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Envelopes for multivariate linear regression with linearly constrained coefficients

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  • R. Dennis Cook
  • Liliana Forzani
  • Lan Liu

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.

Suggested Citation

  • R. Dennis Cook & Liliana Forzani & Lan Liu, 2024. "Envelopes for multivariate linear regression with linearly constrained coefficients," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(2), pages 429-446, June.
    • Handle: RePEc:bla:scjsta:v:51:y:2024:i:2:p:429-446
    DOI: 10.1111/sjos.12690
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    References listed on IDEAS

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    1. Z. Su & G. Zhu & X. Chen & Y. Yang, 2016. "Sparse envelope model: efficient estimation and response variable selection in multivariate linear regression," Biometrika, Biometrika Trust, vol. 103(3), pages 579-593.
    2. Michael G. Kenward, 1987. "A Method for Comparing Profiles of Repeated Measurements," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(3), pages 296-308, November.
    3. R. Dennis Cook & Xin Zhang, 2015. "Foundations for Envelope Models and Methods," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 599-611, June.
    4. Lexin Li & Xin Zhang, 2017. "Parsimonious Tensor Response Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1131-1146, July.
    5. R. Dennis Cook & Bing Li & Francesca Chiaromonte, 2007. "Dimension reduction in regression without matrix inversion," Biometrika, Biometrika Trust, vol. 94(3), pages 569-584.
    6. Tapio Nummi & Laura Koskela, 2008. "Analysis of growth curve data by using cubic smoothing splines," Journal of Applied Statistics, Taylor & Francis Journals, vol. 35(6), pages 681-691.
    7. R. Dennis Cook & Zhihua Su, 2013. "Scaled envelopes: scale-invariant and efficient estimation in multivariate linear regression," Biometrika, Biometrika Trust, vol. 100(4), pages 939-954.
    8. Zhihua Su & R. Dennis Cook, 2011. "Partial envelopes for efficient estimation in multivariate linear regression," Biometrika, Biometrika Trust, vol. 98(1), pages 133-146.
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