Meta-Analytic Interval Estimation for Bivariate Correlations Douglas G. Bonett Iowa State University The currently available meta-analytic methods for correlations have restrictive assumptions. The fixed-effects methods assume equal population correlations and exhibit poor perfor- mance under correlation heterogeneity. The random-effects methods do not assume correla- tion homogeneity but are based on an equally unrealistic assumption that the selected studies are a random sample from a well-defined superpopulation of study populations. The random- effects methods can accommodate correlation heterogeneity, but these methods do not perform properly in typical applications where the studies are nonrandomly selected. A new fixed-effects meta-analytic confidence interval for bivariate correlations is proposed that is easy to compute and performs well under correlation heterogeneity and nonrandomly selected studies. Keywords: confidence interval, fixed effects, random effects, research synthesis The purpose of this article is to describe an interval estimation procedure for a bivariate correlation using infor- mation obtained from m different studies with no participant belonging to more than one study. In each of the m studies, a random sample is obtained from the study population of interest. In each study, a correlation is computed from a random sample of the study population. The sample corre- lation estimates the unknown value of the correlation ( i ) in the specified study population of study i. The correlation estimate from study i may be used to make a statement regarding the possible values of  i with some specified level of confidence. Statements of this type are called confidence intervals. For a given level of confidence, the width of the confidence interval depends on the sample size, and narrow confidence intervals often require large sample sizes. If measurements are costly or participants are difficult to ob- tain, a single researcher may not have the resources that are needed to obtain a sufficiently large sample. One solution to this problem was recognized over 100 years ago by Karl Pearson, who combined correlation estimates from five dif- ferent studies to obtain a more precise estimate of the correlation between inoculation for typhoid fever and mor- tality (Pearson, 1904). The practice of combining estimates from multiple studies, also called meta-analysis (Glass, 1976), is now a standard statistical tool in psychology, medicine, and the social sciences. Several different meta-analytic methods for combining sample correlations from m different studies have been proposed. These methods may be classified into two general categories: random-effects (RE) methods and fixed-effects (FE) methods. The RE methods assume that m study pop- ulations have been randomly sampled from a specific and clearly defined superpopulation that consists of a large number of study populations. A Gaussian distribution is assumed for the set of all  i values in the superpopulation. The researcher’s goal is to estimate the mean and standard deviation of the Gaussian distribution. Two RE methods for correlations are described by Hedges and Vevea (1998) and Hunter and Schmidt (2004). These methods are referred to as HV-R and HS-R, respectively. Computational formulas for the HV-R and HS-R methods are given in Appendix A. The HV-R and HS-R methods have been recommended because they do not make the unrealistic assumption that all  i values are identical. However, unless the m study popu- lations are randomly sampled from a specific superpopula- tion, the researcher cannot make a statistical inference from the m study populations to the superpopulation. A random sample of studies has the property that every possible subset of m study populations in the superpopulation has exactly the same probability of being included in the meta-analysis. Additionally, to obtain a random sample of studies, the selection of any particular study into the meta-analysis sample must not differentially alter the probabilities of any other study from the superpopulation being selected into the meta-analysis sample. In a typical meta-analysis, the m studies selected for analysis are likely to have been pub- lished sequentially over time, with each study intentionally designed to be similar or dissimilar to one or more previous studies, and this would be inconsistent with a random se- lection process. The random selection assumption of the RE methods is a critical assumption that, as noted by Hedges Correspondence concerning this article should be addressed to Douglas G. Bonett, Department of Statistics, Iowa State Univer- sity, Ames, IA 50011. E–mail: dgbonett@iastate.edu Psychological Methods 2008, Vol. 13, No. 3, 173–181 Copyright 2008 by the American Psychological Association 1082-989X/08/$12.00 DOI: 10.1037/a0012868 173