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
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