greater v io la t ion o f the assumption o f c i rcular i ty than the numerator variable wi th human kinetics research data. F o r example, when a ratio variable is used for deflation i n our f ie ld , on ly the numerator variable is expected to be affected by the treatment, and the variat ion o f the denominator is usual ly 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 denominator variable usual ly have a s imi la r variance pattern (e.g., waist /hip girth, nondominant /dominant strength). In addi t ion, the s imulat ion program was unable to generate the populat ion data when p xi X2=0.9, because 92 the der ived covariance matr ix for the ratio variable was not posi t ive definite. Therefore, the subsequent invest igat ion on the sampl ing characteristics of e xi/X2 w i l l not inc lude this condi t ion , and the results reported for the condi t ion e x i=1.0 and eX2=0.7 are based on on ly two correlation condi t ions (p x i X 2=0.7 and p x i X 2=0.5) . In the condi t ion e x i=1.0 and e x 2 =0.7, the results show that £ x i / x 2 decreased when the ratio o f the coefficients o f variation of the component variables decreased f rom 2.0 and 1.0 to 0.5 when p x i X 2=0.7, and decreased constantly f rom V x i / V x 2 = 2 . 0 to 0.5 for Pxix2=0.5. T h e pattern o f e xi/X2 over the three variat ion condit ions is the opposite f rom the pattern in the £ x i = 0 . 7 and £X2=1.0 condi t ion . The results also show that e xi/X2 increased when p x i X 2 decreased in a l l three variat ion condi t ions. T h i s effect of p x i X 2 on E x i / X 2 was the same in both e condi t ions (e x ] =0.7 and e x 2 =1.0, and £ x i = 1 . 0 and e x 2 =0.7). G i v e n the covariance structure and the interrelationship o f the component variables shown i n the methods and procedures chapter, the results indicate that the populat ion e o f a ratio variable (e x v X 2) is a function o f e x i and eX2, the correlat ion between the component variables (p x [ X 2) , and the relative variat ion o f the component variables ( V x i / V X 2 ) . I f the matrices of the component variables have no v io la t ion , or the same degree of v io la t ion , o f the assumption o f c i rcular i ty (e x i=e x 2 ) , the results indicate that the magnitude o f the e xi/X2 is v i r tual ly the same as that o f e x i and e x 2 , regardless the level o f p x l x 2 and V x i / V x 2 . Howeve r , when the component variables have different degrees o f v iola t ion o f the assumption of circular i ty, one can not assume that the structure o f the covariance matr ix o f the ratio variable is a s imple function of the component variables. The results indicate that E x i / X 2 has a negative relationship wi th p x i X 2 and V X ] / V X 2 , when 93 only the covariance matrix o f the numerator variable violates the assumption o f c i rcular i ty . That is , £ x i / X 2 tends to have smaller magnitudes when p x i x 2 and V x i / V x 2 are h igh , and greater magnitudes when p x i x 2 and V x ] / V x 2 are l ow. If on ly the covariance matr ix o f the denominator variable violates the assumption of c ircular i ty , £ x i / x 2 has a negative relat ionship wi th p x l x 2 but a posi t ive relationship wi th V x i / V x 2 . The Effect of exi, V x i /V X 2 and p x i x 2 on the Magnitude of P(Xi/X2)y Researchers in human kinetics usual ly measure raw score variables in repeated measurement experiments and have some knowledge o f the magnitudes o f the correlat ion between variables in repeated trials. Howeve r , the correlat ion between two ratio variables in a repeated measures design (i.e., p(Xi/x2)ij, i^ j ) may not be the same as the correlation between raw score variables (i.e., P(Xi)ij or P(x2)ij, i ^ j ) . In this section, I report how certain factors affect the correlation between ratio variables in repeated measurement situations. G i v e n the designed covariance structure and the interrelationship o f the component variables, Table 5-2 presents the inter-trial correlations o f the ratio variable as a function o f £ x i and £ x 2 , the correlation between the component variables ( p x i x 2 ) , and the relative variat ion o f the component variables ( V x i / V x 2 ) . The unshaded panels o f Table 5-2 show the assigned inter-trial correlations o f the component variables for the different £ x i and £ x 2 , and V x i / V x 2 condit ions. In the condit ions £ x l=£ x 2=0.7, £ xi=0.7 and £x2=1.0, and £ xi=1.0 and £X2=0.7, the correlations for the variable(s) for w h i c h £Xi=0.7 var ied f rom high for adjacent trials correlation to l o w for the most distant trials (i.e., p xu , xi5). The 9 4 Table 5-2 Intertrial Corre la t ions o f Ra t io Var iables for Spec i f ic Condi t ions o f the Componen t Var iab les V x i / V X 2 = 2 . 0 VxlA^x2=1.0 VxiA^x 2=0.5 Input: 8x1=8x2=1.0 Py: Numera tor 0 . 8 7 0 . 8 7 0 . 8 5 Denomina to r 0 . 8 2 0 . 8 1 0 . 8 3 Ratio: o.sy 0 . S 4 0 . 8 7 0 . S 3 0 . 8 7 0 . 8 3 Input: 6x2=8x2=0.7 Pij: Numera tor 0 . 8 7 - 0 . 6 6 0 . 8 7 - 0 . 6 0 0 . 8 5 - 0 . 6 0 Denomina tor 0 . 8 2 - 0 . 4 9 0 . 7 9 - 0 . 4 5 0 . 8 3 - 0 . 4 5 Ratio: Pxii,>_:i=0.7 0 . 8 8 - 0 . 7 0 0 . 8 2 - 0 . 5 2 0 . S 0 - 0 . 4 2 Input: 6x1=0.7, eX2=1.0 Py: Numera tor 0 . 8 7 - 0 . 6 6 0 . 8 7 - 0 . 6 0 0 . 8 5 - 0 . 6 0 Denomina to r 0 . 8 2 0 .81 0 . 8 3 Ratio: 0 . S 7 - 0 . 6 4 0 . 8 3 - 0 . 7 3 0 . S 0 - 0 . S 6 Input: 8xi=1.0, 6x2=0.7 Pij: Numera tor 0 . 8 7 0 . 8 7 0 . 8 5 Denomina to r 0 . 8 2 - 0 . 4 9 0 . 7 9 - 0 . 4 5 0 . 8 3 - 0 . 4 5 Ratio: P M I ^ I=0 .7 0 . S S - 0 . 9 4 O . S 2 - 0 . 5 6 0 . S 0 - 0 . 4 0 95 correlat ion matrix for these condit ions approximated a s implex pattern. The shaded panels are the calculated inter-trial correlations o f the der ived ratio variables for the data in a f ive trials design populat ion for vary ing condit ions o f e xj and £ x 2 , and the relative magnitudes o f the coefficients o f variat ion of the component variables. The ful l correlat ion matrices for the condi t ion £ x i = £ x 2 = 1 . 0 are presented i n Table 5-3 and the correlat ion matrices for the condi t ion p x i X 2=0.7 in the other eps i lon condit ions are presented i n Table 5-4. The other correlation matrices are in A p p e n d i x D . In general, as shown i n shaded panels o f Table 5-2, the relative variat ion o f the numerator and denominator variables affects the correlations among the ratio variables in the f ive trials (p y i y j , where y i=x i i / x 2 i and i^ j ) . That is , p y; yj decreased when V x i / V x 2 decreased in the condi t ion e x l =e x 2 =1.0 . V a r y i n g p x i x 2 f rom 0.9 to 0.5 had no substantial effect on the magnitude o f the correlations in the p y i y j matrices. T h i s was true for a l l three £ condit ions ( E x 1 = £ x 2 = 0 . 7 , £ x i= 0 . 7 and e x 2 =1.0, and £ x i = 1 . 0 and e x 2 =0.7), thus on ly the results for the p xi=0.7 condi t ion are presented i n Table 5-4 and summar ized in the shaded panels o f Table 5-2. W h e n the component variables have the same degree o f v io la t ion o f the assumption of c i rcular i ty (i.e., e x i = £ x 2 = 0 . 7 ) , and the correlations f o l l o w a s implex pattern, the correlation matrices o f the ratio variable exhibi t a decreasing pattern but do not fo l l ow the s implex pattern exact ly for al l V x j / V x 2 condit ions. F o r example , i f the correlations f o l l o w the s implex pattern (given V x i / V x 2 = 2 . 0 and p y i y 2 =0.88) , the magnitudes o f the correlations should f o l l o w the pattern: p y i y 3=p y i y 2 2 =0.77 , p y i y 4=p y i y 2 3 = 0 . 6 8 , and P y i y 5 = P y i y 2 4 = 0 . 6 0 . Howeve r , in the condi t ion £ x l = £ x 2 = 0 . 7 , the magnitude of the io c- r- t- o CO CO 00 CO o O O O O rH ro ro ro ro o CO CO CO CO o O O O O rH o o o o o CO CO CO CO o O O O O rH r- r » r- o CO CO CO o O O O rH * r- c- o >, co co o d o O H ro ro ro o CO CO CO o ^ O O O rH ro ro O >, 00 00 o '—' • t Q . 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O o o 00 o CN O CO o 00 o CO o rH O CO o o o ro o C O o o 00 1 1 o o o 1 1 < a: o o r^r^t^r~ o oi oi ai cf\ o CO CO CO CO O CO CO CO CO o O O O O r H O O O O r H [- r~ r~ c- o «d< o CO CO CO CO O CO CO CO CO o O O O O r H O O O O r H in in in in o O N ON co cn o co co co co o r> c- c- r- o O O O O r H O O O O r H o ON ON cn o o o in cn 'in o ON ON Ol o o 00 00 00 o 00 00 CO o 00 00 CO o 00 00 00 o o rH ^ o O o rH 3 ° o o rH ^ o o o H iH o o o rH o o rH 3 r- r- o It CO CO o ttOOH ON ON O >, CO 00 O Q . O O rH r~ r~ o oo oo o QL O O rH ; n ^ ) o ^ CO CO o d o O H 3 in in o ji{ co oo o Q . O O rH JJ O N cn o >, r- p- o Q . O O rH t- o 00 o O rH O ON O CO O O rH ON t- O CO o O rH o ro O CO o O rH Ol in o co o O rH i n ON O t- o O rH ON 97 Tab le 5-4 The Corre la t ion Mat r i ces o f Ra t io Var i ab le Populat ions (p*_ix2=0.7) exi=eX2=1.0 e x l=eX 2=0.7 eXL=0.7, EX2=1.0 EXI=1.0, eX2=0.7 VXL/VX2=2.0 0.88 0.87 0.87 0.87 1.00 0.88 0.77 0.74 0.70 1.00 0.87 0.74 0.69 0.64 1.00 0.88 0.91 0.94 0.94 1.00 0.87 0.87 0.87 1.00 0.88 0.78 0.74 1.00 0.88 0.74 0.69 1.00 0.88 0.92 0.92 1.00 0.87 0.87 1.00 0.88 0.77 1.00 0.87 0.74 1.00 0.88 0.89 1.00 0.87 1.00 0.87 1.00 0.86 1.00 0.89 1.00 1.00 1.00 1.00 wx2 =1.0 0.83 0.83 0.83 0.83 1.00 0.82 0.71 0.62 0.52 1.00 0.83 0.79 0.76 0.73 1.00 0.82 0.76 0.73 0.56 1.00 0.83 0.83 0.83 1.00 0.82 0.71 0.63 1.00 0.83 0.79 0.76 1.00 0.82 0.76 0.58 1.00 0.83 0.83 1.00 0.82 0.71 1.00 0.83 0.78 1.00 0.82 0.61 1.00 0.83 1.00 0.80 1.00 0.81 1.00 0.65 1.00 1.00 1.00 1.00 VXL/VX2=0.5 0.80 0.80 0.80 0.80 1.00 0.80 0.60 0.50 0.42 1.00 0.80 0.82 0.84 0.86 1.00 0.80 0.58 0.47 0.40 1.00 0.80 0.80 0.79 1.00 0.74 0.61 0.50 1.00 0.80 0.82 0.84 1.00 0.73 0.59 0.47 1.00 0.80 0.79 1.00 0.77 0.57 1.00 0.80 0.82 1.00 0.77 0.56 1.00 0.80 1.00 0.71 1.00 0.80 1.00 0.71 1.00 1.00 1.00 1.00 98 correlations (p y i y 2=0.88 , p y ] y 3=0.77 , p y i y 4 = 0 . 7 4 , and p y l y 5 = 0 . 7 0 ) decreased s lower than w o u l d be the case in a s implex pattern. The results show two different patterns i n the correlation matrices o f the ratio variables for the two condit ions £ x i=0.7 and EX2=1.0, and e x i=1.0 and £ X 2=0.7 (see the two shaded panels f rom the bottom of Table 5-2). In the condi t ion e x i=0.7 and eX2=1.0, the decreasing correlat ion pattern in the correlation matr ix disappeared when the denominator variable had the higher relative variat ion ( V x i / V x 2 = 0 . 5 ) . In this case the correlations among the ratio variables actually increased s l ight ly as the distance between trials increased: p y i y 2=0.80 , p y ] y 3=0.82 , p y i y 4 = 0 . 8 4 , and p y i y 5 = 0 . 8 6 . In the condi t ion £ x i=1.0 and 8X2=0.7 the decreasing correlation pattern also disappeared, but when the numerator variable (not the denominator variable) had the higher relative variation ( V x i / V X 2 = 2 . 0 ) . T h i s study provides some details o f the effect o f the characteristics o f components on the correlation between ratio variables. The results indicate that both the relative variat ion and the relative £ level o f the two component variables affect the correlations among ratio variables in a repeated trials design. W h e n the variat ion o f the denominator variable increases relative to the numerator variable (i.e., V x i / V x 2 decreases), the correlat ion among the ratio variables decreases i n the condit ions E x i=e X2=1.0 and £xi=£x2=0.7 . If both the numerator and denominator variables meet the assumption o f c i rcular i ty (E XI=E X2=1.0), a constant correlation pattern is shown i n the correlat ion matr ix of the ratio variables. If both the correlation matrices o f the numerator and denominator variables are designed to fo l low the s implex pattern and have the same leve l o f v io la t ion of the assumption o f c i rcular i ty (E X 1 =E x 2 =0.7) , a decreasing correlation pattern, s imi la r but 99 not identical to a s implex pattern, is shown in the correlation matr ix o f the ratio variables. In the other two e condi t ions, the pattern o f the correlation matrix o f the ratio variable is affected by the combined factor o f relative level of variat ion and e level o f the component variables. If on ly the numerator variable violates the assumption o f c i rcular i ty , a pattern o f increasing correlations is shown when V x i / V x 2 = 0 . 