Sufficient dimension reduction for compositional data.

Recent efforts to characterize the human microbiome and its relation to chronic diseases have led to a surge in statistical development for compositional data. We develop likelihood-based sufficient dimension reduction methods (SDR) to find linear combinations that contain all the information in the compositional data on an outcome variable, i.e., are sufficient for modeling and prediction of the outcome. We consider several models for the inverse regression of the compositional vector or transformations of it, as a function of outcome. They include normal, multinomial, and Poisson graphical models that allow for complex dependencies among observed counts. These methods yield efficient estimators of the reduction and can be applied to continuous or categorical outcomes. We incorporate variable selection into the estimation via penalties and address important invariance issues arising from the compositional nature of the data. We illustrate and compare our methods and some established methods for analyzing microbiome data in simulations and using data from the Human Microbiome Project. Displaying the data in the coordinate system of the SDR linear combinations allows visual inspection and facilitates comparisons across studies.

Variability in gut mucosal secretory IgA in mice along a working day.

OBJECTIVE: To assess the variability of secretory immunoglobulin A (S-IgA) in the lumen and feces of mice along a working day. RESULTS: Mice were maintained under a 12 h light-dark cycle, light period starting at 8 AM. S-IgA was determined in feces and intestinal content (after one or three washes) at three points along the day: at the beginning, in the middle and at the end of the light period (ELP). Significant reduction in the content of S-IgA in the small intestine fluid and in feces was observed at the end of the light cycle, which coincides with the end of a regular working day (8 PM) in any given animal facility. It was also observed that three washes of the small intestine were more effective than one flush to recover a significant higher amount of S-IgA, with the smallest coefficient of variation observed by the ELP. A smaller CV would imply a reduced number of animals needed to achieve the same meaningful results. The results may be useful when designing animal trials for the selection of probiotic candidates based on their capacity of activating S-IgA, since it would imply a more rational use of experimental animals.

Sufficient dimension reduction for censored predictors.

Motivated by a study conducted to evaluate the associations of 51 inflammatory markers and lung cancer risk, we propose several approaches of varying computational complexity for analyzing multiple correlated markers that are also censored due to lower and/or upper limits of detection, using likelihood-based sufficient dimension reduction (SDR) methods. We extend the theory and the likelihood-based SDR framework in two ways: (i) we accommodate censored predictors directly in the likelihood, and (ii) we incorporate variable selection. We find linear combinations that contain all the information that the correlated markers have on an outcome variable (i.e., are sufficient for modeling and prediction of the outcome) while accounting for censoring of the markers. These methods yield efficient estimators and can be applied to any type of outcome, including continuous and categorical. We illustrate and compare all methods using data from the motivating study and in simulations. We find that explicitly accounting for the censoring in the likelihood of the SDR methods can lead to appreciable gains in efficiency and prediction accuracy, and also outperformed multiple imputations combined with standard SDR.

Sufficient dimension reduction for longitudinally measured predictors.

We propose a method to combine several predictors (markers) that are measured repeatedly over time into a composite marker score without assuming a model and only requiring a mild condition on the predictor distribution. Assuming that the first and second moments of the predictors can be decomposed into a time and a marker component via a Kronecker product structure that accommodates the longitudinal nature of the predictors, we develop first-moment sufficient dimension reduction techniques to replace the original markers with linear transformations that contain sufficient information for the regression of the predictors on the outcome. These linear combinations can then be combined into a score that has better predictive performance than a score built under a general model that ignores the longitudinal structure of the data. Our methods can be applied to either continuous or categorical outcome measures. In simulations, we focus on binary outcomes and show that our method outperforms existing alternatives by using the AUC, the area under the receiver-operator characteristics (ROC) curve, as a summary measure of the discriminatory ability of a single continuous diagnostic marker for binary disease outcomes.