Preface |
|
xv | |
1 Survival analysis |
|
1 | (16) |
|
1.1 Special features of survival data |
|
|
1 | (4) |
|
|
2 | (1) |
|
1.1.2 Independent censoring |
|
|
3 | (1) |
|
1.1.3 Study time and patient time |
|
|
3 | (2) |
|
|
5 | (5) |
|
1.3 Survivor, hazard and cumulative hazard functions |
|
|
10 | (4) |
|
1.3.1 The survivor function |
|
|
10 | (2) |
|
1.3.2 The hazard function |
|
|
12 | (1) |
|
1.3.3 The cumulative hazard function |
|
|
13 | (1) |
|
1.4 Computer software for survival analysis |
|
|
14 | (1) |
|
|
15 | (2) |
2 Some non-parametric procedures |
|
17 | (40) |
|
2.1 Estimating the survivor function |
|
|
17 | (8) |
|
2.1.1 Life-table estimate of the survivor function |
|
|
19 | (2) |
|
2.1.2 Kaplan-Meier estimate of the survivor function |
|
|
21 | (3) |
|
2.1.3 Nelson-Aalen estimate of the survivor function |
|
|
24 | (1) |
|
2.2 Standard error of the estimated survivor function |
|
|
25 | (6) |
|
2.2.1 Standard error of the Kaplan-Meier estimate |
|
|
26 | (1) |
|
2.2.2 Standard error of other estimates |
|
|
27 | (1) |
|
2.2.3 Confidence intervals for values of the survivor function |
|
|
28 | (3) |
|
2.3 Estimating the hazard function |
|
|
31 | (5) |
|
2.3.1 Life-table estimate of the hazard function |
|
|
31 | (1) |
|
2.3.2 Kaplan-Meier type estimate |
|
|
32 | (3) |
|
2.3.3 Estimating the cumulative hazard function |
|
|
35 | (1) |
|
2.4 Estimating the median and percentiles of survival times |
|
|
36 | (2) |
|
2.5 Confidence intervals for the median and percentiles |
|
|
38 | (2) |
|
2.6 Comparison of two groups of survival data |
|
|
40 | (10) |
|
|
41 | (3) |
|
|
44 | (4) |
|
|
48 | (1) |
|
2.6.4 Comparison of the log-rank and Wilcoxon tests |
|
|
49 | (1) |
|
2.7 Comparison of three or more groups of survival data |
|
|
50 | (2) |
|
|
52 | (2) |
|
2.9 Log-rank test for trend |
|
|
54 | (2) |
|
|
56 | (1) |
3 The Cox regression model |
|
57 | (74) |
|
3.1 Modelling the hazard function |
|
|
57 | (3) |
|
3.1.1 A model for the comparison of two groups |
|
|
58 | (1) |
|
3.1.2 The general proportional hazards model |
|
|
59 | (1) |
|
3.2 The linear component of the model |
|
|
60 | (5) |
|
3.2.1 Including a variate |
|
|
61 | (1) |
|
|
61 | (1) |
|
3.2.3 Including an interaction |
|
|
62 | (1) |
|
3.2.4 Including a mixed term |
|
|
63 | (2) |
|
3.3 Fitting the Cox regression model |
|
|
65 | (7) |
|
3.3.1 Likelihood function for the model |
|
|
67 | (2) |
|
|
69 | (2) |
|
3.3.3 The Newton-Raphson procedure |
|
|
71 | (1) |
|
3.4 Confidence intervals and hypothesis tests |
|
|
72 | (4) |
|
3.4.1 Confidence intervals for hazard ratios |
|
|
73 | (1) |
|
|
73 | (3) |
|
3.5 Comparing alternative models |
|
|
76 | (7) |
|
3.5.1 The statistic -2 log L |
|
|
77 | (1) |
|
3.5.2 Comparing nested models |
|
|
78 | (5) |
|
3.6 Strategy for model selection |
|
|
83 | (7) |
|
3.6.1 Variable selection procedures |
|
|
84 | (6) |
|
3.7 Variable selection using the lasso |
|
|
90 | (5) |
|
3.7.1 The lasso in Cox regression modelling |
|
|
91 | (1) |
|
|
92 | (3) |
|
|
95 | (4) |
|
3.8.1 Testing for non-linearity |
|
|
96 | (1) |
