SAS macros for estimation of the cumulative incidence functions based on a Cox regression model for competing risks survival data

https://doi.org/10.1016/S0169-2607(03)00069-5Get rights and content

Abstract

When considering competing risks survival data, the cause specific hazard functions are often modelled by the proportional hazards Cox regression model. First, we present how to estimate the parameters in this model when some of the covariates are allowed to have exactly the same effect on several causes of failure. In many cases, the focus is not on the parameter estimates, but rather on the probability of observing a failure from a specific cause for individuals with specified covariate values. These probabilities, the cumulative incidences, are not simple functions of the parameters and they are, so far, not provided by the standard statistical software packages. We present two SAS macros: a SAS macro named CumInc for estimation of the cumulative incidences and a SAS macro named CumIncV for estimation of the cumulative incidences and the variances of the estimated cumulative incidences. The use of the macros is demonstrated through an example.

Introduction

The semiparametric proportional hazards Cox regression model [1] is often chosen when modelling survival data and in a competing risks setup, each of the cause specific hazards may be modelled by the Cox regression model. Usually the estimation is performed by fitting a separate Cox regression model for each cause of failure and the hazard corresponding to a specific cause of failure is analysed considering failures of the other causes as censored observations. However, this procedure is insufficient if some of the covariates are assumed to have exactly the same effect on two or several of the cause specific hazards. Andersen et al. [2] present the technical details of how to solve this problem using the Cox estimation procedures provided in the statistical software packages. The method is reproduced here since the presented SAS macros require the use of this estimation method.

In many cases, interest will be on the probability of observing a specific cause of failure for individuals with a given value of the covariate vector. These probabilities, also referred to as the cumulative incidences, are not simple functions of the parameter values and the chosen covariate value. Unfortunately, the standard statistical software packages do not provide procedures for estimation of the cumulative incidences. We have derived two SAS macros for this purpose. The SAS macro CumInc computes estimates of the cumulative incidences, whereas the SAS macro CumIncV computes both estimates of the cumulative incidences and estimates of the variances of the estimated cumulative incidences. The macros are available as http://www.biostat.ku.dk/∼pka/CumInc.sas, respectively http://www.biostat.ku.dk/∼pka/CumIncV.sas.

The estimators of the cumulative incidences and the estimators of the variances of the estimated cumulative incidences based on the Cox regression model are given in Section 2. In order to be able to use the macros, some preparation of the data is needed. How to do this is described in Section 3. The use of the macros is explained in Section 4 and Section 5 and demonstrated with an example in Section 6.

Section snippets

Estimation of the cumulative incidences

A competing risks model with k causes of failures is considered. Let T>0 denote the failure time, L∈{1,…,k} the cause of failure and Z a time fixed covariate vector. When right censoring is present, as is usual in analysis of survival data, it is not always possible to observe the actual failure time T and the cause of failure L. Instead a right censored failure time T̃ and a censoring/cause of failure indicator J are observed, J=l if a failure of cause l is observed and J=0 if the failure time

Preparation of the data

Before using the SAS macros, the Cox regression model has to be fitted to the data in order to get some of the quantities needed for the computation of the estimated cumulative incidences and the estimated variances of the estimated cumulative incidences. In the competing risks model, this is usually performed by fitting separate models for each cause of failure. The hazard corresponding to a failure of a specific cause is analysed considering failures of other causes as censored observations.

The SAS macro CumInc

When a model based on a single covariate named Covariate is used, the PHREG procedure has to be used in the following way:

PROC PHREG data=StackDat ;
 MODEL Time*D(0)= Covariate ;
 STRATA Strata ;
 BASELINE OUT=ciData
  COVARIATES=cov SURVIVAL=Surv
  /NOMEAN METHOD=CH ;
RUN ;
StackDat is the name of the stacked data set created as described in Section 3, Time is the name of the variable containing the failure times, D is the failure indicator D, Strata is the stratum indicator h and cov is the name of the

The SAS macro CumIncV

The macro CumIncV also computes the estimated variances of the estimated cumulative incidence functions. If a single covariate named Covariate is included in the model, the PHREG procedure has to be used in this way:

PROC PHREG DATA=StackDat
  OUTEST=EstData COVOUT ;
 MODEL Time*D(0)= Covariate ;
 STRATA Strata ;
 OUTPUT OUT=ResData XBETA=Xbeta ;
 BASELINE OUT=ciData
  COVARIATES=cov SURVIVAL=Surv
  XBETA=Xbeta / NOMEAN METHOD=CH ;
RUN ;
StackDat is the name of the stacked data set created as described in Section 3

Example

The data are described in example I.3.1 in Andersen et al. [2] and consists of n=205 observations for patients with malignant melanoma. Two causes of failure are present: (1) death from malignant melanoma; and (2) death from other causes. Two covariates, sex and tumor thickness, are included in the Cox regression analysis. In example VII.2.5 of Andersen et al. [2] it is demonstrated that it is reasonable to assume that sex has exactly the same effect on both causes of failure whereas tumor

Final comments

The macros were developed and tested in SAS version 6.12. The macro CumIncV requires the SAS/IML software for computation of the estimated variances of the cumulative incidences.

The results of the two SAS macros have been checked thoroughly. We have considered data sets without right censoring, without covariates or with a single cause of failure (corresponding to a simple failure time model), for which the estimated cumulative incidences and the estimated variances of the estimated cumulative

Acknowledgements

The development of the SAS macros was supported by the Danish Heart Foundation and a grant (R01 CA54706-07) from the National Cancer Institute.

References (2)

  • D.R. Cox

    Regression models and life-tables (with discussion)

    J. Roy. Stat. Soc. B

    (1972)
  • P.K. Andersen et al.

    Statistical Models Based on Counting Processes

    (1993)

Cited by (98)

  • Predictive Nomogram for Recurrence following Surgery for Nonmetastatic Renal Cell Cancer with Tumor Thrombus

    2017, Journal of Urology
    Citation Excerpt :

    Histological subtypes were categorized according to the 2004 WHO classification. To evaluate the effect of death from other causes we integrated competing risks analysis with univariate and multivariate Cox proportional hazard models to determine putative risk factors for RCC recurrence.9 Significant variables following multivariate competing risks analysis were used to construct a predictive nomogram.

View all citing articles on Scopus
View full text