5 . If on ly the denominator variable violates the assumption o f c i rcular i ty , a pattern o f increasing correlations is shown when V x i / V X 2 = 2 . 0 . T o examine how the characteristics o f the component variables affect the correlat ion pattern o f the ratio variable, the comparat ive values for a l l the statistics and the correlat ion matrices o f the component variables that cou ld contribute to the correlation pattern o f the ratio variable for f ive related condit ions are presented in Table 5-5. The first two columns show the correlat ion matrices of the two component variables and the th i rd c o l u m n shows the correlat ion matr ix for the transformed ratio variable, and the e xi/ X2 value associated wi th that matr ix . In Cond i t ions 1 and 2, the two component variables have the same correlat ion pattern, and the numerator has a higher variat ion relative to the denominator ( V x i / V X 2 = 2 . 0 ) . A s shown in Table 5-5, when p x ] x 2 decreases f rom 0.9 to 0.5, the correlat ion matr ix o f the ratio variable s t i l l exhibits a decreasing pattern, but to a lesser degree. The comparison between Cond i t ions 2 and 3 shows that when the denominator has the constant correlation pattern and large variat ion ( V x l / V X 2 = 0 . 5 ) the decreasing pattern o f the correlation disappears for the ratio variable. The compar ison o f the correlat ion matrices in Condi t ions 1, 2 and 3 indicates that the correlation pattern o f the ratio variable is close to that of the component variable wh ich has larger relative 100 Tab le 5-5 C o m p a r i s o n of the Corre la t ion Mat r i ces o f the Ra t io Var i ab le under Selected Condi t ions Corre la t ion X n j Corre la t ion X 2 y Corre la t ion Y y Conditionl: exi=0.7, e x ) =1.0, p x i x 2 =0.9, V x l / V x 2 = 2 . 0 e xi/x 2=0.59 1.00 0.87 0.76 0.71 0.66 1.00 0.82 0.82 0.82 0.82 1.00 0.89 0.72 0.64 0.58 1.00 0.87 0.76 0.71 1.00 0.82 0.82 0.82 1.00 0.90 0.71 0.64 1.00 0.87 0.75 1.00 0.82 0.82 1.00 0.87 0.71 1.00 0.85 1.00 0.82 1.00 0.82 1.00. 1.00 1.00 Condition :^ ex,=0.7, e x ] =1.0, p x l x 2 =0.5, V x l / V x 2 = 2 . 0 £ xi/ x 2=0.70 1.00 0.87 0.76 0.71 0.66 1.00 0.82 0.82 0.82 0.82 1.00 0.87 0.75 0.71 0.66 1.00 0.87 0.76 0.71 1.00 0.82 0.82 0.82 1.00 0.87 0.75 0.71 1.00 0.87 0.75 1.00 0.82 0.82 1.00 0.87 0.75 1.00 0.85 1.00 0.82 1.00 0.85 1.00 1.00 1.00 Condition3: exi=0.7, e x i = 1.0, p x l x 2 =0.5, V x l / V x 2 = 0 . 5 e xi/ x 2=0.99 1.00 0.85 0.76 0.66 0.60 1.00 0.83 0.83 0.83 0.83 1.00 0.80 0.80 0.81 0.82 1.00 0.85 0.76 0.68 1.00 0.83 0.83 0.83 1.00 0.80 0.81 0.81 1.00 0.85 0.75 1.00 0.83 0.83 1.00 0.81 0.81 1.00 0.83 1.00 0.83 1.00 0.80 1.00 1.00 1.00 Condition4: e xi=1.0, e xi =0.7, p x l x 2 =0.5, V x l / V x 2 = 2 . 0 £xi/ x 2=0.99 1.00 0.87 0.87 0.87 0.87 1.00 0.82 0.65 0.57 0.49 1.00 0.87 0.87 0.88 0.88 1.00 0.87 0.87 0.87 1.00 0.80 0.63 0.57 1.00 0.87 0.87 0.88 1.00 0.87 0.87 1.00 0.80 0.66 1.00 0.87 0.87 1.00 0.87 1.00 0.76 1.00 0.87 1.00 1.00 1.00 Condition5: e x )=1.0, e x i =0.7, p x l x 2 =0.5, V x l / V x 2 = 0 . 5 E xi/ x 2=0.72 1.00 0.85 0.85 0.85 0.85 1.00 0.83 0.65 0.55 0.47 1.00 0.80 0.61 0.51 0.45 1.00 0.85 0.85 0.85 1.00 0.78 0.65 0.55 1.00 0.75 0.62 0.52 1.00 0.85 0.85 1.00 0.80 0.63 1.00 0.78 0.60 1.00 0.85 1.00 0.75 1.00 0.72 1.00 1.00 1.00 101 variat ion. That is , in Cond i t ions 1 and 2, the numerator has the larger variat ion, and the correlation pattern o f the ratio variable is closer to the decreasing pattern o f the numerator. In Cond i t i on 3 the correlation pattern o f the ratio variable is more s imi lar to the denominator due to the larger variation o f the denominator variable. These results seem to indicate that the correlation pattern o f the ratio variable is most ly dependent on the relative variat ion o f the component variables ( V x i / V X 2 ) . T o conf i rm that the correlation pattern o f the ratio variable is more s imi la r to that o f the component variable w h i c h has a large coefficient o f variat ion, the two condi t ions under the condi t ion e x i=1.0 and e x 2=0.7 were also examined (Condi t ions 4 and 5 in Table 5-5). In these two condi t ions, the correlat ion matrices o f the component variables have the opposite pattern f rom Cond i t i on 3. C o n d i t i o n 4 results show that the decreasing pattern o f the ratio variable does not appear when the numerator has a constant correlat ion pattern wi th large variat ion. In C o n d i t i o n 5, when the denominator has larger variat ion and a decreasing correlation pattern, the results show that the ratio variable also exhibits a decreasing correlation pattern w h i c h is s imi la r to that of the denominator. These results conf i rm that the correlation pattern o f the ratio variable is most ly affected by the component variable w h i c h has the larger coefficient of variat ion. A s shown above, the correlation among the der ived ratio variables over t ime is more complex than the component variables. The e level and relative variat ion o f the component variables play important role in the correlation o f ratio variables in a repeated measurement design. Therefore, one should not automatical ly assume that the ratio variable w o u l d show the same correlation pattern as the component variables. Espec ia l ly , in the situation e x i<1.0 and e x 2 =1.0 (a c o m m o n case in human kinet ics research) and the 102 denominator variable has higher var iabi l i ty , the der ived ratio variable w o u l d not exhibi t a decreasing correlation pattern over t ime. Sample Characteristics of A £ x i / X 2 Since £ was introduced by B o x (1954), it has been used as a correction factor in a R M A N O V A to control for probable inflation in type I error rates brought about by heterogeneity o f covariance. F o r any covariance matr ix , the upper and lower l imi t s o f £ are 1.0 and l / ( k - l ) , where k is the number o f trials. Because the populat ion £ is unknown i n practice, it is usual ly estimated f rom a sample covariance matr ix (using the same equation used to compute E, but applying it to a sample covariance matr ix) . It can be shown that given £ = 1 . 0 this estimate of £ ( a E ) always exhibi ts some degree o f downward bias, and this bias becomes larger wi th increasing k. E o m (1993) examined the distr ibution characteristics o f A E . G i v e n normali ty , £ = 1 . 0 , and k=5 in the populat ion, a M o n t e C a r l o s imulat ion showed that the means o f A E (2,000 samples) wi th n=15, 30, 45 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 A £ wi th n=15, 30, 45 were 0.61(Sd=0.09), 0.66(0.08), and 0.67(0.06), respectively ( E o m , 1993). Ra t io variables are c o m m o n l y used in R M A N O V A i n our f ie ld , however, the sampl ing characteristics o f A £ for a covariance matr ix der ived f rom ratio scores, A £ x i / X 2 , are not w e l l understood. H o w the relative variat ion and correlation of the numerator and denominator variables affect A £ x i / X 2 is unknown. Therefore, one of the purposes i n this study was to systematically examine the sampl ing characteristics of A £ x i / x 2 . W i t h the 103 aforementioned characteristics of 8 of the component variables and the characteristics o f the ratio score variable populations in m i n d , using M o n t e C a r l o s imulat ion procedures, we now investigate the sampl ing characteristics o f A E x i / X 2 for different condi t ions. The M o n t e C a r l o s imulat ion was conducted us ing a f ive-tr ial (k=5) repeated measures design wi th three sample sizes (n=15, 30, 45) and 2,000 repetitions f rom each of the s imulated populat ion data sets (N=90,000). The s imula t ion results are summar ized in Tables 5-6, 5-7, and 5-8. These three tables show the distr ibution characteristics o f A E x i / X 2 for each o f the populations. Condition 1 (8x1=8x2=1.0) Table 5-6 clear ly shows that A e x i / X 2 is a downward biased estimator and the degree of bias is v i r tua l ly unaffected by the correlation between the numerator and denominator variables. F o r example , for the condi t ion n=15 and V x i / V x 2 = 2 . 0 , the mean o f A E x i / x 2 was 0.76-0.77 under a l l three correlation condi t ions. H o w e v e r , A e x i / X 2 is affected by the coefficients o f variat ion of the numerator and denominator variables. T h e mean of A £ x i / x 2 decreases when the coefficient o f variat ion o f the denominator variable increases relative to the numerator variable. F o r example, when the sample size was 15 and V x ) / V x 2 decreased from 2.0 to 0.5, the mean o f A e x i / X 2 decreased f rom 0.76 to 0.68. A d d i t i o n a l l y , the estimator became more variable, as evidenced by the increase i n the standard deviat ion o f A s x i / x 2 f rom 0.08 when V x , / V x 2 = 2 . 0 to 0.12 when V x i / V x 2 = 0 . 5 . T h e effect o f this, and o f the characteristics of the component variables, on type I error rates in a R M A N O V A is discussed in a later section. 104 Table 5-6 Dis t r ibu t ion Characterist ics o f A e 1w x? (Exj=e«2 =l-0< M-XJ=75. 0^2=100) Vx,/Vx2=2.0 Vxl/V x 2 = .0 Vxi/Vx2=0.5 Pxlx2 Sample Size A£xl/x2 mean Sd E xl/x2 -Ac fc xl/x2 Ap fcxl/x2 mean Sd exl/x2 " A£xl/x2 Ap mean Sd e xl/x2 " Ap fcxl/x2 0.9 15 .76 (.08) .24 -75 (.08) .25 .68 (.12) .32 30 86 (06) 14 .85 (.06) .15 .78 (.11) .22 45 i 04) ')() ( 04) .83 (.10) .17 0.7 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 0.5 15 .77 i. 08) .23 .74 (-09) .26 (.11) .31 30 .87, (.05) .13 .84 (-06) .16 .79 (.10) .21 45 .91 (.04) .(W .M . (IS, I ^ i l i i i ! .83 (.()')) .17 Note. The population value of e xi / X2 is 1.0 for all conditions. 105 The sample size effect on A £ x i / X 2 is obvious: when the sample size increases, the bias o f A e x i / X 2 decreases. Th i s reduction is approximately 0.10 when n increases from 15 to 30, and 0.04 when n increases f rom 30 to 45. In general, as expected, the sample size effect on A e x i / X 2 o f the ratio variable data is the same as for the component variables (raw score data) reported in literature. Condition 2 (£^=£^=0.7) In this condi t ion , the covariance matrices o f both the numerator and denominator variables exhib i ted moderate violat ions o f the assumption of c i rcular i ty and the intertrial correlat ion f o l l o w e d the s implex pattern for both the numerator and denominator variables. Table 5-7 summarizes the s imulat ion results for this condi t ion , and reveals the d o w n w a r d bias i n A £ x i / X 2- The A £ x i / X 2 had the greatest bias when V x i / V X 2 = 0 . 5 , but the leve l o f V x i / V X 2 had less effect here than in the E X 1 = E x 2 = 1 . 0 condi t ion (see F igure 5-1). T h e standard deviat ion of A £ x i / X 2 increased s l ight ly when V X 2 increased, but also less than the increase shown i n the £ x i = £ X 2 = 1 . 0 condi t ion . G i v e n £ x l = £ x 2 = 0 . 7 and the s implex pattern o f the intertrial correlation for the component variables, the correlation between the numerator and denominator variables (p x ix2) does not seem to have an effect on the sampl ing characteristics of A £ x i / X 2 -Condition 3 (eYi=0.7, £Y7=1.0) In this condi t ion , only the covariance matrix o f the numerator component violated 106 o o i n r -II i r-' d ll X CO II 3 <w l oo o V-» oo •c CO .c U c o r - -a m 3 •a o 3 1 0 < C O K CD SI ' "3 <2 3 IS <? s « to C O < 9 II > T3 C/5 K C O r v C J a bo ON o r -o Os •: O 8 so v o -ir-, o o OS o .CO o 00 so o f -o r -o CD O SO o so sO — i o o , \ ~ 2 :g O O f I O in NO::; so O s >r, m o o , o „ O OS ,00' — i NO ! 0 CN so ,00 , so so [so Os v. c«-O O O os oo ,so , ,\ o o , o — i irs r~ so so ,sq os r~ — i o o —1 O O O r ' l i n I T ) S O S O O so — i o o O Os loo ~> o ! o i/~ r~ S O o o Os ir r~ o o o os oo O O , o r H I / " , so so o m IO O U"> <0 O U"> I-H m TJ- —i rn j^-O S o d d 108 the assumption o f c i rcular i ty . That is, the intertrial correlations of the numerator variable fo l l owed the s implex pattern, whi le those for the denominator were constant for a l l pairs of trials. In a previous section, it was shown that both p x i x 2 and V x i / V x 2 affect the populat ion 8 of the ratio variable when £ x i = 0 . 