Abstract

The usual fixed-effects multivariate approach for synthesizing k sample correlation matrices employs generalized least squares (GLS) with a weight matrix based on the asymptotic covariance matrix of each sample’s Pearson correlations (B. J. Becker, 1992). These covariances depend on the sample size, ni, and the population correlations, ; conventional practice is to replace with the sample’s observed correlations, Ri. In simulations, this approach has exhibited poor estimation and inference properties inferior to univariate weighted least squares (WLS). The present paper reports a Monte Carlo study of eight fixed-effects correlation synthesis methods that differ in three respects: GLS versus WLS; Pearson’s r versus Fisher’s Z; and replaced by Ri versus , an initial estimate. Conditions varied by k (5 to 200) and the data generating model (fixed- vs. random-effects), and the six-element and ni distribution were constant. With respect to per-element estimation and inference, -based methods—especially using r—performed quite well, and Ri-based r methods (particularly GLS) were inferior; the GLS Ri-based Z method performed fairly well, with some exceptions. Methods using and Z controlled Type I error better for homogeneity tests. Practical recommendations and avenues for further investigation (e.g., simultaneous inference, random-effects analysis) are discussed.
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Abstract

Practical meta-analysis of correlation matrices generally ignores covariances (and hence correlations) between correlation estimates. The authors consider various methods for allowing for covariances, including generalized least squares, maximum marginal likelihood, and Bayesian approaches, illustrated using a 6-dimensional response in a series of psychological studies concerning prediction of exercise behavior change. Quantities of interest include the overall population mean correlation matrix, the contrast between the mean correlations, the predicted correlation matrix in a new study, and the conflict between the existing studies and a new correlation matrix. The authors conclude that accounting for correlations between correlations is unnecessary when interested in individual correlations but potentially important if concerned with a composite measure involving 2 or more correlations. A simulation study indicates the asymptotic normal assumption appears reasonable. Because of potential instability in the generalized least squares methods, they recommend a model-based approach, either the maximum marginal likelihood approach or a full Bayesian analysis. Copyright (c) 2008 APA.
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Abstract

Monte Carlo studies of several fixed-effects methods for combining and comparing correlation matrices have shown that two refinements improve estimation and inference substantially. With rare exception, however, these simulations have involved homogeneous data analyzed using conditional meta-analytic procedures. The present study builds on previous evidence about these methods' relative performance by examining their behavior under heterogeneity, which is more realistic in practice. Results based on both conditional and unconditional estimands indicate that of the two refinements, using estimated correlations in conditional (co) variances improves point and interval estimates of mean correlations more than analyzing Fisher Z correlations, despite the latter's superiority for testing homogeneity. Recommended choices among methods are offered.
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Abstract

Numerical approaches to developing accurate and efficient approximations to combined likelihoods of population correlation matrices in meta-analysis under normality assumptions for the data are studied. The likelihood is expressed as a multiple integral over the unit cube in (p-1)-dimensional space, where p is the row and column dimensionality of the correlation matrix. Three types of computation are proposed as ways to calculate the likelihood for any population correlation matrix P. As an application, inference is explored concerning intercorrelations among math, spatial and verbal scores in a SAT exam. Comparisons are made with conventional methods.
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Abstract

The originally proposed multivariate meta-analysis approach for correlation matrices-analyze Pearson correlations, with each study's observed correlations replacing their population counterparts in its conditional-covariance matrix performs poorly. Two refinements are considered: Analyze Fisher Z-transformed correlations, and substitute better estimates of correlations in the conditional covariances. Fixed-effects methods with and without each refinement were examined in a Monte Carlo study; number of studies and the distribution of within-study sample sizes were varied. Both refinements improved element-wise point and interval estimates, as well as Type I error control for homogeneity tests, especially with many small studies. Practical recommendations and suggestions for future methodological work are offered. An appendix describes how to transform Fisher-Z (co)variances to the Pearson-r metric.
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Abstract

