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I'm working with time-to-event data that is right-censored for two different groups. I have administrative data that gives me information on the outcomes and associated covariates. There are a couple of categorical variables that definitely have an effect on the outcome, but they are not in the administrative data. However, I have some survey results that give me estimates on the distributions of the categorical variables for the two groups.

For example, say the two groups are Group 1 and Group 2, and I have a covariate $W$ with "success" probabilities as

$W | \text{Group 1} \sim \text{Bernoulli(0.9)}$

and

$W | \text{Group 2} \sim \text{Bernoulli(0.3)}$

Is there a way to incorporate this survey data into a "proportional hazards"-like model? I'm pretty sure there's something related to Bayesian statistics here, but I'm not sure what I should be looking for here. If there's another approach I'd be all ears!

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This might best be handled by a Bayesian approach that models the W values directly, but that's beyond my expertise.

You might consider treating this as a missing-data problem to be handled with multiple imputation. See for example Stef van Buuren's book. You generate multiple copies of the data set, with each copy containing W values from a separate set of draws from those Bernoulli distributions. You fit each of those imputed data sets separately, then combine the results in a way that takes both the within-data-set and the among-data-set variances into account.

Whether this is done via Bayesian analysis or multiple imputation, the reliability of your results will depend heavily on the quality of your assumptions about the Bernoulli distributions. There also are likely to be relationships between W and the other covariates in your model. Omitting those relationships in the imputation or the modeling might pose problems, particularly with survival analysis.

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