Cover image for Nonparametric statistical methods for complete and censored data
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Nonparametric statistical methods for complete and censored data
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New York : Chapman & Hall/CRC, 2004
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9781584883197
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30000010062154 QA278.8 D47 2004 Open Access Book Book
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Summary

Summary

Balancing the "cookbook" approach of some texts with the more mathematical approach of others, Nonparametric Statistical Methods for Complete and Censored Data introduces commonly used non-parametric methods for complete data and extends those methods to right censored data analysis. Whenever possible, the authors derive their methodology from the general theory of statistical inference and introduce the concepts intuitively for students with minimal backgrounds. Derivations and mathematical details are relegated to appendices at the end of each chapter, which allows students to easily proceed through each chapter without becoming bogged down in a lot of mathematics.

In addition to the nonparametric methods for analyzing complete and censored data, the book covers optimal linear rank statistics, clinical equivalence, analysis of block designs, and precedence tests. To make the material more accessible and practical, the authors use SAS programs to illustrate the various methods included.

Exercises in each chapter, SAS code, and a clear, accessible presentation make this an outstanding text for a one-semester senior or graduate-level course in nonparametric statistics for students in a variety of disciplines, from statistics and biostatistics to business, psychology, and the social scientists.

Prerequisites: Students will need a solid background in calculus and a two-semester course in mathematical statistics.


Author Notes

M. M. Desu is Emeritus Professor in the Department of Biostatistics, School of Public Health and Health Professions, University at Buffalo, State University of New York, Buffalo, USA.