|
3.8.2 Modelling non-linearity |
|
|
97 | (1) |
|
3.8.3 Fractional polynomials |
|
|
98 | (1) |
|
3.9 Interpretation of parameter estimates |
|
|
99 | (8) |
|
3.9.1 Models with a variate |
|
|
99 | (1) |
|
3.9.2 Models with a factor |
|
|
100 | (4) |
|
3.9.3 Models with combinations of terms |
|
|
104 | (3) |
|
3.10 Estimating the hazard and survivor functions |
|
|
107 | (9) |
|
3.10.1 The special case of no covariates |
|
|
110 | (1) |
|
3.10.2 Some approximations to estimates of baseline functions |
|
|
110 | (6) |
|
3.11 Risk adjusted survivor function |
|
|
116 | (4) |
|
3.11.1 Risk adjusted survivor function for groups of individuals |
|
|
117 | (3) |
|
3.12 Explained variation in the Cox regression model |
|
|
120 | (5) |
|
3.12.1 Measures of explained variation |
|
|
122 | (1) |
|
3.12.2 Measures of predictive ability |
|
|
123 | (1) |
|
|
124 | (1) |
|
3.13 Proportional hazards and the log-rank test |
|
|
125 | (3) |
|
|
128 | (3) |
4 Model checking in the Cox regression model |
|
131 | (40) |
|
4.1 Residuals for the Cox regression model |
|
|
131 | (11) |
|
4.1.1 Cox-Snell residuals |
|
|
132 | (1) |
|
4.1.2 Modified Cox-Snell residuals |
|
|
133 | (2) |
|
4.1.3 Martingale residuals |
|
|
135 | (1) |
|
|
136 | (1) |
|
4.1.5 Schoenfeld residuals |
|
|
137 | (1) |
|
|
138 | (4) |
|
4.2 Assessment of model fit |
|
|
142 | (10) |
|
4.2.1 Plots based on the Cox-Snell residuals |
|
|
142 | (3) |
|
4.2.2 Plots based on the martingale and deviance residuals |
|
|
145 | (2) |
|
4.2.3 Checking the functional form of covariates |
|
|
147 | (5) |
|
4.3 Identification of influential observations |
|
|
152 | (8) |
|
4.3.1 Influence of observations on a parameter estimate |
|
|
153 | (2) |
|
4.3.2 Influence of observations on the set of parameter estimates |
|
|
155 | (3) |
|
4.3.3 Treatment of influential observations |
|
|
158 | (2) |
|
4.4 Testing the assumption of proportional hazards |
|
|
160 | (8) |
|
4.4.1 The log-cumulative hazard plot |
|
|
161 | (2) |
|
4.4.2 Use of Schoenfeld residuals |
|
|
163 | (1) |
|
4.4.3 Tests for non-proportional hazards |
|
|
164 | (2) |
|
4.4.4 Adding a time-dependent variable |
|
|
166 | (2) |
|
|
168 | (1) |
|
|
169 | (2) |
5 Parametric proportional hazards models |
|
171 | (50) |
|
5.1 Models for the hazard function |
|
|
171 | (6) |
|
5.1.1 The exponential distribution |
|
|
172 | (1) |
|
5.1.2 The Weibull distribution |
|
|
173 | (4) |
|
5.2 Assessing the suitability of a parametric model |
|
|
177 | (1) |
|
5.3 Fitting a parametric model to a single sample |
|
|
178 | (3) |
|
5.3.1 Likelihood function for randomly censored data |
|
|
180 | (1) |
|
5.4 Fitting exponential and Weibull models |
|
|
181 | (11) |
|
5.4.1 Fitting the exponential distribution |
|
|
182 | (4) |
|
5.4.2 Fitting the Weibull distribution |
|
|
186 | (2) |
|
5.4.3 Standard error of a percentile of the Weibull distribution |
|
|
188 | (4) |
|
5.5 A model for the comparison of two groups |
|
|
192 | (7) |
|
5.5.1 The log-cumulative hazard plot |
|
|
192 | (2) |
|
|
194 | (5) |
|
5.6 The Weibull proportional hazards model |
