The metafor Package

The metafor package implements various meta-analytic models, methods, and techniques that have been described in the literature. The links below demonstrate how the models, methods, and techniques described in the respective references can be applied via the metafor package. The items are organized by topic (therefore, articles covering multiple topics may be listed multiple times). Alternatively, you can jump to the references at the bottom of the page for the list of articles in alphabetical order.

Note: If an example does not work properly, try installing the development version of the metafor package as described here.

The following items correspond to books on meta-analysis. The analyses described in the books are reproduced using the metafor package.

• Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis. Chichester, UK: Wiley.

• Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Thousand Oaks, CA: Sage.

• Cooper, H., Hedges, L. V., & Valentine, J. V. (Eds.) (2019). The handbook of research synthesis and meta-analysis (3rd ed.). New York: Russell Sage Foundation.

• Rothstein, H. R., Sutton, A. J., & Borenstein, M. (Eds.) (2005). Publication bias in meta-analysis: Prevention, assessment and adjustment. Chichester, UK: Wiley.

The articles below cover the standard equal-, fixed-, random-, and mixed-effects (meta-regression) models for meta-analysis.

• Berkey, C. S., Hoaglin, D. C., Mosteller, F., & Colditz, G. A. (1995). A random-effects regression model for meta-analysis. Statistics in Medicine, 14(4), 395-411.

• Normand, S. T. (1999). Meta-analysis: Formulating, evaluating, combining, and reporting. Statistics in Medicine, 18(3), 321-359.

• Raudenbush, S. W., & Bryk, A. S. (1985). Empirical Bayes meta-analysis. Journal of Educational Statistics, 10(2), 75-98.

• Raudenbush, S. W. (2009). Analyzing effect sizes: Random effects models. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 295–315). New York: Russell Sage Foundation.

• van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. Statistics in Medicine, 21(4), 589-624.

• Viechtbauer, W. (2007). Accounting for heterogeneity via random-effects models and moderator analyses in meta-analysis. Zeitschrift für Psychologie / Journal of Psychology, 215(2), 104-121.

Multivariate/multilevel meta-analytic models can be used to account for non-independent sampling errors and/or true effects (e.g., due to the inclusion of multiple treatment studies, multiple endpoints, or other forms of clustering).

• Berkey, C. S., Hoaglin, D. C., Antczak-Bouckoms, A., Mosteller, F., & Colditz, G. A. (1998). Meta-analysis of multiple outcomes by regression with random effects. Statistics in Medicine, 17(22), 2537-2550.

• Gleser, L. J., & Olkin, I. (2009). Stochastically dependent effect sizes. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 357–376). New York: Russell Sage Foundation.

• Konstantopoulos, S. (2011). Fixed effects and variance components estimation in three-level meta-analysis. Research Synthesis Methods, 2(1), 61-76.

• van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. Statistics in Medicine, 21(4), 589-624.

The conditional logistic model (also called hypergeometric-normal model) can be used to meta-analyze odds ratios (obtained from 2×2 table data).

• Stijnen, T., Hamza, T. H., & Ozdemir, P. (2010). Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29(29), 3046-3067.

• van Houwelingen, H. C., Zwinderman, K. H., & Stijnen, T. (1993). A bivariate approach to meta-analysis. Statistics in Medicine, 12(24), 2273-2284.

• van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. Statistics in Medicine, 21(4), 589-624.

The article below describes and illustrates Peto's (one-step) method for meta-analyzing (log) odds ratio.

• Yusuf, S., Peto, R., Lewis, J., Collins, R., & Sleight, P. (1985). Beta blockade during and after myocardial infarction: An overview of the randomized trials. Progress in Cardiovascular Disease, 27(5), 335-371.

The articles below describe the meta-analysis of proportions via various methods.

• Miller, J. J. (1978). The inverse of the Freeman-Tukey double arcsine transformation. American Statistician, 32(4), 138.

• Stijnen, T., Hamza, T. H., & Ozdemir, P. (2010). Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29(29), 3046-3067.

The articles below describe the meta-analysis of incidence rates and incidence rate ratios.

• Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern epidemiology (3rd ed.). Philadelphia: Lippincott Williams & Wilkins.

• Stijnen, T., Hamza, T. H., & Ozdemir, P. (2010). Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29(29), 3046-3067.

The use of the Mantel-Haenszel method for meta-analyzing risk differences, risk ratios, and odds ratios (for 2×2 table data) and for meta-analyzing incidence rate differences and incidence rate ratios (for two-group person-time data) is illustrated in the following article.

• Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern epidemiology (3rd ed.). Philadelphia: Lippincott Williams & Wilkins.

The article below discusses the calculation of effect size measures for pretest posttest control group designs.

• Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386.

Below are articles that compare various estimators for the amount of (residual) heterogeneity and/or describe methods for obtaining confidence intervals thereof.

• DerSimonian, R., & Kacker, R. (2007). Random-effects model for meta-analysis of clinical trials: An update. Contemporary Clinical Trials, 28(2), 105-114.

• Raudenbush, S. W. (2009). Analyzing effect sizes: Random effects models. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 295–315). New York: Russell Sage Foundation.

• Viechtbauer, W. (2005). Bias and efficiency of meta-analytic variance estimators in the random-effects model. Journal of Educational and Behavioral Statistics, 30(3), 261-293.

• Viechtbauer, W. (2007). Confidence intervals for the amount of heterogeneity in meta-analysis. Statistics in Medicine, 26(1), 37-52.

The articles below illustrate/discuss the calculation of best linear unbiased predictions (BLUPs) (also called empirical Bayes estimates).

• Raudenbush, S. W., & Bryk, A. S. (1985). Empirical Bayes meta-analysis. Journal of Educational Statistics, 10(2), 75-98.

• van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. Statistics in Medicine, 21(4), 589-624.

Instead of assuming normally distributed true effects, one can use mixture models to model heterogeneity in the true effects in a more flexible manner.

• van Houwelingen, H. C., Zwinderman, K. H., & Stijnen, T. (1993). A bivariate approach to meta-analysis. Statistics in Medicine, 12(24), 2273-2284.

• van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. Statistics in Medicine, 21(4), 589-624.

Berkey, C. S., Hoaglin, D. C., Mosteller, F., & Colditz, G. A. (1995). A random-effects regression model for meta-analysis. Statistics in Medicine, 14(4), 395-411.

Berkey, C. S., Hoaglin, D. C., Antczak-Bouckoms, A., Mosteller, F., & Colditz, G. A. (1998). Meta-analysis of multiple outcomes by regression with random effects. Statistics in Medicine, 17(22), 2537-2550.

Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis. Chichester, UK: Wiley.

Cooper, H., Hedges, L. V., & Valentine, J. V. (Eds.) (2019). The handbook of research synthesis and meta-analysis (3rd ed.). New York: Russell Sage Foundation.

DerSimonian, R., & Kacker, R. (2007). Random-effects model for meta-analysis of clinical trials: An update. Contemporary Clinical Trials, 28(2), 105-114.

Gleser, L. J., & Olkin, I. (2009). Stochastically dependent effect sizes. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 357–376). New York: Russell Sage Foundation.

Konstantopoulos, S. (2011). Fixed effects and variance components estimation in three-level meta-analysis. Research Synthesis Methods, 2(1), 61-76.

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Thousand Oaks, CA: Sage.

Miller, J. J. (1978). The inverse of the Freeman-Tukey double arcsine transformation. American Statistician, 32(4), 138.

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386.

Normand, S. T. (1999). Meta-analysis: Formulating, evaluating, combining, and reporting. Statistics in Medicine, 18(3), 321-359.

Raudenbush, S. W., & Bryk, A. S. (1985). Empirical Bayes meta-analysis. Journal of Educational Statistics, 10(2), 75-98.

Raudenbush, S. W. (2009). Analyzing effect sizes: Random effects models. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 295–315). New York: Russell Sage Foundation.

Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern epidemiology (3rd ed.). Philadelphia: Lippincott Williams & Wilkins.

Rothstein, H. R., Sutton, A. J., & Borenstein, M. (Eds.) (2005). Publication bias in meta-analysis: Prevention, assessment and adjustment. Chichester, UK: Wiley.

Stijnen, T., Hamza, T. H., & Ozdemir, P. (2010). Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29(29), 3046-3067.

van Houwelingen, H. C., Zwinderman, K. H., & Stijnen, T. (1993). A bivariate approach to meta-analysis. Statistics in Medicine, 12(24), 2273-2284.

van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. Statistics in Medicine, 21(4), 589-624.

Viechtbauer, W. (2005). Bias and efficiency of meta-analytic variance estimators in the random-effects model. Journal of Educational and Behavioral Statistics, 30(3), 261-293.

Viechtbauer, W. (2007). Confidence intervals for the amount of heterogeneity in meta-analysis. Statistics in Medicine, 26(1), 37-52.

Viechtbauer, W. (2007). Accounting for heterogeneity via random-effects models and moderator analyses in meta-analysis. Zeitschrift für Psychologie / Journal of Psychology, 215(2), 104-121.

Yusuf, S., Peto, R., Lewis, J., Collins, R., & Sleight, P. (1985). Beta blockade during and after myocardial infarction: An overview of the randomized trials. Progress in Cardiovascular Disease, 27(5), 335-371.