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Disagreement : estimation of relative bias or discrepancy rate Ma, Ping Hang 1987

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DISAGREEMENT: ESTIMATION OF RELATIVE BIAS OR DISCREPANCY RATE by PING HANG MA B.Sc, The University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (The Department of Statistics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1987 © Ping Hang Ma, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date , ^ , A B S T R A C T Not only basic research in sciences, but also medicine, law, and manu-facturing need statistical techniques, including graphics, to assess disagree-ment. For some items or individuals i = l,2,---,n suppose that pairs (Xi,Y{) denote each item's measurements by two distinct methods or by two observers, or X, and Yi may be initial and repeat measurement scores, with discrepancy Di = X{ — Y{. Disagreement may be characterized by location and scale parameters of discrepancy distributions. The present work primarily addresses estimation of central tendency — relative bias or median discrepancy (or discrepancy rate in some instances). Most previous literature on "agreement" or "reliability" instead concerns A', Y correlation, which can be regarded as the complement of discrepancy variance. (There is ambiguity or confusion about concepts of "reliability" in the literature of various applications.) Discrepancies D\, D2, • • •, Dn in practice often violate assumptions of standard statistical models and methods that have been commonly applied in studies of agreement. In particular, both Xi and Y{ generally incorporate measurement errors. Further, these two measurement error distributions for the ith item need not be the same; and both distributions could depend on the magnitude ^, of the item being measured. Hence, for example, discrepancy Di could have variance proportional to the size of the item; 11 and in general £ ) l 5 Z?2, • • •, Dn are not identically distributed. Finally, the selection of items i = 1,2, • • •, n often is not random. To estimate median discrepancy, we consider nonparametric confidence intervals corresponding to Student t test, sign test, Wilcoxon signed rank test, or other permutation tests. Several criteria are developed to compare the performance of one procedure relative to another, including expected ratio of confidence interval lengths (related to Pitman asymptotic relative efficiency of tests) and relative variability of interval lengths. Theoretical calculations and Monte Carlo simulation results suggest different procedural preferences for random sampling from different distributions.. For discrepancies distributed non-identically, but symmetrically about a common median value, mixture sampling is used as an approximate model. This approach is related to a "random walk" (rather than random sample) model of Z?i, • • •, Dn proposed particularly for discrepancies between counting processes. We also emphasize graphic methods, especially plots of difference of Y — X versus average (X + Y)/2, for exploratory analysis of discrepancy data and to choose appropriate statistical models and numerical methods. Various data sets are analyzed as examples of the methodology. 111 T A B L E OF C O N T E N T S Abstract ii Table of Contents iv List of Tables vii List of Data Sets viii List of Figures ix Acknowledgements x 1. Introduction 1 2. Survey of "Agreement" in Research Literature 5 2.1. Correlation 6 2.2. Intraclass Correlation Coefficient 8 2.3. Kappa 9 2.4. Regression 12 2.5. Weighted Least-Squares Analysis 14 2.6. Analysis of Variance and General Linear Model 16 2.7. Pairwise Difference and Graphical Approach 18 2.8. Questions Beyond Altman and Bland 19 3. Independent, Identically Distributed Discrepancies: A Simple Model 21 3.1. Mean and t Procedures 22 3.2. Median and Sign Procedures 25 3.3. The Hodges-Lehmann Estimator and Signed Rank Procedures 28 iv 3.4. Discussion: The Permutation Perspective 31 3.5. Bootstrap Method 32 4. Evaluating Relative Performance of Confidence Interval Procedures . 34 4.1. Relative Efficiency of Statistical Tests 34 4.2. Relative Efficiency of Confidence Interval Procedures 36 4.3. Other Criteria of Relative Performance 38 4.4. Evaluation of Performance Criteria for Special Distributions: Monte Carlo Results 40 4.4.1. Comparsion of Asymptotic and Finite-Sample Results ..42 4.4.2. Examples: New Criteria May Be Decisive 43 4.5. Conclusion 47 4.6. Epilogue: Adaptive Procedures 48 5. Independent, Non-Identically Distributed Discrepancies: A General Model 50 5.1. Graphical Methods for Discrepancies 51 5.2. Discrepancy for Two Counting Processes: Random Walk Model 54 5.3. Permutation Procedures for Non-Identically Distributed Observations 56 5.4. Mixture Sampling Approximation to Non-Identically Distributed Data 57 5.5. Summary Guide to Estimation of Median Discrepancy for Real Data 60 6. Discrepancy or Discrepancy Rate? 63 7. Applications: Examples of Analyses of Discrepancy Data 65 v i Bibliography 104 Appendix I. Bootstrap Method 109 Appendix II. Efficacy Calculations and ARE's for Standard Distributions and Mixtures 119 Appendix III. Tail-Weight Adaptive Nonparametric Procedures 125 vi LIST OF T A B L E S 1. Efficacies and Pitman asymptotic relative efficiency (ARE) comparisons of Student t (T), sign (5), and Wilcoxon signed rank (W) procedures 75 2. Pitman asymptotic relative efficiency (ARE) comparisons of Student t (T), sign (5), and Wilcoxon signed rank (W) procedures (results using theoretical efficacies) 76 3. Asymptotic (n —• oo) ratios of lengths and squared lengths of confidence intervals (y ' l /ARE, and 1/ARE, respectively, where ARE's are Pitman asymptotic relative efficiencies) of Student t (T), sign (5), and Wilcoxon signed rank (W) procedures (results using theoretical efficacies) 77 4. Comparisons of T, 5, and W confidence intervals: average ratios of interval lengths and squared lengths (Monte Carlo simulation results) 78 5. Comparisons of T, 5, and W confidence intervals: ratios of standard deviations of interval lengths (Monte Carlo simulation results) 79 6. Confidence interval coverages and length comparisons (Monte Carlo simulation results) SO 7. Point estimate and confidence interval for median discrepancy rate of "old" logging counts (n = 166 batches) using T, 5, and W procedures SI S. Ratios of lengths and squared lengths for confidence intervals in Table 7 SI 9. Point estimate and confidence interval for median discrepancy rate of "new" logging counts (n = 86 batches — with 7 outliers deleted) using T, 5, and W procedures 82 10. Ratios of lengths and squared lengths for confidence intrevals in Table 9 82 vn LIST OF DATA SETS 1.1. Source counts and destination counts for 166 batches of "old" logs . . 83 1.2. Source counts and destination counts for 93 batches of "new" logs . . 86 2. Fuse burning times (seconds) measured by two observers for 30 powder train fuses 88 3. Systolic blood pressures (mm Hg) by two methods in 25 patients . . . 89 4. Spinal curvature (angle, in degrees) by Ferguson method and Cobb method in 26 patients 90 5. Cutaneous oxygen levels (mm Hg) in 50 newborn infants measured in two positions 91 6. Tobacco moisture content in 15 samples measured by two devices . . 92 viii LIST OF F I G U R E S 1. Scatter plot for "old" logs (Data Set 1.1) 93 2. Scatter plot for "new" logs (Data Set 1.2) . 93 3. Average-difference plot for "old" logs (Data Set 1.1) 94 4. Average-difference plot for "new" logs (Data Set 1.2) 94 5. Difference versus square root of average count for "old" logs (Data Set 1.1) 95 6. Difference versus square root of average count for "new" logs (Data Set 1.2) 95 7. Normal probability plot for "old" logs (Data Set 1.1) 96 8. Normal probability plot for "new" logs (Data Set 1.2) 96 9. Normal probability plot for subset of "new" logs: 7 outliers are deleted 97 10. Scatter plot for fuse burning times (Data Set 2) : . . . 98 11. Average-difference plot for fuse burning times (Data Set 2) 98 12. Scatter plot for systolic blood pressures (Data Set 3) 99 13. Average-difference plot for systolic blood pressures (Data Set 3) . . . 99 14. Scatter plot for spinal curvature (Data Set 4) 100 15. Average-difference plot for spinal curvature (Data Set 4) 100 16. Residual plot for spinal curvature (Data Set 4): plot residual (of regressing Cobb on Ferguson) versus Ferguson . . . 101 17. Scatter plot for oxygen level and position (Data Set 5) 102 18. Average-difference plot for oxygen level and position (Data Set 2) . . 102 19. Scatter plot for tobacco moisture content (Data Set 6) 103 20. Average-difference plot tobacco moisture content (Data Set 6) . . . . 103 i x A C K N O W L E D G E M E N T S I would like to thank Dr. Ned Glick for his guidance, assistance and encouragement in producing this thesis, as well as suggesting the topic. I am indebted to Dr. Jonathan Berkowitz for his useful comments and careful reading of this work, in addition to his concern throughout the years. Encouragement and support from other members of the Department of Statistics also are gratefully appreciated. I thank attorneys James W. Peters, Brian J. Wallace, and E. J. Gouge for discussions of the logging data in Example 7.1, originally analyzed by Dr. Glick. I owe my wife, May-Moon, and my son, Lok-Chun, for their patience and support, while I spent days away from home. This work received financial support in part from Dr. Glick's forest industry consulting, as well as my teaching and research assistanceships in the Department of Statistics, University of British Columbia. x 1. I N T R O D U C T I O N Medicine, manufacturing, and research in sciences all require counting of items or measuring amounts of substances being studied. Therefore, it is not surprising that great effort and time are devoted to evaluating measure-ment methodologies, from intra-observer and inter-observer perspectives, and comparing distinct measurement methods. For instance, Beeler (1986) found that as many as one-third of all papers published in American Journal of Clinical Pathology are method comparison studies. Developments of new or "improved" methods may offer operational advantages, such as in speed, cost, convenience, etc. Consequently, there are many contexts in which the statis-tician requires techniques for comparing repeated or paired measurements or comparing one measurement process to another. This thesis is particularly concerned with measurement scales that are continuous or that permit inte-ger values over a large range, although there is some consideration of ordinal categoric ratings. Suppose (Xi, Yi), (X2, Y2), • • • , (Xn, Yn) are the pairwise measurements made by two measuring methods or by two observers or at two distinct occasions on n items. For instance (Xi,Yi) may be the counts of red blood cells in the ith blood specimen by two cell counting devices or the finishing 1 times of the ith racer recorded by two observers. (Other examples are given in Chapter 7.) Then define their discrepancies D\, D2, • • •, Dn by Di = Xi - Yi, i = 1,2, The term "discrepancy" or "disagreement" reflects the possibility of error in both Xi and Yi observations: if Hi denotes the "true" value associated with the ith item, then represent the two measurements, respectively, as Xi = fii + £i and Y; = /X{ + 6\, where e,- and are random errors (with respective biases Eei and E6i). Thus, no available measurement process is assumed to be absolutely accurate. Agreement or disagreement between paired measurements can be charac-terized by at least two aspects — relative bias and reliability, or precision. "Relative bias" refers to the mean of the probability distribution of discrep-ancy Di: E(Di) = E(Xi - Y) = E{Xi-Hi)-E(Yi-iii) = E(el)-E(6i) = bias of Xi — bias of Y{ — relative bias of Xi and Yj. 2 Median or other central location parameter, rather than expectation of £>i, also may be of interest. "Reliability" or "precision" or "reproducibility" usually refers to the predictability of one measurement, given the other. This issue essentially relates to the variance of discrepancy distribution — or, equivalently, to the correlation coefficient, p(Xi,Y{), of the joint distribution the Xi and Y{ measurements: Var(Di) = Var(Xi - Y{) = Var(Xi) + Var(Yi) - 2Cov(Xi,Yi) = Var(Xi) + VariYi) ~ 2[p(Xu Yi)]y/Var(Xl)y/Var(Yt). Note that if Var(Xi) = Var(Y{) = cr2, as in several models considered below, then Var(Dt) = 2a2[l - p(Xi,Yi)). But Var(Xi) and Var(Y}), and hence variance of the discrepancy distribution, may be in any functional relationship with the variances may be, for instance, proportional to m or to / / 2 , etc. This thesis is mostly concerned with relative bias — assessment of central tendency for discrepancy distributions. Emphasis is on graphical methods and estimation procedures, including confidence intervals, for the mean or median of discrepancy distribution. Motivation comes partly from 3 data analysis (by Professor Ned Glick) in litigation where substantial dollar costs were claimed in proportion to an alleged relative bias or discrepancy rate in measuring quantities of wood; see Chapter 7. There appears to be little statistical literature directly related to the relative bias aspect of agreement or disagreement (although there is much literature on "reliability" as noted in Chapter 2, and much general theory related to location parameter estimation and confidence interval issues, as discussed in later chapters). The most directly relevant material seems to be Altman and Bland (1983) and Bland and Altman (1986), and some responses to their works — partly published while this thesis was in progress. 