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My data set consists of a factor that is very sparse, and thus I am modeling it as a random effect intercept in a mixed effect models; there are too many levels to model it as a fixed effect. I am interested in determining "significant" contrasts between specific levels of this random intercept.

As an example, consider the following model in R:

library(lme4)
data(Pastes)

m <- lmer(strength ~ cask + (1 | batch), data=Pastes)

I am interested in measuring the "significance" of the difference between specific batch levels to answer, e.g., is the deviance in batch A significantly different (higher or lower) than the deviance in batch B?

My initial thought is to bootstrap this model, taking the estimated random effect deviances for batches A and B for each bootstrapped sample, and contrast the estimates (though I do not know what underlying distribution these contrasts should follow, if any, in order to determine "significance"). Am I on the right path with this thinking? Or is such a comparison not reasonable in a mixed effects model? Any help would be greatly appreciated.

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