Dealing with large unaffected cohort in parametric survival analysis

[brontidon]

Yes I think that clarified it! so your hazard at for a given individual is a function of age, and onset age?

You should still be able to use a Poisson process. the ‘non parametric’ portion of the Cox model comes from the baseline hazard having a very flexible formulation. It’s only called a semi-parametric model because the baseline hazard is multiplied by an exponential transformation of the covariates (your parametric part!). Interestingly, you are NOT limited to linear relationships-but that works for most applications (there is nothing really stopping you from throwing a gaussian process in there or whatever)

@ddg.palmer

Given I have the number of people first getting the condition at each age and the population size, it seems I have all I need to pass to the Poisson likelihood.

I hope this clarifies things. If not, it might be a sign that there’s something fundamental that I’ve misunderstood.

So to confirm-your panel data is organized as a matrix of subjects 1….n and time measurements 1…j for each individual, so you have at most n x j rows? Note under piecewise approximations to the survival curve that generally this is robust to censoring (the partial likelihood under many formulations is just factored into the (interval specific) individual likelihood when an event occurs when it does not occur-therefore you can have less than n x j rows)

To make sure we’re on the same page, can you provide a brief outline of your generative model?

@ddg.palmer

I’ve been using the word susceptibility to try to take in a slightly larger concept than frailty and avoid using frailty for things that aren’t (perhaps there’s a generic term I don’t know). Essentially it takes the place of frailty, so each model has the form hazard(t | susceptibility) = susceptibility * hazard(t). For two of my ‘susceptibilities’, these are frailty distributions, but the mixture-cure formulation and no frailty (ie susceptibility = 1) cases come under this umbrella and aren’t strictly frailty models.

I should disclaim I usually don’t see frailty presented this way (but this is because I generally see it associated with….the Cox and other parametric/semi parametric model). Frailty to me implies we are affording the likelihood of a single individual at a time t to be scaled by some constant if they belong to a sub-group of interest). But I should also disclaim I’ve seen it used…in a non precise manner