4 2. S U R V E Y OF " A G R E E M E N T " IN R E S E A R C H L I T E R A T U R E As discussed in the Introduction, there are at least two aspects of agreement or disagreement between measurements — namely, relative bias and reliability; or equivalently, location and scale parameters of the dis-crepancy distribution. There seems to be little explicit attention to relative bias in research literature; a notable exception is Bland and Altman (1986) using the graphical approach proposed by Altman and Bland (1983). There is much literature on reliability using techniques such as correlation, intr-aclass correlation coefficient, Cohen's kappa, linear regression, general linear model, etc. For review of statistical methods in reliability studies, see Lan-dis and Koch (1975, Parts I and II). But many applications in reliability literature, discussed in this chapter, either misinterpret these techniques or are based on assumptions that may be invalid, for example, assuming that measurement errors have the same variance for different observers and for all items measured. Also the two aspects of agreement often are confused, for instance, in research literature of medicine or behavioural sciences. In particular, the term "agreement" often is used as a synonym for reliability, neglecting relative bias. Altman and Bland (1983) suggested that such confusion may arise 5 "because virtually all introductory courses and textbooks in statistics are method-based rather than problem-based" — that is, correlation is a promi-nently used elementary method, but the problem(s) of assessing agreement may be nowhere mentioned. "A further reason for poor methodology is", according to Altman and Bland (1983), "the tendency for researchers to imitate what they see in other published papers". A related issue is that many discussions of "agreement" (or of "reliability") calculate some quanti-tative statistics(s) without clearly indicating why one should be interested, nor what may be the practical implications or applications. Altman and Bland (1983) noted that relative bias is a more important issue than reliability in a published comparison of two methods for measuring systolic blood pressure. In another example, mentioned in the last chapter, substantial dollar costs were claimed in proportion to an alleged relative bias or discrepancy rate in counting and measuring volumes of wood. 2.1. C O R R E L A T I O N Some research workers tacitly — and incorrectly — assume that repeata-bility, usually evaluated in terms of high correlation or "significant" linear regression, implies low relative bias. Such fallacy is common in elementary 6 statistics and has been discussed, for instance, by Freedman, Pisani and Purves (1978). Considering the Skeels-Skodak study for intelligence scores of adopted children, their adoptive mothers, and their biologic mothers, Freed-man, Pisani and Purves (1978, pp.139-141) noted that correlation may be stronger between children and their biologic mothers than between children and their adoptive mothers, but that the average score for these children could be much closer to the average of adoptive mothers. In terms of linear regression to predict children's scores from their biologic mothers' scores, the intercept may be large, although the slope is close to one and is "highly significant". The biologic mothers may "predict" their children's scores in the sense of explaining a large fraction of variation in the dependent variable; t but the distribution of scores for the children could be systematically shifted with respect to the distribution of their biologic mothers' scores. Correlation coefficient also has been incorrectly interpreted as a per-centage of agreement. Cassidy, Triplett and LaDuca (1985) studied the Factor VIII inhibitors in blood, evaluating agreement between two measuring methods and between two laboratories. Because all their pairwise correlation coefficients are roughly equal to 0.9, the authors concluded that "these values indicate approximately 90% agreement for each comparison". 7 Other researchers seem to interpret squared correlation as an agreement scale. Because correlation coefficient usually is close to one in reliability studies, Rawles (1986) suggested squaring to "spread" out the "cramped" "meaningful range". In the present context, both X and Y measurements are subject to (non-degenerate) errors; this implies that the expectation of the sample correlation coefficient always is less than one. This phenomenon sometimes is called "attenuated correlation". See Altman and Bland (1983) or Fleiss (1986, pp.3-4). Such components-of-variance perspective also shows that correlation depends on the mechanism by which objects or "items" are selected, and is not an intrinsic property of the measurement procedures. Indeed, in many cases, items to compare measurement methods or to assess agreement between ratings are not drawn by any random procedure, but arbitrarily or deliberately; and a great range of scores usually would lead to a high correlation, regardless of relative bias between measurements [Altman and Bland (1983)]. 2.2. I N T R A C L A S S C O R R E L A T I O N C O E F F I C I E N T In literature of behavioural sciences and elsewhere, intraclass correlation 8 coefficient (ICC) has been outstandingly used to measure reliability or re-peatability of measurement procedures, with two or more observations per item; see Gulliksen (1950), Ebel (1951), Guilford (1954), Haggard (1958), Hoyt and Krishnaiah (1960), Winer (1962), Hoffman (1963), and others. Bartko (1966) showed that, for pairs, ICC and the usual Pearson correlation coefficient estimate the same parameter. He also showed that the ICC applies in a linear model in which "item" is a random effect; but using fixed effect data — items that are arbitrarily chosen — ICC does not resolve the problem of correlation depending on the item selection mechanism, as discussed in last section. 2.3. K A P P A Cohen (1960) introduced a kappa statistic as a measure of inter-rater agreement for categoric data. Later Cohen (1968) generalized to a weighted kappa, which allows the relative seriousness of each disagreement to be quantified. For full discussion of kappa, see Chapter 13 of Fleiss (1981). Suppose two raters (or measuring methods) separately classified n items on an L-point scale; the resulting data can be summarized in an L x L contingency table, or, equivalently, an array of observed proportions, such 9 that pij denotes the proportion of subjects classified into i category by the 1st rater and into jth category by the 2 n d rater. Since agreement requires raters to classify a given subject identically into the same category, one simple index of agreement is estimated by L L 1=1 j=l where {onij, i,j = 1,2, • • • , £ } are a set of non-negative weights, assigned according to the seriousness of disagreement (and independently of the data actually collected). Originally Cohen (1960) took u>a = 1 (corresponding to agreement) and u)ij — 0 for i j (any disagreement), so that p0 = YL^=\Pa- ^n general, we require that una = 1, 0 < u>ij < 1 for i y£ j, and Ulij = OJji. See Feldman, Klein, and Honingfeld (1972), and Cicchetti (1976) for different choices of Wjj. Cohen takes account of chance-expected agreement. If we assume in-dependence between ratings by the two raters, the expected agreement 10 proportion is estimated by L L t'=l j=l where pi. = X^ = 1Pifc, a n d P.j = 5Z*=i Pfcj- Then p0(u) - pe(u) represents the observed excess agreement beyond chance, and 1 — pe(w) indicates the maximum possible excess agreement beyond chance. Cohen proposed a measure of agreement, adjusting for the agreement due to chance: the weighted kappa statistic is po(w) -Pe(w) k(u) 1 - pe(uj) —pe(u>) which ranges from ;—- to 1, with the lower value depending on the 1 - P e O ) marginal distributions. Note that only for the special case where pe(u}) = — does k(u>) range from —1 to 1. In general, o k(u>) > 0 indicates better than chance agreement; o K(U>) = 0 indicates exactly chance agreement; o k(u) < 0 indicates poorer than chance agreement; o k(u>) — 1 indicates perfect agreement. Correspondences have been established between weighted kappa and the 11 Pearson correlation coefficient, and between weighted kappa and the intraclass correlation coefficient (ICC). Cohen (1968) has shown that, assuming the marginal distributions are the same (i.e., pi, = p,i for i = 1,2, • • • , £ ) and using the set of weights the weighted kappa is precisely equal to the Pearson correlation coefficient calculated on integer-valued categories. And for these same weights Uij, Fleiss and Cohen (1973) have shown that, under a random-effect model, the estimate of ICC differs from k(u) by a term involving the factor — and hence is asymptotically equal to k(u>). Thus, weighted kappa is equivalent to correlation and ICC, and hence does not relieve us from the problems noted in previous sections. 2.4. R E G R E S S I O N Linear regression analysis, which is another commonly used approach in comparison study, should be used with caution. Note that comparison of paired measurements in the present context is very different from the calibration problem, in which a set of measurements are compared with and adjusted to the known true measurements, made by a standard, precise method. Misunderstanding the desirable question would lead to an inappropriate analysis. If measurement (X) were free of error, we might fit for given data a "best" line Y = a + (3X, using least-squares regression. Then we might argue that this regression line should go through the origin and have a slope of one, unless there is some systematic bias. Hence we might interpret the intercept and slope — specifically, the quantities a — 0 and (3—1, respectively — as the constant error (or relative bias) and "proportional error" [Cassidy, Triplett and LaDuca (1985), and Rawles (1986)]. However, since both sets of measurements are subject to error, nec-essarily E(/3) < 1 and E(a) > 0 [Altman and Bland (1983)]. Thus, the usual regression analysis would give misleading results: both relative bias and "proportional error" are expected to be non-zero, no matter how well the two sets of measurements agree. Techniques have been developed, for computing a consistent estimate of the slope of the line relating two variables, when both are subject to errors. In particular, distinct methods were developed by Bartlett (1949), Deming 13 (1943), and Mandel (1964); also see survey papers by Madansky (1959) and Mandel (19S4) and bibliography for Chapter 1 of Draper and Smith (1981). Once a slope estimate 0 is obtained, we can estimate the intercept by a = Y — f3X. But these approaches and ordinary least-squares regression all assume that measurement errors i. follow a Guassian distribution, and ii. are identically distributed, regardless of the sizes of items measured. These two assumptions (especially the latter) in general do not hold in the present context. For instance, if we consider two counting processes, it is unlikely to have a discrepancy as great as 50 items for a shipment of size 100, but would be more likely for a shipment of size 1000. Thus, the variance of discrepancy for two counting processes would likely depend on the sizes of items measured; and the X, Y scatter plot would be heteroscedastic ("hetero" means "different", "scedastic" means "scatter" [Freedman, Pisani and Purves (1978, p.178)]). 2.5. W E I G H T E D L E A S T - S Q U A R E S A N A L Y S I S If there were no error in the X measurements, then weighted least-14 squares regression would be appropriate for heteroscedastic data. In the weighted least-squares analysis to fit Y = a + 0X, the sum of squares to be minimized is n YjUiiY-a-pX,)2, t=i where usually u>i = — If the set of weights {a;,-, z = l,2, were known, then the solution would be Hi=i Vi(Xi - i u ) 2 where A w = -=W , and Yu = -=k1r But the variance cr2 and hence Wi usually are unknown. Estimation would require iteration, using u>{ = (a + flxi)-1 or U{ = (ct + /5a;,)-2, etc. Notable works include Jacquez, Mather and Crawford (1968), Bement and Williams (1969), and Amemiya (1973). But weighted least-squares, like ordinary least-squares regression, still may be inappropriate if the X{ are subject to error. 15 2.6. A N A L Y S I S O F V A R I A N C E A N D G E N E R A L L I N E A R M O D E L Suppose Wij denotes measurement of object i made by method j , (i = 1,2, •••,n, and j = 1,2), and Hi denotes the true but unknown value for the object i. Then the general linear model relating Wij to Hi 1S Wij = Oj + fljHi + £,;, where a.j and /3j are parameters that jointly describe the measurement bias for method j , and where e,j is a random error in measuring object i with method j . It is assumed throughout that eij M ~ P A/*(0,CTJ). This model includes the previous discrepancy model as a special case with Yi — Wi2 — a2 + Pi2 + £i2, and Di =Xi-Yi = Wn - Wi2 = ( a i - a2) + (0i - 02)Hi + O i l - e,-2)-o The model is said to have common precision if <r| is the same for all j-o The measuring method j is said to be unbiased if otj = 0, and 0j = 1. 16 o The measuring method j is said to have a constant bias if aj ^  0, but Pj = I- -o Two methods j and k are said to have a constant relative bias if aj ^  ak, but pj = l= plk. o The method j is said to have a nonconstant bias if (3j ^  1. o Two methods j and k are said to have nonconstant relative bias if Two cases have been studied: i. a fixed-effect model where Hi a r e n ° t randomly selected, and ii. a random-effect model where Hi a r e randomly selected from some Guas-sian population. Then various linear model techniques can be employed to estimate relative bias (called "contrast") cti — a2. Notable works include Grubbs (1948) and Thompson (1963). But most linear model methods assume common precision, aj the same for all j. 17 More importantly, in the usual linear model, the distribution of error £ij does not depend on /i,-. 2.7. P A I R W I S E D I F F E R E N C E A N D G R A P H I C A L A P P R O A C H Altman and Bland (1983) criticized various techniques used in reliability literature and also argued that many of these studies should be more interested in relative bias. Noting that the usual X, Y scatter plot is more relevant to correlation than to study of relative bias between the paired measurements, Altman and Bland relied on the "average-difference plot", which is. a graph of pairwise difference (or discrepancy) against the average of the pair (estimate of the true measurement). One advantage of this plot is that it exhibits any trend relating discrepancy and size of measurement in a clear manner. Similar plots have been used by other statisticians, as discussed in the next chapter. Further, Altman and Bland (1983) suggested using Pearson correlation X + Y coefficient between discrepancy (Y — X) and average —— (or sum X + Y) to check for equality of the total variance of the two sets of measurements. This is based on the following results. If Var(Xi) — Var(Yi), then Cov[(Xi + Yi),(Y, - Xi)] = 0; or equivalently, if Cov[(Xi + YJ), (Y{ - Xt)} # 0, then 18 Var(Xi) # Var(Yi). When there is no clear relationship between discrepancy and average, Altman and Bland (1983 and 1986) suggested using the normal percentile to construct a 95% confidence interval for relative bias: (5 — 1.965,73 + 1.965), where Essentially, the central limit theorem is applied here. In the following chapter, we consider also nonparametric confidence intervals for median dis-crepancy. Altman and Bland (1983) proposed using transformation of the data if the "average-difference plot" indicates any relationship between the discrep-ancy and the average. However, no example has been shown. Indeed, an appropriate transformation may not be obvious. Also, if the discrepancies are symmetrically distributed, transformation that destroys symmetry may not be desirable. 