  Motivated by genome-wide association studies, we consider a standard linear
model with one additional random effect in situations where many predictors
have been collected on the same subjects and each predictor is analyzed
separately. Three novel contributions are (1) a transformation between the
linear and log-odds scales which is accurate for the important genetic case of
small effect sizes; (2) a likelihood-maximization algorithm that is an order of
magnitude faster than the previously published approaches; and (3) efficient
methods for computing marginal likelihoods which allow Bayesian model
comparison. The methodology has been successfully applied to a large-scale
association study of multiple sclerosis including over 20,000 individuals and
500,000 genetic variants.
