Multi-state model: how to calculate whether a sample sub-population is significantly different from the parameterised general population

Question

If anyone can explain this, or point me to the right sources to be reading, I would be very grateful. I have got a little lost in reading all the articles I find when I google the subject. My problem is:

1. I have a multi-state model with transition intensities between a 3 state model: healthy, sick, dead. I call this the 'population model'.
2. I have sample time-to-sick data (around 3000 records), which are left-truncated and right-censored. I call this the 'sample model'.
3. I want to parameterise a multi-state model for the 'sample model'.

I can think of two approaches, but do not know which is best, or how to perform the analysis:

Idea 1: use a Bayesian approach with the 'population model' as the prior. (I am not sure what to use as the likelihood function).

Idea 2: a. test whether the sample data differs from the parameterised model at a significant level of 5%, and

b(i). if there isn't a difference, use the unadjusted 'population model' to model the 'sample model', or 

b(ii). if there is a statistically significant difference, somehow scale or adapt the transition intensities to more accurately model the sample data.

Is there a recommended way to approach this; perhaps even a package in R?

Thank you in advance.

Answer 1

Why not follow up on both ideas, or at least on things closely related to both ideas?

You seem to have enough sample data to model those data independently and then evaluate how well the model coefficients agree with those in the "population model." That might be the simplest way to both "parameterise a multi-state model for the 'sample model'" and use your sample data as a validation set for the "population model."

Details depend on the type of model (parametric vs. semi-parametric) you're working with. The Cox-model semi-parametric analysis provided by the standard coxph() function in R nicely handles multi-state models with arbitrary transitions, based on left-truncated/right-censored data, as explained in a multi-state vignette and in Chapter 8 of Therneau and Grambsch. The flexsurv package allows for parametric multi-state modeling, as explained in a vignette.

Using the "population model" to provide priors for a Bayesian model would tend to undercut any use of your "sample data" as a validation set for the "population model." If you do want to use a Bayesian approach, the likelihoods are just standard Cox partial likelihoods or parametric full likelihoods, evaluated for each transition with the corresponding covariates. The R Survival task view has many entries for Bayesian survival modeling, although I can't say whether any handle multi-state models.

Evaluating the performance of the "population model" on your sample data would be a useful parallel evaluation. For a fair test, you should use the performance criteria applied in originally developing that model.