2.8. Q U E S T I O N S B E Y O N D A L T M A N A N D B L A N D There is a serious issue not much dealt with even by Altman and Bland. D = X-Y, and i = i 19 For paired measurements A r, Y, scatter plot often shows heteroscedasticity — plots described in Chapters 5 and 7 demonstrate the issue dramatically. Some reliability techniques allow for heteroscedasticity (using weighted least-squares regression, for example); but the normal confidence interval above does not. More precisely, consider paired measurements Xi and YJ;, and the corre-sponding discrepancy £ \ , such that the variance Var(Di) = Var(Xi) + Var{Yi) - 2Cov{Xi,Yi) is a function of the true magnitude /i;; then how should we inter-pret a "sample" variance of D\, D2, • • •, Dn, or a "sample" correlation for (A"i, Yi), (X2, Y2), • • • , (Xn, K„), if the magnitudes n\ ? ^ 2 , • • • > have been ar-bitrarily or intentionally (but not randomly) selected? This thesis, using a perspective related to permutation tests, tries to estimate the relative bias without any assumption about the mechanism by which the items measured are chosen. 20 3. I N D E P E N D E N T , I D E N T I C A L L Y D I S T R I B U T E D D I S C R E P A N C I E S : A S I M P L E M O D E L Suppose ( X i , Yi), (X2, Y2), • • •, (Xn, Yn) are the pairwise observations ob-tained by two raters or two measuring processes on n objects. Then their discrepancies D\, D2, • • •, Dn are given by Di = Xi-Yi, i = 1,2, •••,!*. As discussed earlier, agreement or disagreement may be characterized primarily by a central location parameter such as the mean (that is, relative bias) or the median of the discrepancy distribution. In various contexts, we may wish to estimate this location parameter, say 6; or to provide a confidence interval; or to test a hypothetical value of this parameter (usually 9 = 0 would be of interest). Our statistical concern here is comparison of observers or of measurement methods — not the "true" magnitudes (say ^1, /J.2, • • •, fin) of the particular objects being measured, nor the separate distributions of X{,Y{ measurements (with, say, A'j = m + £,• and Yi = m + Si for some errors £i,6"i). But, in general, the underlying distribution of discrepancy D{ could depend on the magnitude /i, being measured. For instance, the standard de-viation of the discrepancy distribution may be proportional to the magnitude of the object being measured or to the square root of that magnitude, etc. The simplest model, which is the subject of this chapter, would assume that the discrepancy distribution is not a function of the quantity being mea-sured. This assumption, together with the independence assumption, models Dy, -D2, • • •, Dn as independent and identically distributed observations. This assumption holds, in particular, if the (X{,Yi) are independent and identically distributed random vectors. Even if the observed discrepancies £>i, Z?2, • • •,-On are independent and identically distributed, the underlying distribution, in general, is still un-known and may be in any shape. But an estimator, confidence interval or hypothesis test for the central location may be more or less efficient, relative to some other method, depending on whether the unknown distribution is symmetric or skewed, whether it is light-tailed or heavy-tailed, and so on. In this chapter, we consider alternative (or competitive) estimators, etc. 3.1. M E A N A N D t P R O C E D U R E S The expected value or mean of a distribution is the parameter most 2 2 often used to characterize central tendency. And the usual unbiased esti-mate of the distribution mean is the sample mean. Given a sample of discrepancies, D\, D2, • • •, DN, the sample mean, D, is defined to be D = mean of {Di, D2, • • •, DN} 1 N • - 5 > • rt —<* n J=l We would also like to construct a confidence interval (or interval esti-mate) for the distribution mean; hence, we need the distribution as well as the expected value of the sample mean D. If the discrepancies are normally distributed, then the sample mean, D, also will be normally distributed. However, since we need to use the sample variance * 2 = S > - * » s i=l to estimate the unknown variance of the normal discrepancies, a confidence interval for the location parameter of the discrepancy distribution is obtained based on the Student t distribution, with n — 1 degrees of freedom. Hence, a 1 — 2a symmetric confidence interval, {6LOW,9UP), is given by S S GLOW = D — tan-i—= and 6yp — D + tQ n_! —=, where tan_1 is the upper 100c* percentile point (or denotes the 100(1 — a) ordinary percentile) of the t distribution, with degrees of freedom n — 1. For large n, percentile i a ,n- i can be approximated by za, the corresponding percentile of the standard normal distribution. (In particular, zQ — 1.96 if a = 0.025.) To test the hypothesis Ho:0 = So versus Hi : d # 6>0, at 2a level of significance, we construct the t statistic and compare it with ta>n-i. We would reject HQ and conclude that 8 is significantly different from 9Q at the 2a level of significance if and only if If the underlying discrepancy distribution is not normal, but is at least symmetric, then the t test can be regarded as a permutation test, discussed below, although the nominal significance level would not be exact. Even without symmetry, the central limit theorem still would provide approximate normality for the sampling distribution of D, assuming only that the discrepancy distribution has finite variance. Thus, asymptotically, the situation would be the same as the previous case, and the confidence interval and hypothesis testing can be based on the t distribution as before. There is considerable literature on how non-normality affects the t statistic and confidence intervals. Notable are works by E. S. Pearson and Adyanthaya (1929), Geary (1936, 1947), Gayen (1949), Efron (1969), E. S. Pearson and Please (1975), and Cressie (1980). This literature indicates that asymmetry (or skewness) of the underlying distribution affects the distribution of t more than does the kurtosis (heavy- or light-tailedness). In the present context, the underlying distribution characterizes difference between two measuring processes, X and Y. If measurements X and Y have distributions of the same shape, but shifted — that is, if the processes have different biases, but the same variance — then the distribution of difference, X — Y", must be symmetric [Pratt and Gibbons (1981, p. 147)]. Thus, the t confidence interval and the t test for the discrepancies tend to be robust for inference with respect to measurement discrepancies. 3.2. M E D I A N A N D S I G N P R O C E D U R E S The median is another well known parameter characterizing the central location of a distribution. By definition, the median of the distribution of D is a point d such that Prob(D < d) < i < Prob(D < d). 25 Notice that the median, in general, is not uniquely defined. However, if the underlying distribution is continuous, then the median is unique and can be defined as that value d such that Prob(D <d) = i . For symmetric distribution, median and mean are the same value, the symmetry point (provided that the expectation exists and is finite). One simple estimator of the distribution median would be the sample median. Given a sample of discrepancies, D\, D2, • • •, Dn, the sample median, .D, is given by D = median of {D\,D2, • • • ,Dn} ^(s±iy if n is odd; 1 _ _ . j D(f) + £ >(f+i). if n is even. 2 Here, denotes the ith order statistic of D\, D2, • • •, Dn. Furthermore, a confidence interval can be obtained based on order statistics. Suppose ' D\, D2, Dn are the observed discrepancies; then a 1 — 2a symmetric confidence interval (0LOW,6UP) is given by 0LOW = D^n + \-ba) a n d 6UP = D(bay where ba is the upper 100a percentile point of the binomial distribution with sample size n and p = —. That is, bQ is the value such that Prob(B > ba) = a, where 5 ~ Bin ^n, 0 [Hollander and Wolfe (1973, pp.48-49)]. This binomial percentile point can be obtained from tables of the binomial distribution or of the incomplete beta function [e.g., Harvard (1955) or Owen (1962)]. For large n, the integer bQ can be approximated by n fn bQ « 2 + 1 + * a V 4 ' where za is the standard normal percentile (defined before). The value on the right hand side, in general, is not an integer, so in practice the closest integer is used. This gives a large-sample approximate confidence interval for the median discrepancy [Hollander and Wolfe (1973, p.49)]. If the problem of interest is to test the hypothesis Ho:8 = 60 versus Hx : 6 ± 0O, then we define the sign statistic B = >0O), where the indicator function > *> = {I:« D ( < 2: and reject iJo (to conclude that 9 is significantly different from 9Q) a t the lot level of significance if either B > ba or B < n — bQ [Hollander and Wolfe (1973, p.40)]. In computing the sign statistic B, above, 1(1?, > 9Q) has not been defined when Di — 8Q. We can avoid this difficulty, in theory, by assuming continuous distribution. In practice, measurements are not always sufficiently precise to avoid zeros, even if the distribution is continuous. For methods of handling zeros in the sign test, see Hemelrijk (1952), Putter (1955), Noether (1967), Krauth (1973), and Pratt and Gibbons (1981, pp.97-104). 3.3. T H E H O D G E S - L E H M A N N E S T I M A T O R A N D S I G N E D R A N K P R O C E D U R E S The procedures discussed below compromise between the t and sign pro-cedures: the underlying distribution should be symmetric (or approximately symmetric), but normality is not needed. And, as noted in Section 3.1, the discrepancies would be symmetrically distributed if the distributions of X and Y have the same shape and differ only by a location shift. For a sequence of observations D\, -D2, • • •, Dn, define the set of Walsh n(n + 1) averages, the m = quantities Di + Dj , i,j = 1,2, & t < j Then the corresponding Hodges-Lehmann statistic D is defined to be the sample median of these Walsh averages, that is D — median of | — — i , j = 1,2, • • •, n & i < j ^ . If the underlying discrepancy distribution is symmetric, then the Hodges-Lehmann statistic (the Walsh median) estimates this centre (= median = mean) [Hollander and Wolfe (1973, p.33)].. Moreover, a symmetric confidence interval for the symmetry point can be based on the Walsh averages: for confidence level 1 — 2a the interval (&LOW,0UP) is given by OLOW = W{M + 1 _ W A ) and 6UP = W / W ay where W^) < W(2) 5; •"' ^ W(m) denote the ordered Walsh averages, with 77(77. + 1) m = , and wQ is the upper 100a percentile point of the Wilcoxon signed rank statistic, whose exact distribution is available in tables [e.g., Owen (1962) or Pearson and Hartley (1972)]. For large n, the integer wa can be approximated by wr n(n + l) ln(n + l)(2n + 1) -^r- + l+Zav 2i ' w here zQ is the standard normal percentile and the right hand side is rounded to the closest integer value [Hollander and Wolfe (1973, pp.35-36)]. To test the hypothesis H0:6 = e0 versus Hx : 0 ^ 0O, we define the Wilcoxon signed rank statistic n W = ^2RiI(Dt> 0O), where i?{ denotes the rank of \Di\ in the ranking from least to greatest of absolute values \D\ |, |Z>21, • • •, \Dn\, and the indicator function We would reject i J 0 (to conclude that 6 is significantly different from 60) at the 2ct level of significance if either W > wa or W < m — wa [Hollander and Wolfe (1973, p.28)]. Zero values may be a practical problem for the signed rank procedure (as for the sign test). Also non-zero ties (two or more observations which 30 have the same magnitude) can cause complications for Wilcoxon signed rank procedures. For present purposes, "midranks" as defined by Lehmann (1975) may be used when ties render "rank" ambiguous. For discussion of zeros and ties, see Conover (1973), Cureton (1967), Pratt (1959), and Rahe (1974). 3.4. D I S C U S S I O N : T H E P E R M U T A T I O N P E R S P E C T I V E Obviously, normality is the most restrictive assumption considered above; and the ordinary median estimator and the sign test are the least restricted procedures, not even requiring symmetry. The signed rank procedures are intermediate. If the underlying distribution is symmetric, then all three approaches considered above — that is: the t test, sign test, and signed rank test — are permutation (or randomization) procedures, corresponding to different score functions. For a random vector D = (D\, D2, • • •, Dn), a permutation test statistic or "generalized Student's statistic" [Efron (1969)] has the form Sn = ^Ui, where the vector U = g(U), and D D U = \D\\ v T ^ T 31 and j is a symmetry preserving transformation of the unit n-sphere into itself. For instance, if y n(n + 1) is defined on the positive orthant S+ = { ( f n " ' * > fn) : & > O^XlLi £% = where Ri is the rank of among {£j, £2, • • •, £ n } a n d 9 maps every orthant into itself in a similar fashion, then Sn is the Wilcoxon signed rank statistic. The permutation perspective is important since, as pointed out in Sec-tion 3.1, the discrepancy distribution will be symmetric if the X and Y distributions have the same shape, but possibly differ by a location shift. Choice among different permutation procedures should be based on rela-tive performance as discussed in the next chapter. 3.5. B O O T S T R A P M E T H O D Efron's "bootstrap", related to Tukey's "jackknife", provides a general method to construct nonparametric estimators or confidence intervals [Efron (1979)]. The bootstrap method can be applied to estimate any parameter characterizing central tendency; but the bootstrap estimators of distribution mean and median turn out to be essentially the same as the ordinary sample mean and sample median. Hence, the general theory of the bootstrap is not utilized in the remainder of this thesis. (But discussion of the bootstrap method is provided in Appendix I.) The bootstrap has been modified to utilize partial knowledge about the underlying distribution of interest: but the "smoothed bootstrap" [Efron (1981)], the "parametric bootstrap" [Efron (1985)] and the "Bayesian boot-strap" [Rubin (1981)], will not be discussed here. 33 4. E V A L U A T I N G R E L A T I V E P E R F O R M A N C E OF C O N F I D E N C E I N T E R V A L P R O C E D U R E S Choice among competing statistical procedures should be based on their relative performance. In statistical literature, relative efficiency is most often defined in the hypothesis testing context. This chapter first reviews relative efficiency of tests and an equivalent definition of relative efficiency based on expected ratio of squared lengths of confidence intervals. Also, we propose two other criteria of relative performance: probability that one procedure produces a shorter confidence interval than the other procedure, and the relative variability of confidence interval lengths. (These criteria are easier to interpret than other relative efficiency definitions, such as by Bahadur, Hodges-Lehmann, or Chernoff, etc.) 4.1. R E L A T I V E E F F I C I E N C Y OF STATISTICAL TESTS Relative efficiency of two statistical tests (e.g., sign test and signed rank test) in general would depend on i. the specified significance level, ii. the alternative hypothesis value, 34 iii. the sample size, and iv. the form of the underlying distribution. Pitman (1948) defined an asymptotic relative efficiency (ARE) which depends only on the form of the underlying distribution, but requires sym-metry. Essentially, Pitman efficiency of test procedure T2 with respect to Ti , denoted here by ARE(2,1), is equivalent to the limiting ratio of sample sizes, — , such that both tests achieve equal power against a sequence of n2 alternatives that are "close" to and approaching the null hypothesis [Randies and Wolfe (1979, pp.142-144)]. Note that, if ARE(2,1) is the efficiency of T2 relative to 7\, then ARE(1,2) = -. Pitman efficiency can be ARE(2,1) represented as a ratio of efficacies, defined in Appendix II and evaluated in Table 1 for permutation tests applied to familiar distributions. Based on Pitman efficiency, statistical literature (notably literature on nonparametric methods) gives the following general recommendations for the t test (T), sign test (5), and. Wilcoxon signed rank test (W) [Randies and Wolfe (1979, pp.166-168)]. o T is optimal for normal distribution and performs well for other distributions with moderate tails (e.g., logistic distribution); 35 o T is preferable to S and is comparable to W for distributions with light tails (e.g., uniform distribution); o T is inferior to both 5 and W for distributions with heavy tails (e.g., Cauchy or double exponential distribution); o S is preferable for distributions with very heavy tails; o W is intermediate between S and T, and therefore is "robust" in the sense of being a good compromise. However, there are few guidelines for a mixture of normals or for a contaminated normal distribution, which will be of interest in the next chapter. 