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377 20 MODEL-BASED META-ANALYSIS BETSY JANE BECKER Florida State University C O N T E N T S 20.1 What Are Model-Based Research Syntheses? 378 20.1.1 Model-Driven Meta-Analysis and Linked Meta-Analysis 378 20.1.2 Average Correlation Matrices as a Basis for Inference 379 20.1.3 The Standardized Multiple-Regression Model 379 20.1.4 Slope Coefficients, Standard Errors, and Tests 380 20.2 What We Can Learn from Models, But Not from Bivariate Meta-Analyses 380 20.2.1 Partial Effects 380 20.2.2 Indirect Effects 381 20.2.2.1 Mediator Effects 382 20.2.3 Model Comparisons 383 20.2.4 We Can Learn What Has Not Been Studied 383 20.2.5 Limitations of Linked Meta-Analysis 384 20.3 How to Conduct a Model-Based Synthesis 384 20.3.1 Problem Formulation 384 20.3.2 Data Collection 384 20.3.3 Data Management 385 20.3.4 Analysis and Interpretation 387 20.3.4.1 Distribution of the Correlation Vector r 387 20.3.4.2 Mean Correlation Matrix Under Fixed Effects 388 20.3.4.3 Test of Homogeneity, with H01: ␳1 . . .␳k or ␨1 . . .␨k 389 20.3.4.4 Estimating Between-Studies Variation 390 20.3.4.5 Random-Effects Mean Correlation 390 20.3.4.6 Test of No Association, with H02: ␳0 391 20.3.4.7 Estimating Linear Models 391 20.3.4.8 Moderator Analyses 392 20.4 Summary and Future Possibilities 393 20.5 Acknowledgments 394 20.6 References 394 378 SPECIAL STATISTICAL ISSUES AND PROBLEMS 20.1 WHAT ARE MODEL-BASED RESEARCH SYNTHESES? In this chapter I introduce model-based meta-analysis and update what is known about research syntheses aimed at examining models and questions more complex than those addressed in typical bivariate meta-analyses. I begin with the idea of model-based (or model-driven) metaanalysis and the related concept of linked meta-analysis, and describe the benefits (and limitations) of model-based meta-analysis. In discussing how to do a model-based meta-analysis, I pay particular attention to the practical concerns that differ from those relevant to a typical bivariate meta-analysis. To close, I provide a brief summary of research that has been done on methods for modeldriven meta-analysis since the first edition of this volume was published. 20.1.1 Model Driven Meta-Analysis and Linked Meta-Analysis The term meta-analysis was coined by Gene Glass to mean the analysis of analyses (1976). Originally, Glass applied the idea to sets of results arising from series of independent studies that had examined the same (or very similar) research questions. Many meta-analyses have looked at relatively straightforward questions, such as whether a particular treatment or intervention has positive effects, or whether a particular predictor relates to an outcome. However, linked or model-driven meta-analysis techniques can be used to analyze more complex chains of events such as the prediction of behaviors based on sets of precursor variables. Suppose a synthesist wanted to draw on the accumulated literature to understand the prediction of sport performance from three aspects of anxiety. For now, let us refer to the anxiety measures as X1 through X3 and the outcome as Y. The synthesist might want to use metaanalysis to estimate a model such as the one shown in figure 20.1. I will use the term model to mean “a set of postulated interrelationships among constructs or variables ” (Becker and Schram 1994, 358). This model might be represented in a single study as Ŷj b0 b1 X1j b2 X2j b3 X3j for person j. Model-based meta-analysis provides one way of summarizing studies that could inform us about this model. Suppose now that the synthesist thinks that the Xs may also influence each other, and wants to examine whether X1 and X2 affect Y by way of the mediating effect of X3, as well as directly. We can also examine a model like the one shown in figure 20.2 using model-based meta-analysis. The ideas I cover in this chapter have been called by two different names—linked meta-analysis (Lipsey 1997) and model-driven meta-analysis (Becker 2001). The estimation and data-analysis methods used have been variously referred to as estimating linear models (Becker 1992b, 1995), meta-analytic structural equation modeling , called MASEM by Mike Cheung and Wai Chan (2005) and MA-SEM by...

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