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.