We present a novel statistical technique; the sparse reduced rank regression (sRRR) model
which is a strategy for multivariate modelling of high-dimensional imaging responses and
genetic predictors. By adopting penalisation techniques, the model is able to enforce sparsity
in the regression coefficients, identifying subsets of genetic markers that best explain
the variability observed in subsets of the phenotypes. To properly exploit the rich structure
present in each of the imaging and genetics domains, we additionally propose the use of
several structured penalties within the sRRR model. Using simulation procedures that accurately
reflect realistic imaging genetics data, we present detailed evaluations of the sRRR
method in comparison with the more traditional univariate linear modelling approach. In
all settings considered, we show that sRRR possesses better power to detect the deleterious
genetic variants. Moreover, using a simple genetic model, we demonstrate the potential
benefits, in terms of statistical power, of carrying out voxel-wise searches as opposed to
extracting averages over regions of interest in the brain. Since this entails the use of phenotypic
vectors of enormous dimensionality, we suggest the use of a sparse classification
model as a de-noising step, prior to the imaging genetics study. Finally, we present the
application of a data re-sampling technique within the sRRR model for model selection.
Using this approach we are able to rank the genetic markers in order of importance of association
to the phenotypes, and similarly rank the phenotypes in order of importance to
the genetic markers. In the very end, we illustrate the application perspective of the proposed
statistical models in three real imaging genetics datasets and highlight some potential
associations.