4.2. R E L A T I V E E F F I C I E N C Y O F C O N F I D E N C E I N T E R V A L P R O C E D U R E S In estimation or data analysis context, Pitman ARE can be interpreted in terms of lengths of confidence intervals. Suppose Liyn and Li,n a r e the respective lengths of confidence intervals for corresponding to tests T\ and T2, respectively, and both based on the same sample of size n. If T\ 36 produces a confidence interval with expected length less than that produced by T 2 , then we say Tj is more efficient than T 2 . It can be shown that, under suitable conditions, ( j converges to ARE(2,1) in probability, as n oo [Pratt and Gibbons (1981, p.376)]. It follows that Eg L\,n L2,n ARE(2,l) , or equivalently Eg (jj^) — + y/ARE(2,l) In confidence interval context, it seems natural to consider lengths rather than squared lengths. Thus, the asymptotic expectation of •— (suppressing L2 notational dependence on n) is an important criterion of relative performance. Pratt (1961) provided another interpretation for ARE: relative probability of including a false value. Pratt showed that E9O (8UP - 9Low) = I Pe0 (BLOW <0< 6UP)d6 Je = j Pe0(0Low<0<0up)d6, where {0LOW-,8\JP) is a confidence interval for 8, and 80 is the true (but unknown) value of 8. Notice that the last integral gives the probability of including a particular false value and "averages" over all possible false values [Pratt and Gibbons (1981, p.50)j. 37 Recall that Pitman ARE provides an asymptotic comparison. How applicable are these asymptotic results for the finite sample size n? Monte Carlo results (see below) show that, in general, the finite-sample behaviour is reasonably close to asymptotic results; but anomalies may arise. 4.3. O T H E R C R I T E R I A O F R E L A T I V E P E R F O R M A N C E Instead of comparing lengths in expectation, we can compare them in probability: that is, consider the probability that procedure Ti produces a confidence interval shorter than that produced by T2. If L\ < L2 with probability much greater than 0.5, i.e., if Prob0 (Li < L2) > 0.5, then procedure T x may be preferred to T2 even if their ARE is close to 1. Besides considering which confidence interval is shorter in expectation or in probability, we would also like to have an interval whose length has relatively small variance. Thus, relative variability (or inversely, stability) of confidence interval lengths provides another criterion of performance. If the standard deviation of L\ is much less than that of L2, i.e., if SD(L2) ^ ' 38 we would conclude that T\ performs better than T2. Unluckily, exact theoretical results for the probability and relative vari-ability criteria are very difficult, if not impossible, to obtain. For instance, if the confidence intervals related to the sign test and to the Wilcoxon signed rank test are compared, difficult integrals based on certain joint distributions of order statistics are required [Sarhan and Greenberg (1962)]. In order to study these three criteria for specific distributions, Monte Carlo simulations are needed to approximate the theoretical exact results. In summary, we would prefer Tj to T2 if ii. P(Li < L2) > 0.5, and ... SDiLx) But, in Monte Carlo studies below, there are examples in which one criterion favours Ty while another favours T2. Also, there are examples of distributions for which 39 o E{L1/L2)^l and SD(Li)/SD(L2) « 1, but P(L a < L 2 ) > 0.5, or o E(L1/L2)&1 and P(Lj < L 2 ) « 0.5, but SD(Ly)/SD(L2) < 1. So it is clear that the three criteria, above, do not imply one another; and, in particular, both of our new criteria may have practical importance: to choose between two procedures when relative efficiency is approximately equal to 1. 4.4. E V A L U A T I O N O F T H E T H R E E P E R F O R M A N C E C R I T E R I A F O R S P E C I A L D I S T R I B U T I O N S : M O N T E C A R L O R E S U L T S In order to examine the relevance of asymptotic results and to eval-uate the above criteria for specific distributions, we consider the following Monte Carlo studies. One thousand random samples, each of size n = 32, are generated from each of eleven distributions: standard normal A/"(0,1), uniform( —1,1), Cauchy (or t with one degree of freedom), equal-proportion mixtures of four and five normals, where each component is jV(0,i2) with i = 1,2,3,4 and i — 1,2,3,4,5, and six contaminated standard normals. (Mixed and contaminated distributions are useful to approximate various sit-uations in which the assumption of "identically distributed observations" is 40 not valid. Efficiency results for mixtures also demonstrate importance of the new performance criteria.) Notice that the normal, uniform, and Cauchy are examples of distributions with medium, light, and heavy tails, respectively. Since the three criteria of relative performance do not depend on the dis-tribution parameters (except for the mixtures of normals and contaminated normals), there is no loss of generality in considering standard normal, uni-form, and Cauchy. For interval estimation corresponding to the exact sign test (not using normal approximation), exact 95% coverage in general is not attainable (without randomization); but for sample size n = 32, coverage of 94.98% is an attainable level. Also, n = 32 is close to sizes of some real data sets considered below. For the distributions just described, we consider confidence intervals cor-responding to the t test (T), the sign test (5) and the Wilcoxon signed rank test (W). All three confidence interval procedures are available in the Minitab (version 5) statistics package [Ryan, Joiner and Ryan (1985)], but for present purposes have been programmed in the "S" statistics environ-ment under UNIX [Becker and Chambers (1984)]. For these three confidence interval procedures, asymptotic relative efficiency (ARE) and simulation re-sults (for 77 = 32) are shown in Tables 2 - 6 . (Note that Table 2 gives ARE(W,S) while ARE(S, W) = A R E ^ W s^ i s t h e W : 5 squared length ra-41 tio, given by Table 3; etc.). Table 3 gives length and squared length ratios for finite samples (n = 32), results directly comparable to the asymptotic ratios in Table 2. Tables 4 and 5 compare intervals using variability and probability criteria for finite samples. Table 5 also gives actual coverages of the nominal 95% confidence intervals (each entry based on 1000 samples, with sample size n = 32). 4.4.1. C O M P A R I S O N OF A S Y M P T O T I C A N D F I N I T E - S A M P L E RESULTS o In the Monte Carlo experiment with 1000 replications, most of the actual coverages are within 1% of the nominal coverage, 95%, except for the T interval with Cauchy distribution. Observed coverage of T for Cauchy distribution (98%) confirms that T is conservative for a long-tailed distribution [Benjamini (1983)]. o In general, if one procedure is preferred to another by Pitman's asymptotic relative efficiency criterion, then the same preference holds for n — 32. But the "advantage" may be consistently and considerably less (or more) for finite samples than asymptotically. For instance: "advantage" of W over S is less for n = 32 than 42 asymptotically; "advantage" of T over S is less for n = 32 than asymptotically; and "advantage" of W over T is less for n = 32 than asymptotically, for most distributions. o Reversals are technically possible, an example being the mixture of four normals: asymptotically W is more efficient than S (the length ratio L\v/Ls —* \/0.9693 < 1 as n —• oo); but for n — 32, the Monte Carlo sample mean of (Lw/Ls) = 1.0402. 4.4.2 E X A M P L E S : N E W C R I T E R I A M A Y B E D E C I S I V E Recall that, in addition to the usual definition of relative efficiency as expected ratio of squared interval lengths, we wish to consider the probability that one procedure produces an interval shorter than the other and the relative variability of interval lengths. In the simulation results, we evaluate not only the sample mean of the ratio of lengths, but also the percentage of times such that one interval is shorter than the other, and the ratio of sample standard deviations of lengths. The following examples call attention to criteria other than usual relative efficiency. 43 E X A M P L E 1A: Probability criterion may be critical. Consider W and T for uniform distribution. The Monte Carlo experi-ment with 1000 replications gives 1 3 ( ^ * 1 . 0 9 * 1 , and £ £ ^ « 1 . 1 9 « 1 , \LT J SD(LT) which do not suggest strong preference for T. But since LT < L\y f ° r 968 out of 1000 simulated samples, T would be preferred by the probability criterion. E X A M P L E IB: Probability criterion may be critical. Consider W and T for the mixture of five normals. The Monte Carlo experiment with 1000 replications gives i s f ^ * 0.92*1, and ^ 4 « 1.01 » 1, \LTJ SD{LT) which do not suggest strong preference for W. But since LT < Lw only for 232 out of 1000 samples, W would be preferred by this criterion. These examples suggest that the probability criterion is more sensitive than the expectation criterion (usual relative efficiency) with respect to the shape of the distribution tails. 44 E X A M P L E 2A: Variability criterion may be critical. If W and S are compared for the mixture of five (or four) normals, simulation results give E (j^J ~ 1-06 (or 1.04) « 1, and LW < LS for 470 (or 5 1 8 ) , « 50% of 1000 samples. But since, for both mixtures, SD(Lw) the relative variability of interval lengths clearly favours W, although the other two criteria do not provide any clear preference. E X A M P L E 2B: Variability criterion may be critical. For the mixture of five normals E (j^j ~ i - 1 8 ' a n d LT < LS for only 337 out of 1000 samples, so these two criteria slightly favour S. But SD(LT) SD(Ls) 45 0.64 < 1 and supports use of T. We might accept slightly greater length in order to reduce variability of our confidence interval. E X A M P L E 2C: Variability criterion may be critical — performance of W versus T for contaminated normals. As discussed in Section 4.1, based on Pitman efficiency, T is slightly preferable to W for [pure] normal distributions, with ARE(W, T) = 0.95 or, equivalently, —— —* 1.02; and simulation results for n = 32 give However, normality is a strong assumption — in practice, a small percentage of contaminant is common (e.g., a small percentage of measurements with large errors; or a small percentage of time when a process is "out of control"). For example, consider simulation results for the standard normal contam-inated with 5% of Af(0,16): LT LT < Lw for 731 out of 1000 samples, and SD(LW) SD (LT) 1.13. and 46 LT < L\v for 393 out of 1000 samples, so these two criteria slightly favour W. But since SD (LW) SD (LT) ~ 0.49 < 1 the relative variability of interval lengths strongly favours W. 4.5. D I S C U S S I O N Pitman ARE, which is equivalent to the limiting ratio of squared lengths of two confidence interval procedures, is not the only measure of relative performance. Fortunately, in our examples, the asymptotic efficiency generally is similar to the corresponding expected ratio for finite sample size. But, in addition to expectation of the length ratio, we may wish to consider a) the probability that this length ratio exceeds one, and b) the relative variability of lengths of the two confidence intervals. In Monte Carlo simulation, these three criteria of relative performance generally complement each other, but there are instances which spotlight the differences. Overall, signed rank confidence interval W is generally good: even in the worst case, the longer intervals produced by W have expected length not much longer nor much more variable than the competitive procedures. 47 4.6. E P I L O G U E : A D A P T I V E P R O C E D U R E S The foregoing discussion is from a traditional perspective in which the statistician's choosing of a statistical procedure (from among T, 5, and W confidence interval procedures, or from any other collection) is separated from application and computation. But there have been several suggestions that would formally unify these aspects of statistical analysis. In principle, we could o calculate several confidence intervals (e.g., T, S, and W intervals) for a given data set; and then o calculate some auxiliary statistic(s) and invoke a formal decision rule based on such statistic to select and report one of the available intervals. The simplest such suggestions have been the following. o Adaptive procedures of Randies and Hogg (1973): if one method is generally good for data from light-tailed distributions and an-other method is good for heavy-tailed distributions, then calculate a statistic that quantifies tail weight and adopt a formal cut-off value. This idea obviously generalizes from two competitors to several. See 48 Appendix III. o "Legalized cheating" proposed by Efron (1969): select the first of two available confidence intervals if and only if their length ratio —- < 1. Efron suggests appropriate adjustment of confidence level, working with sign and t procedures. These ideas are intriguing and we hope to consider them in the future — but not in the remainder of this thesis. They present both practical and theoretical problems. Adaptive procedures introduce additional decision parameters; and also they may be computationally difficult. At present, popular statistical com-putation packages, such as SAS, BMDP, and SPSS do not even provide S and W confidence intervals, corresponding to the sign and Wilcoxon signed rank tests; and Minitab made these confidence intervals available only in 19S5 — using normal approximation to Wilcoxon confidence interval [Ryan, Joiner and Ryan (1985, p.290)]. Little is known about relative performance criteria to choose among adaptive options. And little is known about behaviour of such procedures when data are not identically distributed, as considered in the next chapter. 