7 and e x 2 =1.0. In this condi t ion , e x i / x 2 tends to have a smaller magnitude when p x ! x 2 and V x i / V x 2 are h igh, and a greater magnitude when p x i x 2 and V x i / V x 2 are l o w . W i t h the aforementioned characteristics o f the populat ion e x i / x 2 i n m i n d , we now examine the characteristics of A £ x i / x 2 i n this condi t ion . A s can be seen i n Table 5-8, the bias o f A e x i / x 2 becomes more pronounced when the coefficient o f variat ion o f the denominator variable is h igh ( V x i / V x 2 = 0 . 5 ) . F o r example, the mean o f A e x i / x 2 was lowest (0.53-0.61) and the deviat ion e x i / x 2 - A e x i / x 2 was only 0.06-0.09 for the V x i / V x 2 = 2 . 0 condi t ion when n=15, whereas the mean o f A s x i / x 2 was sl ight ly larger (0.60-0.69) but the deviation was much larger (0.26-0.30) for the V x i / V x 2 = 0 . 5 condi t ion . F igure 5-2 illustrates this trend. In contrast to Cond i t ions 1 and 2, the magnitude o f the correlation between the numerator variable and the denominator variable affects the magnitude o f A £ x i / X 2 in Cond i t i on 3. A s shown i n Table 5-8, when the correlat ion p x i x 2 decreases, the difference between A £ x i / x 2 and £ x i / x 2 tends to increase, although by on ly a smal l amount (a difference of 0.03-0.04 between the p x i x 2 =0 .9 and p x i x 2 =0 .5 condi t ions when n=15). There appears to be no V x i / V x 2 by p x i x 2 interaction effect, as the bias o f A £ x i / x 2 increases when the V x 2 increases, regardless the level o f the correlation p x i x 2 . 109 o o II c-J| co r-' ©' 4 o •c CO i u c o 00 va « - C -Q to eg -j-i H Q o II to < tO IS II <a l> II jo a 3 S x EO t0 < 3 II I " a 3 CO 2 K Q. as so {.as -oo.: O VO O O o o oc — i r~ oo co cn O O o o v o r -•n i n o m CO T l -as o v o C N C I — t> vo ' O vo r- oo O —< t— CN — i o Os r-- \o o o o —' o •* r-- oo oo ON r - t CN o o t o o \ r - so O O ! © - ; r-» C CN CO vo VO >/-> O m -H cn o — cn C M — O o Os 30 cn vo : p - oo O CN 00» CN . I - H ! O Os oo vO O O , o <-> C\ oo r- r-- oo Ov Os m co O O l O o o o r -o o o vo vo -VO i n O m - H C O T f Fi gure 5-2. A £ x i / x 2 in the condi t ion eX|—0.7 and £ X 2—1.0. I l l Summary of Characteristics of A e x i / x 2 and e x i / x 2 M o n t e C a r l o s imulat ion procedures were used to investigate how the characteristics o f the numerator and denominator variables affect the c i rcular i ty o f the ratio score populat ion covariance matrix (the effect on the populat ion e xi/ X 2) and o f the sampl ing characteristics of A £ x i / x 2 in the different condit ions. A s indicated i n the methods and procedures chapter, the invest igation was based on a ratio variable w i th a specific covariance structure that is s imi la r to that observed in human kinet ics research. The ratio variable populat ion was constructed f rom populations o f the component variables w i th specific distr ibution characteristics. That is , the numerator and denominator variables were restricted to a bivariate normal distr ibut ion, constant wi th in- t r ia l correlat ion between the numerator and denominator variables ( p x i x 2 = 0 . 9 , 0.7, 0.5), and constant intertrial correlation (e xj=1.0, i = l , 2 ) or a s implex pattern of the intertrial correlations (e xi <1.0). F o l l o w i n g summary o f the f indings are based on the above condi t ions . 1. T h i s study indicates that i f the component variables meet the assumption o f c i rcular i ty (e x i=e x 2 =1.0), the covariance matrix o f a ratio variable also meets the assumption (e xi / x 2 =1.0) , and the downward bias o f A e x i / x 2 has a pattern s imi la r to what is k n o w n to ho ld for raw score variables. In the condi t ion e x l =e x 2 =0.7, e x i / x 2 is v i r tua l ly the same as that o f the component variables, and the downward bias of A E x ] / x 2 is less than in the condi t ion e x i=e x 2 =1.0. The populat ion e value o f a ratio variable is affected by the relative variation o f the component variables ( V X ] / V x 2 ) when on ly the numerator variable has a moderate v iola t ion o f the assumption of c i rcular i ty (e.g., e x i=0.7 and e x 2 =1.0). The covariance matrix of a ratio score populat ion tends to have a more homogeneous 112 covariance structure and higher bias level o f A e when the coefficient o f variat ion o f the numerator variable decreases. 2. The correlat ion between the numerator and denominator variables does not affect the populat ion 8 nor the sampl ing characteristics o f A e x i / x 2 i f the two component variables do not violate the assumption o f circular i ty (i.e., e x i=e X2=1.0), or i f they have the same moderate degree o f v io la t ion e x i=e X2=0.7. H o w e v e r , i f the numerator variable violates the assumption o f c i rcular i ty and the denominator variable does not (e.g., e x i=0.7 and e x 2 =1.0), E x ] / X 2 increases when the correlation between the numerator and denominator variable (p x i x 2 ) decreases, especial ly when V x ] ^ V X 2 . The bias o f A e x i / X 2 is more serious in the condi t ion e x i=0.7 and EX2=1.0 than in the condit ions e x i=e x 2 =1.0 and 8 x i = 8 X 2 = 0 . 7 , especial ly when p x ] x 2 is l o w . A s shown above, A e x i / x 2 is affected by a combinat ion o f the characteristics o f the component variables ( V x i / V X 2 , p x i x 2 , 8 x i , and e X 2) . I f a ratio variable is used, given certain mean differences, and magnitudes of e x i and eX2 o f the component variables, the results indicate that the magnitude o f A e x i / X 2 w o u l d be substantially affected by V x i / V X 2 and p x i X 2 -Thus , it w o u l d affect the type I error rates in the R M A N O V A . The details o f these effects are shown in 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 previous two sections, the effect o f using ratio variables on the magnitude of 8 x i / X 2 has been presented. Because noncircular i ty (e<1.0) affects the type I error rate in 113 a one-way R M A N O V A (Winer , 1991), the impl ica t ions o f us ing ratio variables on type I error rates can be indicated by l o o k i n g at the values o f E x i / X 2 in the different condi t ions. Based on the characteristics o f £ xi/X2, this section discusses how the correlation and relative variat ion between the numerator and denominator variables affect the type I error rate i n a one-way R M A N O V A . G i v e n that the numerator and denominator variables have no v io la t ion , or the same level o f v io la t ion , o f the assumption o f c i rcular i ty (e.g., E XI=E X2=1.0, or £ x i = £ x 2 = 0 . 7 ) , bivariate normal dis t r ibut ion, and the restricted covariance structure i n this study, it seems that us ing a ratio variable results in no change i n the populat ion value o f £ xi/X2. That is £xi/x2=l-0 when E XI=E X2=1.0, and Ex\/X2~0-7 when E XI=E X2=0.7, regardless the levels o f p x i X 2 and V x i / V x 2 (see Table 5-1). Thus , i f the covariance matrices o f both the component variables do not violate the assumption o f c ircular i ty , we may conclude that the empi r i ca l type I error rate o f a ratio variable i n a one-way R M A N O V A w o u l d be close to the nomina l l eve l . N o substantial effect on the type I error rate is expected f rom the correlation between the numerator and denominator variables ( p x i x 2 ) and the relative variat ion o f the component variables ( V x i / V X 2 ) . If the component variables have v io la ted the assumption o f c i rcular i ty at the same level (e.g., E X I=E x 2 =0.7) , the inf la t ion o f the type I error rate using a ratio variable is expected to be equal to that o f its components, but Pxix2 and V x i / V X 2 w o u l d have little addit ional effect. The results o f this study suggest that i f the numerator and denominator variables have the same level o f v io la t ion on c i rcular i ty (£xi=£x2), Pxix2 and V x i / V x 2 w o u l d have no effect on the type I error rate in R M A N O V A test w i th the ratio variable data. However , i f only one o f the component variables violates the assumption o f 114 circular i ty (e.g., e x i=0.7 and eX2=1.0, or e x ] =1.0 and eX2=0.7), p x ) x 2 and V x ) / V x 2 of the component variables affect the level o f v io la t ion of the assumption o f c i rcular i ty for the ratio variable (see Tab le 5-1). The f o l l o w i n g statements are based on the condi t ion £ x i = 0 . 7 and eX2=1.0. T h e results indicate that e x i / X2 has l o w values when p x i X 2 is h igh and V x i / V x 2 = 2 . 0 , and h igh values when pX]X2 is l o w and V x i / V X 2 = 0 . 5 . Therefore, i f on ly the numerator variable violates the assumption o f c i rcular i ty in a ratio R M A N O V A , it is expected that there is a greater empir ica l type I error rate when V x ) > V X 2 , and a smaller empi r ica l type I error rate when V x i < V X 2 . A d d i t i o n a l l y , the higher the correlat ion between the numerator and denominator variables, the greater the empi r ica l type I error rate. Monte Carlo Simulation Investigation on Type I error Rates The inf la t ion o f the type I error rate for the F test on the trials effect i n a one-way R M A N O V A is related to the level o f the v io la t ion o f c i rcular i ty in a populat ion covariance matr ix w h i c h is measured by 8. In practice, the populat ion epsi lon 8 is usual ly not k n o w n and it has to be estimated by the sample eps i lon A e . However , the eps i lon estimate A e based on a sample is a biased estimator o f populat ion e. Therefore, the characteristics o f the distr ibution o f the sample A e need to be investigated to show the impact o f A e on the type I error rate in a R M A N O V A . A s shown i n previous sections, the 8 xi/ X2 estimate A e x i / X 2 , is a biased estimator and the bias is affected by p x i X 2 and V x i / V x 2 , especial ly when only one of the component variables violates the assumption o f circular i ty . In this M o n t e Ca r lo study, I investigated 115 how the interactive characteristics o f the numerator and denominator variables affect the empir ica l type I error rates when using ratio variables i n a one-way R M A N O V A design (k=5). Table 5-9 summarizes the general pattern of type I error rates for the three £ condit ions across the levels of a (.01, .05, .10), V x i and V X 2 , and p x i X 2- T h e values in each b lock i n Table 5-9 were the observed type I error rates based on the average o f the observed type I error rates of the three sample sizes (n=15, 30, 45) and 2,000 repetitions f rom each o f the s imulated populat ion data sets (N=90,000). The general pattern o f the empi r i ca l type I error rates was repeti t ively demonstrated across the three correlation condit ions (p xix2=-9, .7, .5) and relative variat ion condi t ions when £ x i = £ X 2 = 1 . 0 (see Tab le 5-9). The correlation p x i x 2 does not have substantial effect on the type I error rate i n the condi t ion £ x i = £ X 2 = 1 . 0 , w i t h the differences between the observed type I error rates over the three p x i X 2 condi t ions never exceeding 0.003. A d d i t i o n a l l y , there is no consistent pattern o f the difference between the three correlat ion condit ions. There is a slight decrease in the type I error rate when the variat ion o f the numerator variable decreases. T h e largest discrepancy occurs in the condit ions p x i x 2 = 0 . 9 , oc=0.05 and p x i X 2=0.5 , a=0.10 , where the type I error rate under condi t ion V x i / V x 2 = 0 . 5 is 0.008 less than it is under V x i / V x 2 = 2 . 0 . T h i s difference represents a 15% and 8% difference, respectively. Howeve r , these differences are smal l and indicate that there is no substantial deviat ion o f the empir ica l type I error rate f rom the nomina l level when a ratio variable is used i f the component variables do not violate the assumption o f circular i ty . A s shown in Table 5-9, the type I error rate is inflated when there are heterogeneous covariances of the component variables (i.e., £ x i = £ x 2 = 0 . 7 , E x i=0.7 and 116 Table 5-9 E m p i r i c a l T y p e I E r r o r Rates V x l / V x 2 = 2 . 0 V x l / V x 2 = 1 . 0 V x l / V x 2 = 0 . 5 e oc= .01 .05 .10 .01 .05 .10 .01 .05 .10 e x l - e x 2 - l • 0 Pxlx2= • 9 . 010 . 053 . 097 . 012 . 048 . 099 . 008 . 045 .094 Pxlx2= • 7 . 010 . 051 . 099 . O i l . 