11.7.3 Time-Dependent Covariates

A powerful feature of the proportional hazards model is its ability to handle time-dependent covariates , predictors whose value may change over time. For example, suppose we measure a patient’s blood pressure every week over the course of a medical study. In this case, we can think of the blood pressure for the i th observation not as xi , but rather as xi ( t ) at time t .

Because the partial likelihood in (11.16) is constructed sequentially in time, dealing with time-dependent covariates is straightforward. In particular, we simply replace xij and xi′j in (11.16) with xij ( yi ) and xi′j ( yi ), respectively; these are the current values of the predictors at time yi . By contrast, time-dependent covariates would pose a much greater challenge within the context of a traditional parametric approach, such as (11.13).

One example of time-dependent covariates appears in the analysis of data from the Stanford Heart Transplant Program. Patients in need of a heart transplant were put on a waiting list. Some patients received a transplant, but others died while still on the waiting list. The primary objective of the analysis was to determine whether a transplant was associated with longer patient survival.

A naïve approach would use a fixed covariate to represent transplant status: that is, xi = 1 if the i th patient ever received a transplant, and xi = 0 otherwise. But this approach overlooks the fact that patients had to live long enough to get a transplant, and hence, on average, healthier patients received transplants. This problem can be solved by using a time-dependent covariate for transplant: xi ( t ) = 1 if the patient received a transplant by time t , and xi ( t ) = 0 otherwise.