Goodness-of-Fit Tests and

Model Validity

C. Huber-Carol N. Balakrishnan M.S. Nikulin M. Mesbah Editors

Birkhäuser Boston • Basel • Berlin

Contents

Preface Contributors List of Tables List of Figures

PART I: HISTORY AND FUNDAMENTALS

1 Karl Pearson and the Chi-Squared Test D. R. Cox

1.1 Karl Pearson 1857-1937: Background to the Chi-Squared Paper 3

1.2 K. P : After Chi-Squared 5 1.3 The 1900 Paper 5 1.4 Importance of the Chi-Squared Test 6

References 8

2 Karl Pearson Chi-Square Test—The Dawn of Statistical Inference C. R. Rao

2.1 Introduction 9 2.2 Large Sample Criteria: The Holy Trinity 11

2.2.1 Likelihood ratio criterion 11 2.2.2 Wald test 12 2.2.3 Rao's score test 12

2.3 Specification Tests for a Multinomial Distribution 13 2.3.1 Test of a simple hypothesis 13 2.3.2 Tests of a composite hypothesis 14 2.3.3 Test for goodness-of-fit in a subset of cells 2.3.4 Analysis of chi-square 17 2.3.5 Some applications of the chi-square test 18

2.4 Other Tests of Goodness-of-Fit 18

v

VI

2.5 Specification Tests for Continuous Distributions 20 References 22

3 A p p r o x i m a t e M o d e l s 25 Peter J. Huber

3.1 Models 25 3.2 Bayesian Modeimg 27 3.3 Mathematical Statistics and Approximate Models 29 3.4 Statistical Signincance and Relevance 31 3.5 Composite Models 32 3.6 The Role of Simulation 38 3.7 Summary Conclusions 40

References 40

P A R T IL C H I - S Q U A R E D T E S T

4 Par t i t ion ing the Pearson-Fisher Chi -Squared Goodness -o f -F i t Stat i s t ic 45 G. D. Rayner

4.1 Introduction 45 4.2 Neyman Smooth Goodness-of-Fit Tests 46

4.2.1 Smooth goodness-of-fit tests for categorized data 47

4.2.2 Partit ioning the Pearson-Fisher chi-squared statistic 48

4.3 Constructing the Pearson-Fisher Decomposition 49 4.4 Simulation Study 50 4.5 Results and Discussion 51

References 55

5 Stat is t ica l Tests for N o r m a l Family in P r e s e n c e of Out ly ing Observat ions 57 Aicha Zerbet

5.1 The Chi-Squared Test of Normality in the Univariate Case 57 5.1.1 Example: Analysis of the data of Milliken 59

5.2 Bol'shev Test for Outliers 59 5.2.1 Stages of applications of the test of Bol'shev 60 5.2.2 Example 2: Analysis of the da ta of Daniel (1959) 60

5.3 Power of the Chi-Squared Test 61 References 63

Contents

6 Chi -Squared Test for the Law of A n n u a l D e a t h R a t e s : Case w i t h Censure for Life Insurance Fi les Leo Gerville-Reache

6.1 Introduction 65 6.2 Chi-Squared Goodness-of-Fit Test 66

6.2.1 Statistics with censure 66 6.2.2 Goodness-of-fit test for a composite hypothesis 67

6.3 Demonstration 68 References 69

P A R T II I : G O O D N E S S - O F - F I T T E S T S F O R

P A R A M E T R I C D I S T R I B U T I O N S

7 Shapiro-Wilk T y p e Goodness -of -Fi t Tests for Normal i ty : A s y m p t o t i c s Rev i s i t ed Pranab Kumar Sen

7.1 Introduction 73 7.2 Preliminary Notion 74 7.3 SOADR Results for BLUE and LSE 77 7.4 Asymptotics for W* 81 7.5 Asymptotics Under Alternatives 85

References 87

8 A Test for Exponent ia l i ty B a s e d on Spacings for Progress ive ly Type - I I Censored D a t a N. Balakrishnan, H. K. T. Ng, and N. Kannan

