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Overview

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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```COX REGRESSION
Overview	9
Application examples	10
In medicine	10
In social science	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
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
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