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Proportional Hazards Regression

By: John O'Quigley

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Hardcover | 27 February 2008

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564 Pages

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23.39 x 15.6 x 3.18

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Proportional hazards models and their extensions (models with ti- dependent covariates, models with time dependent regression co- cients, models with random coe?cients and any mixture of these) can be used to characterize just about any applied problem to which the techniques of survival analysis are appropriate. This simple obser- tion enables us to ?nd an elegant statistical expression for all plausible practical situations arising in the analysis of survival data. We have a single unifying framework. In consequence, a solid understanding of the framework itself o?ers the statistician the ability to tackle the thorniestofquestionswhichmayarisewhendealingwithsurvivaldata. The main goal of this text is not to present or review the very s- stantial amount of research that has been carried out on proportional hazards and related models. Rather, the goal is to consider the many questions which are of interest in a regression analysis of survival data (prediction, goodness of ?t, model construction, inference and int- pretation in the presence of misspeci?ed models) from the standpoint of the proportional hazards and the non-proportional hazards models.

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"The book is clearly intended to be student-friendly. Each chapter begins with a section called Summary and a following one called motivation; each chapter ends with some exercises and class projects. ... It is very carefully written, with detailed explanation and discussion everywhere. ... I believe that the book can be thoroughly recommended to the student starting his research in the field and to the practitioner who needs to understand some of the theory." (Martin Crowder, International Statistical Review, Vol. 76 (3), 2008)

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Non-Fiction Science Biology, Life Sciences Medicine Medicine in General Public Health & Preventive Medicine Epidemiology & Medical Statistics Mathematics Probability & Statistics Economics Econometrics

