# Survival Analysis Expected Time To Event - Time-Dependent Covariates Cox PH

My question is around using Cox PH to obtain expected time to event with time dependent covariates.

Suppose you have a non-negative random variable, T, representing the time until the occurrence of an event. You have time dependent covariates and decide to model the hazard rate via Cox PH i.e.

$$\lambda(t \vert \pmb{x}(t)) = \lambda_0 e^{B_1x_1(t) + ... + B_pX_p(t)}$$

For example, you are modelling time to lung disease (apologies for the bleak example), and your time dependent feature is smoking status (i.e. 1 if smoker, 0 if not -> can change over time), with other features such as weight i.e.

$$\lambda(t \vert \pmb{x}(t)) = \lambda_0 e^{B_1*weight + B_2*smoking status}$$

Given a dataset you then find a maximum partial likehood estimate for your parameters. Going forwards, you obtain new data for a person (midway through life i.e. time dependent features not know for all t), and you want to estimate the expected or mean value of T i.e. However in order to do this you need to integrate the survival function to obtain the expected value i.e.: $$\mu = \int_0^\infty S(t) dt$$

In order to do this using your model, you need to specify a trajectory for the time dependent features in order to generate the survival curve at all times (which is unknown). My question is:

• What is best practice in this scenario when using Cox PH to obtain expected time to event with time dependent covariates? For example, do people tend to generate different forward looking scenarios for covariate trajectories? I have also seen examples where people feed in what is available from the partial survival curve into a separate regression model for time to event.

Any insight would be greatly appreciated.

## 1 Answer

As Therneau and Grambsch say in "Modeling Survival Data: Extending the Cox Model," Springer (2000), defining the covariates as a function of time is tricky (p. 278):

Examples where the covariate path is guaranteed are the exception, however. A major concern ... is whether the hypothetical path represents any patient at all. Survival curves based on a time-dependent covariate must be used with extreme caution.

First, I don't know that there is a single "best practice." As Therneau and Grambsch note on page 277, some even argue that survivor curves based on hypothetical time-dependent covariate paths are meaningless. There's also a risk of inadvertently engaging in survivorship bias. Think instead about what you want to demonstrate as possible outcomes to your audience, based on what might make sense in practice from your understanding of the subject matter.

Second, are you even sure that illustrating particular survival patterns based on time-dependent covariate values will be more useful to your audience than patterns based on values fixed at time = 0? After all, if proportional hazards hold, the future hazard just changes proportionately after a change in a time-dependent covariate.

Third, your choice of smoking status as an example covariate points up another potential problem. A Cox model uses the instantaneous values of time-dependent covariates to estimate relative hazards at event times. The development of lung disease typically depends on an extended history of smoking, so you might need to devise a more complicated time-varying covariate to capture the correct influence of smoking or similar covariate.

Finally, think about whether you want to use mean survival as your outcome measure. Unless the last observation for your Cox model was an event, you won't have a survival curve out to infinite time and will be stuck with a restricted mean survival taken out (at most) to the last time point, making practical interpretation potentially even more difficult. A median survival, or some other representative survival quantile, might be preferable.