## Modeling Survival Data: Extending the Cox ModelThis is a book for statistical practitioners, particularly those who design and analyze studies for survival and event history data. Its goal is to extend the toolkit beyond the basic triad provided by most statistical packages: the Kaplan-Meier estimator, log-rank test, and Cox regression model. Building on recent developments motivated by counting process and martingale theory, it shows the reader how to extend the Cox model to analyse multiple/correlated event data using marginal and random effects (frailty) models. It covers the use of residuals and diagnostic plots to identify influential or outlying observations, assess proportional hazards and examine other aspects of goodness of fit. Other topics include time-dependent covariates and strata, discontinuous intervals of risk, multiple time scales, smoothing and regression splines, and the computation of expected survival curves. A knowledge of counting processes and martingales is not assumed as the early chapters provide an introduction to this area. The focus of the book is on actual data examples, the analysis and interpretation of the results, and computation. The methods are now readily available in SAS and S-Plus and this book gives a hands-on introduction, showing how to implement them in both packages, with worked examples for many data sets. The authors call on their extensive experience and give practical advice, including pitfalls to be avoided. Terry Therneau is Head of the Section of Biostatistics, Mayo Clinic, Rochester, Minnesota. He is actively involved in medical consulting, with emphasis in the areas of chronic liver disease, physical medicine, hematology, and laboratory medicine, and is an author on numerous papers in medical and statistical journals. He wrote two of the original SAS procedures for survival analysis (coxregr and survtest), as well as the majority of the S-Plus survival functions. Patricia Grambsch is Associate Professor in the Division of Biostatistics, School of Public Health, University of Minnesota. She has collaborated extensively with physicians and public health researchers in chronic liver disease, cancer prevention, hypertension clinical trials and psychiatric research. She is a fellow the American Statistical Association and the author of many papers in medical and statistical journals. |