Table of Contents

1 Procedures for a single samplep. 1
1.1 Introductionp. 1
1.2 Binary responsep. 1
1.2.1 Estimation of success probabilityp. 2
1.2.2 Testing one-sided hypotheses about [theta]p. 4
1.2.3 P-values for one-sided testsp. 6
1.2.4 Power function of one-sided testsp. 7
1.2.5 Sample sizep. 8
1.2.6 Testing a two-sided hypothesis about [theta]p. 10
1.2.7 Confidence intervals for [theta]p. 10
1.3 Complete data on continuous responsesp. 14
1.3.1 Point estimation of the medianp. 14
1.3.2 Sign test for testing a simple null hypothesis about the medianp. 15
1.3.3 Estimation of the cdfp. 17
1.3.4 Estimation of survival functionp. 19
1.3.5 Point estimation of population percentilesp. 20
1.3.6 Confidence intervals for percentilesp. 21
1.3.7 Kolmogorov's goodness-of-fit testp. 22
1.3.8 Confidence band for the population distribution functionp. 27
1.3.9 A plotting procedurep. 28
1.4 Procedures for censored datap. 30
1.4.1 Kaplan-Meier estimate of the survival functionp. 31
1.4.2 Estimation of the quartilesp. 34
1.5 Appendix A1: Mathematical supplementp. 35
A1.1 Binomial cdf expressed as a beta integralp. 35
A1.2 Union intersection principlep. 36
A1.3 Distribution of the rth order statisticp. 37
A1.4 Confidence intervals for percentilesp. 38
A1.5 Delta methodp. 39
A1.6 Relation between percentiles of Kolmogorov testsp. 41
1.6 Appendix B1: Computer programsp. 41
B1.1 (1 - [alpha]) quantile and [alpha] quantile of Bin(n, [theta]) distributionp. 41
B1.2 Sample size calculationp. 43
B1.3 Confidence limits for [theta] using beta percentilesp. 45
B1.4 Large sample confidence limits for [theta] (Ghosh's method)p. 46
B1.5 Critical values for a two-sided test for the medianp. 47
B1.6 Critical values for a two-sided test for a quantilep. 49
B1.7 K-S goodness of fitp. 51
B1.8 Kaplan-Meier estimationp. 55
1.7 Problemsp. 57
1.8 Referencesp. 60
2 Procedures for two independent samplesp. 63
2.1 Introductionp. 63
2.2 Two-sample problem with binary responsesp. 63
2.2.1 Testing the homogeneity hypothesisp. 64
2.2.2 Fisher's exact testp. 66
2.2.3 Establishing clinical equivalencep. 69
2.2.4 Confidence interval for the risk difference [Delta] = [theta subscript 1] - [theta subscript 2]p. 70
2.2.5 Confidence interval for the risk ratio [psi] = ([theta subscript 1]/[theta subscript 2])p. 71
2.2.6 Designing a parallel studyp. 73
2.3 Studies with categorical responsesp. 74
2.4 Methods for continuous responsesp. 76
2.4.1 Precedence tests--control median test (Mathisen's test)p. 77
2.4.2 Combined sample percentile tests: Mood's median testp. 82
2.4.3 Wilcoxon-Mann-Whitney procedurep. 86
2.4.4 Analysis of proportional hazards modelp. 97
2.4.5 Smirnov testp. 100
2.4.6 P-P plot for the two-sample problemp. 104
2.4.7 Confidence interval for the difference between medians without shift assumptionp. 105
2.5 Linear rank statistics for the two-sample problemp. 107
2.5.1 Location model (shift model)p. 109
2.5.2 Proportional hazards modelp. 112
2.5.3 Scale modelp. 112
2.6 Analysis of censored datap. 114
2.6.1 Gehan's Wilcoxon testp. 114
2.6.2 Logrank testp. 116
2.6.3 Tarone and Ware testp. 118
2.6.4 Testing for equivalence with censored datap. 120
2.7 Asymptotic relative efficiency (Pitman efficiency)p. 122
2.8 Appendix A2: Mathematical supplementp. 128
A2.1 Derivation of the conditional distribution of A given T = tp. 128
A2.2 Maximum likelihood estimation in the case of clinical equivalencep. 129
A2.3 Koopman's interval for the ratio of two binomial [theta]'sp. 130
A2.4 Calculation of exact P-values for the problem of Section 2.3: Extension of Fisher's exact testp. 132
A2.5 Some models that induce stochastic orderingp. 133
A2.6 The null distribution of T[subscript a]p. 137
A2.7 Confidence interval for [Delta] from Mathisen's testp. 139
A2.8 A class of distribution-free statisticsp. 139
A2.9 The null distribution of Vp. 141
A2.10 Confidence interval for [Delta] from Mood's median testp. 141
A2.11 Null distribution of the rank vectorp. 142
A2.12 Mean and variance of linear rank statisticsp. 146
A2.13 Motivation for the definition of U*[subscript XY] as in (2.78)p. 147
A2.14 Two properties of midranksp. 147
A2.15 Confidence interval for [Delta] from the WMW testp. 149
A2.16 Score test statistic for the PH modelp. 150
A2.17 Expectation of V[subscript (i,N)], of Section 2.5p. 151
A2.18 Asymptotic distribution of X[subscript (k)] the kth order statistic of a random sample of size np. 152
A2.19 Proof of (2.101)p. 153
2.9 Appendix B2: Computer programsp. 154
B2.1 Fisher's test for a 2 X 2 tablep. 154
B2.2 Testing for clinical equivalencep. 155
B2.3 Sample size for one-sided testp. 157
B2.4 Analysis of a 2 X 3 tablep. 158
B2.5 Wilcoxon procedure for complete datap. 159
B2.6 Wilcoxon test for ordered categorical datap. 160
B2.7 Confidence interval for [Delta] from the WMW testp. 162
B2.8 Savage testp. 164
B2.9 The Smirnov testp. 165
B2.10 Wilcoxon and logrank tests for censored datap. 168
2.10 Problemsp. 170
2.11 Referencesp. 172
3 Procedures for paired samplesp. 177
3.1 Introductionp. 177
3.2 Analysis of paired binary responsesp. 177
3.2.1 McNemar's large sample test for the equality of marginal distributionsp. 178
3.2.2 Exact test for equality of marginal distributionsp. 180
3.2.3 Testing for clinical equivalencep. 181
3.2.4 Confidence interval for the difference [Delta]p. 182
3.2.5 Sample size for equivalence trialsp. 183
3.2.6 Estimation of the ratio of marginal probabilitiesp. 184
3.3 Complete data on continuous responsesp. 186
