## Bootstrapping in Multilevel Models

We note, however, that even as the number of bootstrap replications tends to infinity, the estimate of the population density function that is used to generate the bootstrap samples is the empirical “plug in” one derived from the actual sampled observations by placing mass points (e.g., equal probabilities) at each one. In other words the sample is assumed to be a reasonable representation of the population. Thus, with nonparametric bootstrapping, we do not have exact inference. This does not carry over to the parametric case that we describe below, where the model-based (assumed) population distribution is used for sampling: we shall return to this case later. In fact, in some situations the nonparametric bootstrap can perform very badly, for example, in small or moderate samples where the statistic of interest is the smallest or largest value, say of a set of higher level residuals in a multilevel model.