  We consider a problem of data integration. Consider determining which genes
affect a disease. The genes, which we call predictor objects, can be measured
in different experiments on the same individual. We address the question of
finding which genes are predictors of disease by any of the experiments. Our
formulation is more general. In a given data set, there are a fixed number of
responses for each individual, which may include a mix of discrete, binary and
continuous variables. There is also a class of predictor objects, which may
differ within a subject depending on how the predictor object is measured,
i.e., depend on the experiment. The goal is to select which predictor objects
affect any of the responses, where the number of such informative predictor
objects or features tends to infinity as sample size increases. There are
marginal likelihoods for each way the predictor object is measured, i.e., for
each experiment. We specify a pseudolikelihood combining the marginal
likelihoods, and propose a pseudolikelihood information criterion. Under
regularity conditions, we establish selection consistency for the
pseudolikelihood information criterion with unbounded true model size, which
includes a Bayesian information criterion with appropriate penalty term as a
special case. Simulations indicate that data integration improves upon,
sometimes dramatically, using only one of the data sources.
