I know that under typical conditions, the coefficients of a Cox model, the coefficients are asymptotically normally distributed. I plan to weight my Cox model by inverse probability of treatment weights. Should I be concerned that the distribution of the coefficients will no longer by normally distributed? If it is not normally distributed, what distribution does it become?
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In short, as long as none of the observations dominate the estimation of the regression coefficients, or none of the covariate effects are too extreme (i.e. binary covariates that perfectly sort outcomes), then the estimated coefficients should be approximately normally distributed. This same rule of thumb applies to the unweighted Cox-PH model.
With the probability weights, the new aspect of potential "observation domination" can come in the form a small number of observations having the majority of the probability weights. This is very conservative, but if $p_{max} = \arg \max_j \frac{p_j}{\sum p_i}$, then you can think of the effective sample size as $\tilde n = \frac{1}{p_{max}}$. Better yet, you can use bootstrapping to get an idea of whether the estimator is approximately normal.
