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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.
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Envelopes for multivariate linear regression with linearly constrained coefficients
Abstract 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.
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- Award ID(s):
- 1916013
- PAR ID:
- 10479342
- Publisher / Repository:
- Wileys
- Date Published:
- Journal Name:
- Scandinavian Journal of Statistics
- ISSN:
- 0303-6898
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Summary Envelopes have been proposed in recent years as a nascent methodology for sufficient dimension reduction and efficient parameter estimation in multivariate linear models. We extend the classical definition of envelopes in Cook et al. (2010) to incorporate a nonlinear conditional mean function and a heteroscedastic error. Given any two random vectors $${X}\in\mathbb{R}^{p}$$ and $${Y}\in\mathbb{R}^{r}$$, we propose two new model-free envelopes, called the martingale difference divergence envelope and the central mean envelope, and study their relationships to the standard envelope in the context of response reduction in multivariate linear models. The martingale difference divergence envelope effectively captures the nonlinearity in the conditional mean without imposing any parametric structure or requiring any tuning in estimation. Heteroscedasticity, or nonconstant conditional covariance of $${Y}\mid{X}$$, is further detected by the central mean envelope based on a slicing scheme for the data. We reveal the nested structure of different envelopes: (i) the central mean envelope contains the martingale difference divergence envelope, with equality when $${Y}\mid{X}$$ has a constant conditional covariance; and (ii) the martingale difference divergence envelope contains the standard envelope, with equality when $${Y}\mid{X}$$ has a linear conditional mean. We develop an estimation procedure that first obtains the martingale difference divergence envelope and then estimates the additional envelope components in the central mean envelope. We establish consistency in envelope estimation of the martingale difference divergence envelope and central mean envelope without stringent model assumptions. Simulations and real-data analysis demonstrate the advantages of the martingale difference divergence envelope and the central mean envelope over the standard envelope in dimension reduction.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1111/sjos.12690
