  Large crossed data sets, described by generalized linear mixed models, have
become increasingly common and provide challenges for statistical analysis. At
very large sizes it becomes desirable to have the computational costs of
estimation, inference and prediction (both space and time) grow at most
linearly with sample size. Both traditional maximum likelihood estimation and
numerous Markov chain Monte Carlo Bayesian algorithms take superlinear time in
order to obtain good parameter estimates. We propose moment based algorithms
that, with at most linear cost, estimate variance components, measure the
uncertainties of those estimates, and generate shrinkage based predictions for
missing observations. When run on simulated normally distributed data, our
algorithm performs competitively with maximum likelihood methods.