051 . 098 . 009 . 044 . 093 Pxlx2 = • 5 . 010 . 051 .100 . 010 . 049 . 099 . 010 . 044 . 092 Ex2=0.7 Pxlx2 = • 9 . 020 .064 .116 . 021 . 067 . 114 . 019 . 060 .113 Pxlx2= • 7 . 021 . 066 . 116 .023 . 067 . 112 . 020 . 066 .115 Pxlx2= • 5 . 022 . 066 . 115 . 023 . 065 . 114 . 019 . 066 . 112 0.7 ex2=1.0 Pxlx2= - 9 . 030 . 073 .123 . 015 . 059 . 113 . 014 . 060 . I l l Pxlx2= • 7 . 024 . 072 .118 . 013 . 057 . 105 . 010 . 049 . 096 Pxlx2= • 5 . 022 . 067 . 112 . 014 . 055 . 104 . 010 . 045 . 093 117 £x2=l-0), and is affected by p x i X 2 and V x i / V x 2 in the latter condi t ion . T o clear ly show these effects and their relat ionship to e x i / x 2 and A e x i / x 2 , detailed results based on a=0.05 wi th n=45 are presented i n Table 5-10. The results in Table 5-10 indicate that the type I error rate is consistently inflated when the component variables have the same degree o f v io la t ion on the assumption o f c i rcular i ty (i.e., e x i=e x 2 =0.7) . F o r example , the mean o f A £ x i / x 2 ranged between 0.65 and 0.70 (n=45), and the actual type I error rate ranged between 0.061 and 0.068 (a=0.05). There is no clear effect pattern o f p x j x 2 and V x i / V x 2 on the type I error rate, w h i c h is to be expected based on the f indings o f the populat ion e x i / x 2 i n the previous section (it was shown that e x i / x 2 seems independent o f p x i x 2 and V x i / V x 2 ) . The lowest panel o f Table 5-10 shows the effect o f p x i x 2 and V x i / V x 2 on the empir ica l type I rate in the condi t ion e x i=0.7 and e x 2 =1.0. T h e results suggest that when only the numerator variable violates the assumption of c ircular i ty , the empi r i ca l type I error rate has a posi t ive relationship wi th p x i x 2 and V x l / V x 2 . F o r example , based on a=0.05 and n=45 in the condi t ion £ x i = 0 . 7 and 8 x 2=1.0, the greatest type I error rate in this study was 0.073 when p x i x 2 = 0 . 9 and V x i / V x 2 = 2 . 0 , and the smallest type I error rate was 0.041 when p x i x 2 = 0 . 5 and V x i / V x 2 = 0 . 5 . These results are expected, based on the differences in the populat ion value o f e x i / x 2 for the different condit ions o f V x i / V x 2 and P x i x 2 - That is, the higher the value o f V x i / V x 2 and the lower the value of p x i X 2 , the lower the magnitude of 8x)/x2 and the higher the type I error rate. 118 m u H CO CD lH o W i—i CD OJ o •c 1 W c o Q. w C H O H—» o o vn JL ^ d ll CN X > co o 8s 3 CM X i — i X > o OJ c n co <n in q ^ o CN II CN X m x > co < O CO i-H II c co CD II ro in oo ID ON rH ID ID ID in o o o o o O o o O ro ro ro in in m o o ro 00 00 00 ID ID ID co 00 oo oo ON ro ro ro ID CO l> r - r - ID ID ID ID r~ 00 ON ON 00 00 00 o ON ID ID ID in in m VO ID VD O O o CM CN] ID ID ON r~ 00 ON ON rH rH rH ON O ON oo 00 o in CN in ID ID ID VO in in O o O o o o o o o O ON ON o 00 rH ro ON 00 00 l> ID ID oo 00 00 IT) in 00 ID in 00 o ON 00 00 00 ID ID ID r - 00 in in ro CN <H o rH rH f - e-» r - ID ID ID r - t - t-~ o o o ro rH rH t~- rH rH r - 00 ON ON rH rH rH 00 ro rH ID ID CO CN CO in in m ID ID ID o ID o o o o o O o O o o o rH r~ r~ r - CN ON ON CfN ID ID ID m ID VD ID ID [> in in in VO O in OO 00 00 ID ID ID in VO VD ID ID r- o rH rH ro r- rH f- r - r - ID ID ID m in VD O o o r - r - r - ON o in io rH rH rH o o o • • r~ ON in o ON r- in ON r- in II II ll ll . c*» II II II t> o II II II C N CN CN X CN CN CN CN CN CN X X X CO X X X o rH X X X <-t rH rH H rH rH H rH X X X • II rH X X X X II II X X X Q. Q. Q. Q. Q. Q. rH X CN X Q. Q. CL CO CO CO CD ) H g CD HH CD •*-» o r^ 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 A e and Huynh-Fe ld t ~e are sometimes used to protect against a probable inf lat ion in the type I error rates in the within-subject F test whenever the A e o f a sample is less than 1.0 (the definit ions for A e and ~e are in the 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 many others have shown that, in general, v io la t ion o f the assumption o f c i rcular i ty (e<1.0) results in the art if icial inf lat ion o f the F values for the within-subject ma in effects, thus inf lat ing the type I error rate. A s indicated in the chapter on methods and procedures, i n practice, the populat ion e is unknown and the sample A 8 is used to estimate the popula t ion e, and an A e-adjusted F test may be conducted to correct the inflat ion o f the type I error rate. H u y n h and Fe ld t (1970, 1976) indicated that the A e-adjusted F test is negat ively biased and w o u l d make the F test too conservative. T h e y showed that the bias is most serious when a populat ion 8 is above 0.75, especial ly when the sample size is smal l . Thus a less biased estimate, ~e, was suggested (see F o r m u l a (19)). H u y n h and Fe ld t also indicated that the ~e-adjusted F test may result in an inflated type I error rate when ~e is less than 0.75, and thus a more conservative approach (e.g., A e) is needed i n such situations. In this study, we investigated how the bias o f A e x i / x 2 and the type I error rate are affected by p x ( x 2 , V x i / V x 2 , and e x i and e x 2 . T o investigate the sampl ing characteristics o f A e x i / x 2 , the s imulat ion results shown i n Tables 5-6, 5-7, and 5-8 are expanded and graphical ly presented in Figures 5-3, 5-4, and 5-5. Figures 5-3, 5-4, and 5-5 show the mean of the 2,000 A e x i / x 2 values wi th plus and minus two standard deviations in each o f the s imulated condit ions. 120 A s shown in F igure 5-3, even though £ x i / X 2=1.0, the values o f A £ x i / X 2 were considerably smaller than unity for a l l the levels o f p x i X 2 and V x i / V X 2 , w h i c h indicates biased estimation o f £ x i / X 2- F o r example, the lower bound o f A £ x i / X 2 was much lower than 0.75 for n=15 in a l l three V x i / V x 2 condi t ions. W h e n the sample size increased, the lower bound increased substantially, especial ly i n the condit ions V x i / V x 2 = 2 . 0 and V x l / V x 2 = 1 . 0 . G i v e n £ x i / X 2=1.0, when a within-subject F test is conducted, there should be no correction. H o w e v e r , because the populat ion epsi lon is not k n o w n i n practice, A £ x i / X 2 w i l l result i n a "false" correction o f the F test in a l l cases o f F igure 5-3, and ~£ xi/X2 i n many o f them. F o r n=30 and 45 A £ x i / X 2 is usually greater than 0.75 and thus ~£ adjustment w o u l d usual ly be applied, result ing in only a slight bias. B u t for n=15, because A £ x i / X 2 is quite often smaller than 0.75, we w o u l d use the B o x A £ adjustment in the F test and that w i l l cause a considerable decrease in type I error rate. In the condi t ion V X ] / V X 2 = 0 . 5 , because the lower bound is much lower than in the other two condi t ions, the A £ x y X 2 adjustment w o u l d be used for those low A £ x i / X 2 values ( A £ x i / X 2<0.75) , and an serious over adjustment in the F test is expected. M o r e subjects in a sample i n the condi t ion V x i / V x 2 = 0 . 5 are needed to reduce the bias o f the A £ x i / X 2 , and even wi th n=45 there is s t i l l considerable bias and large sampl ing var iabi l i ty . F igure 5-4 shows the range o f A £ x i / X 2 +/- 2sd in the condi t ion £ x l = £ x 2 = 0 . 7 . T h e results show that the var iabi l i ty of A £ x i / X 2 increases when the variation o f the denominator variable increases. The upper bound o f A £ x i / X 2 was approximately 0.8 for a l l the variat ion condi t ions, but the lower bound was lower when V X 2 increased. A s shown i n a previous section, i f £ x i = £ x 2 = 0 . 7 , the values of £ x i / x 2 ranged f rom 0.70 to 0.73 in different V x i / V x 2 1 2 1 aieiujisg uo|isd3 122 OTCOr^COWTtCOCM d d d d d d d d a ) e u i j ) S 3 u o i j s d g 123 and p x ix2 condi t ions. The results show that some values o f A£ xi/ x 2 were equal or very close to the £ x ] / x 2 and some values o f A s x i / X 2 over- or under-estimated e x i / x 2 by a considerable degree. A n adjusted F test, using a E or ~£, w i th the A£ xi/ x 2 equal or close to the £xi/x2 w o u l d appropriately estimate the populat ion £xi/x2, and appropriately adjust the degrees o f freedom in the F test to protect the type I error rate. H o w e v e r , as shown i n F igure 5-4, there are more values of A£ xi/ x 2 lower than e xi/ x 2, thus an over-correction is more l i ke ly than an under-correction in the condi t ion e x i=e x 2 =0.7, result ing in an over ly conservative F test. F igure 5-5 shows the range o f A £ x i / x 2 +/- 2sd in the condi t ion £xi=0.7 and £ x 2 = 1 . 0 . In this condi t ion , it has been shown that the magnitudes o f £ x i / x 2 are different over the different levels of p x i x 2 and V x i / V x 2 . The £ x i / x 2 had low values (i.e., l ower than 0.7) when V x i / V x 2 = 2 . 0 and high values (i.e., higher than 0.85) when V x l / V x 2 = 1 . 0 or V x i / V x 2 = 0 . 5 . In the condit ions V x ( / V x 2 = 1 . 0 and V x i / V x 2 = 0 . 5 , the magnitudes o f E X I / x 2 were high and substantial bias in A£ xi/ x 2 was shown. Therefore, the A£ xi/ x 2 adjustment conducted in most samples (especially when n=15 where A £ x i / x 2 < 0 . 5 0 sometimes) w o u l d overcorrect the degrees o f freedom associated wi th the F test, resulting in type I error rate considerably less than the nomina l leve l . In the condi t ion V x | / V x 2 = 2 . 0 , the over-correct ion is also more evident than the under-correction because more samples have lower A £ X ] / x 2 values. In general, g iven £ xi/ x 2=1.0, no adjustment should be needed, however, i n pratice the adjusted F test w o u l d be used in a l l the samples and result in F tests that were too conservative. If £ x i and/or £ x 2 are not equal to 1.0, using an adjusted F test in a R M A N O V A may result in over-correction in some condit ions and under-correction in other 124 125 condit ions. Table 5-11 shows the summary for the results o f £ x i = £ x 2 = 0 . 7 , and £ x i = 0 . 7 and £ X 2=1.0 condi t ions . Because s imi lar patterns were shown i n different p x ) X 2 condi t ions , only the results for p x i X 2=0.7 (n=15) are presented in Tab le 5-11. In Table 5-11, an "appropriate correc t ion" is defined as the adjusted F tests based on the magnitudes o f A £ x i / x 2 i n the range £ xi/ X2+/-0.05, an "under-correct ion" as an insufficient correct ion o f the F test occurr ing because A £ x i / x 2 exceeds £ x i / x 2 by more than 0.05, and an "over-correct ion" as too conservative an adjustment in the F tests result ing f rom the condit ions where A £ x i / x 2 is less than £ x i / X 2 by more than 0.05. The results o f the adjusted F tests wi th an ~ £ x i / X 2 adjustment i n the different condit ions was compared to the results wi th the A £ x i / X 2 adjustment. G i v e n that the mean of A £ x i / X 2 is smaller than £ x i / X 2 in a l l cases in Tab le 5-11, the A £ x i / x 2 adjustment is l i ke ly to show an over-correction in the F test. W h e n the A £ x i / X 2 adjustment is used, the conservative F tests w o u l d result i n type I error rate seriously less than the nomina l level in the condi t ion £ x i = 0 . 7 and £ x 2 = 1 . 0 . That is , the percent o f over-correction was 58.7%, 95 .1%, and 98 .6% for the three V x i / V x 2 condi t ions, respectively. T h e results also indicate that us ing an ~ £ x i / x 2 correction can not substantially increase the probabi l i ty o f a correct adjustment in the F test i n the two condi t ions 8 x l = E x ] = 0 . 7 and £ x i = 0 . 7 and £ X 2=1.0. A l t h o u g h it can not increase the percent o f appropriate corrections in the F test, as shown i n Table 5-11, the ~ £ x i / X 2 adjustment substantially reduces the percent o f the over-correction and increases the percent o f the under-correction. 126 Table 5-11 A Summary o f the Effect o f V x j / V x ? on the T y p e I E r r o r Rates (px ix2=0-7 , a=.Q5, n=15) %am E x i £ x 2 V x i / V x 2 e x i / x 2 A £ x i / x 2 Type I Appropriate Under- Over-Error Correction' 1 Correction Correction A e ~e A e ~e A e ~e 0 7 0 7 2 0 0 70 0 61( 09) . 061 27 0 25 6 6 0 49 6 67 0 24 8 1 0 0 71 0 62 ( 10) . 065 26 4 22 2 8 1 50 0 65 5 27 8 0 5 0 72 0 58( 11) . 067 16 4 21 9 4 2 34 8 79 4 43 3 0 7 1 0 2 0 0 64 0 57 ( 09) . 068 32 1 26 1 9 2 50 0 58 7 23 9 1 0 0 91 0 71 ( 09) . 