## Electronic Theses and Dissertations

1989

Dissertation

Ph.D.

#### Department

Mathematics and Statistics

Shoukri, Mohamed M.,

Statistics.

CC BY-NC-ND 4.0

#### Abstract

Inferences for familial (intraclass and interclass) correlations are considered for unbalanced familial data from multivariate normal populations. The most commonly used familial correlation is that between siblings (without regard to gender), called intraclass or sib-sib correlation coefficient. The expressions for the large sample biases and variances of several point estimators of the intraclass correlation are derived and the sampling properties of these estimators are compared for a wide variety of unbalanced designs. It is recommended that the Karlin's individual estimator should be used for the small number of groups with a severe degree of unbalancedness. However, for a large number of groups, the Karlin's empirical estimator is recommended provided that the true value of intraclass correlation is less than or equal to 0.5. Several procedures for testing that the intraclass correlation is equal to a specified value are derived and compared by extensive Monte Carlo studies. The Neyman's C($\alpha$) (or partial score) and modified F-ANOVA procedures are shown to be consistently more powerful than the other procedures for the said hypotheses. Estimation procedures, based on the maximum likelihood and the ANOVA methods, for intraclass correlations in multiple samples are discussed. In order to test the homogeneity of the intraclass correlations in multiple samples, several procedures are derived and compared in terms of their empirical powers. The use of a test based on Fisher's variance stabilizing transformation is recommended for small values of the common intraclass correlation and of the Neyman's C($\alpha$) test for moderate values of the common intraclass correlation is recommended. The maximum likelihood estimation of sibling correlations (brother-brother, sister-sister, and brother-sister correlations) is considered next and it is shown that the estimates of the parameters can be obtained by numerical maximization of a function of fewer parameters. The expressions for the large sample variances and covariances of the estimators are derived and the procedures to test the significance of these correlations are discussed. The procedures are illustrated by using a published arterial blood pressures dataset from the literature. Using a linear model approach, procedures to find the maximum likelihood estimates of five familial correlations (mother-brother, mother-sister, brother-brother, sister-sister and brother-sister correlations) and other parameters are developed. The expressions for the asymptotic variances and covariances of estimators are derived. Procedures for testing the significance of the above familial correlations are also presented and the methodologies are illustrated on the previously mentioned epidemiological data sets.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1989 .M535. Source: Dissertation Abstracts International, Volume: 52-11, Section: B, page: 5915. Co-Supervisors: Mohamed M. Shoukri; Derrick S. Tracy. Thesis (Ph.D.)--University of Windsor (Canada), 1989.

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