  In this paper we study a bootstrap strategy for estimating the variance of a
mean taken over large multifactor crossed random effects data sets. We apply
bootstrap reweighting independently to the levels of each factor, giving each
observation the product of independently sampled factor weights. No exact
bootstrap exists for this problem [McCullagh (2000) Bernoulli 6 285-301]. We
show that the proposed bootstrap is mildly conservative, meaning biased toward
overestimating the variance, under sufficient conditions that allow very
unbalanced and heteroscedastic inputs. Earlier results for a resampling
bootstrap only apply to two factors and use multinomial weights that are poorly
suited to online computation. The proposed reweighting approach can be
implemented in parallel and online settings. The results for this method apply
to any number of factors. The method is illustrated using a 3 factor data set
of comment lengths from Facebook.
