  We consider the application of a popular penalised regression method, Ridge
Regression, to data with very high dimensions and many more covariates than
observations. Our motivation is the problem of out-of-sample prediction and the
setting is high-density genotype data from a genome-wide association or
resequencing study. Ridge regression has previously been shown to offer
improved performance for prediction when compared with other penalised
regression methods. One problem with ridge regression is the choice of an
appropriate parameter for controlling the amount of shrinkage of the
coefficient estimates. Here we propose a method for choosing the ridge
parameter based on controlling the variance of the predicted observations in
the model.
  Using simulated data, we demonstrate that our method outperforms subset
selection based on univariate tests of association and another penalised
regression method, HyperLasso regression, in terms of improved prediction
error. We extend our approach to regression problems when the outcomes are
binary (representing cases and controls, as is typically the setting for
genome-wide association studies) and demonstrate the method on a real data
example consisting of case-control and genotype data on Bipolar Disorder, taken
from the Wellcome Trust Case Control Consortium and the Genetic Association
Information Network.