054 4 6 4 8 0 3 47 6 95 1 47 6 0 5 0 96 0 67 ( 11) . 055 1 3 17 1 0 1 15 9 98 6 67 0 3 Appropr ia te correct ion refers to percent of times an adjustment i n the F test w o u l d be "appropriate" defined as | A£ xi/x2-£xi/x2 |<=0.05. Under-correc t ion is defined as insufficient adjustment w h i c h occurs when A £ x i /x2>£ x i /x2+0.05, and over-correct ion is defined as too conservative adjustment w h i c h occurs when A £ x i / X 2 < £ x i / x 2 - 0 . 0 5 . 127 In general, the over-correction o f the F test is more pronounced in the two condit ions e xi=e x 2 =1.0, and e x i=0.7 and e xi=1.0, than in the condi t ion e xi=e x 2 =0.7. The most severely over-corrected F test w o u l d be in the condi t ion V x ) / V x 2 = 0 . 5 and e x i=0.7 and 8 xi=1.0. The results indicate that large bias and extreme var iabi l i ty o f A£ xi/ x 2 w o u l d be expected when V x i < V x 2 , especial ly wi th smal l sample size (the sample size equal or smaller than 15 w h i c h is very c o m m o n i n some fields in human kinetics research). If V x i < V x 2 in the component variable data, it is quite possible that a researcher c o u l d get a very l o w A e x i / x 2 and the over-corrected F test w o u l d make the within-subject F test too conservative. In this case, the ~ e x i / x 2 adjustment may be an alternative choice but it may result i n an under-correction. T h e results suggest that, when a ratio variable is used, i f the denominator variable has lower variat ion than the numerator variable and sample size is large, the r i sk o f the over-correction 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 THE INVESTIGATION The purposes o f this dissertation were twofold . The first was to use a practical procedure to evaluate four deflation models for ratio scores, and to examine the va l id i ty and re l iab i l i ty o f c o m m o n l y used ratio variables i n human kinetics research. T h e second was to investigate how the statistical characteristics o f the component variables affect the assumptions and results o f a R M A N O V A test. Th i s chapter presents a summary and impl ica t ions o f the findings o f this study. Empirical Ratio Data Study In this invest igation it was shown that the s imple ratio mode l c o m m o n l y used for deflation purposes d i d not appropriately deflate the effect of the denominator for the two empi r i ca l data sets. The l inear regression mode l wi th an intercept ( L R M ) and the nonl inear regression model without an intercept ( N L R M l ) seemed equal ly preferable for deflation purposes in these data. These results i m p l y that an op t imal deflation model useful for a l l ratio variables may not exist, and different models should be appl ied to a data set to determine the appropriate deflation mode l . It is recommended that the procedures developed i n this study be used to determine the best deflation model for c o m m o n l y used ratio variables in our f ie ld . T o obtain the best ratio variable for deflation purposes, one should fit each o f the four deflation models to the data and evaluate the va l id i ty o f the mode l using five criteria; (a) zero correlat ion between a der ived ratio variable and the denominator variable, (b) no curvi l inear relat ionship between the der ived ratio and the denominator in 129 the scatterplots, (c) equali ty o f the estimated expected value o f the mode l and calculated empi r i ca l mean o f the der ived ratio data, (d) h igh R , and (e) h igh re l iab i l i ty o f the der ived ratio data. It was shown that the re l iab i l i ty o f a ratio variable is strongly affected by not on ly wi th in and between trial correlations, but also by the relative variat ion o f the component variables ( V x i / V X 2 ) . I f the coefficients o f variat ion o f the component variables i n two trials are the same, the results show that the re l iab i l i ty of the ratio variable is a function o f on ly the w i t h i n and between trial correlat ion o f the component variables, and is not affected by the coefficients o f variat ion. H o w e v e r , g iven that the w i th in and between trial correlations o f the component variables do not change, unequal coefficients o f variat ion o f the numerator and denominator variables may result i n the high re l iab i l i ty for the ratio variable, and the effect is most pronounced when the denominator variable has the smaller coefficient of variat ion ( V x i > V X 2 ) . It impl ies that researchers should compute the re l iab i l i ty o f the der ived ratio scores, and not assume that strong rel iabi l i t ies i n the component measures automatical ly lead to strong re l iabi l i ty i n the ratio measures. Simulation Investigation Characteristics of Evi/v2, pv\v^ and A e x i / X2 The magnitude o f epsi lon o f a ratio variable, e x i / X 2, and its sample estimate, A £ x i / x 2 , were shown to be affected not on ly by the magnitudes o f epsi lon o f the component variables e x i and eX2, but also by the relative variat ion V x i / V X 2 and the correlat ion (p x i X 2) between the component variables. The nature o f these relationships can be summarized as fo l lows . 130 Characteristics of e^^i. I f the component variables have the same level o f eps i lon ( E X I = £ x 2 ) , a ratio variable has v i r tua l ly the same epsi lon value as the component variables. H o w e v e r , the magnitude o f epsi lon o f a ratio variable £ x i / X 2 is strongly affected by the relative variat ion and the correlat ion o f the component variables i f the covariance matr ix o f the numerator variable or o f the denominator variable has a heterogeneous structure (£ x i or £ x 2 is smaller than unity) . If the numerator variable violates the assumption o f c i rcular i ty and the denominator does not (E X I=0.7 and £ X 2=1.0) , the results show that £ x i / X 2 has a negative relat ionship wi th V x i / V x 2 and p x i X2- W h e n V x i / V X 2 and p x i X 2 decrease the populat ion epsi lon o f the ratio variable £ x i / X 2 increases. F igure 6-1 shows these f indings. W h e n on ly the denominator variable violates the assumption o f c i rcular i ty ( £ x i = 1 . 0 and £ X 2=0.7) , the pattern o f £ x i / X 2 over the three variat ion condit ions is the opposite f rom the condi t ion £ x i= 0 . 7 and £ x 2 = 1 . 0 , but the effect o f p x i x 2 on £ x i / x 2 is the same i n the two epsi lon condi t ions . That is , i n the condi t ion £ x i = 1 . 0 and £ x 2 = 0 . 7 , when V x i / V x 2 increases and Pxix2 decreases the populat ion epsi lon o f the ratio variable £ x i / X 2 increases. Characteristics of p vi vi. T h e results show that the magnitudes i n the correlat ion matrices o f the ratio variable are ma in ly affected by the between trial correlat ion o f the numerator and o f the denominator variables, and the relative variat ion o f the two component variables ( V x ] / V x 2 ) . V a r y i n g p x l x 2 has no substantial effect on the magnitude of the correlations i n the p yi yj matrices. W h e n £ x i = £ x 2 = 1 . 0 , the correlat ion matrix o f the ratio variable has a constant pattern and the magnitudes o f p y i y j decrease when V x ) / V x 2 decreases from 2.0 to 0.5. 131 Figure 6-1. The £ x i/ X2 values and the empi r ica l type I error rates in the condi t ion 8 x i=0.7 and 6x2=1.0 (0=0.05 , n=45). Vx1 /Vx2 = 0.5 Vx1/Vx2=0.5 132 W h e n 8 xi=eX2=0.7 and the correlations of the component variables fo l l ow a s implex pattern, the correlat ion matrices of the ratio variable exhibi t a s lower decreasing pattern than the s implex pattern i n a l l V x i / V X 2 condi t ions. In the two condit ions e xi=0.7 and eX2=1.0, and 8xi=1.0 and eX2=0.7, the results indicate that the correlation pattern o f the ratio variable is more s imi la r to that o f the component variable (numerator or denominator) w h i c h has a larger coefficient of variat ion than the other component. Characteristics of A e v i / V ? , . The value o f eps i lon estimated f rom a sample, A e x i / X 2 , is a biased estimator o f the populat ion epsi lon o f the ratio variable e xi / X 2 , and the bias varies according to the condit ions o f e xi, eX2, V x i / V X 2 , and p xi X 2 -W h e n exi=8X2=1.0, the bias of Ae xi / X 2 is affected by the coefficient o f variat ion o f the component variables, but the correlation p x i x 2 does not have substantial effect on the bias i n this condi t ion . That is , i f V xi / V X 2 > 1 . 0 , the mean values of Ae xi / X 2 for n=15 are between 0.74 and 0.77 (Sd= 0.08-0.09), and the mean bias is between 0.23 to 0.26. W h e n the variat ion o f the denominator increases (V X ] /V X 2=0 .5 ) , the bias o f Ae xi / X 2 is more serious, and the mean value o f Ae xi / X 2 is 0.68 (Sd=0.12), and the mean bias is 0.32. W h e n 8x1=8x2=0.7, the bias o f Ae xi / x2 is also more serious when V x i / V X 2 has a l o w value ( V x i / V X 2 = 0 . 5 ) , but the effect o f V x i / V x 2 is less than in the condi t ion e xi=eX2=1.0. The correlation p x t x 2 does not have a substantial effect in this condi t ion . The largest mean bias o f A e x i / x 2 is 0.14 and the standard deviat ion o f A e x i / x 2 is 0.11 ( V x i / V x 2 = 0 . 5 , n=15). If the numerator variable violates the assumption o f c i rcular i ty and the denominator does not (e.g., e xi=0.7 and eX2=1.0), both V x i / V x 2 and p x i x 2 affect the bias of 133 A Exi/x2- That is , the bias o f a E X I / x 2 increases when V x i / V X 2 and p x ! x 2 decrease. The bias o f A Exi/x2 is more pronounced in the condi t ion £ x i = 0 . 7 and e x 2 =1.0 than the other two condit ions ( E x 1 = £ x 2 = 1 . 0 and £ X I=E x 2 =0.7) . A m o n g the cases considered, the largest bias o f A £ x i / x 2 was 0.3 and the standard deviat ion o f A £ x i / x 2 is 0.11 ( V x i / V x 2 = 0 . 5 , n=15). If a ratio variable is used, g iven certain mean differences, and magnitudes o f £ x i and £ X 2 o f the two component variables, the results indicate that the magnitudes o f £ x i / X 2 and A £ x i / X 2 are substantially affected by V x i / V x 2 and p x i x 2 . W h e n the magnitudes o f E X I / X 2 and A E x i / X 2 vary, the type I error rates o f the R M A N O V A tests is affected. T h e impact o f using ratio variables on the type I error rate i n a R M A N O V A test is summar ized 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. The s imulat ion invest igat ion was designed to examine how the characteristics o f the component variables affect the type I error rate o f the F test i n a one-way R M A N O V A based on the transformed ratio variable. The advantage o f this study is that we c o u l d investigate the impact o f us ing the ratio variable i n a one-way R M A N O V A , g iven that the characteristics o f the ratio variable populat ion and the related characteristics o f the component variables were k n o w n . In the M o n t e C a r l o s imula t ion invest igat ion, results show that the naive F tests i n a f ive-tr ial one-way R M A N O V A on a ratio variable result i n a type I error rate that is close to the nomina l l eve l i n the condi t ion E x i = £ x 2 = l - 0 ( £ x i / x 2 = l - 0 ) , regardless on the condi t ion o f V x l / V x 2 and p x i X 2- The type I error rate o f the F test exhibits inflat ion i n the condi t ion E X I = £ X 2 = 0 . 7 ( £ x i / X 2 = 0 . 7 ) , but there is no clear effect o f V x l / V x 2 and p x i x 2 . If o n l y the numerator variable violates the assumption o f c i rcular i ty (e xi=0.7 and £ x 2 = 1 . 0 ) , the type I error rate o f the F test increases 134 when V x l / V x 2 and p x i x 2 increase. Because the value o f e x i / x 2 indicates the level of v io la t ion o f c i rcular i ty in the populat ion covariance matrix o f the ratio variable, the high leve l o f the inflat ion o f the type I error rate i n the F test should correspond to the l o w value o f Exi/x2- A s shown i n the first graph i n F igure 6-1, E xi/X2 decreases when V x l / V x 2 and p x ix2 increase. Therefore, it is expected that the inflat ion o f the type I error rate is more severe when V x l / V x 2 and p x i x 2 increase (see F igure 6-1). Sample Estimates Aevi/v7 and Type I Error Rates. Because the value o f e x i / x 2 is not k n o w n i n practice, i n R M A N O V A designs, the B o x epsi lon A e and H - F ~e are used to protect against a probable inf la t ion i n the type I error rates i n the within-subject F test whenever A 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 sampl ing characteristics o f A e x i / x 2 was investigated and the effects of A e x i / X 2 and H - F ~£ xi/ X2 i n the R M A N O V A were investigated. The results indicate that over-adjustment i n the A e-adjusted F test is more pronounced i n the two condit ions e xi= e x 2=1.0, and e xi=0.7 and e x 2=1.0. T h e large bias and extreme var iabi l i ty o f A 8-adjusted F tests w o u l d be expected when V x l / V x 2 = 0 . 