Statistical Science

The Synthesis of Regression Slopes in Meta-Analysis

Betsy Jane Becker and Meng-Jia Wu

Full-text: Open access


Research on methods of meta-analysis (the synthesis of related study results) has dealt with many simple study indices, but less attention has been paid to the issue of summarizing regression slopes. In part this is because of the many complications that arise when real sets of regression models are accumulated. We outline the complexities involved in synthesizing slopes, describe existing methods of analysis and present a multivariate generalized least squares approach to the synthesis of regression slopes.

Article information

Statist. Sci., Volume 22, Number 3 (2007), 414-429.

First available in Project Euclid: 2 January 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Generalized least squares multivariate meta-analysis regression slopes


Becker, Betsy Jane; Wu, Meng-Jia. The Synthesis of Regression Slopes in Meta-Analysis. Statist. Sci. 22 (2007), no. 3, 414--429. doi:10.1214/07-STS243.

Export citation


  • Amemiya, Y. and Fuller, W. A. (1984). Estimation for the multivariate errors-in-variables model with estimated error covariance matrix. Ann. Statist. 12 497--509.
  • Ashenfelter, O., Harmon, C. and Oosterbeek, H. (1999). A review of estimates of the schooling/earnings relationship, with tests for publication bias. Labour Economics 6 453--470.
  • Baker, C. B., Tweedie, R., Duval, S. and Woods, S. W. (2003). Evidence that the SSRI dose response in treating major depression should be reassessed: A meta-analysis. Depression and Anxiety 17 1--9.
  • Bini, L. M., Coelho, A. S. G. and Diniz-Filho, J. A. F. (2001). Is the relationship between population density and body size consistent across independent studies? A meta-analytical approach. Revista Brasileira de Biologia 61 1--6.
  • Card, D. and Krueger, A. B. (1995). Time-series minimum-wage studies: A meta-analysis. Amer. Econom. Rev. (AEA Papers and Proceedings) 85 238--243.
  • Coleman, J. S., Campbell, E. Q., Hobson, C. J., Mcpartland, J., Mood, A. M. and Weinfeld, F. D. (1966). Survey of Equal Educational Opportunity. U.S. Department of Health, Education and Welfare, Washington, DC.
  • Crouch, G. I. (1995). A meta-analysis of tourism demand. Ann. Tourism Research 22 103--118.
  • Crouch, G. I. (1996). Demand elasticities in international marketing: A meta-analytical application to tourism. J. Business Research 36 117--136.
  • Doucouliagos, C. (2005). Publication bias in the economic freedom and economic growth literature. J. Economic Surveys 19 367--387.
  • Doucouliagos, C. and Paldam, M. (2006). Aid effectiveness on accumulation: A meta study. Kyklos 59 227--254.
  • Farley, J. U., Lehmann, D. R. and Sawyer, A. (1995). Empirical marketing generalization using meta-analysis. Marketing Science 14 G36--G46.
  • Greenland, S. (1987). Quantitative methods in the review of epidemiologic literature. Epidemiologic Reviews 9 1--30.
  • Greenland, S. and Longnecker, M. P. (1987). Methods for trend estimation from summarized dose-response data, with applications to meta-analysis. Amer. J. Epidemiology 135 1301--1309.
  • Hanushek, E. A. (1974). Efficient estimators for regressing regression coefficients. Amer. Statist. 28 66--67.
  • Hanushek, E. A. (1989). The impact of differential expenditures on school performance. Educational Researcher 18 45--65.
  • Hedges, L. V. (1988). The meta-analysis of test validity studies: Some new approaches. In Test Validity (H. Wainer and H. I. Braun, eds.) 191--212. Hillsdale, Lawrence Erlbaum, NJ.
  • Hedges, L. V., Laine, R. D. and Greenwald, R. (1994). Does money matter? A meta-analysis of studies of the effects of differential school inputs on student outcomes. Educational Researcher 23 5--14.
  • Hedges, L. V. and Olkin, I. (1985). Statistical Methods for Meta-Analysis. Academic Press, New York.
