Document Type


Date of Degree

Summer 2019

Access Restrictions

Access restricted until 09/04/2021

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Aloe, Ariel M.

First Committee Member

LeBeau, Brandon

Second Committee Member

Mitchell, Colleen

Third Committee Member

Jorgensen, Palle

Fourth Committee Member

Darcy, Isabel K.


One single primary study is only a little piece of a bigger puzzle. Meta-analysis is the statistical combination of results from primary studies that address a similar question. The most general case is the random-effects model, in where it is assumed that for each study the vector of outcomes T_i~N(θ_i,Σ_i ) and that the vector of true-effects for each study is θ_i~N(θ,Ψ). Since each θ_i is a nuisance parameter, inferences are based on the marginal model T_i~N(θ,Σ_i+Ψ). The main goal of a meta-analysis is to obtain estimates of θ, the sampling error of this estimate and Ψ.

Standard meta-analysis techniques assume that Σ_i is known and fixed, allowing the explicit modeling of its elements and the use of Generalized Least Squares as the method of estimation. Furthermore, one can construct the variance-covariance matrix of standard errors and build confidence intervals or ellipses for the vector of pooled estimates. In practice, each Σ_i is estimated from the data using a matrix function that depends on the unknown vector θ_i. Some alternative methods have been proposed in where explicit modeling of the elements of Σ_i is not needed. However, estimation of between-studies variability Ψ depends on the within-study variance Σ_i, as well as other factors, thus not modeling explicitly the elements of Σ_i and departure of a hierarchical structure has implications on the estimation of Ψ.

In this dissertation, I develop an alternative model for random-effects meta-analysis based on the theory of hierarchical models. Motivated, primarily, by Hoaglin's article "We know less than we should about methods of meta-analysis", I take into consideration that each Σ_i is unknown and estimated by using a matrix function of the corresponding unknown vector θ_i. I propose an estimation method based on the Minimum Covariance Estimator and derive formulas for the expected marginal variance for two effect sizes, namely, Pearson's moment correlation and standardized mean difference. I show through simulation studies that the proposed model and estimation method give accurate results for both univariate and bivariate meta-analyses of these effect-sizes, and compare this new approach to the standard meta-analysis method.


Hierarchical Model, Meta-Analysis, Minimum Covariance Estimator, Random-Effects


xiii, 128 pages


Includes bibliographical references (pages 106-112).


Copyright © 2019 Roberto C. Toro Rodriguez

Available for download on Saturday, September 04, 2021