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A constrained multivariate linear model is a multivariate linear model with the columns of its coefficient matrix constrained to lie in a known subspace. This class of models includes those typically used to study growth curves and longitudinal data. Envelope methods have been proposed to improve the estimation efficiency in unconstrained multivariate linear models, but have not yet been developed for constrained models. We pursue that development in this article. We first compare the standard envelope estimator with the standard estimator arising from a constrained multivariate model in terms of bias and efficiency. To further improve efficiency, we propose a novel envelope estimator based on a constrained multivariate model. We show the advantage of our proposals by simulations and by studying the probiotic capacity to reduced Salmonella infection.

Magnetic resonance imaging (MRI) is a technique that scans the anatomical structure of the brain, whereas functional magnetic resonance imaging (fMRI) uses the same basic principles of atomic physics as MRI scans but image metabolic function. A major goal of MRI and fMRI study is to precisely delineate various types of tissues, anatomical structure, pathologies, and detect the brain regions that react to outer stimuli (e.g., viewing an image). As a key feature of these MRI‐based neuroimaging data, voxels (cubic pixels of the brain volume) are highly correlated. However, the associations between voxels are often overlooked in the statistical analysis. We adapt a recently proposed dimension reduction method called the envelope method to analyze neuoimaging data taking into account correlation among voxels. We refer to the modified procedure the envelope chain procedure. Because the envelope chain procedure has not been employed before, we demonstrate in simulations the empirical performance of estimator, and examine its sensitivity when our assumptions are violated. We use the estimator to analyze the MRI data from ADHD‐200 study. Data analyses demonstrate that leveraging the correlations among voxels can significantly increase the efficiency of the regression analysis, thus achieving higher detection power with small sample sizes.

When multiple measures are collected repeatedly over time, redundancy typically exists among responses. The envelope method was recently proposed to reduce the dimension of responses without loss of information in regression with multivariate responses. It can gain substantial efficiency over the standard least squares estimator. In this paper, we generalize the envelope method to mixed effects models for longitudinal data with possibly unbalanced design and time‐varying predictors. We show that our model provides more efficient estimators than the standard estimators in mixed effects models. Improved accuracy and efficiency of the proposed method over the standard mixed effects model estimator are observed in both the simulations and the Action to Control Cardiovascular Risk in Diabetes (ACCORD) study.