5 especia l ly w i t h smaller sample size (e.g., n=15). If the H - F ~e correct ion is used, it reduces the r i sk o f over-adjustment but increases the r isk o f under-adjustment. In summary, this dissertation investigated the va l id i ty and re l iab i l i ty o f some c o m m o n l y used ratio variables in human kinet ics research, and the effect o f us ing ratio variables on the c i rcular i ty assumption o f the covariance matr ix , type I error rates i n R M A N O V A tests. It shows that different models should be used to derive an appropriate deflation mode l i n empir ica l research. A c o m m o n l y used deflation mode l for a l l ratio 135 variables may not exist. T h e results indicate that high re l iab i l i ty o f the component variables does not necessari ly result in high re l iab i l i ty o f the transformed ratio variable. Thus , when a ratio variable is used the re l iabi l i ty should be examined based on the ratio variable data. S imula t ion results show that the characteristics o f the two component variables (exi, ex2, V xi/V x 2, and p xi x 2) s trongly affect the c i rcular i ty o f the covariance matrix o f the ratio variable, and the type I error rate i n R M ANOVA tests. In general, the mean A E x i / x 2 exhibi ted the greatest bias and the largest standard deviat ion, result ing in a serious inflat ion o f type I error rate in the condi t ion V xi/V x 2=0.5, regardless the condit ions ex), ex2, and p x i x 2 . I f homogenei ty o f the denominator variable (small V x 2) and large sample size are present, it may reduce l i k e l i h o o d o f bias i n A £ x i / x 2 and protect the type I error rate. 136 References Albrech t , G . H . (1978). S o m e comments on the use o f ratios. Systematic Z o o l o g y , 27(1), 67-71. Albrech t , G . H . , G e l v i n , B . , & Har tman, S. E . (1993). Rat ios as a size adjustment in morphometries. 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Statist ical Pr inc ip les in Exper imenta l Des ign . (3rd ed.). N e w Y o r k : M c G r a w - H i l l . Y u l e , G . (1910). O n the interpretation o f correlat ion between indices or ratios. Journal of the R o y a l Statistical Society , 73, 644-647. 142 APPENDICES Appendix A Approximating p xii.x2j Using the Attenuation Correlation Function In the classic measurement mode l , two variables w o u l d be expected to have uncor rec ted errors, and errors on either variable w o u l d be uncorrelated w i t h true scores i n either variable. The observed scores o f two variables can be expressed as X i = T ] + e i , X 2 = T 2 + e 2 , where T; is a true score and ej is the error ( i= l ,2 ) . The correlation between the two observed variables can be shown as fo l lows : n _ <?„,2 _ l / J V X ( r i + e 1 ) ( T 2 + e 2 ) Pxlx2 ~ axl°x2 GxPx2 °xl°x2 Under the postulated mode l , a l l terms but the first in the numerator are zero, so the formula can be written as Px\x2 ®T\T2 °x\Gx2 This shows that the covariance o f observed scores is equal to the covariance o f true scores aT I T2=pxix2(7xiC?x2=crxix2- Because the re l iab i l i ty is defined as true variance d i v i d e d by observed variance p x x=G T 2/o x 2 ( p x x refers to the re l iab i l i ty for a variable X ) , i f there were no errors present, the true correlation between two variables X ] and X 2 ( p T m) w o u l d be as fo l lows : 143 _ ®T\TI ®xix2 PT\TI ~ ~ I / 0Ty0T2 JpxlxlOxlJpx2x2Ox2 ^°x\x2 1^x^x2) _ Px\x2 VPx\x\ ^Px2x2 V'Px\x\ -\Px2x2 T h i s formula is spoken o f as the "correc t ion" for attenuation, but it is real ly an estimate of the magnitude o f a correlation i f two variables were made perfectly rel iable (Nunna l ly , 1978 ,p215-221) . In this study, the attenuation fo rmula was employed to estimate the correlat ion between the numerator and denominator variables i n different trials p x i i , X 2j: Px X PTUTU =~J=^'r (A-1) V Px^ -\J Px2x2 where p T l i > T 2 j is true correlation (measured without error) between X j in the i th trial and X 2 i n the j th tr ial , p x i i,x2j is the observed populat ion correlat ion between X i i n the i th trial and X 2 i n the j th t r ia l , and p x i x i and p x 2 x 2 are the re l iab i l i ty coefficients o f X i and X 2 , respectively. T h e re l iab i l i ty coefficients o f X i and X 2 were der ived f rom the defined covariance matr ix o f the component variables in each o f the control led condi t ions . A s shown above, this investigation assumed that the wi th in trial correlat ion between the numerator and denominator variables was the same, and the variance o f the component variables was constant, over al l f ive trials. Therefore, the best approximat ion for the true correlat ion between the numerator in trial i and the denominator i n trial j w o u l d be the correlat ion between the two variables in the same trial . That is , PTii,T2j=Pxii,x2i- Therefore, F o r m u l a ( A - l ) can be rewritten to show the observed correlat ion between the numerator and denominator variables in different trials: (A -2 ) Appendix B Covariance Matrices Used to Generate the Population Data 145 Exi=Ex2=1.0, V x l / V x 2 = 2 . 0 , p x l x 2 = 0 . 9 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 131 582 110 996 110 996 110 996 110 996 110 996 131 582 110 996 110 996 110 996 110 996 110 996 131 582 110 996 110 996 110 996 110 996 110 996 131 582 110 996 110 996 110 996 110 996 110 996 131 582 6x1=6x2=1.0, V x l / V x 2 = 1 . 0 , p x l x 2 = 0 . 9 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 162 000 135 942 135 942 135 942 135 942 135 942 162 000 135 942 135 942 135 942 135 942 135 942 162 000 135 942 135 942 135 942 135 942 135 942 162 000 135 942 135 942 135 942 135 942 135 942 162 000 e x i=e x 2 =1.0, V x l / V x 2 = 0 . 5 , p x l x 2 = 0 . 9 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 131 042 109 675 109 675 109 675 109 675 109 675 131 042 109 675 109 675 109 675 109 675 109 675 131 042 109 675 109 675 109 675 109 675 109 675 131 042 109 675 109 675 109 675 109 675 109 675 131 042 131 582 110 996 110 996 110 996 110 996 110 996 131 582 110 996 110 996 110 996 110 996 110 996 131 582 110 996 110 996 110 996 110 996 110 996 131 582 110 996 110 996 110 996 110 996 110 996 131 582 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 162 000 135 942 135 942 135 942 135 942 135 942 162 000 135 942 135 942 135 942 135 942 135 942 162 000 135 942 135 942 135 942 135 942 135 942 162 000 135 942 135 942 135 942 135 942 135 942 162 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 131 042 109 675 109 675 109 675 109 675 109 675 131 042 109 675 109 675 109 675 109 675 109 675 131 042 109 675 109 675 109 675 109 675 109 675 131 042 109 675 109 675 109 675 109 675 109 675 131 042 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 146 Exl=E X 2=1.0, V x l / V x 2 = 2 . 0 , p xix2=0.7 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 102 341 86 330 86 330 86 330 86 330 86 330 102 341 86 330 86 330 86 330 86 330 86 330 102 341 86 330 86 330 86 330 86 330 86 330 102 341 86 330 86 330 86 330 86 330 86 330 102 341 e x ,=e x 2 =1.0, V x l / V x 2 = 1 . 0 , P x , x 2 = 0 . 7 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 126 000 105 732 105 732 105 732 105 732 105 732 126 000 105 732 105 732 105 732 105 732 105 732 126 000 105 732 105 732 105 732 105 732 105 732 126 000 105 732 105 732 105 732 105 732 105 732 126 000 e x ,=e x 2 =1.0, V x l / V x 2 = 0 . 5 , p x i x 2 = 0 . 7 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 101 922 85 302 85 302 85 302 85 302 85 302 101 922 85 302 85 302 85 302 85 302 85 302. 101 922 85 302 85 302 85 302 85 302 '• 85 302 101 922 85 302 85 302 85 302 '• 85 302 85 302 101 922 102 341 86 330 86 330 86 330 86 330 86 330 102 341 86 330 86 330 86 330 86 330 86 330 102 341 86 330 86 330 86 330 86 330 86 330 102 341 86 330 86 330 86 330 86 330 86 330 102 341 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 126 000 105 732 105 732 105 732 105 732 105 732 126 000 105 732 105 732 105 732 105 732 105 732 126 000 105 732 105 732 105 732 105 732 105 732 126 000 105 732 105 732 105 732 105 732 105 732 126 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 101 922 85 302 85 302 85 302 85 302 85 302 101 922 85 302 85 302 85 302 85 302 85 302 101 922 85 302 85 302 85 302 85 302 85 302 101 922 85 302 85 302 85 302 85 302 85 302 101 922 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 147 6x1=6x2=1.0, V x l / V x 2 = 2 . 0 , p x ] x 2 = 0 . 5 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 73 101 61 664 61 664 61 664 61 664 61 664 73 101 61 664 61 664 61 664 61 664 61 664 73 101 61 664 61 664 61 664' 61 664 61 664 73 101 61 664 61 664 61 664 61 664 61 664 73 101 6*1=6*2=1.0, V x l / V x 2 = 1 . 0 , p x i x 2 = 0 . 5 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 90 000 75 523 75 523 75 523 75 523 75 523 90 000 75 523 75 523 75 523 75 523 75 523 90 000 75 523 75 523 75 523 75 523 75 523 90 000 75 523 75 523 75 523 75 523 75 523 90 000 e x i=8 x 2 =1.0, V x l / V x 2 = 0 . 5 , p x i x 2 = 0 . 5 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 72 801 60 930 60 930 60 930 60 930 60 930 72 801: 60 930 60 930 60 930 60 930 60 930 72 801 60 930 60 930 60 930 60 930 60 930 72 801 60 930 60 930 60 930 60 930 60 930 72 801 73 101 61 664 61 664 61 664 61 664 61 664 73 101 61 664 61 664 61 664 61 664 61 664 73 101 61 664 61 664 61 664 61 664 61 664 73 101 61 664 61 664 61 664 61 664 61 664 73 101 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 90 000 75 523 75 523 75 523 75 523 75 523 90 000 75 523 75 523 75 523 75 523 75 523 90 000 75 523 75 523 75 523 75 523 75 523 90 000 75 523 75 523 . 75 523 75 523 75 523 90 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 72 801 60 930 60 930 60 930 60 930 60 930 72 801 60 930 60 930 60 930 60 930 60 930 72 801 60 930 60 930 60 930 60 930 60 930 72 801 60 930 60 930 60 930 60 930 60 930 72 801 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 148 e x l =e x 2 =0.7 , V x l / V x 2 = 2 . 0 , p x l x 2 = 0 . 9 225 000 195 000 170 000 160 000 150 000 195 000 225 000 196 000 170 000 160 000 170 000 196 000 225 000 195 000 170 000 160 000 170 000 195 000 225 000 192 000 150 000 160 000 170 000 192 000 225 000 131 582 110 996 92 398 83 656 75 568 110 996 131 582 109 844 90 896 83 656 92 398 109 844 131 582 109 564 92 398 83 656 90 896 109 564 131 582 105 818 75 568 83 656 92 398 105 818 131 582 e x l =e x 2 =0.7 , V x l / V x 2 = 1 . 0 , p x l x 2 = 0 . 9 135 000 117 000 104 000 95 000 81 000 117 000 135 000 116 000 104 000 95 000 104 000 116 000 135 000 116 000 102 000 95 000 104 000 116 000 135 000 111 000 81 000 95 000 102 000 111 000 135 000 162 000 134 188 116 097 101 923 84 954 134 188 162 000 133 613 116 097 101 923 116 097 133 613 162 000 133 613 114 975 101 923 116 097 133 613 162 000 130 012 84 954 101 923 114 975 130 012 162 000 e x l =e x 2 =0.7, V x l / V x 2 = 0 . 5 , p x [ x 2 = 0 . 