# Wild Cluster Bootstrap and GLM

I am a Stata user and am trying to determine the most appropriate method for improving statistical inference in GLM (generalized linear models) applications with very few clusters (in one study G=29 and in another G=4 - I am aware of Webb's 2014 advice on 6-point weights for studies with fewer than 11 clusters). The clusters are unbalanced and I've got heteroskedasticity.

Please take as given that I need to use GLM rather than OLS for these studies.

Is it appropriate to use the wild cluster bootstrap procedure with GLMs? Recent explanations of the wild method indicate that the first step is to be completed using OLS, and Stata packages (e.g., CLUSTSE and CGMWILDBOOT) note restrictions to -regress- commands. It seems to me, however, that I should be ok using Wild cluster bootstrap with GLM because the error component of the GLM is additively separable.

Any thoughts or advice on where to look for further instruction on combining GLM and wild cluster bootstrap will be greatly appreciated!

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• – Nick Cox Dec 2 '15 at 0:46

Consider logistic regression for simplicity. Your outcome $y_i$ is either 0 or 1. So unless the predicted probability is $\hat p_i=1/[1+\exp(-x_i'\hat\beta)]=0.5$, flipping the residual will produce an outcome that is out of range: if $y_i=1$ and $\hat p_i=0.7$, then the wildly-bootstrapped values are $y_i^*=1$ and $y_i^*=0.4$, the latter not being an appropriate value for the logistic regression.
You can probably devise a wild-bootstrap-like routine that would produce $y_i^*=1$ with probability $\hat p_i$ and $y_i^*=0$ with probability $1-\hat p_i$; this would essentially be a parametric bootstrap procedure. I am not entirely sure it is worth much as your model may be misspecified, and the model-predicted probabilities may miss the true ones by a mile and a quarter. Moreover, generalizations to cluster wild bootstrap are absolutely weird: the wild bootstrap version would be to either retain all the exiting patterns of 0 and 1 within the cluster, or flipping all of them to the opposite -- but you need to flip them in a way that preserves the predicted probabilities somehow, and I am not sure I see how this is at all feasible.
Note also that the bootstrap is an asymptotic procedure. Expecting it to produce good results for $n=4$ is naive.