|
|
199 | (9) |
|
|
200 | (1) |
|
5.6.2 Standard error of a percentile in the Weibull model |
|
|
201 | (2) |
|
5.6.3 Log-linear form of the model |
|
|
203 | (2) |
|
5.6.4 Exploratory analyses |
|
|
205 | (3) |
|
5.7 Comparing alternative Weibull models |
|
|
208 | (7) |
|
5.8 Explained variation in the Weibull model |
|
|
215 | (1) |
|
5.9 The Gompertz proportional hazards model |
|
|
216 | (2) |
|
|
218 | (1) |
|
|
219 | (2) |
6 Accelerated failure time and other parametric models |
|
221 | (54) |
|
6.1 Probability distributions for survival data |
|
|
221 | (4) |
|
6.1.1 The log-logistic distribution |
|
|
222 | (1) |
|
6.1.2 The lognormal distribution |
|
|
222 | (2) |
|
6.1.3 The gamma distribution |
|
|
224 | (1) |
|
6.1.4 The inverse Gaussian distribution |
|
|
225 | (1) |
|
|
225 | (2) |
|
6.3 Accelerated failure model for two groups |
|
|
227 | (5) |
|
6.3.1 Comparison with the proportional hazards model |
|
|
228 | (3) |
|
6.3.2 The percentile-percentile plot |
|
|
231 | (1) |
|
6.4 The general accelerated failure time model |
|
|
232 | (4) |
|
6.4.1 Log-linear form of the accelerated failure time model |
|
|
234 | (2) |
|
6.5 Parametric accelerated failure time models |
|
|
236 | (7) |
|
6.5.1 The Weibull accelerated failure time model |
|
|
236 | (3) |
|
6.5.2 The log-logistic accelerated failure time model |
|
|
239 | (1) |
|
6.5.3 The lognormal accelerated failure time model |
|
|
240 | (1) |
|
|
241 | (2) |
|
6.6 Fitting and comparing accelerated failure time models |
|
|
243 | (7) |
|
6.7 The proportional odds model |
|
|
250 | (5) |
|
6.7.1 The log-logistic proportional odds model |
|
|
253 | (2) |
|
6.8 Some other distributions for survival data |
|
|
255 | (1) |
|
6.9 Flexible parametric models |
|
|
256 | (12) |
|
6.9.1 The Royston and Parmar model |
|
|
259 | (3) |
|
6.9.2 Number and position of the knots |
|
|
262 | (1) |
|
|
262 | (4) |
|
6.9.4 Proportional odds models |
|
|
266 | (2) |
|
6.10 Modelling cure rates |
|
|
268 | (2) |
|
6.11 Effect of covariate adjustment |
|
|
270 | (2) |
|
|
272 | (3) |
7 Model checking in parametric models |
|
275 | (20) |
|
7.1 Residuals for parametric models |
|
|
275 | (3) |
|
7.1.1 Standardised residuals |
|
|
275 | (1) |
|
7.1.2 Cox-Snell residuals |
|
|
276 | (1) |
|
7.1.3 Martingale residuals |
|
|
277 | (1) |
|
|
277 | (1) |
|
|
277 | (1) |
|
7.2 Residuals for particular parametric models |
|
|
278 | (6) |
|
7.2.1 Weibull distribution |
|
|
279 | (1) |
|
7.2.2 Log-logistic distribution |
|
|
279 | (1) |
|
7.2.3 Lognormal distribution |
|
|
280 | (1) |
|
7.2.4 Analysis of residuals |
|
|
280 | (4) |
|
7.3 Comparing observed and fitted survivor functions |
|
|
284 | (3) |
|
7.4 Identification of influential observations |
|
|
287 | (4) |
|
7.4.1 Influence of observations on a parameter estimate |
|
|
287 | (1) |
|
7.4.2 Influence of observations on the set of parameter estimates |
|
|
288 | (3) |
|
7.5 Testing proportional hazards in the Weibull model |
|
|
291 | (1) |
|
|
292 | (3) |
8 Time-dependent variables |
|
295 | (24) |
|
8.1 Types of time-dependent variables |
|
|