3.3.1 Sign test for complete paired datap. 187
3.3.2 Wilcoxon signed rank testp. 188
3.3.3 Rank transformed t-testp. 192
3.3.4 Confidence interval for [Delta] corresponding to Wilcoxon signed rank testp. 192
3.3.5 Analysis of cross-over designsp. 193
3.4 Asymptotic relative efficiencyp. 194
3.5 Analysis of censored datap. 195
3.5.1 A sign test for censored datap. 195
3.5.2 A generalized signed rank testp. 196
3.5.3 Paired Prentice-Wilcoxon testp. 199
3.6 Appendix A3: Mathematical supplementp. 200
A3.1 Maximum likelihood estimation of [theta subscript 10]p. 200
A3.2 Approximate variance of [phi]p. 200
A3.3 Symmetric property of the distribution of Wp. 201
A3.4 Mean and variance of V[subscript +], under the null hypothesisp. 201
A3.5 Statistic V[subscript +] expressed in terms of Walsh averagesp. 203
A3.6 Some general results about E(V[subscript +])p. 203
A3.7 Confidence interval for [Delta], using Wilcoxon signed rank testp. 204
3.7 Appendix B3: Computer programsp. 205
B3.1 McNemar testp. 205
B3.2 Confidence interval for the ratio [psi]p. 206
B3.3 Confidence interval for risk difference [Delta]p. 208
B3.4 Sign and signed rank proceduresp. 209
B3.5 Rank transformed t-testp. 212
B3.6 Confidence interval for [Delta] difference in meansp. 214
3.8 Problemsp. 216
3.9 Referencesp. 219
4 Procedures for several independent samplesp. 221
4.1 Introductionp. 221
4.2 Discrete responsesp. 222
4.2.1 Binary response studiesp. 222
4.2.2 Categorical data with c categoriesp. 224
4.3 Continuous responses with complete datap. 226
4.3.1 Kruskal-Wallis testp. 226
4.3.2 Savage testp. 228
4.3.3 Mood's median testp. 229
4.3.4 Extension of Mathisen's testp. 230
4.4 Multiple comparison proceduresp. 232
4.4.1 Steel-Dwass procedure based on pairwise rankingsp. 233
4.5 Jonckheere's test for completely ordered alternativesp. 235
4.6 Comparison of several treatments with a controlp. 237
4.6.1 Steel's multiple comparison testp. 238
4.6.2 Spurrier's procedurep. 238
4.6.3 Slivka's control quantile testp. 239
4.6.4 Fligner and Wolfe testp. 239
4.6.5 Chakraborti and Desu testp. 240
4.7 Censored datap. 242
4.8 Appendix A4: Mathematical supplementp. 245
A4.1 Pearson's x[superscript 2] statisticp. 245
A4.2 Derivation of the variance of W[subscript J]p. 247
A4.3 Tukey's studentized range statisticp. 250
A4.4 Null variance of T[subscript FW]p. 251
A4.5 Reformulation of the sum of censored data scoresp. 251
4.9 Appendix B4: Computer programsp. 252
B4.1 Homogeneity of three samplesp. 252
B4.2 Analysis of several independent samplesp. 253
B4.3 Computation of Jonckheere's testp. 256
B4.4 Comparison of survival in three groupsp. 258
4.10 Problemsp. 260
4.11 Referencesp. 263
5 Analysis of block designsp. 267
5.1 Introductionp. 267
5.2 RCB designs with binary responsesp. 267
5.3 RCB designs with continuous uncensored datap. 270
5.3.1 Friedman's testp. 270
5.4 Rank tests for RCB designsp. 273
5.4.1 Median proceduresp. 275
5.4.2 Downton's procedurep. 276
5.5 General block designs with continuous uncensored datap. 278
5.5.1 Proportional cell frequenciesp. 280
5.5.2 Equal block sizesp. 281
5.5.3 Unequal block sizesp. 282
5.5.4 GRCB designsp. 284
5.5.5 Wilcoxon scores procedurep. 285
5.5.6 Blocked comparison of two treatmentsp. 286
5.5.7 Balanced incomplete block (BIB) designsp. 287
5.6 A multiple comparison procedure using Friedman's ranksp. 289
5.7 Page test for ordered alternatives in RCB designsp. 289
5.8 RCB designs with censored datap. 292
5.8.1 Woolson-Lachenbruch rank testsp. 292
5.8.2 Comparing two treatments in blocks (or strata)p. 295
5.9 Appendix A5: Mathematical supplementp. 297
A5.1 Covariance matrix of T of Section 5.3p. 297
A5.2 Derivation of (5.47)p. 297
5.10 Appendix B5: Computer programsp. 298
B5.1 Computation of Friedman's statisticp. 298
B5.2 Analysis of within block ranks for a design with unequal block sizesp. 300
B5.3 Computation of page statisticp. 302
B5.4 Within strata statisticsp. 304
5.11 Problemsp. 307
5.12 Referencesp. 309
6 Independence, correlation, and regressionp. 311
6.1 Introductionp. 311
6.2 Analysis of a bivariate samplep. 311
6.2.1 Test for independence between categorical responsesp. 312
6.2.2 A measure of agreement-[kappa]p. 314
6.3 Testing for correlation between continuous variablesp. 315
6.3.1 Spearman's rank correlation testp. 317
6.3.2 Kendall's taup. 319
6.4 Linear regressionp. 322
6.4.1 Testing a hypothesis about the slope (Theil's test)p. 322
6.4.2 Estimation of the slopep. 323
6.5 Logistic regressionp. 324
6.5.1 Interpretation of [alpha] and [beta]p. 325
6.5.2 Estimation of [alpha] and [beta]p. 325
6.5.3 Logistic regression with several explanatory variablesp. 327
6.6 Procedures for censored datap. 329
6.6.1 Test for independencep. 329
6.6.2 Proportional hazards (PH) modelp. 332
6.7 Appendix A6: Mathematical supplementp. 335
A6.1 Confidence interval for the slopep. 335
A6.2 Maximum likelihood equations for logistic regressionp. 336
6.8 Appendix B6: Computer programsp. 337
B6.1 Test for independencep. 337
B6.2 Spearman's correlation and Kendall's taup. 339
B6.3 Fitting logistic model for Example 6.4 datap. 340
B6.4 Fitting logistic model with several X-variablesp. 342
B6.5 PH regression modelp. 345
6.9 Problemsp. 347
6.10 Referencesp. 349
7 Computer-intensive methodsp. 351
7.1 Introductionp. 351
7.2 Permutation tests and randomization testsp. 351
7.3 Bootstrap methodsp. 354
7.4 Referencesp. 357
Answers to selected problemsp. 359
Subject Indexp. 361
Author Indexp. 365