49 5. N O N - I D E N T I C A L L Y D I S T R I B U T E D D I S C R E P A N C I E S : A G E N E R A L M O D E L The simplest model assumes that discrepancy between two measuring procedures does not depend on the quantity measured, so that observed dis-crepancies are independent and identically distributed. This chapter considers a more general model: discrepancies that are independent, but need not be identically distributed. In particular, the variance of a discrepancy distri-bution may be a function of (for instance, proportional to) the magnitude of the object measured, so that distributions of D\, Di, • • •, Dn differ with respect to scale. But, in order to have a meaningful concept of agreement to estimate or to test, the discrepancy distributions must have some location parameter in common; and this chapter will assume that the non-identical discrepancy distributions have identical means or medians. Again note that these two parameters would be equal if the discrepancies are symmetrically distributed; and that Di = X{ — Y{ will have symmetric distribution whenever the X{ and Yi distributions have the same shape, differing only by some shift (or relative bias). Variance or other scale parameters for X{ and Yi distributions still could be a function of the magnitude of the object measured. (In 50 practice, the assumption of sampling with a common median or mean often may be valid for discrepancy rate — difference between measurements as a percentage of quantity measured — considered in the next chapter.) This chapter first considers empirical (largely graphical) methods to explore or validate distributional assumptions about Di,D2, • • •,Dn, such as symmetry, functional modelling of variance, etc. A "random walk" model is then proposed to provide theoretical basis for approximate normality (and symmetry) of discrepancy distribution when X and Y are two counting processes. Assuming symmetry, choice among the three statistical procedures is discussed by comparing the non-identically distributed case to sampling from a suitable mixture. Finally, a summary guide is provided to estimate median discrepancy for real data. 5.1. G R A P H I C A L M E T H O D S F O R D I S C R E P A N C I E S Scatter plotting of X{ and Yi values and calculation of the usual (Pearson) correlation are commonly used to check for linear relationship and to demonstrate that one variable can predict the other. But when X and Y processes measure the same objects, strong positive correlation (i.e., high reliability) is usual — otherwise the measurements would be useless in 51 practice. For strongly linear X, Y points, visual resolution of discrepancy is poor. Also, the calculated correlation may be misleading: the greater the range of magnitudes measured, the better agreement will appear to be. See Altman and Bland (19S3) and the discussion of correlation above, in Chapter 2. A more effective graphical method for assessing agreement between mea-A -(- Y surmg processes is to plot Y — A against the average measurement — - — [Altman and Bland (1983)]. This leads to an "average-difference plot", which is a variation of the "sum-difference graph" attributed to Tukey by Cleveland (1985, pp.118-23). The average is used here in lieu of the sum in Tukey's graph, because of its obvious interpretation — as a combined estimate of the magnitude measured separately by X and Y. Either the average-difference plot or the sum-difference graph rotates a scatter plot 45° in a clockwise direction and then expands the rotated points vertically to fill the plotting region [Cleveland (1985, p.122)]. Notice that the difference Y{ — Xi and the sum A,- + Yi are uncorrelated random variables if A, and Yi have the same variance (similarly for Xi — Y,- instead Yi — A, , and for average (X,-+ Yi)/2 instead of Xi + Yi) and hence are independent if X,-,Yi are bivariate normal. Or, equivalently, the variances of Xi and Y, cannot be equal if the difference Xi — Yi and the sum A,- + Yi are not uncorrelated. It is sometimes useful also to plot absolute value \X — Y\ versus average or sum. For some data it may be useful to plot the abscissa on a non-linear scale, such as logarithm or square root of X + Y. (See the first example in Chapter 7.) Using these plots, we can visually assess the range of measurement and check for symmetry of discrepancy; and we can see if there is any trend, for instance, whether and how magnitude of discrepancy increases with the (estimated) magnitude of the object measured. If D\, D2, • • •, Dn are independent and identically distributed, then verti-cal scatter in the average-difference plot should be about the same over any horizontal interval, regardless of the interval's position along the abscissa. And if the data are symmetric, then scatter in the average-difference plot should be symmetric about a horizontal line. (We can also check symmetry using a q-q plot or normal probability plot of the sample distribution, as well as a histogram.) That is, symmetric, identically distributed data tend to fill a rectangle in the average-difference plot. If instead the standard deviation of the discrepancy distribution is proportional to the magnitude of measurement, 53 then the average-difference plot will exhibit a "shotgun" pattern out in a triangle, symmetric about a horizontal line. — spread Similarly, if the standard deviation of the discrepancy distribution is proportional to the square root of the magnitude measured, then the average-difference plot will diverge like a root function; or a plot of difference versus the square root of the average will fan out linearly. (This plot was used by by Professor Ned Glick for the logging data in Chapter 7.) The following section develops a corresponding theoretical model. 5.2. D I S C R E P A N C Y B E T W E E N T W O COUNTING P R O C E S S E S : R A N D O M W A L K M O D E L Suppose X and Y are integer counts of the same lot of items; for instance, in data analysis considered in Chapter 7, X and Y may be counts by two inspectors of the same batch of logs or of cells on the same laboratory slide. Then the discrepancy X — Y may be considered as a sum (over items) of random differences. Such sequence is equivalent to "steps" in a "random walk" — the walk observed only once, after an unknown number of steps (corresponding to the number of items in the batch). This model was suggested by Professor Ned Glick in analysis of the • 54 logging data considered in Chapter 7. As shown here, this model implies that variance of discrepancy should be proportional to batch size. Consider a batch containing B items. For each item or piece in the batch, the counting process X may miss that piece, may count it correctly once, or may double count it, etc., with some (unknown) probability distribution whose expectation is p and whose variance is r. Then the X count can be treated as the sum of these successive contributions: X = ]Cf=i -Pfc, where Pk is the contribution of the kth piece in the lot; and If the piecewise contributions Pk are independent, then Notice that X is a binomial random variable, with parameters B and p, if there are no multiple countings — that is, if necessarily each Pk = 0 or 1. Moreover, if X is sum of independent and identically distributed piece counts, then the central limit theorem implies approximate normality — X ~ Af(Bp, Br) approximately, if the batch size B is large. Similarly, a second count Y is approximately normal, Af(Bq, Bs), where q and 5 are the piecewise expectation and variance for the process Y. 55 Thus, discrepancy D = X — Y is approximately normal, M(Bp — Bq,v), where variance v = Var(X - Y) = Var(X) + Var(Y) - 2Cov(X, Y) = Br + Bs-.2Cov{X,Y) = B[r + s-2p(X,Y)y/rl}. In particular, the random walk model implies that discrepancy between two counting processes would be symmetric, with variance proportional to batch size. 5.3. P E R M U T A T I O N P R O C E D U R E S F O R N O N - I D E N T I C A L L Y D I S T R I B U T E D O B S E R V A T I O N S Three approaches already considered for assessing median discrepancy in the independent and identically distributed case are: the Student t test, the sign test, and Wilcoxon signed rank test — and the three corresponding confidence interval procedures. As noted in Chapter 3, symmetry implies that all three approaches are permutation procedures corresponding to distinct score functions. It is well known that the two nonparametric approaches, sign and signed rank procedures, remain valid for non-identically distributed observations. (The sign test and corresponding confidence interval do not 56 even require symmetry.) [Pratt and Gibbons (1981, p.87 and p.155)]. Efron (1969) also showed that the t test, and hence the t confidence interval procedure, remain valid (and conservative) for non-identically distributed symmetric observations. Thus, under the symmetry assumption, all three approaches still can be applied for non-identically distributed observations. But relative performance criteria to choose among these permutation procedures have been developed (see Chapter 4) only for the independent and identically distributed case. 5.4. M I X T U R E S A M P L I N G A P P R O X I M A T I O N T O N O N - I D E N T I C A L L Y D I S T R I B U T E D D A T A In order to make the relative performance criteria applicable, one might hope to generalize the criteria for non-identical distributions; unfortunately, there is no clear generalization yet. An alternative, developed below, in-volves modelling (or approximating) a sequence of non-identically distributed observations by an independent and identically distributed sequence from a mixture. Suppose that the data D\, D2, • • •, Dn combine m observations from one distribution and n — m observations from a second distribution. That is, 57 suppose that .Di, £ > 2 , • • • ,-D n are an arbitrary permutation of D\, D'2, • • •, D'm, D ' J n + 1 , • • •, D'„, where D[,D2, • • •,D'm are m independent and identi-cally distributed observations from F' and D'^nJrl, D'L+2, • • •, D'^ are n — m independent and identically distributed observations from F". Then the com-bined data are not identically distributed; but, with high probability, the data look like an independent and identically distributed sequence drawn TTi 72 — TYl from a mixture of F' and F" with proportions — and , respectively. n n For sampling from a mixture, — would be the expected rather than the n actual fraction from F' (with the distinction vanishing as n gets large and the observed proportion converges to its expected value); and for mixture sampling, the permutation of the 7J>i, D2, • • •, Dn would be rigorously random rather than arbitrary. But, if it is difficult in principle to distinguish whether independent data D\,D2,- • • ,Dn are the product of simple random sampling from a mixture or of another sampling process involving non-identical distributions (with common point of symmetry), then it seems reasonable to base estimation of the symmetry point on" procedures appropriate for the simple mixture sampling. Obviously, this idea generalizes from a mixture of two symmetric distributions to a mixture with three, four, or many components. A sufficiently rich mixture could approximate the random walk model for 58 discrepancies between counting processes. Fisher (1955) questioned whether real data ever correspond to "repeated sampling from the same population" and described that assumption as one of the "products of the statistician's imagination". Fisher might have used conditioning (on the observed proportions, etc.) to argue that inference for independent and identically distributed mixture sampling should be the same as for a permutation of non-identically distributed observations. On the other hand, our perspective regards independent and identically distributed sampling from a mixture as a mathematically tractable approximation to the general case. Notice that the confidence interval based on the mixture would be conservative, because the variance of data from a mixture distribution is greater than that of corresponding data from deterministically non-identical distributions. Suppose U is distributed as a mixture of k symmetric distribu-tions with densities fjj^, fjj , • • •, fjjk, having a common point of symmetry (assume E(U-) = 0, for i = 1,2, without loss of generality) and with weights (or expected proportions) u>x, u2, • • •, ojk, where 0 < c j , < 1 and 5Zi=i ui ~ 1- Then the probability density function of 17, is given by 59 fu(u) = S i = i u i f T J i ^ u ) - ^ follows that /oo roo fc fc /• oo u2fu(u)du = / u 2 ^  u>ifTj.(u)du = ^ a;,- / u2fjj.(u)du -oo J — oo j_ j J— oo k where <7 2 = Var(rj,) = J^^u2fjj.(u)du. And suppose that V comes from a fixed permutation of independent data from non-identical densities / f / i ' fu2' " '' -fa*: w i t h a c t u a l proportions , u>2, • • •, ojk. Then F ~ E ! = i ^ ^ and ( fc \ fc fc fc ^UiUx =^2u2Var(Ut) = ^ u 2 a 2 < ^ T C J . C T 2 = Var(U). 1=1 / :=1 1=1 j ' = l 5.5. S U M M A R Y G U I D E T O E S T I M A T I O N O F M E D I A N D I S C R E P A N C Y F O R R E A L D A T A Monte Carlo simulations have been used to compare performances of the three permutation procedures for particular mixtures (using the performance criteria of Chapter 4). Note that the Wilcoxon signed rank methods perform well for a great variety of normal contaminations or mixtures. In summary, the following steps are recommended for estimation of median discrepancy for real data. 60 Use graphical methods, especially, the average-difference plot for first evaluation of simple assumptions and models. If the data clearly are not symmetric, then sign procedures may be the only valid option. If, however, we can regard the data as symmetric, then check whether discrepancy variance seems to be constant or a function of the size of measurement. If the average-difference plot is consistent with constant variance and independent and identically distributed data, then use normal probability plot, tail-weight statistic, etc., to decide whether the distribution has light, moderate, or heavy tails — and accordingly choose among the permutation procedures (t, sign, or signed rank). If discrepancy variance is not constant, but increases with the magnitude measured — especially for discrepancies between integer counts — consider mixture models. The signed rank confidence interval is robust in senses noted above. But if length of the signed rank confidence interval greatly exceeds 61 length of the t or sign confidence interval, then it may be worth-while to consider (by fresh Monte Carlo simulations) specialized non-normal mixture models. 