Meta-analysis, the analysis of analyses, is a standard statistical procedure for summarizing research findings in behavioral sciences. The Hunter-Schmidt approach (Hunter & Schmidt, 1990) and the Hedges-Olkin (Hedges & Olkin, 1985) approach are the two most frequently used procedures. When dealing with multivariate cases, for instance, correlation matrices, these procedures are applied to individual correlation coefficients of the correlation matrix separately. A better approach, however, is to use the generalized least squares approach, which was proposed for multivariate effect sizes (e.g., Becker, 1992, 1995; Hedges & Olkin, 1985). (PsycINFO Database Record (c) 2007 APA, all rights reserved)
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Abstract

Researchers are becoming interested in combining meta-analytic techniques and structural equation modeling to test theoretical models from a pool of studies. Most existing procedures are based on the assumption that all correlation matrices are homogeneous. Few studies have addressed what the next step should be when studies being analyzed are heterogeneous and the search for moderator variables for homogeneous subgroup analysis fails. Cluster analysis is proposed and evaluated in this article as an exploratory tool to classify studies into relatively homogeneous groups. Simulation studies indicate that using Euclidean distance on raw correlation coefficients or U-transformed scores with the complete linkage or Ward's minimum-variance methods will provide satisfactory results. (PsycINFO Database Record (c) 2007 APA, all rights reserved)
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Abstract

A recent multilaboratory study to investigate phenotypic and genotypic markers for alcoholism is described. A preliminary investigation of the reliability of electroencephalographic data amassed at six laboratories was undertaken; data from each laboratory could be usefully summarized with sample correlation matrices. A graphical method for depicting these correlation matrices is presented. The Larntz-Perlman procedure for assessing the equality of correlation matrices can immediately be incorporated into this graphical technique. The experimental data from the preliminary investigation are illustrated with the graphical method, which readily conveys large amounts of correlational information and reveals meaningful patterns in the electroencephalographic responses.
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Abstract

Developmental psychologists use correlation matrices both as tools for summarizing the relations among sets of variables and as input to multivariate statistical procedures. This article reviews methods for testing whether 2 or more correlation matrices are different from each other. Methods are illustrated for evaluating the similarity of 2 independent correlation matrices, such as those obtained from boys and girls, and 2 dependent correlation matrices, such as those obtained in longitudinal research by using multiple measures. Applications of the models to data from the published developmental literature are provided.
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Abstract

In statistical meta-analysis paradigm dealing with the techniques of pooling of evidence across several studies, the notion of effect size is fundamental and it is necessary to perform appropriate tests for the equality of population effect sizes arising out of these studies before performing any meta-analysis. A well known test for homogeneity in this context is based on Cochran's chi-square statistic. However Cochran's test might be inaccurate in testing homogeneity for some effect sizes in the sense of not maintaining the stipulated Type I errors. In this dissertation, three such scenarios are identified: (1) testing homogeneity of standardized mean difference, (2) testing homogeneity of correlations and correlation matrices, and (3) testing homogeneity of proportions with low event rates, and in each case the severe inadequacy of Cochran's homogeneity test is demonstrated. Apppropriate bootstrap procedures for testing homogeneity hypotheses are suggested, and applications in each problem are indicated. Two additional problems are also dealt with in the dissertation: Mantel-Haenszel and Peto weights in random effects meta-analysis, and extraction methods in statistical meta-analysis.
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Abstract

