Cox regression, which implements the proportional hazards model or duration model, is designed for analysis of time until an event or time between events. If the dependent variable is not time to event but rather is count of events, then a logistic or other model may be appropriate instead. For any given predictor variable, Cox regression results in estimates of how much the predictor increases or decreases the odds of the event occurring and whether time to event is increased or decreased. The central effect size measure is the hazard ratio (a form of odds ratio), which can be used to assess the relative importance of the predictor variables.
In Cox regression, one or more predictor variables, called covariates, are used to predict a status (event) variable. The classic example is time from diagnosis with a terminal illness until the event of death (hence survival analysis). Cox regression is also used for policy adoption/diffusion studies to better understand factors leading to policy adoption (see Jones & Branton, 2005).
There are a wide variety of Cox models beyond the basic model using time-constant variables. Stepwise Cox regression is an automated procedure for exploratory purposes in constructing a model with optimal predictions. Stratified Cox regression is a method used when the same baseline hazard function cannot be assumed for a predictor variable but instead the baseline function must be allowed to vary by level of the categorical predictor. Time-dependent Cox regression handles time-varying predictor variables and comes in two flavors: discrete time-varying and continuous time-varying models. Frailty models extend Cox regression to handle linear mixed modeling situations where observations cluster at the individual level. That is, frailty models handle dependent data such as repeated measures, which would otherwise violate the assumptions of Cox regression. Finally, there are several types of multiple events Cox models, which handle situations where the event of interest may be experienced more than once or where there are multiple event types. All these are discussed in this volume.
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Below is the unformatted table of contents.
COX REGRESSION TABLE OF CONTENTS Overview 9 Application examples 10 In medicine 10 In social science 11 In business 11 Data used in this volume 11 Key terms and concepts 12 Variables 12 Status variable 13 Time variable 13 Covariates 14 Interaction terms 16 Observations 16 Uncensored observations 16 Right-censored observations 17 Right-truncated observations 17 Left-censored observations 18 Left-truncated observations 18 Non-informative censoring 19 "Random censoring" 19 Interval-censored observations 19 Survival function 20 Survival function in SPSS 21 Survival function in Stata 22 Hazard function 22 Hazards 22 Hazard rates 23 Hazard functions 23 Baseline vs. covariate hazard functions 23 Hazard ratios 24 Baseline hazard ratio 24 Hazard ratio with covariates 27 Proportional hazards 32 Partial likelihood methods and why Cox models are semi-parametric 33 Handling tied failure times 33 Cox models 34 Time-constant Cox regression models 34 Time-dependent Cox regression models 34 Frailty models 35 Conditional frailty models 35 Repeated events models 37 Competing risks models 37 Parametric models 38 Time-constant Cox regression in SPSS 38 Example 38 SPSS Options 39 SPSS Plots 40 SPSS Statistical Output 40 The hazard ratio 40 Confidence intervals on the odds ratio 41 Significance 41 Relative risk 42 Likelihood ratio (omnibus) tests 42 Cox regression coefficients 43 Baseline hazard, survival, and cumulative hazard rates 47 Covariate means 51 Pattern plots 52 Saved variables in SPSS 53 Outlier analysis with DfBeta 53 Time-constant Cox regression in Stata 55 Stata setup 55 Stata command syntax 56 Stata statistical output 57 Likelihood ratio test in Stata 57 Cox regression coefficients in Stata 57 Test of equality of survivor functions in Stata 59 Types of variance estimates 59 Time-constant Cox regression in SAS 60 SAS Interface 60 SAS syntax 61 Data setup for SAS 62 Cox regression with tests in SAS 63 SAS syntax 63 SAS model output 64 SAS test output 65 Cox regression in SAS with dummy variables 67 SAS syntax 67 SAS model output 67 SAS test output 68 Testing for proportional hazards 69 SAS syntax 69 SAS model output 70 SAS test output 70 SAS PROC GPLOT: Survival Plot 70 Stepwise Cox Regression 72 Why forced entry results may seem different from stepwise results 72 Stepwise Cox regression In SPSS 72 Overview 72 Entry criterion 74 Removal criteria 74 Omnibus tests 74 Stepwise Cox regression In Stata 75 Overview 75 Stata stepwise options 76 Stepwise Cox regression In SAS 77 Overview 77 Output 78 Stratified Cox Regression 79 Overview 79 Example 79 Testing to see if a stratified model is required 80 Stratified Cox regression in SPSS 81 Overview 81 SPSS output for stratified Cox regression 82 Stratified Cox regression in Stata 85 Stata syntax for stratified Cox regression 85 Stata output 86 Stratified Cox regression in SAS 88 SAS syntax for stratified Cox regression 88 SAS output 88 Time-dependent Cox regression 90 Overview 90 Discrete time-varying models 91 Overview 91 Example 91 Discrete models in Stata 91 Discrete models in SPSS 93 Discrete models in SAS 93 Continuous time-varying models 94 Example 94 Continuous models in Stata 95 Continuous models in SPSS 99 Continuous models in SAS 105 Segmented time-dependent models 107 Frailty models 107 Overview 107 Example 108 Shared frailty models in Stata 108 Overview 108 Syntax 109 Output 109 Shared frailty models in SPSS 111 Shared frailty models in SAS 111 Overview 111 Syntax 111 Output 112 Multiple-events models 113 Overview 113 Example 118 Multiple events models in Stata 119 Overview 119 Syntax 119 Output 120 Multiple events models in SPSS 121 Multiple events models in SAS 121 Overview 121 Syntax 121 Output 122 Assumptions of Cox Regression 123 Assumption of proportional hazards 123 True starting time 127 Clearly defined events 128 Absence of outliers 128 Sample size and sparse data 128 Proper model specification 129 Few ties 130 Independent observations 131 Not applying single-event models to multiple event data 132 Exogenous covariates 132 Factor invariance 132 Baseline distribution of survival times 132 Hazard rate linearity 133 Log linearity 133 No high multicollinearity 133 Random sampling 133 No censoring patterns 134 Frequently Asked Questions 134 Why can't the researcher just use OLS or logistic regression to analyze time until event data? 134 Couldn't I use Poisson or logistic regression instead of event history models when analyzing time to event? 135 When would one use a parametric event history analysis model rather than a Cox model? 136 Describe data setup for Cox regression 137 Time-constant data setup 137 Time-dependent data setup 137 Discontinuous risk intervals 138 Cox data setup in Stata 138 Why is no intercept coefficient reported for Cox models? 142 Since the Cox model does not posit any particular baseline hazard ratio, how can the baseline hazard function be retrieved? 142 Should I always standardize my covariates prior to Cox regression? 142 Does SPSS support Cox regression for multiple events? 143 Does SPSS support multilevel Cox regression? 143 Can I use Cox regression with non-random samples? 144 What is a segmented time dependent Cox model? 144 Bibliography 146 Acknowledgments 150 Pagecount: 152