How to parameterize covariates in parametric time to event model (lognormal distribution) I have a parametric time to event model for survival data and I found that a lognormal distribution has the lowest objective function value for a base model. Below is the hazard function for the lognormal parametric model:
$$ h(t) =\frac{f(t)}{S(t)}=\frac{e^{-\frac12\cdot\left(\frac{\log(t)-\mu}{\sigma}\right)^2}\cdot \frac{1}{t\cdot\sigma\cdot\sqrt{2\pi}}}{\left(1-\Phi\left[\frac{\log(t)-\mu}{\sigma}\right]\right) }$$
I have incorporated covariates into the model in two different ways:

*

*I added covariate effects to the "$\mu$" term.

*I added covariate effects to the entire hazard as indicated below:

$$ h(t) =\frac{f(t)}{S(t)}=\frac{e^{-\frac12\cdot\left(\frac{\log(t)-\mu}{\sigma}\right)^2}\cdot \frac{1}{t\cdot\sigma\cdot\sqrt{2\pi}}}{\left(1-\Phi\left[\frac{\log(t)-\mu}{\sigma}\right]\right)} \cdot Covariates $$
Are there any issues with parameterizing covariates like I did for the second option? The second option results in a lower objective function value and the visual predictive checks are superior compared to parameterizing the covariate effects on the "μ" term. I think that the second parameterization effectively shifts the hazard up or down depending on the covariate value, so it seems reasonable to me. I'd appreciate any thoughts on this.
 A: Method 1 uses a lognormal distribution as the basis for an accelerated failure time model. Incorporating covariate effects into the location parameter means that changes in covariate values stretch or shrink the time axis. That maintains the overall form of the resulting survival curves as lognormal. That's a typical way of constructing a lognormal survival model. See these course notes for a succinct explanation.
Method 2 starts with a lognormal baseline survival distribution and then multiplies the hazard function by a function of covariate values. That places a proportional hazards assumption on the covariate effects. You can certainly do that. As discussed on this page, that approach is implemented for example in the phreg() function of the R eha package, although that's "on an experimental basis at the moment."
A potential drawback, also noted there, is that the lognormal distribution family isn't closed under a proportional hazards model. Thus the resulting survival curves for non-baseline covariate values won't be strictly lognormal.
If you want to keep a consistent distribution family for all covariate values under a proportional hazards assumption, the Weibull distribution is a standard choice. If you don't need a parametric form, then a Cox survival regression can be followed by extraction of an empirical baseline cumulative hazard function.
I'd recommend that you not put too much emphasis on  having "found that a lognormal distribution has the lowest objective function value for a base model." There's a risk that a lognormal might represent a very good fit to your current data set but not necessarily one that will extend the best to new data, particularly as you seem to be in a proportional hazards rather than accelerated failure time situation with respect to covariate effects.