Corrections for attenuation have long been used to adjust sample correlations for measurement error. Current research synthesis (meta-analysis) procedures involve synthesizing correlational data. The synthesis of multivariate correlation data raises several statistical questions including correcting for measurement error. The focus of this work was to discover the most statistically sound method for correcting multivariate data for attenuation which accounts for dependence among the correlations and reliabilities. Multivariate corrections from 'errors-in-variables' regression analysis were examined, as was an existing multivariate correction from educational literature. These methods were compared to the traditional univariate correction for attenuation. Simulated and exact comparisons were made of corrected correlations and their resulting variance-covariance matrices. All the methods examined produced similar corrected correlations. Even the simple univariate correction yielded corrected correlations that were good estimates of the population correlation. In addition, a variety of approximations to the population variance-covariance matrices of the corrected correlations were associated with these correction methods. A variance estimator derived in this dissertation and based on large-sample theory yielded the best estimates of variation when compared to the empirical sampling distribution. A related variance estimation method based on the correction from Fuller and Hidiroglou (1978) also gave similar results to the large-sample theory method, but relied on raw data, which are often unavailable in most research syntheses. An example illustrated the results of a multivariate synthesis using the new procedure. The results of this example showed more variability in the average correlations and larger homogeneity test statistics when compared to previous analyses of the same studies. Overall, if corrections are to be applied, the univariate correction, used with the l (PsycINFO Database Record (c) 2002 APA, all rights reserved)
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Abstract

The problem of combining information to estimate standardized partial regression coefficients in a linear model is considered. A combined estimate obtained from the pooled correlation matrix is proposed, and its large sample distribution is obtained. This estimate can be generalized to address situations in which not every study measures every variable. The method can be used to synthesize estimates of coefficients that cannot be estimated in any single study because no study obtains all of the necessary correlations. It can also be generalized to handle random effects models in which correlation parameters vary across studies. Notation and a model for the results of a series of studies examining intercorrelations among a set of "p+1" variables are provided. An estimator of the pooled correlation matrix and its standard error are derived, and a set of procedures for determining whether a common population correlation matrix underlies the set of empirical results is provided. How the pooled correlation matrix can be used to estimate the standardized coefficients of a regression model for variables measured in a series of studies is shown. Data from a synthesis of relationships among school science variables are used to illustrate the procedures. (RLC)
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Abstract

Consider the meta-analytic problem of combining sample correlation matrices across k independent studies. Available methods accomplish this under fixed- and random-effects models. The conventional fixed-effects approach, which employs generalized least squares, has been improved appreciably by two refinements: using initial estimates of the common correlations ( ) instead of observed correlations (ri) in the conditional covariance matrices that serve as (inverse) weight matrices, and analyzing Fisher Z-transformed correlations instead of Pearson correlations. The present paper extends these refinements to the random-effects case where both the between-studies mean ( ) and covariance-component matrix ( ) of study correlations ( i) are estimated. One choice concerns using estimates of versus i in the conditional covariance matrices. These options, along with the Fisher-Z modification, were evaluated in a Monte Carlo study. Other factors included k and mean within-study sample size. Random-effects data generated from a six-element mean correlation matrix were analyzed by conventional and refined methods, each of which yields estimates of and . In terms of element-wise estimation and inference, all five refined methods outperformed the conventional approach, especially with many small studies. The superior method varied, however, according to the focal parameter (i.e., vs. ). Practical recommendations are discussed.
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Abstract

Numerical approximations are important research areas for dealing with complicated functional forms. Techniques for developing accurate and efficient calculation of combined likelihood functions in meta-analysis are studied. The first part of the thesis introduces a B-spline approximation for making a parsimonious model in the simplest case(2-dimensional case) of correlation structure. Inference about the correlation between vitamin C intake & vitamin C serum level is developed by using likelihood intervals and the MLE, along with comparison with conventional methods. The second part studies a multivariate numerical integration method for developing a better approximation of the likelihood for correlation matrices. Analyses for (1) intercorrelations among Math, Spatial and Verbal scores in an SAT exam and (2) intercorrelations among Cognitive Anxiety, Somatic Anxiety and Self Confidence from Competitive State Anxiety Inventory (CSAI-2) are explored. Algorithms to evaluate likelihood and to find the MLE is developed. Comparison with two conventional methods (joint asymptotic weighted average method & marginal asymptotic weighted average method) is shown.
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