8.1 Introduction 89 8.2 Progressive Censoring 91 8.3 Test for Exponentiality 92

8.3.1 Null distribution of T 93 8.4 Power Function Approximation and Simulation

Results 95 8.4.1 Approximation of power function 95 8.4.2 Monte Carlo power comparison 97

8.5 Modified EDF and Shapiro-Wilk Statistics 98 8.6 Two-Parameter Exponential Case 99 8.7 Illustrative Examples 100

8.7.1 Example 1: One-parameter exponential case 100 8.7.2 Example 2: Two-parameter exponential case 101

8.8 Multi-Sample Extension 102 8.9 Conclusions 103

References 103

Vll l Contents

9 Goodness-of-Fit Statistics for the Exponential Distribution When the Data are Grouped 113 Sneh Gulati and Jordan Neus

9.1 Introduction 113 9.2 The Model and the Test Statistics 115 9.3 Asymptotic Distribution 116 9.4 Power Studies 119

References 122

10 Characterization Theorems and Goodness-of-Fit Tests 125 Carol E. Marchetti and Govind S. Mudholkar

10.1 Introduction and Summary 126 10.2 Characterization Theorems 127

10.2.1 Entropy characterizations 127 10.2.2 Statistical independence 128

10.3 Maximum Entropy Tests 130 10.4 Four Z Tests 131 10.5 Byproducts: The G-IG Analogies 134

References 137

11 Goodness-of-Fit Tests Based on Record Data and Generalized Ranked Set Data 143 Barry C. Arnold, Robert J. Beaver, Enrique Castillo, and Jose Maria Sarabia

11.1 Introduction 143 11.2 Record Data 144 11.3 Generalized Ranked Set Data 144 11.4 Power 150 11.5 Composite Null Hypotheses 154 11.6 Remarks 156

References 156

PART IV: REGRESSION AND GOODNESS-OF-FIT TESTS

12 Gibbs Regression and a Test of Goodness-of-Fit 161 Lynne Seymour

12.1 Introduction 161 12.2 The Motivation and the Model 162 12.3 Application and Evaluation of the Model 165 12.4 Discussion 169

References 170

13 A CLT for t h e L_2 N o r m of t h e Regres s ion Es t imators U n d e r a -Mix ing: Appl i ca t ion t o G-O-F Tests 173 Cheikh A. T. Diack

13.1 Introduction 173 13.2 Estimators 174 13.3 A Limit Theorem 175 13.4 Inference 177 13.5 Proofs 178

References 183

14 Tes t ing t h e Goodness -of -F i t of a Linear M o d e l in N o n p a r a m e t r i c Regress ion 185 Zäher Mohdeb and Abdelkader Mokkadem

14.1 Introduction 185 14.2 The Test Statistic 186 14.3 Simulations 189

References 193

15 A N e w Test of Linear H y p o t h e s i s in Regres s ion 195 Y. Baraud, S. Huet, and B. Laurent

15.1 Introduction 195 15.2 The Testing Procedure 196

15.2.1 Description of the procedure 197 15-2.2 Behavior of the test under the null

hypothesis 198 15.2.3 A toy framework: The case of a known

variance 198 15.3 The Power of the Test 198

15.3.1 The main result 198 15.3.2 Rates of testing 199

15.4 Simulations 201 15.4.1 The Simulation experiment 201 15.4.2 The testing procedure 202 15.4.3 The test proposed by Horowitz and

Spokoiny (2000) 202 15.4.4 Results of the Simulation study 203

15.5 Proofs 203 15.5.1 Proof of Theorem 15.3.1 203 15.5.2 Proof of Corollary 15.3.1 204 References 206

X Contents

PART V: GOODNESS-OF-FIT TESTS IN SURVIVAL ANALYSIS

AND RELABILITY

16 Inference in Extensions of the Cox Model for Heterogeneous Populations 211 Odile Pons

16.1 Introduction 211 16.2 Non-Stationary Cox Model 212 16.3 Varying-Coefficient Cox Model 219

References 224

17 Assumptions of a Latent Survival Model 227 Mei-Ling Ting Lee and G. A. Whitmore

17.1 Introduction 227 17.2 Latent Survival Model 228 17.3 Data and Parameter Estimation 229 17.4 Model Validation Methods 230 17.5 Remedies to Achieve a Better Model Fit 233

