Least squares and maximum likelihood estimation of sufficient reductions in regressions with matrix-valued predictors

Structured time-dependent inverse regression (STIR)

We propose and study structured time-dependent inverse regression (STIR), a novel sufficient dimension reduction model, to analyze longitudinally measured, correlated biomarkers in relation to an outcome. The time structure is accommodated in an inverse regression model for the markers that can be applied both to equally and unequally spaced time points for each sample. The inverse regression structure also naturally accommodates retrospectively sampled markers, that is, markers measured in case-control studies. We estimate the corresponding linear combinations of the markers, the reduction, using least squares. We show that under additional distributional assumptions the reduction contains sufficient information about the outcome. In extensive simulations the STIR linear combinations perform well in predictive models based on samples of realistic size. A Wald-type test for association of a particular marker with outcome at any time point based on the STIR reduction has better power overall than assessing associations based on logistic or linear regression models that include all longitudinally measured markers as independent predictors. As illustrations we estimate the STIR reductions for a cohort study of diabetes and hyperlipidemia and a case-control study of brain cancer with multiple longitudinally measured biomarkers. We assess the STIR reductions' predictive performance and identify outcome-associated biomarkers.