62 6. D I S C R E P A N C Y O R D I S C R E P A N C Y RATE? In contexts where the items measured range from very small to very large magnitudes, it is often preferable to express discrepancy as a rate: for instance, a discrepancy of 50 for a shipment of size 100 is very different in importance from the same amount of discrepancy for a shipment of size 5000. More importantly, since in some cases the items being measured are not randomly sampled, but arbitrarily or intentionally chosen over a wide range, a discrepancy rate may be a more relevant comparison or more intrinsic characterization to describe the relative bias of two measuring processes. This chapter discusses discrepancy in this relative sense. As in previous chapters, suppose A, = Hi + ei and Yi = Hi + f ° r true magnitude Hi and random errors £,• and 8{. Then the discrepancy is D, = X{ - Yi, and we may denote the discrepancy rate by Hi It follows that E(Rt) = E Di E(Di) and Var(Rt) = Var Hi . 'Ei . Vi J 63 Hi Var(Dj) In particular, if Var(Dt) is directly proportional to magnitude Hii then Var(Ri) is inversely proportional to Hi-Thus, if Hi were known, inference on discrepancy would lead easily to inference on discrepancy rate. But, of course, Hi 1S n°t known — otherwise there would be no need X- + Y-for X, , Yi measurements. One natural solution is to use Hi = —' ~ i n 2 place of the unknown Hi- This leads to a practical issue: how relevant is Ri = El to inference about — ? Hi Hi IX — Y\ Suppose = \R\ <C 1, or, equivalently, |X — Y\ <C H- Then small A* perturbations in the denominator do not substantially alter the ratio. In practice, measurement errors should be small relative to the magnitude measured; and difference between two small errors should be very small. Hence the magnitudes in numerator and denominator of the discrepancy rate are so different that uncertainty in the denominator (due to estimation) usually is irrelevant. 64 7. A P P L I C A T I O N S : E X A M P L E S O F D I S C R E P A N C Y D A T A Several specific contexts for assessing agreement are presented in this chapter. For some examples, discrepancy data are analyzed in detail; for other examples, we just describe the context and note whether discrepancy variance is proportional to the measured magnitude or to its square, etc. E X A M P L E 7.1: Counting logs. This thesis was motivated, in part, by certain questions about counting and "scaling" (measuring volumes) of logs in the British Columbia forest industry. Evidence included data on certain shipments of logs that were counted and "scaled" twice: first at a central facility and again at various destinations. From paired counts, discrepancies can be found by subtraction. Median or relative bias of discrepancy rate was relevant to financial claims. In particular, two data sets are considered here: 166 batches of logs processed prior to change of the central facility's counting and scaling procedure in October 1981 ("old" logs), and 93 batches of logs after October 19S1 ("new" logs). These data are presented in Data Sets 1.1 and 1.2. Generally speaking, these data show that the source counts tended to be slightly below the destination counts in the "old" ' period, but slightly above 65 the destination counts in the "new" period. The "old" log discrepancies range roughly from —200 to 120, with more negative than positive; while the "new" log discrepancies range roughly from —150 to 225, with positive and negative discrepancies more or less balanced. See the average-difference plots in Figures 3 and 4. Several parties were interested in discrepancy rates for these data; and because of the substantial dollar amounts involved, point estimation and confidence intervals were of great concern (while hypothesis testing was less important). Professor Glide's analyses considered data subsets determined by calendar year, species of log, and so on, as well as the "old" and "new" data overall. The scatter plots of source counts versus destination counts for the "old" logs and "new" logs are given in Figures 1 and 2, respectively. Both scatter plots indicate high correlation — with data points tightly along a straight line. But recall that high correlation does not necessarily suggest strong agreement in the present context; see Chapter 2. Figure 1 seems to suggest heteroscedasticity — that the amount of variability of discrepancy increases with the batch sizes for the "old" logs, 66 but the trend in not clear. This phenomenon is even less obvious in Figure 2 for the "new" logs. However, heteroscedasticity is clear in two average-difference plots, Figures 3 and 4, respectively — the "shotgun" pattern, spreading out like a root function, suggests that the variance of discrepancy is proportional to the batch size. This proportionality phenomenon can be displayed more clearly in a plot of difference (X — Y) versus square root suggests a "random walk" model that is compatible with these graphical results and that could be approximated by a mixture of normals which differ only in their variances; see Chapter 5. Since the batch sizes cover a large range (roughly from 100 to 3000) and since the batches are not randomly, but arbitrarily chosen, it would be preferable to consider discrepancy rate rather than the simple difference; see Chapter 6. The normal probability plot of the "old" discrepancy rates, provided in Figure 7, exhibits a certain degree of linearity (except for one outlier, with discrepancy rate roughly 28%) although, as just noted, the average-difference plots, Figures 5 and 6, indicate that the data are not identically normally distributed. This normal probability plot is fairly similar to plots for data simulated from the scale mixtures of four and five normals (mixtures discussed in Chapter 4, although probability plots for simulated of the average see Figure 5 and 6. The context (counting) 67 mixture sampling have not been shown). Hence, for these data (and for subsets of these data), the Wilcoxon signed rank procedure is preferable to its competitors, based on the simula-tion results in Chapter 4. This preference is more-or-less compatible with the tail-weight statistic used by the adaptive procedure mentioned in Appendix III: the statistic Q* = 3.06, while 2.92 is the suggested boundary value between "moderate" and "heavy" tails. For all 166 "old" batches of wood, the Wilcoxon signed rank interval, with 95% confidence, estimates that the median discrepancy rate (source count minus destination count) is negative, with magnitude interval 1.28% to 2.83%; see Table 7. Student t and sign confidence intervals also were calculated for the 166 "old" batches of logs. Tables 7 and 8 show that length ratios of these intervals relative to Wilcoxon signed rank interval are very similar to corresponding interval length ratios for scale mixture of four or five normals studied in Chapter 4. These ratio results provide further support for applying to these data the "random walk" model and the corresponding scale mixture approximation. The normal probability plot of the "new" discrepancy rates, provided in 68 Figure 8, shows that the five smallest and the two largest rates are potential outliers, which would be deleted for further analysis. The probability plot of the remaining rates, given in Figure 9, indicates a complicated mixture, with some skewness to the right. Although simulation results show that the Wilcoxon signed rank procedures are robust over a wide variety of distributions, asymmetry and heavy-tailedness (classified by the Q* tail-weight statistic) make the sign procedure a good choice for the "new" logs. For 86 batches of "new" logs (after deleting outliers), the sign confidence interval, with 96% confidence, estimates that the median discrepancy rate (source count minus destination count) is 0% to 0.34%; see Tables 9 and 10. Note that for interval estimation using the exact sign procedure with sample size n = S6, an exact confidence level 95% is not attainable, and 96% is as close as possible. E X A M P L E 7.2: Fuse-burning times. Grubbs (1948) gave burning times (in seconds) of 30 powder train fuses reported by three observers, say A, B, and C. Since one burning time for observer B was lost, this example only considers data for observers A and C, whose times are provided in Data Set 2. Scatter plot and average-difference 69 plot are provided in Figures 10 and 11, respectively. Notice that although the correlation between burning times recorded by observers A and C is high (0.99), the average-difference plot does show some systematic disagreement between the two observers. Grubbs (1948) used a components-of-variance model, assuming that er-rors are unrelated to the times measured and are identically distributed for all observers. He partitioned variation into two components: due to fuse variation, and due to observer error. However, the average-difference plot, showing a "shotgun" pattern, suggests that the standard deviation of discrepancy may be proportional to the size of measurement, and hence that the validity of Grubbs' assumption is questionable. Indeed, Grubbs did notice that "errors of measurement (e) in some cases increase with increasing magnitude of the characteristic measured (a:)". But he assumed that ux and e are sufficiently independent to insure that limited variations in x are not reflected in the errors of measurement". Such assumption has often been made in literature of agreement prob-lems. This example draws our attention to the need for considering this assumption more seriously and for finding appropriate methods when the 70 assumption does not hold. E X A M P L E 7.3: Systolic blood pressure readings. Systolic blood pressures (in mm Hg) measured by two different methods on 25 patients were used in a textbook example of correlation by Daniel (19S3). This example also was discussed by Altman and Bland (1983); the data are listed in Data Set 3. Scatter plot and average-difference plot are given in Figure 12 and 13, respectively. Note that although the correlation coefficient between readings by the two methods is high (approximately 0.95), this does not imply agreement between the methods, in the sense of low relative bias; in fact, disagreement is clear in the average-difference plot. The average-difference plot also exhibits a "shotgun" pattern and hence calls into question the assumption of error distribution with constant variance. E X A M P L E 7.4: Spinal curvature — angular data. Spinal curvature, which is often used as a clinical assessment of scoliosis, can be described by two angles, viz., the Ferguson angle and the Cobb angle. The data in Data Set 4 come from a study comparing these two angles 71 for n = 26 patients [Robinson and Wade (1983)]. Predictability of one angle from the other angle seems to be the primary interest, but relative bias also would be interesting. Scatter plot and average-difference plot are given in Figures 14 and 15, respectively. This average-difference plot exhibits a pattern like an ordinary X, Y scatter plot and differs from all the average-difference plots consid-ered above. This implies that error variances for Ferguson and Cobb angle measurements are not equal; see Chapter 2 and Altman and Bland (1983). Further study of replicated Ferguson measurements and replicated Cobb mea-surements, on the same patients, likely would show greater reliability (higher correlation) for one method relative to the other, and hence may suggest practical preference for one method. Disagreement between the two angles is clear (even though correlation coefficient is 0.95) — Cobb angle is uniformly greater than the corresponding Ferguson angle. Moreover, relative bias of discrepancy obviously increases with the size of the angle, as shown in the average-difference plot. Hence it is not clear that any estimation methods considered above would be appropriate. And when the Cobb angle is regressed on the Ferguson angle, as by Robinson and Wade (1983), interpretation of the intercept is 72 questionable. Also, since both the Cobb and Ferguson angles are measured with errors, it is inappropriate to apply the usual regression; see Chapter 2. The plot of residuals versus the Ferguson angles, provided in Figure 16, indicates that the residuals do depend on the Ferguson angle. E X A M P L E 7.5: Oxygen levels for newborn infants. This data set comes from a study of newborn infants, comparing a "con-taining" position in a hammock with the supine position, when measuring respiration [Bottos, et al. (1985)]. Oxygen levels (pressure in mm Hg) of 50 babies measured in both positions are given in Data Set 5. Scatter plot and average-difference plot are given in Figures 17 and 18, respectively. The average-difference plot shows that relative bias between oxygen measurements in the two positions may be small (differences approximately symmetric around zero horizontal) but variation of discrepancy obviously increases with the oxygen level. (Note the substantial difference in oxygen for baby number 14 — 55.42 mmHg in supine position, 108.92 mmHg in hammock position.) These observations suggest usage of a nonparametric or robust confidence interval procedure. E X A M P L E 7.6: Tobacco moisture content. 73 These data come from a study of two electrical devices, say A and B, which measure the moisture content of tobacco. Data of 15 tobacco samples are listed in Data Set 6 (adapted from a B.Sc. Special Examination, University of London; no unit specified for "moisture content"). Scatter plot and average-difference plot are shown in Figure 19 and 20, respectively. Again, although the correlation is high (0.996), the average-difference plot suggests that variance of discrepancy increases with the moisture content; however, the trend is not very clear, possibly because of small sample size. 74 Table 1: Efficacies and Pitman asymptotic relative efficiency (ARE) comparisons of Student t (T), sign (S), and Wilcoxon signed rank (W) procedures. Numeric efficacies are for familiar den-sities standardized so that efficacy is 1 for the sign procedure. Hence, the entries also are Pitman asymptotic efficiencies relative to the sign proce-dure. For example, ARE(T,S) = 1.57 for normal distribution. [Adapted from Pratt and Gibbons (1981, p.384)]; see also Appendix II. Distributions W T S Normal (0,2/TT) Uniform (-1,1) Cauchy (0,2/TT) 1.50 1.57 1.00 3.00 3.00 1.00 0.75 0.00 1.00 75 Table 2: Pitman asymptotic relative efficiency (ARE) comparisons of Student t (T), sign (5), and Wilcoxon signed rank (W) proce-dures (results using theoretical efficacies); Relative efficiencies of procedures for normal, Cauchy, and uniform distributions do not depend on location and scale parameters of the distributions; see also Table 1. The listed distributions include standard normal and Cauchy; uniform distri-bution over ( — 1,1); and normals mixed or contaminated: N.Mix(l : 4) denotes an equal-proportions mixture of normals Af(0,i2) for i = 1,2,3,4; CN(2;5) denotes 5% of A/"(0,22) contaminating standard normal A/"(0,1); etc. Pitman ARE Distributions W : S T : S W : T Normal 1.5000 1.5708 0.9549 Uniform 3.0000 3.0000 1.0000 Cauchy 0.7500 0.0000 oo N.Mix(l:5) 0.9568 0.6847 1.3973 N.Mix(l:4) 1.0317 0.7721 1.3363 CN(2;5) 1.4658 1.4369 1.0202 CN(2;1) 1.4930 1.5404 0.9692 CN(4;5) 1.4177 0.9689 1.4632 CN(4;1) 1.4832 1.3866 1.0696 CN(10;5) 1.3803 0.2895 4.7687 CN(10;1) 1.4755 0.8037 1.8358 76 Table 3: Asymptotic (n —• oo) ratios of lengths and squared lengths of confidence intervals (^/ l /ARE and 1/ A R E , respectively, where A R E ' s are Pitman asymptotic relative efficiencies) for Student t (T), sign (5), and Wilcoxon signed rank (W) procedures (results using theoretical efficacies). Asymptotic ratios for normal, Cauchy, and uniform distributions do not depend on location and scale parameters of the distributions. For details of other listed distributions, see Table 2. Distributions Lengths Ratio = yj 1/ARE Sq. Lengths Ratio = 1/ARE W : S T : 5 W : T W : S T : S W : T Normal 0.8165 0.7979 1.0233 0.6667 0.6366 1.0472 Uniform 0.5773 0.5773 1.0000 0.3333 0.3333 1.0000 Cauchy 1.1547 oo 0.0000 1.3333 oo 0.0000 N.Mix(l:5) 1.0224 1.2085 0.8460 1.0452 1.4604 0.7157 N.Mix(l:4) 0.9845 1.1381 0.8650 0.9693 1.2952 0.7483 CN(2;5) 0.8260 0.8343 0.9901 0.6822 0.6960 . 