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### Contents

Introduction | 1 |

12 Overview | 2 |

13 Counting processes | 3 |

Estimating the Survival and Hazard Functions | 7 |

212 Estimating the survival function | 13 |

22 Counting processes and martingales | 17 |

221 Modeling the counting process | 18 |

222 Martingale basics | 19 |

85 Ordered multiple events | 185 |

852 Fitting the models | 187 |

VII | 190 |

VIII | 196 |

IX | 205 |

X | 211 |

86 Multistate models | 216 |

862 Crohns disease | 217 |

23 Properties of the NelsonAalen estimator | 26 |

232 Efficiency | 28 |

24 Tied data | 31 |

The Cox Model | 39 |

32 Stratified Cox models | 44 |

33 Handling ties | 48 |

34 Wald score and likelihood ratio tests | 53 |

341 Confidence intervals | 57 |

35 Infinite coefficients | 58 |

36 Sample size determination | 61 |

361 The impact of strata on sample size | 67 |

37 The counting process form of a Cox model | 68 |

371 Timedependent covariates | 69 |

372 Discontinuous intervals of risk | 74 |

575 Alternate time scales | 75 |

374 Summary | 76 |

Residuals | 79 |

42 Martingale residuals | 80 |

422 Overall tests of goodnessoffit | 81 |

424 Usage | 82 |

43 Deviance residuals | 83 |

45 Score residuals | 84 |

46 Schoenfeld residuals | 85 |

Functional Form | 87 |

511 Stage Dl prostate cancer | 88 |

512 PBC data | 90 |

I | 92 |

II | 95 |

III | 96 |

IV | 99 |

V | 102 |

VI | 107 |

56 Timedependent covariates | 111 |

57 Martingale residuals under misspecified models | 115 |

572 Relation to functional form | 118 |

58 Penalized models | 120 |

582 SPlus functions | 122 |

583 Spline fits | 124 |

59 Summary | 126 |

Testing Proportional Hazards | 127 |

62 Timedependent coefficients | 130 |

63 Veterans Administration data | 135 |

64 Limitations | 140 |

643 Stratified models | 141 |

65 Strategies for nonproportional data | 142 |

66 Causes of nonproportionality | 148 |

Influence | 153 |

72 Variance | 159 |

73 Case weights | 161 |

Multiple Events per Subject | 169 |

82 Robust variance and computation | 170 |

822 Connection to other work | 173 |

83 Selecting a model | 174 |

84 Unordered outcomes | 175 |

842 Diabetic retinopathy study | 177 |

843 UDCA in patients with PBC | 179 |

844 Colon cancer data | 183 |

87 Combination models | 227 |

88 Summary | 229 |

Frailty Models | 231 |

92 Computation | 232 |

93 Examples | 238 |

94 Unordered events | 240 |

942 Familial aggregation of breast cancer | 241 |

95 Ordered events | 243 |

952 Survival of kidney catheters | 245 |

953 Chronic granulotamous disease | 250 |

96 Formal derivations | 251 |

962 EM solution for shared frailty | 252 |

963 Gamma frailty | 253 |

964 Gaussian frailty | 255 |

965 Correspondence of the profile likelihoods | 256 |

97 Sparse computation | 258 |

98 Concluding remarks | 259 |

Expected Survival | 261 |

102 Individual survival Cox model | 263 |

1021 Natural history of PBC | 264 |

1022 Mean survival | 266 |

1024 Timedependent covariates | 268 |

103 Cohort survival population | 272 |

1033 Ederer estimate | 273 |

1034 Hakulinen estimate | 274 |

1035 Conditional expected survival | 275 |

1036 Example | 276 |

104 Cohort survival Cox model | 279 |

1042 Naive estimate | 280 |

1044 Hakulinen and conditional estimates | 281 |

1045 Comparing observed to expected for the UDCA trial | 283 |

Introduction to SAS and SPlus | 289 |

A1 SAS | 290 |

A2 SPlus | 294 |

SAS Macros | 301 |

B2 phlev | 303 |

B3 schoen | 304 |

B4 surv | 305 |

B5 survtd | 307 |

B6 survexp | 308 |

S Functions | 309 |

C3 gamterms | 310 |

C4 plotterm | 311 |

Data Sets | 313 |

D2 Primary biliary cirrhosis | 314 |

D3 Sequential PBC | 316 |

D4 rIFNg in patients with chronic granulomatous disease | 318 |

D5 rhDNase for the treatment of cystic fibrosis | 320 |

Test Data | 323 |

E1 2 Efron approximation | 326 |

E1 3 Exact partial likelihood | 329 |

E2 Test data 2 | 330 |

E22 Efron approximation | 332 |

333 | |

346 | |

### Other editions - View all

Modeling Survival Data: Extending the Cox Model Terry M. Therneau,Patricia M. Grambsch Limited preview - 2013 |

Modeling Survival Data: Extending the Cox Model Terry M. Therneau,Patricia M. Grambsch No preview available - 2014 |

Modeling Survival Data: Extending the Cox Model Terry M. Therneau,Patricia M. Grambsch No preview available - 2010 |

### Common terms and phrases

analysis Andersen-Gill approximation bilirubin cancer censoring Chi-Square Chisq clinical cluster(id coef exp(coef coefficients computation confidence intervals covariate Cox model coxph coxph(Surv(time cumulative hazard data set death default degrees of freedom dfbeta disease distribution edema enum equation example expected survival failure Figure followup frailty model fustat futime hazard function hazard ratio infection iteration jackknife Kaplan-Meier lbili Likelihood ratio linear log(bili log(protime loglikelihood macro martingale residuals method multiple events Nelson-Aalen estimator Newton-Raphson number of events observations output parameter partial likelihood patients PBC data penalized placebo plot Poisson Poisson regression proc phreg proportional hazards model random effect recurrence regression S-Plus sample scale Schoenfeld residuals score test se(coef shows smoothing spline spline standard error status strata stratum survival curve Table temp2 temp3 term time-dependent covariates time2 total number transplant UDCA values variance estimate vector Wald test weights

### Popular passages

Page 345 - Stage Dl prostatic adenocarcinoma: significance of nuclear DNA ploidy patterns studied by flow cytometry. Mayo Clin. Proc., 63: 103-112.

Page 340 - Mantel-Haenszel analysis of litter-matched time-to-response data, With modifications for recovery of interlitter information.

Page 342 - RL Prentice, BJ Williams and AV Peterson, On the regression analysis of multivariate failure time data, Biometrika, vol.

Page 342 - Ricci P, Therneau TM, Malinchoc M et al. A prognostic model for the outcome of liver transplantation in patients with cholestatic liver disease. Hepatology 1997; 25: 672.

Page 342 - M. Pugh, J. Robbins, S. Lipsitz, and D. Harrington, Inference in the Cox proportional hazards model with missing covariate data.

Page 343 - D. Schoenfeld. Chi-squared goodness-of-fit tests for the proportional hazards regression model.