Age as a time-dependent covariate 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.  
 A: 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.
A: 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.
