# 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.

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.