References 235

18 Goodness-of-Fit Testing for the Cox Proportional Hazards Model 237 Karthik Devarajan and Nader Ebrahimi

18.1 Introduction 237 18.2 Goodness-of-Fit Testing for the Cox PH Model 240 18.3 Comparison of the Proposed Goodness-of-Fit Test

with Existing Methods 242 18.4 Illustration of the Goodness-of-Fit Test using

Real-Life Data 249 18.5 Concluding Remarks 250

References 251

19 A New Family of Multivariate Distributions for Survival Data 255 Shulamith T. Gross and Catherine Huber-Carol

19.1 Introduction 255 19.2 Frailty Models: An Overview 255 19.3 The Model 257 19.4 An Application to Skin Grafts Rejection 261

19.4.1 Description of the data 261 References 264

Contents X I

20 Discrimination Index, the Area Under the ROC Curve 267 Byung-Ho Nam and Ralph B. D'Agostino

20.1 Introduction 268 20.2 Nonparametric Confidence Interval for Area under

the ROC Curve 269 20.2.1 Discrimination in logistic regression 269 20.2.2 Estimation of the shift parameter A under

the shift model 271 20.2.3 Confidence interval for the area under the

ROC curve 272 20.3 Extension of C Statistic to Survival Analysis 273

Appendix 277 References 279

21 Goodness-of-Fit Tests for Accelerated Life Models 281 Vilijandas Bagdonavicius and Mikhail S. Nikulin

21.1 Introduction 281 21.2 Generalized Sedyakin's Model 282 21.3 Alternatives to the GS Model 286

21.3.1 Proportional hazards model 286 21.3.2 Model including influence of

switch-up's of stresses on reliability 287 21.4 Test Statistic for the GS Model 287 21.5 Asymptotic Distribution of the Test Statistic 288 21.6 The Test 293 21.7 Consistency and the Power of the Test Against

Approaching Alternatives 293 References 296

PART VI: GRAPHICAL METHODS AND

GENERAL GOODNESS-OF-FIT TESTS

22 Two Nonstandard Examples of the Classical Stratification Approach to Graphically Assessing Proportionality of Hazards 301 Niels Keiding

22.1 Introduction 301 22.2 Some Approaches to Testing Proportionality of

Hazards 302 22.3 "Proportionality" in Discrete-Time Regression for

Retro-Hazard 303

XU Contents

TIA The Renewal Assumption in Modulated Renewal Processess 304 References 308

23 Association in Contingency Tables, Correspondence Analysis, and (Modified) Andrews Plots 311 Ravindra Khattree and Dyanand N. Naik

23.1 Introduction 311 23.2 (Modified) Andrews Plots in Correspondence

Analysis 313 23.3 Some Examples 314 23.4 Modified Andrews Plots and Rao's Correspondence

Analysis 320 23.5 Conclusions 325

References 325

24 Orthogonal Expansions and Distinction Between Logistic and Normal 327 Carles M. Cuadras and Daniel Cuadras

24.1 Introduction 327 24.2 Orthogonal Expansion in Principal Components 328 24.3 Maximum Correlation for the Logistic Distribution 331 24.4 Distinction Between Logistic and Normal 333

References 339

25 Functional Tests of Fit 341 Denis Bosq

25.1 Introduction 341 25.2 Behaviour of ||Tn|| in Distribution 342 25.3 Consistency of FTF Tests and Rate of Convergence 344 25.4 Adjacent Hypothesis 346 25.5 Choosing a Kernel 347 25.6 Local Efficiency of FTF Tests 348 25.7 Indications Concerning the Proofs 351 25.8 Simulations 352

References 355

26 Quasi Most Powerful Invariant Tests of Goodness-of-Fit 357 Gilles R. Ducharme and Benoit Frichot

26.1 Introduction 357 26.2 Laplace Approximation 358 26.3 Quasi Most Powerful Invariant Test 359

References 360

Contents xm

PART VII: MODEL VALIDITY IN QUALITY OP LIFE

27 Test of Monotonicity for the Rasch Model 365 Jean Bretagnolle

27.1 Results of the Literature 365 27.2 Extension of Hoeffding Result 366 27.3 A Questionnaire Model 366 27.4 Simulations about the Level in the

