Age as a time-dependent covariate

Question

I have some confusion around time-dependent covariates / coefficients. I'm trying to run a cox regression with age as a time-dependent covariate. Lets say the variable is age-at-first-drivers-license (yrs) predicting first crash, among numerous other predictors. Once splitting the file time spans many years for a given id, due to another time-dependent co-variate in the dataset -

[id 1 1 1; age 20 20 20; time 5 10 15]

If I want the effect of age to vary time should I enter age as is, or should it be updated at every interval, age+elapsed time -

[id 1 1 1; newage 20 25 30; time 5 10 15]

I'm basically trying to let the effect of age vary over time, and estimate the time-dependent coefficient. I think my confusion lies in the fact that age and time are perfectly correlated. It seems odd to update it. I hope I've made some sense.

Apologies for the matrix notation.

Answer 1

Let $X(t)$ denote the time-dependent covariate representing age at time $t$.

For subject $i$ we thus have $X_i(t) = X_i(0) + t$ where $X_i(0)$ is age at baseline.

For simplicity purpose we consider a model with only one variable.

The Cox regression is hazard-based as follows, for subject $i$:

\begin{align*} h(t \mid X_i(t)) &= h_0(t) \text{exp} ( \beta X_i(t) ) \\ &= h_0(t) \text{exp} ( \beta (X_i(0) + t ) ) \\ &= h_0(t) \text{exp} ( \beta X_i(0) + \beta t ) \end{align*}

Usually $\beta$ is estimated via a partial likelihood :

let $T_1 < \dots < T_K$ the $K$ observed failure time. This partial likelihood is written :

\begin{align*} L(\beta) &= \prod _{i=1}^K \frac{e^{\beta X_i(T_i)}}{\sum_{j : T_j \geq T_i} e^{ \beta X_j(T_i)} } \\ &= \prod _{i=1}^K \frac{e^{\beta T_i + \beta X_i(0)}}{\sum_{j : T_j \geq T_i} e^{ \beta T_i + \beta X_j(0)} } \end{align*}

As $ e^{a+b} = e^a e^b$ the terms $e^{\beta T_i}$ cancelled each other so that the partial likelihood is actually

\begin{align*} L(\beta) = \prod _{i=1}^K \frac{e^{\beta X_i(0)}}{\sum_{j : T_j \geq T_i} e^{\beta X_j(0)} } \end{align*}

That is the estimation of $\beta$ for age as a time-dependent covariate will be the same as for age included as a baseline covariate.

Answer 2

Your variable is "age at first drivers license", thus is does not vary with time and is not a time-dependent covariate so you shouldn't update it.

You estimate the time-varying effect of age by including an interaction between (constant) age and a variable of time.

My answers here and here might help with implementation.