Correlations among coefficients in random coefficient linear regression models
Abstract
Random coefficient linear regression models have been employed in economics, medical and psychological sciences and other fields to study repeated measurements, correlated measurements and cross sectional measurements. When the measurement is longitudinal and the independent variable is time, the model is referred to as the growth curve model. In this research correlation between two coefficients, partial correlations and multiple correlations are defined. Estimators of these parameters are derived from estimators of the covariance matrix of random coefficients in the model. Unbiased estimators, maximum likelihood estimators and restricted maximum likelihood estimators via EM are given. A computational EM algorithm is described. The standard errors for estimators of the correlations are important. For the case where each subject has the same design matrix and a simple linear regression model with homogeneity of variance of the random error the asymptotic distribution of the estimator of the correlation between the intercept and the slope is derived with its asymptotic variance. The variance of the estimator is approximated by using the $\delta$ method to the order $1\over n.$ The jackknife method is used to obtain approximate variances, and the bootstrap method is used to approximate variances of estimators in general models. While the unbiased estimator of the covariance matrix of random coefficients can be negative with positive probability, the maximum likelihood estimator and restricted maximum likelihood estimator via EM are guaranteed to be nonnegative. Bayes analysis by means of a Gibbs sampler is used to study posterior distributions of these correlations. The methods are illustrated using real data on the growth of children. The results answer the question of whether or not catch-up growth is present in this population.
Degree
Ph.D.
Advisors
McCabe, Purdue University.
Subject Area
Statistics
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