  In the AGEMAP genomics study, researchers were interested in detecting genes
related to age in a variety of tissue types. After not finding many age-related
genes in some of the analyzed tissue types, the study was criticized for having
low power. It is possible that the low power is due to the presence of
important unmeasured variables, and indeed we find that a latent factor model
appears to explain substantial variability not captured by measured covariates.
We propose including the estimated latent factors in a multiple regression
model. The key difficulty in doing so is assigning appropriate degrees of
freedom to the estimated factors to obtain unbiased error variance estimators
and enable valid hypothesis testing. When the number of responses is large
relative to the sample size, treating the estimated factors like observed
covariates leads to a downward bias in the variance estimates. Many ad-hoc
solutions to this problem have been proposed in the literature without the
backup of a careful theoretical analysis. Using recent results from random
matrix theory, we derive a simple, easy to use expression for degrees of
freedom. Our estimate gives a principled alternative to ad-hoc approaches in
common use. Extensive simulation results show excellent agreement between the
proposed estimator and its theoretical value. Applying our methodology to the
AGEMAP genomics study, we found an order of magnitude increase in the number of
significant genes. Although we focus on the AGEMAP study, the methods developed
in this paper are widely applicable to other multivariate models, and thus are
of independent interest.