295 | (1) |
|
8.2 A model with time-dependent variables |
|
|
296 | (6) |
|
8.2.1 Fitting the Cox model |
|
|
297 | (3) |
|
8.2.2 Estimation of baseline hazard and survivor functions |
|
|
300 | (2) |
|
8.3 Model comparison and validation |
|
|
302 | (2) |
|
8.3.1 Comparison of treatments |
|
|
303 | (1) |
|
8.3.2 Assessing model adequacy |
|
|
303 | (1) |
|
8.4 Some applications of time-dependent variables |
|
|
304 | (2) |
|
|
306 | (10) |
|
8.6 Counting process format |
|
|
316 | (1) |
|
|
317 | (2) |
9 Interval-censored survival data |
|
319 | (26) |
|
9.1 Modelling interval-censored survival data |
|
|
319 | (3) |
|
9.2 Modelling the recurrence probability in the follow-up period |
|
|
322 | (3) |
|
9.3 Modelling the recurrence probability at different times |
|
|
325 | (7) |
|
9.4 Arbitrarily interval-censored survival data |
|
|
332 | (10) |
|
9.4.1 Modelling arbitrarily interval-censored data |
|
|
332 | (2) |
|
9.4.2 Proportional hazards model for the survivor function |
|
|
334 | (3) |
|
9.4.3 Choice of the step times |
|
|
337 | (5) |
|
9.5 Parametric models for interval-censored data |
|
|
342 | (1) |
|
|
343 | (1) |
|
|
344 | (1) |
10 Frailty models |
|
345 | (36) |
|
10.1 Introduction to frailty |
|
|
345 | (3) |
|
|
346 | (1) |
|
10.1.2 Individual frailty |
|
|
346 | (1) |
|
|
347 | (1) |
|
10.2 Modelling individual frailty |
|
|
348 | (4) |
|
10.2.1 Frailty distributions |
|
|
349 | (2) |
|
10.2.2 Observable survivor and hazard functions |
|
|
351 | (1) |
|
10.3 The gamma frailty distribution |
|
|
352 | (4) |
|
10.3.1 Impact of frailty on an observable hazard function |
|
|
353 | (1) |
|
10.3.2 Impact of frailty on an observable hazard ratio |
|
|
354 | (2) |
|
10.4 Fitting parametric frailty models |
|
|
356 | (7) |
|
|
357 | (6) |
|
10.5 Fitting semi-parametric frailty models |
|
|
363 | (3) |
|
10.5.1 Lognormal frailty effects |
|
|
363 | (2) |
|
10.5.2 Gamma frailty effects |
|
|
365 | (1) |
|
10.6 Comparing models with frailty |
|
|
366 | (6) |
|
10.6.1 Testing for the presence of frailty |
|
|
366 | (6) |
|
10.7 The shared frailty model |
|
|
372 | (5) |
|
10.7.1 Fitting the shared frailty model |
|
|
373 | (1) |
|
10.7.2 Comparing shared frailty models |
|
|
374 | (3) |
|
10.8 Some other aspects of frailty modelling |
|
|
377 | (2) |
|
|
377 | (1) |
|
10.8.2 Correlated frailty models |
|
|
378 | (1) |
|
10.8.3 Dependence measures |
|
|
378 | (1) |
|
10.8.4 Numerical problems in model fitting |
|
|
378 | (1) |
|
|
379 | (2) |
11 Non-proportional hazards and institutional comparisons |
|
381 | (24) |
|
11.1 Non-proportional hazards |
|
|
381 | (2) |
|
11.2 Stratified proportional hazards models |
|
|
383 | (6) |
|
11.2.1 Non-proportional hazards between treatments |
|
|
385 | (4) |
|
11.3 Restricted mean survival |
|
|
389 | (4) |
|
11.3.1 Use of pseudo-values |
|
|
391 | (2) |
|
11.4 Institutional comparisons |
|
|
393 | (10) |
|
11.4.1 Interval estimate for the RAFR |
|
|
397 | (3) |
|
11.4.2 Use of the Poisson regression model |
|
|
400 | (2) |
|
11.4.3 Random institution effects |
|
|
402 | (1) |
|
|
403 | (2) |
