Sufficient dimension reduction for high dimensional longitudinally measured biomarkers

Ruth Pfeiffer
Biostatistics Branch
National Cancer Institute, NIH, HHS
Bethesda, MD 20892-7244

In many practical applications one encounters predictors that are matrix valued. For example, in cohort studies conducted to study diseases, multiple biomarkers are often measured longitudinally during follow up. It is of interest to assess the associations of these repeated multivariate marker measurements with outcome to aid understanding of biological underpinnings of disease, and to use marker combinations for diagnosis and disease classification. Sufficient dimension reduction (SDR) aims to find a low dimensional transformation of predictors that preserves all of most of their information about a particular outcome. In earlier work we developed nonparametric SDR methods to combine several markers that are measured longitudinally using information on correlations over time and across markers (Pfeiffer et al., Statistics in Medicine, 2012). Here, we propose least squares and maximum likelihood based SDR approaches to estimate optimal combinations for longitudinally measured markers, i.e. matrix valued predictors. We assume a linear model for the inverse regression of the predictors as a function of the outcome variable, and model the mean using a matrix that is the Kronecker-product (two-dimensional tensor) of two sub-matrices, one that captures the association of markers and the outcome over time and one that captures the associations of the outcome with the different markers. These model-based approaches improve efficiency compared to nonparametric methods. We derive computationally fast least squares algorithms building on results of Van Loan and Pitsianis (1993) and show in simulations that they lead to estimates close in efficiency to maximum likelihood estimates for practically relevant sample sizes. The methods are illustrated using biomarker and imaging data. This is joint work with Wei Wang and Efstathia Bura Keywords: inverse regression; Kronecker product; non-linear dimension reduction