  Pan-omics, pan-cancer analysis has advanced our understanding of the
molecular heterogeneity of cancer, expanding what was known from single-cancer
or single-omics studies. However, pan-cancer, pan-omics analyses have been
limited in their ability to use information from multiple sources of data
(e.g., omics platforms) and multiple sample sets (e.g., cancer types) to
predict important clinical outcomes, like overall survival. We address the
issue of prediction across multiple high-dimensional sources of data and
multiple sample sets by using exploratory results from BIDIFAC+, a method for
integrative dimension reduction of bidimensionally-linked matrices, in a
predictive model. We apply a Bayesian hierarchical model that performs variable
selection using spike-and-slab priors which are modified to allow for the
borrowing of information across clustered data. This method is used to predict
overall patient survival from the Cancer Genome Atlas (TCGA) using data from 29
cancer types and 4 omics sources. Our model selected patterns of variation
identified by BIDIFAC+ that differentiate clinical tumor subtypes with markedly
different survival outcomes. We also use simulations to evaluate the
performance of the modified spike-and-slab prior in terms of its variable
selection accuracy and prediction accuracy under different underlying
data-generating frameworks. Software and code used for our analysis can be
found at https://github.com/sarahsamorodnitsky/HierarchicalSS_PanCanPanOmics/ .