12 Competing risks |
|
405 | (24) |
|
12.1 Introduction to competing risks |
|
|
405 | (1) |
|
12.2 Summarising competing risks data |
|
|
406 | (8) |
|
12.2.1 Kaplan-Meier estimate of survivor function |
|
|
407 | (2) |
|
12.3 Hazard and cumulative incidence functions |
|
|
409 | (1) |
|
12.3.1 Cause-specific hazard function |
|
|
409 | (1) |
|
12.3.2 Cause-specific cumulative incidence function |
|
|
410 | (3) |
|
12.3.3 Some other functions of interest |
|
|
413 | (1) |
|
12.4 Modelling cause-specific hazards |
|
|
414 | (5) |
|
12.4.1 Likelihood functions for competing risks models |
|
|
415 | (3) |
|
12.4.2 Parametric models for cumulative incidence functions |
|
|
418 | (1) |
|
12.5 Modelling cause-specific incidence |
|
|
419 | (3) |
|
12.5.1 The Fine and Gray competing risks model |
|
|
419 | (3) |
|
|
422 | (6) |
|
|
428 | (1) |
13 Multiple events and event history modelling |
|
429 | (28) |
|
13.1 Introduction to counting processes |
|
|
429 | (6) |
|
13.1.1 Modelling the intensity function |
|
|
430 | (1) |
|
13.1.2 Survival data as a counting process |
|
|
431 | (2) |
|
13.1.3 Survival data in the counting process format |
|
|
433 | (1) |
|
13.1.4 Robust estimation of the variance-covariance matrix |
|
|
434 | (1) |
|
13.2 Modelling recurrent event data |
|
|
435 | (8) |
|
13.2.1 The Anderson and Gill model |
|
|
436 | (1) |
|
13.2.2 The Prentice, Williams and Peterson model |
|
|
437 | (6) |
|
|
443 | (5) |
|
13.3.1 The Wei, Lin and Weissfeld model |
|
|
443 | (5) |
|
13.4 Event history analysis |
|
|
448 | (7) |
|
13.4.1 Models for event history analysis |
|
|
449 | (6) |
|
|
455 | (2) |
14 Dependent censoring |
|
457 | (14) |
|
14.1 Identifying dependent censoring |
|
|
457 | (1) |
|
14.2 Sensitivity to dependent censoring |
|
|
458 | (5) |
|
14.2.1 A sensitivity analysis |
|
|
459 | (2) |
|
14.2.2 Impact of dependent censoring |
|
|
461 | (2) |
|
14.3 Modelling with dependent censoring |
|
|
463 | (7) |
|
14.3.1 Cox regression model with dependent censoring |
|
|
464 | (6) |
|
|
470 | (1) |
15 Sample size requirements for a survival study |
|
471 | (16) |
|
15.1 Distinguishing between two treatment groups |
|
|
471 | (1) |
|
15.2 Calculating the required number of deaths |
|
|
472 | (7) |
|
15.2.1 Derivation of the required number of deaths |
|
|
474 | (5) |
|
15.3 Calculating the required number of patients |
|
|
479 | (5) |
|
15.3.1 Derivation of the required number of patients |
|
|
480 | (3) |
|
15.3.2 An approximate procedure |
|
|
483 | (1) |
|
|
484 | (3) |
A Maximum likelihood estimation |
|
487 | (4) |
|
A.1 Inference about a single unknown parameter |
|
|
487 | (2) |
|
A.2 Inference about a vector of unknown parameters |
|
|
489 | (2) |
B Additional data sets |
|
491 | (8) |
|
B.1 Chronic active hepatitis |
|
|
491 | (1) |
|
B.2 Recurrence of bladder cancer |
|
|
492 | (1) |
|
B.3 Survival of black ducks |
|
|
492 | (3) |
|
B.4 Bone marrow transplantation |
|
|
495 | (1) |
|
B.5 Chronic granulomatous disease |
|
|
495 | (4) |
Bibliography |
|
499 | (22) |
Index of Examples |
|
521 | (2) |
Index |
|
523 | |