Multivariate Repeated Measures Linear Models with Normal and Skew Normal Errors Characterized by Patterned Covariance Structures

Date

2021

Authors

Opheim, Timothy

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Abstract

The popularity of the classical general linear model (CGLM) is attributable mostly to its ease of fitting and validating; however, the CGLM is inappropriate for correlated observations such as repeated measurements. In this dissertation, we develop linear models for {dollar}N>1{dollar} multivariate repeated measurements ({dollar}p{dollar}-variate observations measured {dollar}n{dollar} times) characterized by an exchangeable covariance structure (Arnold 1979) and multivariate normal errors or multivariate skew normal errors of the Azzalini and Dalla Valle (1996) type. We also develop such models under the assumption of an identifiable separable covariance structure {dollar}\bar{lcub}\boldsymbol{lcub}\Omega{rcub}{rcub} \otimes \boldsymbol{lcub}\Sigma{rcub}{dollar} with {dollar}\bar{lcub}\boldsymbol{lcub}\Omega{rcub}{rcub} = \bar{lcub}\boldsymbol{lcub}\Omega{rcub}{rcub}\left(\boldsymbol{lcub}\rho{rcub}\right){dollar} representing a patterned correlation matrix or any matrix with one of its diagonal elements restricted to unity and {dollar}\boldsymbol{lcub}\Sigma{rcub}{dollar} an unstructured variance-covariance matrix. In addition to general derivations for the separable structure, we explore three specific choices for {dollar}\bar{lcub}\boldsymbol{lcub}\Omega{rcub}{rcub}{dollar}: the MA(1), AR(1), and the compound symmetric correlation matrix. For each linear model considered, with some exceptions, we detail concise methods to obtain the MLEs of the models' parameters, develop exact model-building tests (covariate selection) of hypotheses where possible, supplementing the impossibilities thereof with large sample results, and perform Monte Carlo simulations to ascertain the bias of the slope parameters, the multimodality of the profile log-likelihoods and the applicability of the large sample results. For the models considered with multivariate normal errors, the estimated bias of the slope parameters appears nugatory for sample sizes as small as {dollar}N=10{dollar} and the estimated probability of a multimodal profile log-likelihood is rare. For the models with multivariate skew normal errors, the bias of the slope parameters still appears to be practically nonexistent for moderately small sample sizes; however, the multimodality of the profile log-likelihood is rampant.

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Keywords

Statistics, Statistical physics, Applied mathematics

Citation

Department

Management Science and Statistics