Detection of Gene×Gene Interactions by Multistage Sparse Low-Rank Regression

Researchers in biological sciences nowadays often encounter the curse of
high-dimensionality. A serious consequence is that many traditional statistical
methods fail to fit for high-dimensional models. The problem becomes even
more severe when the interest is in interactions between variables, as there will
be p(p−1)/2 interaction terms with p variables. To improve the performance,
in this thesis we model the interaction effects utilizing its matrix form with
a low-rank structure. A low-rank model for symmetric matrix then greatly
reduces the number of parameters required, and hence, increases the stability
and quality of statistical analysis. Individual hypothesis tests are then carried
out on each interaction effect to wash out insignificant interactions. A low-
rank matrix, however, is not necessarily sparse. We thus impose a sparsity
constraint in the second stage to select interactions.
Due to the extremely high-dimensionality for gene×gene interactions, a
single-stage method is not adequately flexible enough for variable selection.
Our sparse low-rank approach for interactions is a modification of a multi-
stage screen-and-clean procedure byWasserman and Roeder (2009) andWu et
al. (2010). We replace their mere sparsity constraint by combining a low-rank
structure and a sparsity constraint to the interactions. In simulation studies,
we show that the proposed low-rank approximation-aided screen and clean
procedure often can achieve higher power and higher selection-consistency
probability.