Survival Analysis – Proportional Hazards Assumption in Cox Models with Time-Dependent Covariates & Recurrent Event Models

Question

This question has been asked here before but in each thread I found, there was some disagreement about whether the proportional hazards (PH) assumption needs to be met for cox models with time dependent covariates. Could someone please explain either way?

What about recurrent event models like Andersen-Gill (AG) and Prentice-williams-Petersen (PWP), etc.? Does the PH assumption need to be met for these models as well?

If yes, would Schoenfeld's test be an appropriate test to check for this?

Answer 1

If you are fitting a Cox model, you are making an assumption about a particular type of proportionality in hazards: at each event time the relative hazards among individuals at risk are proportional to functions of covariate values that are in place at that time. That assumption might never be met perfectly, but it's important to evaluate the importance of possible violations.

The confusion comes from a terminological dispute over what should be considered a "proportional hazards model." For example this answer on the page you link cites a text that claims, when there are time-varying covariates, "the hazard of death at time t is no longer proportional to the baseline hazard, and the model is no longer a proportional hazards model." There is similar terminological restriction of the term "proportional hazards model" on this page and underlying this question.

I think that restriction of the phrase "proportional hazards model" to time-invariant covariates is unnecessary, misleading, and incorrect. The very first equation (equation 3.1) for a Cox model in Therneau and Grambsch allows for time varying covariate values $X(t)$:

The Cox model specifies the hazard for individual i as: $$\lambda_i(t) = \lambda_0(t) e^{X_i(t) \beta} $$ where $\lambda_0$ is an unspecified nonnegative function of time called the baseline hazard, and $\beta$ is a p x 1 column vector of coefficients.

That is precisely the proportional hazards assumption at each event time. At any event time, the hazard $\lambda_i(t)$ for an individual i who is at risk is assumed proportional to a function of covariate values in place at that time, relative to the covariate values assumed for the baseline hazard $\lambda_0(t)$.

Whether or not you choose to call a model with time-varying covariates a "proportional hazards model," you still are making that specific assumption about proportionality of hazards when you fit such data with a Cox model. You certainly can test that assumption with time-varying covariate values. The answers to this question go into more detail, including some of the pitfalls of trying to model with time-varying covariate values that go beyond violations of proportional hazards. In Chapter 8, Therneau and Grambsch use the cox.zph() function based on Schoenfeld residuals to test the proportional hazards assumption for recurrent-event models.

So the assumption of proportional hazards should be evaluated when you are fitting Cox regression models, whether or not you choose to call them "proportional hazards models."