Assessment of the Proportional Hazards Model

The proportional hazards model specifies that the hazard function for the failure time T associated with a column covariate vector takes the form

where is an unspecified baseline hazard function and is a column vector of regression parameters. Lin, Wei, and Ying (1993) present graphical and numerical methods for model assessment based on the cumulative sums of martingale residuals and their transforms over certain coordinates (such as covariate values or follow-up times). The distributions of these stochastic processes under the assumed model can be approximated by the distributions of certain zero-mean Gaussian processes whose realizations can be generated by simulation. Each observed residual pattern can then be compared, both graphically and numerically, with a number of realizations from the null distribution. Such comparisons enable you to assess objectively whether the observed residual pattern reflects anything beyond random fluctuation. These procedures are useful in determining appropriate functional forms of covariates and assessing the proportional hazards assumption. You use the ASSESS statement to carry out these model-checking procedures.

For a sample of n subjects, let be the data of the ith subject; that is, represents the observed failure time, has a value of 1 if is an uncensored time and 0 otherwise, and is a p-vector of covariates. Let and . Let

Let be the maximum partial likelihood estimate of , and let be the observed information matrix.

The martingale residuals are defined as

where .

The empirical score process is a transform of the martingale residuals:

Checking the Functional Form of a Covariate

To check the functional form of the jth covariate, consider the partial-sum process of :

Under that null hypothesis that the model holds, can be approximated by the zero-mean Gaussian process

where are independent standard normal variables that are independent of , .

You can assess the functional form of the jth covariate by plotting a small number of realizations (the default is 20) of on the same graph as the observed and visually comparing them to see how typical the observed pattern of is of the null distribution samples. You can supplement the graphical inspection method with a Kolmogorov-type supremum test. Let be the observed value of and let . The p-value is approximated by , which in turn is approximated by generating a large number of realizations (1000 is the default) of .

Checking the Proportional Hazards Assumption

Consider the standardized empirical score process for the jth component of

Under the null hypothesis that the model holds, can be approximated by

where is the jth component of , and are independent standard normal variables that are independent of , .

You can assess the proportional hazards assumption for the jth covariate by plotting a few realizations of on the same graph as the observed and visually comparing them to see how typical the observed pattern of is of the null distribution samples. Again you can supplement the graphical inspection method with a Kolmogorov-type supremum test. Let be the observed value of and let . The p-value is approximated by , which in turn is approximated by generating a large number of realizations (1000 is the default) of .