We propose methods to estimate sufficient reductions of matrix-valued predictors for regression or classification. We assume that the first moments of the predictor matrix given the response can be decomposed into a row and column component via a Kronecker product structure. We obtain least squares and maximum likelihood estimates of the sufficient reductions of the matrix predictors, derive statistical properties of the resulting estimates and present fast computational algorithms with assured convergence. The performance of the proposed approaches in regression and classification is compared in simulations. We illustrate the methods on neuroimaging data.

Motivation: Comparisons of microbiome communities across populations are often based on pairwise distance measures (beta-diversity). Standard analyses (principal coordinate plots, permutation tests, kernel methods) require access to primary data if another investigator wants to add or compare independent data. We propose using standard reference measurements to simplify microbiome beta-diversity analyses, to make them more transparent, and to facilitate independent validation and comparisons across studies. Results: Using stool and nasal reference sets from the Human Microbiome Project (HMP), we computed mean distances (actually Bray-Curtis or Pearson correlation dissimilarities) to each reference set for each new sample. Thus, each new sample has two mean distances that can be plotted and analyzed with classical statistical methods. To test the approach, we studied independent (not reference) HMP subjects. Simple Hotelling tests demonstrated statistically significant differences in mean distances to reference sets between all pairs of body sites (stool, skin, nasal, saliva and vagina) at the phylum, class, order, family and genus levels. Using the distance to a single reference set was usually sufficient, but using both reference sets always worked well. The use of reference sets simplifies standard analyses of beta-diversity and facilitates the independent validation and combining of such data because others can compute distances to the same reference sets. Moreover, standard statistical methods for survival analysis, logistic regression and other procedures can be applied to vectors of mean distances to reference sets, thereby greatly expanding the potential uses of beta-diversity information. More work is needed to identify the best reference sets for particular applications. Availability and implementation: https://github.com/NCI-biostats/microbiome-fixed-reference. Supplementary information: Supplementary data are available at Bioinformatics online.

MOTIVATION: In genomic studies, thousands of features are collected on relatively few samples. One of the goals of these studies is to build classifiers to predict the outcome of future observations. There are three inherent steps to this process: feature selection, model selection and prediction assessment. With a focus on prediction assessment, we compare several methods for estimating the 'true' prediction error of a prediction model in the presence of feature selection. RESULTS: For small studies where features are selected from thousands of candidates, the resubstitution and simple split-sample estimates are seriously biased. In these small samples, leave-one-out cross-validation (LOOCV), 10-fold cross-validation (CV) and the .632+ bootstrap have the smallest bias for diagonal discriminant analysis, nearest neighbor and classification trees. LOOCV and 10-fold CV have the smallest bias for linear discriminant analysis. Additionally, LOOCV, 5- and 10-fold CV, and the .632+ bootstrap have the lowest mean square error. The .632+ bootstrap is quite biased in small sample sizes with strong signal-to-noise ratios. Differences in performance among resampling methods are reduced as the number of specimens available increase. SUPPLEMENTARY INFORMATION: A complete compilation of results and R code for simulations and analyses are available in Molinaro et al. (2005) (http://linus.nci.nih.gov/brb/TechReport.htm).

MOTIVATION: We introduce simple graphical classification and prediction tools for tumor status using gene-expression profiles. They are based on two dimension estimation techniques sliced average variance estimation (SAVE) and sliced inverse regression (SIR). Both SAVE and SIR are used to infer on the dimension of the classification problem and obtain linear combinations of genes that contain sufficient information to predict class membership, such as tumor type. Plots of the estimated directions as well as numerical thresholds estimated from the plots are used to predict tumor classes in cDNA microarrays and the performance of the class predictors is assessed by cross-validation. A microarray simulation study is carried out to compare the power and predictive accuracy of the two methods. RESULTS: The methods are applied to cDNA microarray data on BRCA1 and BRCA2 mutation carriers as well as sporadic tumors from Hedenfalk et al. (2001). All samples are correctly classified.