0.9802 CN(2;1) 0.8184 0.8057 1.0158 0.6698 0.6492 1.0318 CN(4;5) 0.8399 1.0159 0.8267 0.7054 1.0321 0.6835 CN(4;1) 0.8211 0.8492 0.9669 0.6742 0.7212 0.9349 CN(10;5) 0.8512 1.8587 0.4579 0.7245 3.4546 0.2097 CN(10;1) 0.8232 1.1154 0.7380 0.6777 1.2442 0.5447 77 Table 4: Comparisons of T, 5, and W confidence intervals: average ratios of interval lengths and squared lengths (Monte Carlo sim-ulation results). Each table entry is based on 1000 samples with n = 32, and all intervals have nominal 95% confidence; see Table 2 for descriptions of the listed distributions. For example, among 1000 standard normal sam-ples, 0.8900 was the average W : S length ratio (dividing length of the Wilcoxon interval by length of the interval corresponding to the sign test, for each sample). Mean of Lengths Ratio Mean of Sq. Length s Ratio Distributions W : 5 T : 5 W : T W : 5 T:S W : T Normal 0.8900 0.8704 1.0314 0.8345 0.8103 1.0684 Uniform 0.7233 0.6614 1.0905 0.5598 0.4648 1.1916 Cauchy 1.3522 11.7286 0.3647 2.0823 1478.381 0.1941 N.Mix(l:5) 1.0636 1.1843 0.9164 1.1968 1.5240 0.8510 N.Mix(l:4) 1.0402 1.1485 0.9258 1.1485 1.4510 0.8691 CN(2;5) 0.8921 0.8930 1.0105 0.8374 0.8530 1.0276 CN(2;1) 0.8866 0.8715 1.0263 0.8280 0.8103 1.0580 CN(4;5) 0.9123 1.0487 0.9116 0.8746 1.2300 0.8574 CN(4;1) 0.8909 0.9094 0.9992 0.8360 0.8953 1.0105 CN(10;5) 0.9326 1.7199 0.6795 0.9170 3.9835 0.5413 CN(10;1) 0.8941 1.0847 0.9243 0.8421 1.5144 0.9026 78 Table 5: Comparisons of T, 5, and W confidence intervals: ratios of standard deviations of interval lengths (Monte Carlo simulation results). Each entry is based on 1000 samples with n = 32; see Table 2 for descriptions of the listed distributions. For example, the first entry shows that SD of Wilcoxon interval lengths, divided by SD of interval lengths for the sign procedure, gives a ratio 0.4696, for normal samples. Distributions Ratio of SD of Lengths W(SD) : S(SD) T(SD) : S(SD) W(SD) : T(SD) Normal 0.4696 0.4156 1.1299 Uniform 0.2428 0.2040 1.1904 Cauchy 1.2385 110.1337 0.0112 N.Mix(l:5) 0.6479 0.6418 1.0094 N.Mix(l:4) 0.6451 0.6408 1.0067 CN(2;5) 0.4787 0.4960 0.9651 CN(2;1) 0.4538 0.4188 1.0835 CN(4;5) • 0.5361 1.1005 0.4871 CN(4;1) 0.4608 0.6441 0.7155 CN(10;5) 0.6028 3.5477 0.1699 CN(10;1) 0.4696 2.0320 0.2311 79 Table 6: Confidence interval coverages and length comparisons (Monte Carlo simulation results). For each distribution listed below (all are symmetric about zero; see Table 2 for descriptions) 1000 samples were simulated, each with sample size n = 32; and, for each sample, confi-dence interval estimates of distribution median were calculated using Stu-dent t, sign, and Wilcoxon signed rank procedures (denoted respectively as T, S, W) with nominal confidence level 95%. Table entries: a) for each of the three procedures, the percentage of samples (from each distribution) for which the calculated intervals included zero; and b) the percentage of sam-ples for which the length of one interval was shorter than another (T < S, etc.). % Coverage % Shorter Length Distributions T S W T<S T<W W<S Normal 94.6 94.4 95.3 77.2 73.1 75.9 Uniform 95.5 93.8 96.1 95.5 96.8 92.5 Cauchy 98.0 93.8 96.1 1.0 0.6 17.7 N.Mix(l:5) 94.3 95.0 94.7 33.7 23.2 47.0 N.Mix(l:4) 95.5 94.9 94.6 39.5 27.1 51.8 CN(2;5) 94.6 94.4 95.0 74.1 63.3 77.2 CN(2;1) 94.7 94.4 95.4 77.9 70.6 78.4 CN(4;5) 95.6 94.4 94.9 54.2 39.3 74.2 CN(4;1) 95.3 94.4 95.2 72.4 62.7 78.1 CN(10;5) 96.9 94.4 94.6 28.1 21.6 71.1 CN(10;1) 96.1 94.4 95.3 62.1 55.0 78.1 80 Table 7: (Example 7.1). Point estimate and confidence interval for median discrepancy rate of "old" logging counts ( n = 166 batches) using T , 5, and W procedures. See Data Set 1.1. Procedure Pt Est Confidence Level Interval Endpoints Length T -0.0269 95.00% (-0.0356,-0.0182) 0.0174 S. -0.0225 94.80% (-0.0283,-0.0128) 0.0154 W -0.0255 95.00% (-0.0338,-0.0177) 0.0161 Table 8: (Example 7.1). Ratios of lengths and squared lengths for confidence intervals in Table 7. Procedures Length Ratio Squared Length Ratio W:S 1.0446 1.0911 T:S 1.1291 1.2748 W:T 0.9251 0.8559 81 Table 9: (Example 7.1). Point estimate and confidence interval for median discrepancy rate of "new" logging counts (n = 86 batches — with 7 outliers deleted) using T, S, and W procedures. See Data Set 1.2. Confidence Interval Procedure Pt Est Level Endpoints Length T 0.0038 95.00% (-0.00244,0.01003) 0.01247 S 0.0005 96.01% ( 0.00000,0.00343) 0.00343 w 0.0018 95.00% (-0.00266,0.00746) 0.01012 Table 10: (Example 7.1). Ratios of lengths and squared lengths for confidence intervals in Table 9. Procedures Length Ratio Squared Length Ratio W:S 2.9459 .8.6783 T:S 3.6311 13.1850 W:T 0.8113 0.65S2 82 Data Set 1.1: (Example 7.1). Source counts and destination counts for 166 batches of "old" logs. Batch Source Destination Number Count Count 1 1068 1116 2 623 624 3 1655 1644 4 672 683 5 21 19 6 2402 2398 7 551 547 8 1026 1065 9 148 164 10 2489 2503 11 2272 2455 12 850 844 13 409 398 14 729 719 15 1027 1006 16 548 546 17 240 242 18 395 386 19 456 458 20 112 118 21 2437 2614 22 1963 1977 23 587 587 24 2516 2719 25 765 768 26 506 511 27 2161 2209 28 1547 1567 29 867 866 30 638 662 Batch Source Destination Number Count Count 31 302 312 32 630 650 33 595 659 34 580 647 35 375 369 36 575 577 37 1172 1200 38 589 608 39 696 690 40 529 539 41 117 113 42 418 468 43 745 762 44 1066 1169 45 1199 1302 46 687 656 47 1233 1240 48 1029 1052 49 846 852 50 519 503 51 1898 1997 52 1475 1438 53 883 957 54 1319 1329 55 887 946 56 1028 1075 57 1176 1208 58 929 925 59 1392 1391 60 544 574 • • • continued below 83 Data Set 1.1: (Example 7.1). ••• continued Batch Source Destination Number Count Count 61 467 503 62 171 186 63 571 584 64 129 123 65 309 311 66 974 1098 67 1105 1267 68 891 871 69 1355 1387 70 1314 1371 71 848 924 72 631 654 73 904 894 74 705 713 75 717 702 76 883 882 77 1349 1347 78 1132 1052 79 870 933 80 894 894 81 1013 1051 82 696 803 83 850 874 84 2436 2501 85 291 289 86 1071 1054 87 865 832 88 1217 1234 89 1074 1112 90 855 872 Batch Source Destination Number Count Count 91 1014 1060 92 446 468 93 642 677 94 498 592 95 646 538 96 526 565 97 35 33 98 516 550 99 587 625 100 19 19 101 865 869 102 1065 1092 103 910 934 104 298 322 105 947 947 106 1046 1067 107 405 394 108 644 651 109 668 641 110 610 647 111 1005 1000 112 277 291 113 675 689 114 649 667 115 1433 1596 116 868 908 117 1269 1181 118 900 893 119 277 294 120 364 377 • • • continued below 84 Data Set 1.1: (Example 7.1). ••• continued Batch Source Destination Number Count Count 121 891 915 122 775 777 123 666 630 124 1085 1184 125 1329 1436 126 320 328 127 610 606 128 327 434 129 930 886 130 2445 2616 131 725 638 132 797 945 133 2162 2456 134 835 920 135 1362 1377 136 772 797 137 838 838 138 581 566 139 885 924 140 1411 1301 141 534 538 142 1033 1134 143 1158 1250 Batch Source Destination Number Count Count 144 1004 1055 145 1054 1006 146 864 899 147 449 437 148 438 441 149 273 243 150 585 668 151 734 746 152 597 630 153 443 479 154 423 443 155 562 603 156 900 913 157 977 1005 158 1045 1073 159 696 735 160 513 519 161 817 906 162 504 564 163 1201 1280 164 564 573 165 1209 1302 166 236 256 85 Data Set 1.2: (Example 7.1). Source for 93 batches of "new" logs. counts and destination counts Batch Source Destination Number Count Count 1 2833 2622 2 507 508 3 376 382 4 158 160 5 486 485 6 623 638 7 779 768 8 639 661 9 951 950 10 465 482 11 1035 1032 12 841 806 13 641 718 14 670 668 15 534 529 16 590 583 17 720 719 18 1011 1008 19 880 906 20 711 733 21 476 495 22 617 631 23 522 540 24 395 401 25 111 128 Batch Source Destination Number .Count Count 26 596 594 27 783 762 28 555 587 29 2489 2481 30 2693 2663 31 654 653 32 875 872 33 2344 2392 34 380 384 35 271 267 36 354 352 37 509 582 38 721 603 39 477 480 40 849 838 41 347 363 42 690 668 43 341 322 44 686 699 45 753 763 46 859 859 47 876 902 48 611 603 49 770 777 50 408 408 . . . continued below 86 Data Set 1.2: (Example 7.1). ••• continued Batch Source Destination Number Count Count 51 612 599 52 171 168 53 964 1118 54 327 318 55 609 641 56 229 252 57 541 544 58 607 639 59 522 512 60 266 281 61 164 164 62 653 633 63 252 252 64 361 343 65 385 381 66 238 237 67 1504 1392 68 221 208 69 691 648 70 1464 1246 . 71 862 868 72 613 595 Batch Source Destination Number Count Count 73 984 990 74 546 543 75 265 265 76 351 351 77 254 253 78 396 365 79 486 487 80 758 744 81 496 479 82 296 296 .83 467 • 467 84 238 229 85 654 661 86 303 304 87 306 307 88 297 297 89 1881 1791 90 2158 2158 91 295 297 92 2524 2564 93 300 287 Data Set 2: (Example 7.2). Fuse burning times (seconds) measured by two observers for 30 powder train fuses. [Grubbs (1948)]. Sample Number Observer Sample Number Observer A B A B 1 10.10 10.07 16 9.74 9.74 2 9.98 9.90 17 10.32 10.34 3 9.89 9.86 18 9.86 9.86 4 9.79 9.70 19 10.01 10.03 5 9.67 9.65 20 9.65 9.65 6 9.89 9.83 21 9.50 9.50 7 9.82 9.79 22 9.56 9.55 8 9.59 9.59 23 9.54 9.54 9 9.76 9.72 24 9.89 9.88 10 9.93 9.92 25 9.53 9.51 11 9.62 9.64 26 9.52 9.53 12 10.24 10.24 27 9.44 9.45 13 9.84 9.86 28 9.67 9.67 14 9.62 9.63 29 9.77 9.78 15 9.60 9.65 30 9.86 9.86 88 Data Set 3: (Example 7.3). Systolic blood pressures (mm Hg) by two methods in 25 patients. [Daniel (1983)]. Patient Number Method I II 1 132 130 2 138 134 3 144 132 4 146 140 5 148 150 6 152 144 7 158 150 8 130 122 9 162 160 10 168 150 11 172 160 12 174 178 13 180 168 14 180 174 15 188 186 16 194 172 17 194 182 18 200 178 19 200 196 20 204 188 21 210 180 22 210 196 23 216 210 24 220 190 25 220 202 89 Data Set 4: (Example 7.4). Spinal curvature (angle, in degrees) by Ferguson method and by Cobb method in 26 patients. [Robinson and Wade (1983)]. Patient Number Meth od Ferguson Cobb 1 73 97 2 66 90 3 60 .88 4 50 67 5 48 70 6 47 63 7 45 55 8 43 50 9 43 48 10 40 65 11 40 64 12 38 47 13 37 52 14 37 49 15 36 60 16 36 48 17 33 41 18 30 45 19 30 40 20 29 45 21 29 39 22 28 . 42 23 28 37 24 27 39 25 27 35 26 21 28 90 Data Set 5: (Example 7.5). Cutaneous oxygen levels (mmHg) in 50 newborn infants measured in two positions. [Bottos, et al. (1985)]. Infant Number Position Infant Number Position Hammock Supine Hammock Supine 1 80.67 93.83 26 77.46 96.75 2 56.13 69.08 27 60.96 54.04 3 95.17 103.58 28 74.33 69.46 4 66.42 68.88 29 52.67 71.83 5 • 77.42 67.83 30 .52.96 58.67 6 55.92 59.50 31 71.50 62.88 7 61.79 60.50 32 56.96 55.21 8 65.92 68.71 33 66.67 59.79 9 65.71 65.54 34 67.58 72.75 10 67.30 75.33 35 69.92 77.71 11 77.17 69.67 36 86.29 85.00 12 71.67 67.13 37 55.67 54.33 13 85.00 77.79 38 64.25 76.58 14 108.92 55.42 39 71.71 75.50 15 52.71 57.59 40 71.13 85.83 16 66.00 62.67 41 72.63 85.54 17 75.83 78.83 42 50.58 87.54 18. 66.83 64.04 43 49.29 56.S8 19 76.04 70.50 44 82.83 79.75 20 67.71 56.63 45 88.58 80.13 21 72.00 77.21 46 58.95 61.96 22 69.96 71.75 47 54.17 63.83 23 87.71 72.75 48 49.96 50.00 24 82.33 76.38 49 80.25 61.17 25 84.63 79.83 50 60.96 56.8S 91 Data Set 6: (Example 7.6). Tobacco moisture content in 15 samples measured by two devices. [Adapted from a B.Sc. Special Examination, University of London]. Sample Device Number A B 1 12.0 10.1 2 12.1 13.5 3 7.5 8.5 4 8.0 9.6 5 16.0 16.8 6 24.5 23.6 7 5.0 4.9 8 47.9 47.8 9 43.1 46.7 10 38.2 38.3 11 69.0 64.8 12 11.8 12.0 13 20.0 17.5 14 57.6 55.2 15 15.0 14.8 92 Figures 1 & 2: Counting logs (Data Set 1) Scatter plot: destination vs source count, "old" logs, n = 166 O 3 O O c g V-• CO c w CD Q O r-o CO o o in eg o o o CvJ O o m o o o o o in J L 500 1000 1500 2000 2500 3000 Source count (No. logs) Scatter plot: destination vs source count, "new" logs, n = 93 CO o 6 2 8 £ c o CO CD D o o co o o in CM o o o (M o o m o o o o o in S J L 500 1000 1500 2000 2500 3000 Source count (No. logs) 93 Figures 3 & 4: Counting logs (Data Set 1) c o •*-* CO c to CD • CD O o .52. o c CO Q. CU o to o o CO o o CM O o o o o o CM O O CO Average-difference plot, "old" logs, n = 166 500 1000 1500 2000 2500 3000 Average count (No. logs) g CO _c CO d> TJ I CD O o to o C CO Q. CD O CO b o o CO o o CM O O o o o o CM O O CO Average-difference plot, "new" logs, n = 93 500 1000 1500 2000 2500 3000 Average count (No. logs) 94 Figures 5 & 6: Counting logs (Data Set 1) Diff. vs sq. root of avg. count, "old" logs, n = 166 CO c to CD •o CD 3 O -52->% o c CO CL CD t O to o o CO o o CM O o o o o o CM O O CO 10 20 30 40 50 60 Square root of average count o CO CD "O I CD O i _ 3 O c CO CL CD O CO O o o o CVJ o o CO Diff. vs sq. root of avg. count, "new" logs, n = 93 o CO o o C\J o o o 10 20 30 40 50 60 Square root of average count 95 Figures 7 & 8: Counting logs (Data Set 1) CO v— >« O c CO CL CD i O w C\J p o CM O CO O Normal probability plot, "old" logs, n = 166 Normal standard units CD CO u. >. O c CO CL CD b CO LO o o in o in Normal probability plot, "new" logs, n = 93 -1 1 Normal standard units 96 Figures 9: Counting logs (subset of Data Set 1.2 — 7 outliers are deleted) Normal probability plot, o "new" logs, n = 86 Normal standard units 97 Figures 10 & 11: Fuse burning times (Data Set 2, n = 30 ) Scatter plot to C o o CD tO GO i CD £ CD CO X) o o CM O O O co CO 9.4 9.6 9.8 10.0 10.2 10.4 m >» o c CC Q. CD t O to CD O O CM O O CM O CD O O O O by Observer A (seconds) Average-difference plot 9.4 9.6 9.8 10.0 10.2 10.4 Average time ((B + A)/2) 98 Figures 12 & 13: Systolic blood pressures (Data Set 3, n = 25 ) Scatter plot X E E T3 O O CM CM O O CM O oo o CD O O CM 120 140 160 180 200 220 Method I (mm Hg) Average-difference plot CO CL CD O o in zz. o 5 V m o CM in CM o CO 120 140 160 180 200 220 Average ((I + ll)/2) 99 Figures 14 & 15: Spinal curvature (Data Set 4, n = 26 ) Scatter plot 20 40 60 80 100 by Ferguson method (angle, in degrees) Average-difference plot 20 30 40 50 60 70 80 90 Average angle ((Ferguson + Cobb)/2) 100 Figures 16: Spinal curvature (Data Set 4, n = 26 ) Residual plot in in m m 20 30 40 50 60 70 80 Ferguson method (angle, in degrees) 101 Figures 17 & 18: Oxygen level and position (Data Set 5, n = 50 ) Scatter plot X E E, c o 'to o Q . CD c Q . 