Bootstrapping under the null
LIMO tools estimate the null distribution using bootstraps. This means that for a given statistical test, data are nullified and resampled, re-estimating each time the test statistics and p-values. From there, the different techniques to control the type 1 family-wise error rate can be used.
In a GLM, we fit a model (X) to the data (Y) such Y=XB+e. A simple and efficient way to obtain the null is to resample Y at random such the link between X and Y is broken - possibly centering as well each condition so that the means do not differ if by chance the resampling attributes the right Y with the right X.
When using weights, it is not obvious what the best solution is as the null should reflect the mismatch of the model to the data - this can be done using (and resampling with Y) the weights from the observed data or recomputing weights each time. Comparison of the two approaches showed that using the weights from the observed data is a little conservative but, on average, closer to the nominal level and this is therefore what is used in limo_glm_boot.
Figure 1. Average error rate for WLS under the null using alternative computations for weights