I want to generate predictions from a fitted logistic regression model that reflect the uncertainty of the model (within a classic frequentist framework). To clarify, my objective is not to characterize the predictive uncertainty as such, but to get a set of reasonable predictions which are fair to the uncertainty about the coefficients of the predictors included in the model .
My approach is as follows:
for each group of interest, defined by a combination of values of the predictors;
draw values for each regression coefficient from a normal distribution with a mean – the point estimate of the respective regression coefficient, and a standard deviation – the respective standard error of the regression coefficient; plug the resulting values in the inverse logit formula to get the prediction; and repeat. Would that be a valid approach? Would it work for mixed-effects (multilevel) models as well where the predictors are modeled as random effects?
Best Answer
In the frequentist framework it suffices to compute the standard error of $X\hat{\beta}$, get a confidence interval from this for $X\beta$, and use the logistic transformation of those two confidence limits to get a confidence interval for $P$ for an individual with characteristics $X$. I believe an adaptation of this approach will work for mixed effects models but have not done that myself.