  Multimodal data, where different types of data are collected from the same
subjects, are fast emerging in a large variety of scientific applications.
Factor analysis is commonly used in integrative analysis of multimodal data,
and is particularly useful to overcome the curse of high dimensionality and
high correlations. However, there is little work on statistical inference for
factor analysis based supervised modeling of multimodal data. In this article,
we consider an integrative linear regression model that is built upon the
latent factors extracted from multimodal data. We address three important
questions: how to infer the significance of one data modality given the other
modalities in the model; how to infer the significance of a combination of
variables from one modality or across different modalities; and how to quantify
the contribution, measured by the goodness-of-fit, of one data modality given
the others. When answering each question, we explicitly characterize both the
benefit and the extra cost of factor analysis. Those questions, to our
knowledge, have not yet been addressed despite wide use of factor analysis in
integrative multimodal analysis, and our proposal bridges an important gap. We
study the empirical performance of our methods through simulations, and further
illustrate with a multimodal neuroimaging analysis.