9 53 000 45 000 40 000 35 000 32 000 45 000 53 000 45 000 40 000 36 000 40 000 45 000 53 000 45 000 40 000 35 000 40 000 45 000 53 000 44 000 32 000 36 000 40 000 44 000 53 000 131 042 109 675 91 782 78 975 70 177 109 675 131 042 106 299 91 782 80 095 91 782 106 299 131 042 108 000 90 000 78 975 91 782 108 000 131 042 103 402 70 177 80 095 90 000 103 402 131 042 131 582 110 996 92 398 83 656 75 568 110 996 131 582 109 844 90 896 83 656 92 398 109 844 131 582 109 564 92 398 83 656 90 896 109 564 131 582 105 818 75 568 83 656 92 398 105 818 131 582 95 000 78 000 62 000 54 000 47 000 78 000 95 000 76 000 60 000 54 000 62 000 76 000 95 000 76 000 62 000 54 000 60 000 76 000 95 000 72 000 47 000 54 000 62 000 72 000 95 000 162 000 134 188 116 097 101 923 84 954 134 188 162 000 133 613 116 097 101 923 116 097 133 613 162 000 133 613 114 975 101 923 116 097 133 613 162 000 130 012 84 954 101 923 114 '975 130 012 162 000 240 000 190 000 160 000 135 000 110 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 188 000 110 000 135 000 160 000 188 000 240 000 131 042 109 675 91 782 78 975 70 177 109 675 131 042 106 299 91 782 80 095 91 782 106 299 131 042 108 000 90 000 78 975 91 782 108 000 131 042 103 402 70 177 80 095 90 000 103 402 131 042 400 000 330 000 260 000 220 000 190 000 330 000 400 000 310 000 260 000 220 000 260 000 310 000 400 000 320 000 250 000 220 000 260 000 320 000 400 000 300 000 190 000 220 000 250 000 300 000 400 000 149 e x i = £ x 2 = 0 . 7 , V x l / V x 2 = 2 . 0 , p x l x 2 = 0 . 7 225 000 195 000 170 000 160 000 150 000 195 000 225 000 196 000 170 000 160 000 170 000 196 000 225 000 195 000 170 000 160 000 170 000 195 000 225 000 192 000 150 000 160 000 170 000 192 000 225 000 102 341 86 330 71 865 65 066 58 775 86 330 102 341 85 434 70 697 65 066 71 865 85 434 102 341 85 216 71 865 65 066 70 697 85 216 102 341 82 303 58 775 65 066 71 865 82 303 102 341 £xi=e X 2=0.7, V x l / V x 2 = 1 . 0 , p x l x 2 = 0 . 7 135 000 117 000 104 000 95 000 81 000 117 000 135 000 116 000 104 000 95 000 104 000 116 000 135 000 116 000 102 000 95 000 104 000 116 000 135 000 111 000 81 000 95 000 102 000 111 000 135 000 126 000 104 368 90 297 79 273 66 075 104 368 126 000 103 921 90 297 79 273 90 297 103 921 126 000 103 921 89 425 79 273 90 297 103 921 126 000 101 120 66 075 79 273 89 425 101 120 126 000 £xi=e X 2=0.7, V x l / V x 2 = 0 . 5 , p x l x 2 = 0 . 7 53 000 45 000 40 000 35 000 32 000 45 000 53 000 45 000 40 000 36 000 40 000 45 000 53 000 45 000 40 000 35 000 40 000 45 000 53 000 44 000 32 000 36 000 40 000 44 000 53 000 101 922 85 302 71 386 61 425 54 582 85 302 101 922 82 677 71 386 62 296 71 386 82 677 101 922 84 000 70 000 61 425 71 386 84 000 101 922 80 424 54 582 62 296 70 000 80 424 101 922 102 341 86 330 71 865 65 066 58 775 86 330 102 341 85 434 70 697 65 066 7 1 865 85 434 102 341 85 216 71 865 65 066 70 697 85 216 102 341 82 303 58 775 65 066 71 865 82 303 102 341 95 000 78 000 62 000 54 000 47 000 78 000 95 000 76 000 60 000 54 000 62 000 76 000 95 000 76 000 62 000 54 000 60 000 76 000 95 000 72 000 47 000 54 000 62 000 72 000 95 000 126 000 104 368 90 297 79 273 66 075 104 368 126 000 103 921 90 297 79 273 90 297 103 921 126 000 103 921 89 425 79 273 90 297 103 921 126 000 101 120 66 075 79 273 89 425 101 120 126 000 240 000 190 000 160 000 135 000 110 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 188 000 110 000 135 000 160 000 188 000 240 000 101 922 85 302 71 386 61 425 54 582 85 302 101 922 82 677 71 386 62 296 71 386 82 677 101 922 84 000 70 000 61 425 71 386 84 000 101 922 80 424 54 582 62 296 70 000 80 424 101 922 400 000 330 000 260 000 220 000 190 000 330 000 400 000 310 000 260 000 220 000 260 000 310 000 400 000 320 000 250 000 220 000 260 000 320 000 400 000 300 000 190 000 220 000 250 000 300 000 400 000 150 E x l = e x 2 = 0 . 7 , V x l / V x 2 = 2 . 0 , p x i x 2 = 0 . 5 225 000 195 000 170 000 160 000 150 000 195 000 225 000 196 000 170 000 160 000 170 000 196 000 225 000 195 000 170 000 160 000 170 000 195 000 225 000 192 000 150 000 160 000 170 000 192 000 225 000 73 101 61 664 51 332 46 476 41 982 61 664 73 101 61 025 50 498 46 476 51 332 61 025 73 101 60 869 51 332 46 476 50 498 60 869 73 101 58 788 41 982 46 476 51 332 58 788 73 101 £ x , = e x 2 = 0 . 7 , V x l / V x 2 = 1 . 0 , p x ix 2 =0.5 135 000 117 000 104 000 95 000 81 000 117 000 135 000 116 000 104 000 95 000 104 000 116 000 135 000 116 000 102 000 95 000 104 000 116 000 135 000 111 000 81 000 95 000 102 000 111 000 135 000 90 000 74 549 64 498 56 624 47 196 74 549 90 000 74 229 64 498 56 624 64 498 74 229 90 000 74 229 63 875 56 624 64 498 74 229 90 000 72 229 47 196 56 624 63 875 72 229 90 000 £ x i = £ x 2 = 0 . 7 , V x l / V x 2 = 0 . 5 , p x l x 2 = 0 . 5 53 000 45 000 40 000 35 000 32 000 45 000 53 000 45 000 40 000 36 000 40 000 45 000 53 000 45 000 40 000 35 000 40 000 45 000 53 000 44 000 32 000 36 000 40 000 44 000 53 000 72 801 60 930 50 990 43 875 38 987 60 930 72 801 59 055 50 990 44 497 50 990 59 055 72 801 60 000 50 000 43 875 50 990 60 000 72 801 57 446 38 987 44 497 50 000 57 446 72 801 73 101 61 664 51 332 46 476 41 982 61 664 73 101 61 025 50 498 46 476 51 332 61 025 73 101 60 869 51 332 46 476 50 498 60 869 73 101 58 788 41 982 46 476 51 332 58 788 73 101 95 000 78 000 62 000 54 000 47 000 78 000 95 000 76 000 60 000 54 000 62 000 76 000 95 000 76 000 62 000 54 000 60 000 76 000 95 000 72 000 47 000 54 000 62 000 72 000 95 000 90 000 74 549 64 498 56 624 47 196 74 549 90 000 74 229 64 498 56 624 64 498 74 229 90 000 74 229 63 875 56 624 64 498 74 229 90 000 72 229 47 196 56 624 63 875 72 229 90 000 240 000 190 000 160 000 135 000 110 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 188 000 110 000 135 000 160 000 188 000 240 000 72 801 60 930 50 990 43 875 38 987 60 930 72 801 59 055 50 990 44 497 50 990 59 055 72 801 60 000 50 000 43 875 50 990 60 000 72 801 57 446 38 987 44 497 50 000 57 446 72 801 400 000 330 000 260 000 220 000 190 000 330 000 400 000 310 000 260 000 220 000 260 000 310 000 400 000 320 000 250 000 220 000 260 000 320 000 400 000 300 000 190 000 220 000 250 000 300 000 400 000 151 Exi=0.7, 6x2=1.0, V x l / V x 2 = 2 . 0 , p x , x2=0 .9 225 000 195 000 170 000 160 000 150 000 195 000 225 000 196 000 170 000 160 000 170 000 196 000 225 000 195 000 170 000 160 000 170 000 195 000 225 000 192 000 150 000 160 000 170 000 192 000 225 000 131 582 110 996 103 637 100 543 97 350 110 996 131 582 111 280 103 637 100 543 103 637 111 280 131 582 110 996 103 637 100 543 103 637 110 996 131 582 110 139 97 350 100 543 103 637 110 139 131 582 e x i=0.7, eX2=1.0, V x l / V x 2 = 1 . 0 , px ix2=0 .9 135 000 117 000 104 000 95 000 81 000 117 000 135 000 116 000 104 000 95 000 104 000 116 000 135 000 116 000 102 000 95 000 104 000 116 000 135 000 111 000 81 000 95 000 102 000 111 000 135 000 162 000 135 942 128 167 122 496 113 110 135 942 162 000 135 360 128 167 122 496 128 167 135 360 162 000 135 360 126 929 122 496 128 167 135 360 162 000 132 410 113 110 122 496 126 929 132 410 162 000 6x1=0.7, 6x2=1.0, V x l / V x 2 = 0 . 5 , p x l x 2 = 0 . 9 53 000 45 000 40 000 35 000 32 000 45 000 53 000 45 000 40 000 36 000 40 000 45 000 53 000 45 000 40 000 35 000 40 000 45 000 53 000 44 000 32 000 36 000 40 000 44 000 53 000 123 762 103 581 97 658 91 350 87 348 103 581 123 762 103 581 97 658 92 646 97 658 103 581 123 762 103 581 97 658 91 350 97 658 103 581 123 762 102 424 87 348 92 646 97 658 102 424 123 762 131 582 110 996 103 637 100 543 97 350 110 996 131 582 111 280 103 637 100 543 103 637 111 280 131 582 110 996 103 637 100 543 103 637 110 996 131 582 110 139 97 350 100 543 103 637 110 139 131 582 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 162 000 135 942 128 167 122 496 113 110 135 942 162 000 135 360 128 167 122 496 128 167 135 360 162 000 135 360 126 929 122 496 128 167 135 360 162 '000 132 410 113 110 122 496 126 929 132 410 162 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000- 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 123 762 103 581 97 658 91 350 87 348 103 581 123 762 103 581 97 658 92 646 97 658 103 581 123 762 103 581 97 658 91 350 97 658 103 581 123 762 102 424 87 348 92 646 97 658 102 424 123 762 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 152 exi=0.7, e x 2=1.0, V x l / V x 2 = 2 . 0 , p x l x 2 = 0 . 7 225 000 195 000 170 000 160 000 150 000 195 000 225 000 196 000 170 000 160 000 170 000 196 000 225 000 195 000 170 000 160 000 170 000 195 000 225 000 192 000 150 000 160 000 170 000 192 000 225 000 102 341 86 330 80 606 78 200 75 717 86 330 102 341 86 551 80 606 78 200 80 606 86 551 102 341 86 330 80 606 78 200 80 606 86 330 102 341 85 664 75 717 78 200 80 606 85 664 102 341 exi=0.7, 6x2=1.0, V x l / V x 2 = 1 . 0 , p x i x 2 = 0 . 7 135 000 117 000 104 000 95 000 81 000 117 000 135 000 116 000 104 000 95 000 104 000 116 000 135 000 116 000 102 000 95 000 104 000 116 000 135 000 111 000 81 000 95 000 102 000 111 000 135 000 126 000 105 732 99 686 95 275 87 975 105 732 126 000 105 280 99 686 95 275 99 686 105 280 126 000 105 280 98 722 95 275 99 686 105 280 126 000 102 986 87 975 95 275 98 722 102 986 126 000 exi=0.7, 6x2=1.0, V x l / V x 2 = 0 . 5 , p x l x 2 = 0 . 7 53 000 45 000 40 000 35 000 32 000 45 000 53 000 45 000 40 000 36 000 40 000 45 000 53 000 45 000 40 000 35 000 40 000 45 000 53 000 44 000 32 000 36 000 40 000 44 000 53 000 101 922 85 302 80 424 75 230 71 933 85 302 101 922 85 302 80 424 76 297 80 424 85 302 101 922 85 302 80 424 75 230 80 424 85 302 101 922 84 349 71 933 76 297 80 424 84 349 101 922 102 341 86 330 80 606 78 200 75 717 86 330 102 341 86 551 80 606 78 200 80 606 86 551 102 341 86 330 80 606 78 200 80 606 86 330 102 341 85 664 75 717 78 200 80 606 85 664 102 341 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 126 000 105 732 99 686 95 275 87 975 105 732 126 000 105 280 99 686 95 275 99 686 105 280 126 000 105 280 98 722 95 275 99 686 105 280 126 000 102 986 87 975 95 275 98 722 102 986 126 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 101 922 85 302 80 424 75 230 71 933 85 302 101 922 85 302 80 424 76 297 80 424 85 302 101 922 85 302 80 424 75 230 80 424 85 302 101 922 84 349 71 933 76 297 80 424 84 349 101 922 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 153 exi=0.7, 6x2=1.0, V x l / V x 2 = 2 . 0 , p x ,x2=0.5 225 000 195 000 170 000 160 000 150 000 195 000 225 000 196 000 170 000 160 000 170 000 196 000 225 000 195 000 170 000 160 000 170 000 195 000 225 000 192 000 150 000 160 000 170 000 192 000 225 000 73 101 61 664 57 576 55 857 54 083 61 664 73 101 61 822 57 576 55 857 57 576 61 822 73 101 61 664 57 576 55 857 57 576 61 664 73 101 61 188 54 083 55 857 57 576 61 188 73 101 6x1=0.7, 8x2=1.0, V x l / V x 2 = 1 . 0 , Pxlx2=0.5 135 000 117 000 104 000 95 000 81 000 117 000 135 000 116 000 104 000 95 000 104 000 116 000 135 000 116 000 102 000 95 000 104 000 116 000 135 000 111 000 81 000 95 000 102 000 111 000 135 000 90 000 75 523 7 1 204 68 053 62 839 75 523 90 000 75 200 71 204 68 053 71 204 75 200 90 000 75 200 70 516 68 053 71 204 75 200 90 000 73 561 62 839 68 053 70 516 73 561 90 000 8xi=0.7, 6x2=1.0, V x l / V x 2 = 0 . 5 , p x l x 2 = 0 . 5 53 000 45 000 40 000 35 000 32 000 45 000 53 000 45 000 40 000 36 000 40 000 45 000 53 000 45 000 40 000 35 000 40 000 45 000 53 000 44 000 32 000 36 000 40 000 44 000 53 000 72 801 '60 930 57 446 53 735 51 381 60 930 72 801 60 930 57 446 54 498 57 446 60 930 72 801 60 930 57 446 53 735 57 446 60 930 72 801 60 249 51 381 54 498 57 446 60 249 72 801 73 101 61 664 57 576 55 857 54 083 61 664 73 101 61 822 57 576 55 857 57 576 61 822 73 101 61 664 57 576 55 857 57 576 61 664 73 101 61 188 54 083 55 857 57 576 61 188 73 101 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78 000 95 000 78 000 78 000 78 000 78 000 78' 000 95 000 90 000 75 523 71 204 68 053 62 839 75 523 90 000 75 200 71 204 68 053 71 204 75 200 90 000 75 200 70 516 68 053 71 204 75 200 90 000 73 561 62 839 68 053 70 516 73 561 90 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 195 000 195 000 195 000 195 000 195 000 240 000 72 801 60 930 57 446 53 735 51 381 60 930 72 801 60 930 57 446 54 498 57 446 60 930 72 801 60 930 57 446 53 735 57 446 60 930 72 801 60 249 51 381 54 498 57 446 60 249 72 801 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 330 000 330 000 330 000 330 000 330 000 400 000 154 £xi=1.0 , 8 x 2 =0.7, V x l / V x 2 = 2 . 0 , p x l x 2 = 0 . 7 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 100 879 85 097 75 869 70 805 66 056 85 097 100 879 83 999 74 635 70 805 75 869 83 999 100 879 83 999 75 869 70 805 74 635 83 999 100 879 81 758 66 056 70 805 75 869 81 758 100 879 e x l =1.0, £ x 2 = 0 . 7 , V x l / V x 2 = 1 . 0 , p x i x 2 = 0 . 7 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 117 000 117 000 117 000 117 000 117 000 135 000 124 200 102 877 94 407 86 718 78 278 102 877 124 200 102 877 94 407 86 718 94 407 102 877 124 200 102 877 94 407 86 718 94 407 102 877 124 200 102 334 78 278 86 718 94 407 102 334 124 200 6x1=1.0, e x 2 =0.7, V x l / V x 2 = 0 . 