  • Hedges, L. V. and Vevea, J. L. (1998). Fixed- and random-effects models in meta-analysis. Psychological Methods 3 486--504.
  • Hunter, J. E. and Schmidt, F. L. (2004). Methods of Meta-Analysis: Correcting Error and Bias in Research Findings. Newbury Park, Sage Publications, CA.
  • Ingels, S. J., Scott, L. A., Lindmark, J. T., Frankel, M. R. and Myers, S. L. (1992). NELS 88 First Follow-up: Student Component Data file User's Manual. 1. U.S. Department of Education, Office of Educational Research and Improvement.
  • Jarrell, S. B. and Stanley, T. D. (1990). A meta-analysis of the union-nonunion wage gap. Industrial and Labor Relations Review 44 54--67.
  • Keef, S. P. and Roberts, L. A. (2004). The meta-analysis of partial effect sizes. British J. Math. Statist. Psych. 57 97--129.
  • Lau, R. R., Sigelman, L., Heldman, C. and Babbitt, P. (1999). The effects of negative political advertisements: A meta-analytic assessment. Amer. Political Science Review 93 851--875.
  • Novick, M. R., Jackson, P. H., Thayer, D. T. and Cole, N. S. (1972). Estimating multiple regressions in m groups: A cross-validation study. British J. Math. Statist. Psych. 25 33--50.
  • Olkin, I. (2003). Personal communication.
  • Pang, F., Drummond, M. and Song, F. (1999). The use of meta-analysis in economic evaluation. Discussion paper 173, The University of York, Centre for Health Economics, NHS Centre for Reviews and Dissemination.
  • Peterson, R. A. and Brown, S. P. (2005). On the use of beta coefficients in meta-analysis. J. Appl. Psychology 90 175--181.
  • Raju, N. S., Fralicx, R. and Steinhaus, S. D. (1986). Covariance and regression slope models for studying validity generalization. Applied Psychological Measurement 10 195--211.
  • Raju, N. S., Pappas, S. and Williams, C. P. (1989). An empirical Monte Carlo test of the accuracy of the correlation, covariance, and regression slope models for assessing validity generalization. J. Appl. Psychology 74 901--911.
  • Raudenbush, S. W., Becker, B. J. and Kalaian, H. (1988). Modeling multivariate effect sizes. Psychological Bulletin 103 111--120.
  • Roberts, C. J. (2005). Issues in meta-regression analysis: An overview. J. Economic Surveys 19 295--298.
  • Root, T. L., Price, J. T., Hall, K. R., Schneider, S. H., Rosenzweig, C. and Pounds, J. A. (2003). Fingerprints of global warming on wild animals and plants. Nature 421 57--60.
  • Rose, A. K. and Stanley, T. D. (2005). A meta-analysis of the effect of common currencies on international trade. J. Economic Surveys 19 347--365.
  • Shi, J. Q. and Copas, J. B. (2004). Meta-analysis for trend estimation. Statistics in Medicine 23 3--19.
  • Sidik, K. and Jonkman, J. N. (2005). Simple heterogeneity variance estimation for meta-analysis. Appl. Statist. 54 367--384.
  • Stanley, T. D. (2001). Wheat from chaff: Meta-analysis as quantitative literature review. J. Economic Perspectives 15 131--150.
  • Stanley, T. D. (2005). Beyond publication bias. J. Economic Surveys 19 309--345.
  • Stanley, T. D. and Jarrell, S. B. (1989). Meta-regression analysis: A quantitative method of literature surveys. J. Economic Surveys 3 161--170.
  • Stanley, T. D. and Jarrell, S. B. (1998). Gender wage discrimination bias? A meta-regression analysis. J. Human Resources 33 947--973.
  • Stanley, T. D. and Jarrell, S. B. (2005). Meta-regression analysis: A quantitative method of literature surveys. J. Economic Surveys 19 299--308.
  • Stapleton, J. (1995). Linear Statistical Models. Wiley, New York.
  • Timm, N. H. (2004). Estimating effect sizes in exploratory experimental studies when using a linear model. Amer. Statist. 58 213--217.
  • Walker, G. A. and Saw, J. G. (1978). The distribution of linear combinations of $t$-variables. J. Amer. Statist. Assoc. 73 876--878.
  • Wu, M.-J. and Becker, B. J. (2004). Synthesizing results from regression studies: What can we learn from combining results from studies using large data sets? Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.