# Discrete Time Event History: Include time interactions

I want to fit a discrete hazard model and incorporate time-interactions of multiple terms. Since in the basic model, onle the intercept is time dependent, how do the interpretation and the fitting rpocedure change?

So assume, that the deta-manipulation as outligned in Discrete-Time Event History (Survival) Model in R have been accomplished.

The baseline model is then:

$P(T=t|T \geqslant t,X)=\frac{1}{1+exp(-\alpha_t-\beta X_i)}$

The illustrate this in R, i would type:

reg_surv=glm(Y~ -1 +time+X1+ X2,data=data_long,family=binomial(link='logit'))


Now incorporate time interactions, such that the model is given by:

$P(T=t|T \geqslant t,X)=\frac{1}{1+exp(-\alpha_t-[\sum_{k=1}^{T} 1(k=t) \beta_k X_i])}$

In R this would be accompliced by:

reg_surv=glm(Y~ -1 +time+time:X1+ time:X2,data=data_long,family=binomial(link='logit'))


So the question is very open but what changes in the interpretation of the model?

Especially, can I use estimation as outligned above?

And how does the interaction influence the "baseline" hazard? - Is it allready only $\frac{1}{1+exp(-\alpha_t)}$ for timte $t$ ?