5 , p x l x 2 = 0 . 7 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 101 922 85 302 75 717 69 649 64 726 85 302 101 922 82 677 75 717 69 649 75 717 82 677 101 922 84 000 74 246 69 649 75 717 84 000 101 922 81 333 64 726 69 649 74 246 81 333 101 922 100 879 85 097 75 869 70 805 66 056 85 097 100 879 83 999 74 635 70 805 75 869 83 999 100 879 83 999 75 869 70 805 74 635 83 999 100 879 81 758 66 056 70 805 75 869 81 758 100 879 95 000 78 000 62 000 54 000 47 000 78 000 95 000 76 000 60 000 54 000 62 000 76 000 95 000 76 000 62 000 54 000 60 000. . 76 000 95 000 72 000 47 000 54 000 62 000 72 000 95 000 124 200 102 877 94 407 86 718 78 278 102 877 124 200 102 877 94 407 86 718 94 407 102 877 124 200 102 877 94 407 86 718 94 407 102 877 124 200 102 334 78 278 86 718 94 407 102 334 124 200 240 000 190 000 160 000 135 000 110 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 188 000 110 000 135 000 160 000 188 000 240 000 101 922 85 302 75 717 69 649 64 726 85 302 101 922 82 677 75 717 69 649 75 717 82 677 101 922 84 000 74 246 69 649 75 717 84 000 101 922 81 333 64 726 69 649 74 246 81 333 101 922 400 000 330 000 260 000 220 000 190 000 330 000 400 000 310 000 260 000 220 000 260 000 310 000 400 000 320 000 250 000 220 000 260 000 320 000 400 000 300 000 190 000 220 000 250 000 300 000 400 000 155 £ x i - =1.0, Ex2= 0.7, V x l / V x 2 = 2 . 0 , p x i x 2 = 0 . 5 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 195 000 195 000 195 000 195 000 195 000 225 000 73 101 61 .664 54 977 51 308 47 867 61 664 73 101 60 869 54 083 51 308 54 977 60 .869 73 101 60 869 54 977 51 308 54 .083 60 869 73 101 59 245 47 867 51 308 54 977 59 245 73 101 Ex l = =1.0, £x2= 0 .7 , V x l / V x 2 = 1.0, Pxlx2=0.5 135 000 117 .000 117 000 117 000 117 000 117 000 135 .000 117 000 117 000 117 000 117 000 117 .000 135 000 117 000 117 000 117 000 117 .000 117 000 135 000 117 000 117 000 117 .000 117 000 117 000 135 000 90 000 74 .549 68 411 62 839 56 723 74 549 90 .000 74 549 68 411 62 839 68 411 74 .549 90 000 74 549 68 411 62 839 68 . 4 1 1 74 549 90 000 74 155 56 723 62 .839 68 411 74 155 90 000 E x i=1 .0 , e x 2 =0.7, V x l / V x 2 = 0 . 5 , p x l x 2 = 0 . 5 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 45 000 45 000 45 000 45 000 45 000 53 000 72 801 60 930 54 083 49 749 46 233 60 930 72 801 59 055 54 083 49 749 54 083 59 055 72 801 60 000 53 033 49 749 54 083 60 000 72 801 58 095 46 233 49 749 53 033 58 095 72 801 73 101 61 664 54 977 51 308 47 867 61 664 73 101 60 869 54 083 51 308 54 977 60 869 73 101 60 869 54 977 51 308 54 083 60 869 73 101 59 245 47 867 51 308 54 977 59 245 73 101 95 000 78 000 62 000 54 000 47 000 78 000 95 000 76 000 60 000 54 000 62 000 76 000 95 000 76 000 62 000 54 000 60 000 76 000 95 000 72 000 47 000 54 000 62 000 72 000 95 000 90 000 74 549 68 411 62 839 56 723 74 549 90 000 74 549 68 411 62 839 68 411 74 549 90 000 74 549 68 411 62 839 68 411 74 549 90 000 74 155 56 723 62 839 68 411 74 155 90 000 240 000 190 000 160 000 135 000 110 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 190 000 160 000 135 000 160 000 190 000 240 000 188 000 110 000 135 000 160 000 188 000 240 000 72 801 60 930 54 083 49 749 46 233 60 930 72 801 59 055 54 083 49 749 54 083 59 055 72 801 60 000 53 033 49 749 54 083 60 000 72 801 58 095 46 233 49 749 53 033 58 095 72 801 400 000 330 000 260 000 220 000 190 000 330 000 400 000 310 000 260 000 220 000 260 000 310 000 400 000 320 000 250 000 220 000 260 000 320 000 400 000 300 000 190 000 220 000 250 000 300 000 400 000 156 Appendix C Accuracy of the Estimation Formulas (Ratio Mean, Sd., and Correlation) T o examine the accuracy o f the Pearson approximations for ratio variables (Formulas (4), (5), and (7)), the computed statistics of the nine s imulated ratio variable populat ions shown in the first panel of Table 5-1 (corresponding to e x i = e x 2 = 1.0) are compared wi th the approximations of the mean, standard deviat ion, and correlation based on Formulas (4), (5), and (7). Because the covariance can be expressed as a function o f the magnitudes o f the standard deviat ion and correlation o f the ratio variables, the accuracy o f F o r m u l a (6) was not separately examined. The means and standard deviations, for both the numerator and denominator variables used to create the ratio variable, are shown in Tab le D - l for each o f the nine condit ions o f V x i / V x 2 and p x i x 2 . Table D - 2 presents p - x ) / x 2 and a x i / x 2 calculated from the first trial o f the populat ion data o f the ratio variable, and the approximations ~ | L I X I / x 2 , and ~rj x i / x 2 based on Formulas (4) and (5). The approximations are shown i n the brackets. The values o f the p y i y 2 were calculated between the first trial and second trial o f the ratio populat ion data, and the values o f the ~ p y i y 2 were approximated f rom F o r m u l a (7) based on the characteristics o f the component variables. The subscript y i refers the ratio variable o f the first t r ial , and y 2 refers the second trial in the populat ion of the ratio variable data. A s can be seen in Table D - 2 , although a slight under-approximation exists in the condit ions V x i / V x 2 = 2 . 0 and V x ] / V x 2 = 1 . 0 , the ratio variable means are accurately approximated by F o r m u l a (4). W h e n the variat ion of the denominator variable increased f rom V x ) / V x 2 = 2 . 0 to V x ) / V x 2 = 0 . 5 , the populat ion mean was s l ight ly under-approximated; 157 Table D - l The Popula t ion Characterist ics o f the Component Var iab les (Xu X ? ) v x l/v x 2= =2.0 v x l/v x 2= = 1.0 V x l / V x 2 : =0.5 H x i H x l 1^ x2 Hxl 1^ x2 (CTxl) (0-x2) (Oxl) (Ox2) (CTxl) (<7x2) T r i a l 1 Pxlx2=0.9 75.04 100.03 75.03 100.05 75.02 100.07 (14.98) (9.74) (11.60) (15.49) (7.27) (20.00) Pxlx2=0.7 75.04 100.04 75.03 100.06 75.02 100.08 (14.98) (9.75) (11.60) (15.49) (7.27) (20.01) Pxlx2=0.5 75.04 100.04 75.03 100.06 75.02 100.07 (14.98) (9.75) (11.60) (15.49) (7.27) (20.00) T r i a l 2 Pxlx2=0.9 75.01 100.02 75.00 100.02 75.00 100.04 (15.01) (9.77) (11.63) (15.52) (7.29) (20.04) P x 1x2=0.7 75.01 100.02 75.01 100.04 75.00 100.05 (51.01) (9.78) (11.63) (15.54) (7.29) (20.06) Pxlx2=0.5 75.01 100.03 75.01 100.04 75.00 100.06 (15.01) (9.78) (11.63) (15.54) (7.29) (20.07) Table D-2 Compar i son Between Approx ima t ions r ^ x i / * ? , "Qxu*?, ~Pyiy2> and the Popula t ion Characterist ics (iuu^. o,i<v9. p y i Y 7 ) V * i / V x 2 = 2 . 0 V *i/V x 2=1.0 V x i / V x 2 = 0 . 5 U-xl/x2 C» xl/x2 Pyly2 M l^/x2 O xl/x2 Pyly2 Hxl/x2 C* xl/x2 Pyly2 ( Mnl/x2) ( O xi/x2) (~Pyly2) ( M*l/x2) ( O xl/x2) (~Pvly2) ( M*l/x2) ( 0"xl/x2) (~PylV2) Pxlx2 : =0.9 .744 .092 .891 .752 .054 .834 .769 .110 .792 (.744) (.090) (.900) (.752) (.052) (.840) (.767) (.090) (.814) Pxlx2= =0.7 .747 .113 .875 .756 .094 .831 .772 .133 .797 (.747) (.111) (.876) (.755) (.090) (.838) (.769) (.112) (.820) Pxlx2 : =0.5 .750 .131 .866 .760 .123 .829 .776 .154 .801 (.750) (.130) (.869) (.759) (.116) (.838) (.772) (.130) (.823) 158 the greatest difference between the approximated value and true value was 0.004 (0.5% error). T h e standard deviat ion also shows an under-approximated pattern. Howeve r , the difference between the approximate value and the true value is much higher i n the condi t ion V x i / V x 2 = 0 . 5 than in the other two condi t ions . The approximate standard deviations in the condi t ion V x ) / V x 2 = 0 . 5 underestimated the true values by 0.020, 0.021, and 0.024, and the percent error were 18.2%, 15.8%, and 15.6%, respectively. T h e approximate correlat ion (Formula (7)) between the first trial and the second trial o f the ratio variable result i n overestimate when the variat ion o f the denominator variable increased f rom V x ] / V x 2 = 2 . 0 to V x ] / V x 2 = 0 . 5 . In the condi t ion V x i / V x 2 = 0 . 5 , the overestimations were 0.022, 0.023, and 0.022, and the percent errors were 2 .7%, 2 .9%, and 2.7%, respectively. The results indicate that the three parameters (the mean and standard deviat ion in the first trial and the correlat ion between the two trials) are not w e l l approximated by Formulas (4), (5), and (7) when V x i / V x 2 = 0 . 5 . It can be conc luded that i f the variat ion o f the denominator variable is re lat ively higher than o f the numerator variable ( V X ] / V x 2 = 0 . 5 ) , the approximat ion formulas (Formulas (4), (5), and (7)) may not accurately approximate the populat ion mean, standard deviat ion, and correlat ion of ratio variables. 159 r ~ H LD Lf l O vo r> r- co o O O O O rH CN CN O O O in vo r- co o O O O O rH 'J (N O CI O ^ in vo r-> o O O O O rH O O O rH ^ O O O rH rH CN CO O i n u ) h o O O O rH H h O ?i r> co o H H in o >1 VO 0 O CO O O rH i n II •3 o o | ll $ ll rH ><f w c o id c o o <U H _c CO CU X •c o cu fci o U rH CU Q H Q. O O O rH T"> O O O rH >» r> co o Q. O O rH CN rH M m vo r> co o X <3* in vo r> o » O O O O rH OOOOrH Q. O O O O rH ^ O O O rH r» co o >i r> co o vo h oi o in vo r- o O O O rH h cn o vD h O O O O O rH O O O O rH O O O O rH ' O O O O rH CN rH CN O vo r- co o rH CN O K l I s CO O Q. O O rH « m in o o n in vo co o Q. O O O rH in co o vo r> o ' O O O O rH H o r- vo o K r- r> co o O O O O rH in CN ro o vo r-- oo o ^ O O O rH Tt * rH rH O PICOOIO (Tl O 00 O Q-OOOrH ^ O O O r H M O O c o o O O rH CN OO O Pi r- oo o Q. O O rH CO O CO o •n O O O rH Q. O O rH II cs o 3 O n CN ' i n • o II O O O O rH O H h O in vo o o ^ O O O rH o O >i vo r- o Q. O O rH II Q. rH O O O O <H vo m m o vo r- co o co vo ro co o ro ^  in vo o O O O O rH m r> o ^ O O O rH n 3 c 3 c x 160 io H in in o vo r- r> oo o o o o o rH rH VO 00 rH O c- r> r- oo o O O O O rH CN rH rH O O 00 CO CO 00 o O O O O rH H in [— o r> r- co o O O O rH vo cn ro o rH rH rH O r> 00 o 00 00 00 O o o o rH ^ o o O rH in r> o >, r> oo o oo ro o >, t> co o ^ o o o >, oo oo o c- o 00 o O rH in ro o oo o O rH i n o o 00 o O rH i n II co II col1 e o •a c o U a CO X •a c o fc o O co -C H Ol CM (M CM O CO 00 00 00 o T I O O O O rH 3 CN CN CN O !«j 00 00 GO O Q . O O O rH r j o i r j m o vo vo r~ oo o O O O O rH oi -a1 r> o vo r> oo o O O O rH rH rH rH rH O 00 00 CO 00 o TT O O O O rH X J H H ri O % CO CO 00 o Q. O O O rH ro VO 00 rH O r> r> r> oo o O O O O rH CN CN O 00 00 o CN O 00 o ^ rji CO O >, t- CO o 0 . O O rH rH rH O CO CO O rH O CO o CD Ol CO O h M B O _ O O O rH j C d PI O >, r> oo o ro o co o ro ro ro ro o oo co co oo o TT O O O O rH r^  ro ro ro o ) j CO CO CO o lO r) n o o 00 00 00 00 o o o o o CN O O 00 00 00 o Q . O O O rH ^ O O O r H ro ro o 00 CO o CN O O >, 00 oo o Q . O O rH 00 o O rH I-l £ r H o o " o VD rH VD in o 00 rH CN o o o VO CN O CN CN CN in o o CO in ro o O in rH OS VD t> 00 o in VD r- CO o vo r> r> 00 O r> r- o VD VD r> 00 o OS 00 CO r~ O o O o rH o o o o rH o o o O rH o o o O rH o o o o rH O o o o rH CD r> o rH r> o o r~ vo O r> OS ro o VD m m o CO ro as o r- 00 o VD r> CO o r> r> 00 O r> CO o vo r> 00 o CO 00 r> o ,—> • ,—, ,—. • TT O o o rH o o o rH TT o o o rH _ o o o rH TT O o o rH o o o rH TT >l 3 VD r~ o o o r~ vo o ro o 3 in in o J5ro OS o 00 o ON o r> CO o 00 o K c- CO o £<» r- o a . o o rH Q. O o rH a . O o rH Q . O o rH a . o o rH Q.O o rH o OS o o ro o in o OS o 00 o CO o 00 o 00 o 00 o o o rH o rH o rH o rH o rH o rH <N o II • CM rH rH OS •is rH SC • Q . rH o • o i i o II • CN rH •I , in • « CD O o II • CN rH X o d r H 161 co co o r- o co co co co o O O O O rH r- rH VD CN o vo r- r- co o O O O O rH i n CN o OJ o ^ LT) V D r - O O O O O rH co r - r> o CO CO CO o O O O rH t—i VD CN O r- [-- co o O O O rH rH CN CO O m V D r - o O O O rH •j l> I> o ^ CO CO o >l • *— O O rH Q. Lfl CN O ^ r- co o w O O rH Q. in o ^ V D r - o >l • • • ' O O rH Q. 3 ° K o 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 vo co rH in o m m V D vo o O O O O rH in in ro m o ^ in vo r- o O O O O rH O O O O rH w O O O rH O O O rH * vo r- CJ\ o n m vo o o x • • * • w O O O rH a O O O rH w O O O rH Q. O O O rH rH CO O ^ (Tl CO O s • • O O rH "V VO CN O . H r- co o w O O tH Q. co ro o . j m t o >i • • • w O O <H o. O O O O rH Q.rH 3 ° Q. tH O O O O rH O O O O rH O O O rH O O O rH O O O rH Q. j m in o % CO CO o •H O O rH H X a. r> o CO o CN II <N 0 II j! m • o II CN 

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