Conditional Test Case 368 27.5 Simulations about the Power under HA 369 27.6 Conclusion 369

References 369

28 Validation of Model Assumptions in Quality of Life Measurements 371 A. Hamon, J. F. Dupuy, and M. Mesbah

28.1 Introduction 371 28.2 Classical Theory 372 28.3 SIP Mobility Data (I) 373 28.4 The Rasch Model 375

28.4.1 Goodness-of-fit tests 375 28.4.2 A graphical method 378

28.5 SIP Mobility Data (II) 378 28.6 Conclusion 382

References 382

PART VIII: TESTS OF HYPOTHESES AND ESTIMATION

WITH APPLICATIONS

29 One-Sided Hypotheses in a Multinomial Model 387 Richard M. Dudley and Dominique M. Haughton

29.1 Introduction 387 29.2 Putting Multiple Data Sets Into an i.i.d. Form 388 29.3 Model Selection Criteria 388 29.4 Application to 2 x 2 Contingency Tables 390 29.5 Common Odds Ratio Profile Likelihoods 391 29.6 Jeffreys Priors for Mixture Models 391 29.7 Posterior Probabilities that Models are Best 393 29.8 Data on Long-Term Aspirin Therapy after an MI 394 29.9 Numerical Results 395

29.10 Discussion and Conclusions 396 References 397

XIV Contents

30 A Depth Test for Symmetry 401 Peter J. Rousseeuw and Anja Struyf

30.1 Introduction 401 30.2 Location Depth and Angular Symmetry 402 30.3 A Test for Angular Symmetry 405 30.4 Regression Depth and Linearity of the

Conditional Median 407 References 411

31 Adaptive Combination of Tests 413 Yadolah Dodge and Jana Jureckovd

31.1 Introduction 413 31.2 Adaptive Combination of Estimators 415 31.3 Adaptive Combination of Tests 417

31.3.1 Adaptive combination of F-test and median-type test 420

31.3.2 Adaptive combination of M-test and median-type test 421

References 423

32 Partially Parametric Testing 425 J. C. W. Rayner

32.1 Partially Parametric Inference 425 32.2 5-Sample Smooth Tests for Goodness-of-Fit 426 32.3 Partially Parametric Alternatives to the t-Test 428 32.4 Tests for the Location of Modes 430

References 432

33 Exact Nonparametric Two-Sample Homogeneity Tests 435 Jean-Marie Dufour and Abdeljelil Farhat

33.1 Introduction 435 33.2 Test Statistics 437 33.3 Exact Randomized Permutation Tests 440 33.4 Simulation Study 442 33.5 Conclusion 444

References 447

34 Power Comparisons of Some Nonparametric Tests for Lattice Ordered Alternatives in Two-Factor Experiments 449 Thu Hoäng and Van L. Parsons

34.1 Introduction 449 34.2 Hypotheses and Test Statistics 450 34.3 Test Statistic Power Evaluations 452

Contents XV

34.4 Results and Conclusions 455 Appendix 461 References 461

35 Tests of Independence with Exponential Marginals 463 Paul Deheuvels

35.1 Introduction 463 35.2 Karhunen-Loeve Expansions 465 35.3 Applications to Tests of Independence 470

References 472

36 Testing Problem for Increasing Function in a Model with Infinite Dimensional Nuisance Parameter 477 M. Nikulin and V. Solev

36.1 Introduction 477 36.2 Consistency of the Estimator 9n 483 36.3 Asymptotic Behavior of Kernel Estimators of Densities 486

References 492

37 The Concept of Generalized Asymptotic Deficiency and its Application to the Minimum Discrepancy Estimation 495 M. Akahira

37.1 Introduction 495 37.2 The Concept of Generalized Asymptotic Deficiency 496 37.3 An Application to the Minimum Discrepancy

Estimation 500 References 502

Index 505