Introduction	p. 1
Summary	p. 1
Motivation	p. 1
Objectives	p. 6
Controversies	p. 7
Data sets	p. 10
Use as a graduate text	p. 10
Exercises and class projects	p. 11
Background: Probability	p. 13
Summary	p. 13
Motivation	p. 14
Integration and measure	p. 14
Random variables and probability measure	p. 17
Distributions and densities	p. 19
Expectation	p. 24
Order statistics and their expectations	p. 26
Entropy and variance	p. 32
Approximations	p. 36
Stochastic processes	p. 41
Brownian motion	p. 42
Counting processes and martingales	p. 51
Exercises and class projects	p. 59
Background: General inference	p. 63
Summary	p. 63
Motivation	p. 64
Limit theorems for sums of random variables	p. 64
Functional Central Limit Theorem	p. 68
Empirical distribution function	p. 71
Inference for martingales and stochastic integrals	p. 74
Estimating equations	p. 80
Inference using resampling techniques	p. 88
Explained variation	p. 91
Exercises and class projects	p. 99
Background: Survival analysis	p. 103
Summary	p. 103
Motivation	p. 103
Basic tools	p. 104
Some potential models	p. 112
Censoring	p. 120
Competing risks as a particular type of censoring	p. 124
Exercises and class projects	p. 125
Marginal survival	p. 129
Summary	p. 129
Motivation	p. 129
Maximum likelihood estimation	p. 130
Empirical estimate (no censoring)	p. 136
Empirical estimate (with censoring)	p. 138
Exercises and class projects	p. 148
Regression models and subject heterogeneity	p. 151
Summary	p. 151
Motivation	p. 152
General or nonproportional hazards model	p. 153
Proportional hazards model	p. 154
The Cox regression model	p. 155
Modeling multivariate problems	p. 165
Partially proportional hazards models	p. 172
Non proportional hazards model with intercept	p. 183
Time-dependent covariates	p. 186
Time-dependent covariates and non proportional hazards models	p. 188
Proportional hazards models in epidemiology	p. 189
Exercises and class projects	p. 199
Inference: Estimating equations	p. 203
Summary	p. 203
Motivation	p. 204
The observations	p. 205
Main theorem	p. 207
The estimating equations	p. 219
Consistency and asymptotic normality of [beta]	p. 225
Interpretation for [beta]* as average effect	p. 226
Exercises and class projects	p. 228
Inference: Functions of Brownian motion	p. 231
Summary	p. 231
Motivation	p. 232
Brownian motion approximations	p. 233
Non and partially proportional hazards models	p. 238
Tests based on functions of Brownian motion	p. 239
Multivariate model	p. 249
Graphical representation of regression effects	p. 254
Operating characteristics of tests	p. 258
Goodness-of-fit tests	p. 260
Exercises and class projects	p. 263
Inference: Likelihood	p. 267
Summary	p. 267
Motivation	p. 267
Likelihood solution for parametric models	p. 268
Likelihood solution for exponential models	p. 270
Semi-parametric likelihood solution	p. 275
Other likelihood expressions	p. 280
Goodness-of-fit of likelihood estimates	p. 287
Exercises and class projects	p. 292
Inference: Stochastic integrals	p. 295
Summary	p. 295
Motivation	p. 295
Counting process framework to the model	p. 296
Some nonparametric statistics	p. 298
Stochastic integral representation of score statistic	p. 299
Exercises and class projects	p. 308
Inference: Small samples	p. 311
Summary	p. 311
Motivation	p. 311
Additive and multiplicative models	p. 312
Estimation: First two moments	p. 315
Edgeworth and saddlepoint approximations	p. 316
Distribution of estimating equation	p. 318
Simulation studies	p. 323
Example	p. 327
Further points	p. 329
Exercises and class projects	p. 330
Inference: Changepoint models	p. 331
Summary	p. 331
Motivation	p. 331
Some changepoint models	p. 333
Inference when [gamma] is known	p. 334
Inference when [gamma] is unknown	p. 335
Maximum of log-rank type tests	p. 338
Computational aspects	p. 339
Two groups with crossing hazards	p. 342
Illustrations	p. 347
Some guidelines in model and test selection	p. 354
Exercises and class projects	p. 358
Explained variation	p. 359
Summary	p. 359
Motivation	p. 359
Finding a suitable measure of R[superscript 2]	p. 361
An R[superscript 2] measure based on Schoenfeld residuals	p. 366
Finite sample properties of R[superscript 2] and R[superscript 2 subscript epsilon]	p. 370
Large sample properties	p. 371
Interpretation	p. 375
Simulation results	p. 379
Extensions	p. 381
Theoretical construction for distance measures	p. 386
Isolation method for bias-reduction	p. 390
Illustrations from studies in cancer	p. 395
Exercises and class projects	p. 403
Explained randomness	p. 407
Summary	p. 407
Motivation	p. 407
Information gain and explained randomness	p. 409
Explained randomness in Z given T	p. 411
Approximation of [rho superscript 2 subscript 1] by [rho superscript 2 subscript 2]	p. 416
Simple working approximation of [rho superscript 2 subscript 1] and [rho superscript 2 subscript 2]	p. 417
Multiple coefficient of explained randomness	p. 420
Partially explained randomness	p. 421
Isolation method for bias-reduction	p. 422
Simulations	p. 423
Illustrations	p. 426
Further extensions	p. 429
Exercises and class projects	p. 435
Survival given covariates	p. 437
Summary	p. 437
Motivation	p. 438
Probability that T[subscript i] is greater than T[subscript j]	p. 438
Estimating conditional survival given that Z = z	p. 441
Estimating conditional survival given Z [set membership] H	p. 442
Estimating the variance of S(t[vertical bar]Z [set membership] H)	p. 444
Relative merits of competing estimators	p. 448
Illustrations	p. 448
Generalization under multiplicative relative risk	p. 451
Informative censoring	p. 454
Exercises and class projects	p. 460
Proofs of theorems, lemmas and corollaries	p. 463
Bibliography	p. 477
Index	p. 539
Table of Contents provided by Ingram. All Rights Reserved.

Proportional Hazards Regression

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