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Regression with Linear Predictors.pdf
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Per Kragh Andersen • Lene Theil Skovgaard
Regression with Linear
Predictors
With 171 illustrations by Therese Graversen
1 Introduction ............................................... 1
1.1 Introductory examples and types of outcome................ 3
1.1.1 Introductory examples ............................. 3
1.1.2 Types of outcome ................................. 7
1.2 Covariates.............................................. 9
1.2.1 Categorical covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Quantitative covariates............................. 11
1.3 Link functions .......................................... 13
1.4 Building a regression model............................... 22
1.4.1 The linear predictor and the link function . . . . . . . . . . . . 24
1.4.2 Regression models and their interpretation . . . . . . . . . . . 28
1.5 Further examples........................................ 30
1.6 The scopeof this book and how to read it .................. 37
2 Statistical models .......................................... 43
2.1 Random variables and probability . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.1.1 TheBernoulli distribution.......................... 47
2.1.2 TheBinomial distribution .......................... 48
2.1.3 ThePoisson distribution ........................... 50
2.1.4 TheNormal distribution ........................... 51
2.1.5 Other common distributions ........................ 55
2.1.6 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2 Descriptive statistics..................................... 57
2.2.1 Binary outcome ................................... 57
2.2.2 Quantitative outcome.............................. 59
2.2.3 Survival time outcome ............................. 63
2.3 Statistical inference...................................... 64
2.3.1 Estimation ....................................... 65
2.3.2 Model checking ................................... 73
2.3.3 Hypothesis testing................................. 79
2.3.4 Thelikelihood function............................. 84
X Contents
2.4 Exercises ............................................... 92
3 One categorical covariate .................................. 95
3.1 Binary covariate......................................... 96
3.1.1 Quantitative outcome: t-tests ....................... 97
3.1.2 Binary outcome: (2 ×2)-tables and the chi-square test . . 110
3.1.3 Survival time outcome: the 2-sample logrank test . . . . . . 123
3.2 Categorical covariate with more than two levels . . . . . . . . . . . . . 137
3.2.1 Quantitative outcome: One-way analysis of variance . . . 142
3.2.2 Binary outcome: The 2× (k + 1)-table . . . . . . . . . . . . . . . 157
3.2.3 Survival time outcome: The ( k + 1)-sample logrank test 161
3.3 Exercises ...............................................166
4 One quantitative covariate .................................173
4.1 Linear effect ............................................175
4.1.1 Quantitative outcome: Simple linear regression . . . . . . . . 176
4.1.2 Binary outcome: Simple logistic regression . . . . . . . . . . . . 194
4.1.3 Survival time outcome: Simple Cox regression . . . . . . . . . 201
4.2 Nonlinear effect .........................................210
4.2.1 Dividing the covariate range into intervals . . . . . . . . . . . . 214
4.2.2 Polynomials ......................................221
4.2.3 Other nonlinear models with a linear predictor . . . . . . . . 223
4.3 Exercises ...............................................228
5 Multiple regression, the linear predictor ...................231
5.1 Two covariates: Models without interaction .................234
5.1.1 Two categorical covariates . . . . . . . . . . . . . . . . . . . . . . . . . . 234
5.1.2 One categorical and one quantitative covariate . . . . . . . . 245
5.1.3 Two quantitative covariates . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.2 Two covariates: Models with interaction . . . . . . . . . . . . . . . . . . . . 263
5.2.1 Two categorical covariates . . . . . . . . . . . . . . . . . . . . . . . . . . 264
5.2.2 One categorical and one quantitative covariate . . . . . . . . 273
5.2.3 Two quantitative covariates . . . . . . . . . . . . . . . . . . . . . . . . . 278
5.2.4 Saving degrees-of-freedom ..........................285
5.3 Several covariates........................................287
5.3.1 Models without higher-order interactions . . . . . . . . . . . . . 287
5.3.2 Models with higher-order interactions . . . . . . . . . . . . . . . . 289
5.4 Matched studies.........................................294
5.5 Exercises ...............................................300
6 Model building: From purpose to conclusion ...............303
6.1 General principles for model selection . . . . . . . . . . . . . . . . . . . . . . 304
6.1.1 Identification of covariates..........................305
6.1.2 Model diagrams...................................308
6.1.3 Initial model building..............................311
Contents XI
6.1.4 Strategy of analysis................................314
6.1.5 Model checks and diagnostics . . . . . . . . . . . . . . . . . . . . . . . 317
6.1.6 Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
6.1.7 Interactions.......................................319
6.2 Examples...............................................320
6.2.1 Thevitamin Dexample ............................321
6.2.2 Thesurgeryexample...............................331
6.2.3 ThePBC-3 trial...................................342
6.3 Sample sizedetermination ................................354
6.4 Exercises ...............................................365
7 Alternative outcome types and link functions .............. 367
7.1 Multinomial outcome ....................................367
7.1.1 Ordinal outcome ..................................368
7.1.2 Nominal outcome..................................383
7.2 Count outcome..........................................387
7.3 Quantitative outcome ....................................394
7.4 Binary outcome .........................................403
7.4.1 Alternatives to thelogit link........................404
7.4.2 Case-control studies ...............................409
7.5 Survival time outcome ...................................416
7.5.1 Multiplicative hazard models . . . . . . . . . . . . . . . . . . . . . . . 416
7.5.2 Additive hazard models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
7.5.3 Accelerated failure time models . . . . . . . . . . . . . . . . . . . . . 427
7.6 Exercises ...............................................429
8 Further topics .............................................431
8.1 Multivariate outcome ....................................431
8.1.1 Random effects models.............................433
8.1.2 Marginal models ..................................438
8.1.3 Longitudinal and life history data ...................440
8.2 Errors in covariates......................................447
8.2.1 Regression dilution ................................448
8.2.2 Correction for measurement error in covariates . . . . . . . . 452
A Appendix: Notation.......................................457
B Appendix: Use of logarithms...............................463
C Appendix: Some recommendations.........................473
D Programming inR, SAS and STATA . .......................... 477
References.....................................................483
Index .......................................................... 487
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