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Contributions to Order Statistics. New York: John Wiley & Sons. [74] Thompson, W.A.Jr. (1963). Precision of simultaneous measurement procedures. Journal of the American Statistical Association 58:474-479. [75] Tibshirani, R.J. (1984). Bootstrap confidence intervals. Stanford University, Division of Biostatistics Technical Report No. 91. [76] Winer, B.J. (1962). Statistical Principles in Experimental Design. New York: McGraw-Hill. 108 A P P E N D I X I. B O O T S T R A P M E T H O D 1. G E N E R A L T H E O R Y O F B O O T S T R A P M E T H O D S Suppose we wish to draw inferences about some parameter 8 of a pop-ulation with unknown distribution F based on realization of an independent identically distributed sample Xi = xx, X2 = x2, • • •, Xn = xn from F. It may be convenient to denote the parameter of interest by 0(F). And suppose 8 is an estimator of 9; we also write 9 as 9(X\, X2, • • •, Xn) to indicate that the statistic is a function of Xi, X2, • • •, Xn. Let F be the empirical distribution of the random sample, putting prob-ability mass — on each a^ ; and let X*, X%, • • •, X* denote a random sample n from F, i.e., drawn independently with replacement from {x\, x2, • • •, xn}: , A 2 , • • •, An ~ £ . Call A 7 , A r 2 V - - , X* a "bootstrap sample". Then 9* = 9(X*, JJf*, • • •, X*) estimates 8(F), considering F as fixed, that is, conditioning on the sample values. In theory, inferences about the parameter 8(F) can be based on the dis-tribution of 9 = 8 (Xy, X2, • • •, Xy); and behaviour of 8 can be approximated 109 by behaviour of 9* — d(X*, XVf, • • •, A"*). The distribution of $*, in general, may be difficult to obtain analytically; but it can always be approximated by using a Monte Carlo algorithm, as discussed below. Suppose we know that the probability distribution F is symmetric. In this case, we would symmetrize F. One way to achieve this is to replace F by FSYMI the symmetric probability distribution obtained from F by reflec-tion about the median. That is, FSYM bas probability mass on each 2n — 1 x(i),x(2),- and 2x( m )-x ( 1 . ) ,2x ( m )-a ;(2),-•- ,2a:( m )-X( n), assuming that n is odd and equal to 2m — 1 for convenience [Efron (1979)]. In this case, even though the symmetrized distribution is not a nonparametric maximum likelihood estimate for F, the symmetrized distribution has properties similar to the maximum likelihood estimate, F [Hinkley (1976)]. 2. B O O T S T R A P E S T I M A T O R A N D C O N F I D E N C E I N T E R V A L Recall that bootstrap estimators or confidence intervals for 9 rely on the distribution of 9* = 9 (A*, A^, • • •, A*) — the estimator 9 evaluated at a "bootstrap sample" {A'*, A | , • • •, A*} generated as independent and identically distributed observations from the empirical distribution F. As already noted, in general, the distribution of 9* is hard to find analytically, but can be 110 approximated by a Monte Carlo method. But for the sample mean and the sample median, the bootstrap distribution can be obtained theoretically, without using the Monte Carlo methods. 2.1. B O O T S T R A P P I N G F O R T H E M E A N For the mean, the parameter of interest is 0(F) = E(X). So, 8 = X, the sample mean, is the estimator of 0(F). It can be shown that E+(X*) = X, and Var*(X*) = ^ E (X* ~ X ? = ~ * 2 -i = i Also, the central limit theorem implies that the bootstrap distribution of X* is approximately normal, Af(X,—a2). Thus, by the central limit theorem, the bootstrap interval estimate would essentially be the same as the t interval estimate as derived in Section 3.1. 2.2. B O O T S T R A P P I N G F O R T H E M E D I A N For the median, the parameter of interest is the point 0 such that Prob(X <0)<-< Prob(X < 0). i n . So, the estimator is 6 = A", the sample median. Suppose A"i = X\,X2 = x2, - • •, Xn = xn is the realization of a sample. For convenience, suppose the sample size is odd and equal to 2m — 1, say. Then the sample median estimate of 6(F) is A' = £ ( m ) - Then the bootstrap distribution of 6* is concentrated on the values < X(2) < ••• <,X( n) such that pk = Prob* = x{k) - I H G - ) ^ ) ' ^ ) " " ' - © ® ' ^ ) " " ' } [Efron (1982, p.77)]. Furthermore, Efron showed that the corresponding confidence interval is very close to the classical interval estimate for the median as discussed in Section 3.2 [Efron (1982, pp.80-81)]. 2.3. M O N T E C A R L O E V A L U A T I O N O F T H E B O O T S T R A P D I S T R I B U T I O N F O R A R B I T R A R Y 0* 1. Construct the nonparametric maximum likelihood estimator of F, the empirical distribution F, F : mass — at xi, x2, • • •, xn. n (For emphasis, we could write Fx = F and Fx = F). In the symmetric bootstrap case, replace F by FSY MI FSYM '• mass at xt2), • • •, xin), and 2n — 1 2a;(m) - 2x ( m ) - X(2), • • • ,2x ( m) - x ( n ) , assuming that n is odd and equal to 2m — 1, say, for convenience [Efron (1979)]. Draw a "bootstrap sample" of size n from F, V"* V * v * ''^ r> A 1,A 2,--,A n ~ t and calcalute 0* = 0(X*, X*, • • •, A'*). Independently repeat step 2 B times (for some large B), obtaining "bootstrap replications" b = 1,2, • • •, i? j . Approximate the cumulative distribution function of 9* by the em-pirical cumulative distribution function of b = 1, 2, • • •, I?j: where the indicator function I{6*b<t} = {1' i{ft^ {0, otherwise. 113 [Efron (1982, p.28)]. 2.4. B I A S - C O R R E C T E D B O O T S T R A P E S T I M A T E Statistic 0 need not be unbiased; in general Bias = EF9-9, where EF indicates expectation is taken with respect to the distribution F. This bias can be approximated by Bias* = Ej* - 9; where E* denotes expectation with respect to F; and a Monte Carlo approximation of Bias* is given by B I A S * = B 6=1 = 9* - 6, where 9* is the average of the B bootstrap replications of 9*, j(9*, 9%, • • •, 0*B^ Thus, a bias-corrected bootstrap estimate is given by 9B = 9 - (9* - 9) = 29-9*. 114 However, this bias-corrected estimate would have a larger variance than the original estimate 9 because Var(8B) = Var(9 - Bias*) = Var(9) + Var(Bias*) - 2Cov(9, Bias*), where usually 2Cov(9, Bias*) « 0. (This is a "variance-bias tradeoff"). And the bootstrap estimate of standard error of 9 is equal to cr*(9), which can be estimated by sample standard deviation of the bootstrap replication of 9* : [Efron (1982, p.28)]. b=l 2.5. B O O T S T R A P C O N F I D E N C E I N T E R V A L S Let Fg.(t) = Prob*{9* < t} be the cumulative distribution function of 9*. Note that if the bootstrap distribution is obtained by the Monte Carlo methods, then Fg.(t) is approximated by the empirical cumulative distribution function of the bootstrap replications of 9*, |#*,9*, • • •,9B^ : 6=1 where the indicator function I{9*b < t) = \ 1' i f - *; [ 0, otherwise. 115 2.5.1. T H E P E R C E N T I L E M E T H O D Suppose that 6 — 6 is a pivotal quantity, that is 6-6 ~ H, ( A l ) where H is a distribution not involving 6. Also, suppose that approximately 6* -6 ~ H. (A2) In (A2) the distribution F plays the same role as F in (Ai), "~" indicating the distribution under independent and identical sampling from F. Notice that the second assumption is reasonable because, if F is close to F, then the bootstrap distribution of 8* — 6 will be "close" to that of 6 — 6, as long as #() is a reasonably smooth functional. Finally, assume that H is symmetric about 0. (A3) Then a 1 — 2a symmetric confidence interval (9LOW,6~UP) is given by 6Low = Fr\a) and 6UP = F 7 \ \ - a ) [Efron (19S2, p.78)]. Assumptions (Al) and (A2) can be generalized to g{6)-g{6)~H, (Al') 116 and g(h-90)~H, (A21) where H symmetric about 0 and g() is an unknown, monotone increasing function. Indeed, further knowledge about g() is not necessary since the resultant interval does not depend on g(') [Tibshirani (1984)]. These pro-cedures simply assume the existence of a symmetric pivotal on some other scale. , Under these generalized assumptions, the interval ^i ?r 1(a),JF 1r 1(l — a)J remains valid as a 1 — 2a confidence interval [Tibshirani (1984)]. 2.5.2. T H E B I A S - C O R R E C T E D P E R C E N T I L E M E T H O D If H, the distribution of the pivotal quantity g(8) — g(6), is symmetric about a point, say (j., which does not equal 0, then the percentile interval will be biased and will not have the correct coverage. In order to estimate fj, and hence to derive a bias correction to the percentile interval, we need to assume a parametric form for H. Tibshirani (1984) showed that the bias-corrected percentile interval is robust with respect to the choice of a symmetric pivotal distribution. 117 Suppose H = Af(fi, 1). Define to estimate u,- where $ is the cumulative distribution function of Af(0,1). Then a 1 — 2a bias-corrected percentile interval {BLOW,® UP) is given by OLOW = F71 ($(2 2 a - *«)) and 6W = F T 1 ($(2z0 + ^)) [Efron (1982, p.82)]. Efron (1982, p.86) remarked that the bias-corrected percentile interval should be used with caution, or not at all, when distributional asymmetry is definite. 118 A P P E N D I X II. E F F I C A C Y C A L C U L A T I O N S A N D A R E ' S F O R S T A N D A R D DISTRIBUTIONS A N D M I X T U R E S Recall that for two statistical tests, say T\ and T 2 , Pitman asymptotic relative efficiency (ARE) of Tj with respect to T 2 can be represented as a squared ratio of efficacies: ARE(TUT2) = e/ /(TQ [eff(T2)\ 1 2 where e//(T;) denotes the efficacy of test Ti [Randies and Wolfe (1979, pp.147-149)]. Suppose K is a statistic used by test T for a hypothesis 9 = #o, where 9 is a parameter of a symmetric density. Suppose we reject the hypothesis if K is outside a certain interval. Then the efficacy of T, denoted by e//(T), is defined as ~dEe(i<y2 eff(T) d9 0=00 Vareo(K) [Kotz and Johnson (1982, p.468)]. For a density function fx symmetric about 0, the efficacies of t test 119 (T), sign test (5), and Wilcoxon signed-rank test (W) are given by 1 o-x 2/ x(0), and /oo fx(x)dx, -oo where o~2x = Var(X) = I x2fx{x)dx [Randies and Wolfe (1979, pp.165-J — oo 168)]. These efficacies are well known for particular symmetric families, includ-ing normal, uniform, and Cauchy densities. Note that although efficacy may be a function of family parameter(s) (normal standard deviation cr, etc.), efficacy ratios — that is, Pitman relative efficiencies — are not. Hence, it suffices to evaluate numerically the efficacy of standard normal density, etc., as in Table 1, adapted from Pratt and Gibbons (1981, p.384). Efficacy and ARE also can be calculated for mixtures of normal distribu-tions. Suppose X ~ Hi=i uiXi, where X{ i n ~ p A/"(0,tr2), and OJ, are weights k * 1 such that 0 < u>i < 1, ^2i=1 w\ = 1. Then fx(x) = / ,UJi~==—exP V 27T(7j We need to compute ax, /A'(0), and J^°oofx(x)dx in order to obtain 120 eff(T) = eff(S) = eff(W) = my the efficacies of T, 5, and W for the mixture. /oo fc -OO • i k V2^ateXP\2a2 dx k i=l /A '(0) = E T = ^ - K 0 ) = 4=E7 l (y- ",2 /°° 1 V2^ \ £ t ^ J-°° V2^(o-i/y/2)eXP [2(a2 fc /•OO i 2 E E / -<=i v / a 2 + <Ti 0 0 ^ (<7«-<Ti/v/<T'? + a?) f fc 2 fc = j ^  i ^ v r .fa,'fa;j' -exp i .2 • 72). tix + —x 2 ( ^ ? / ( ^ + ^)) since both integrands are density functions of normals. E X A M P L E 1: Consider a mixture with equal proportions of four normals, 121 Af(0, i2),i = 1,2, 3,4. That is, w,- = \ and a{ = i, for i = 1,2,3,4. Then 4 4 2 2 1 2 15 «'=1 1=1 /x(0) 25 V ^ T T ^ tr.- 4\/27r t 4 \ ^ T T V12 1=1 1=1 /oo i f 4 2 4 25 48 v 7 ^ - ' and 16v/27T (4.8870). Thus, ARE(W, S) = 3 [Ho 2 — -i '3(4.8870)" /x(0) — o 25 = 1.0317, ARE(S,T) = 4axfx(0) = 4 Qfj {l^J = 1-2952, and ARE(W,T) = 1 1 2 fx(x)dx T 2 16\/2T (4.8870) - 1.3363. Recall that, for corresponding confidence intervals, the asymptotic ratio of Ly 1 interval lengths L2 y/ARE(l,2) one for the mixture in this example. ; and note that the ratios all are close to E X A M P L E 2: Consider a mixture with equal proportions of five normals, 122 Af(0,i2),i = 1,2, 3,4,5. That is, Ui = \ and cr,-= i, for i = 1,2,3,4,5. Then <4 5 ^ 5 = EW^?"=?E*'2 = 1 1 ' /x(0) /•oo •/ — oo i=l 1=1 Thus, 1 ^ u>i_ _ 1 ^ 1 _ 1 /137 i f 5 2 5 7^\£? ^ + 2 £ t 7^+^ 2 5 v /^{|t^ + 2 S § \ ^ f + 1 ! 25\/2^ (6.4474) and ARE(W,S) = 3 JZofx(x)dx 2 — -5 "60(6.4474)" /x(0) — o 5(137) AB£;(S,T) = 4<ri./J(0) = 4 ( l l ) . 5 \ ^ T T V 60 J_ = 0.9568, = 1.4604, and ARE(W,T) = 12a2x / fx{x)dx =12(11) .J —oo 25V27T (6.4474) 1.3973. Note that, although this example is very similar to the preceding mixture, numeric results for ARE(W, S) are less than one here. E X A M P L E 3: Consider a standard normal A"(0,1) contaminated with 5% 123 of Af(0,102). That is, u>i = 0.95, UJ2 = 0.05, ax = 1, and ,o2 = 10. Then 2 a2x = ^ 2^i<T2 = 0.95(1) + 0.05(100) = 1.45, y/^iri ai I io 0.051 0.955 1 / (0.95)2 (0.05)2 2(0.95)(0.05)) \/2if\ y/2 10\/2 v/TTTOO j (0.6478). Thus, A R £ ( W , S ) = 3 2 — 1 "0.6478" [ fx(0) \ — o 0.955 = 1.3804, ARE(S,T) = 4a2xf2x(0) = 4(1.45) ARE(W, T) = I2a2x f2x(x)dx 0.955 ' Z7T 0.8419, and T 2 12(1.45) 0.6478 1 2 and = 1.1621. Note that, for this contaminated normal, W becomes preferable to contrasting with the result for pure normal. 124 A P P E N D I X III T A I L - W E I G H T A D A P T I V E N O N P A R A M E T R I C P R O C E D U R E S Given a sample of discrepancies D\,D2, • • •,Dn, Randies and Hogg (1973) defined a tail-weight statistic Q = 10(l7o,o5 - LQ.QS) 0^.50 — 0^.50 where Up (Lp) is the sum of the largest (smallest) n/3 order statistics (fractional items are used if n/3 is not an integer). For instance, if n = 26, ?7(0.05) = 1.3; so Lo.os = D(26) + 0.3£> ( 2 5 ) . Then the underlying distribution will be classified as having light, mod-2 5 5 erate, or heavy tails if Q < 2.OS - - , 2.08 < Q < 2.96 —, or n n n ^ 5.5 Q > 2.96 , respectively. n Randies and Hogg (1973) showed that the Q statistic discussed above is uncorrelated with the Student t, sign, and Wilcoxon signed rank statistics. A zero correlation would give asymptotic independence, because the relevant joint distribution is asymptotically normal; but, independence does not hold for finite samples. In particular, simulation gives confidence intervals with confidence less than the nominal level for n = 18; see Randies and Hogg (1973). 125 To obtain a truly nonparametric adaptive procedure, Randies and Hogg modified the tail-weight statistic Q as l O O t / ^ o V — n i 1=1 where UQ1Q is the sum of the largest 10% values |Di |, |Z)2|, • • •, | D n | , and the classification rule works as above. Notice that Q* is independent of all rank statistics, like the Student t (viewed as an approximation to permutation statistic), sign, and Wilcoxon signed rank statistics, because Q* is a function of the order statistics of \D\\, \D2\, • • •, \Dn\, which are sufficient and complete for a continuous symmetric distribution and because sufficient statistics are independent of every rank statistic [Lehmann (1983, p.40 and p.68)]. 126 

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