Chapter 13 Effect size calculators

Although the meta package can calculate all individual effect sizes for every study if we use the metabin or metacont function, a frequent scenario is that some papers do not report the effect size data in the right format. Especially older articles may often only report results of \(t\)-tests, ANOVAs, or \(\chi^2\)-tests. If enough data is reported, we can also use such outcome formats to calculate effect sizes. This way, we can calculate the effect size (e.g., Hedges’ g) and the Standard Error (SE), which we can then use in a meta-analysis with pre-calculated effect sizes using the metagen function (see Chapter 4.1.1).

Hedges’ g

When dealing with continuous outcome data, it is conventional to calculate the Standardized Mean Difference (SMD) as an outcome for each study, and as your summary measure (Borenstein et al. 2011).

A common format to to calculate the SMD in single trials is Cohen’s d (Cohen 1988). Yet, this summary measure has been shown to have a slight bias in small studies, for which it overestimates the effect (Hedges 1981).

Hedges g is a similar summary measure, but it controls for this bias. It uses a slightly different formula to calculate the pooled variance \(s_{pooled}\), \(s*_{pooled}\). The transformation from d to g is often performed using the formula by Hedges and Olkin (Hedges and Olkin 1985).

\[g \simeq d\times(1-\frac{3}{4(n_1+n_2)-9}) \]

Hedges’ g is commonly used in meta-analysis, and it’s the standard output format in RevMan. Therefore, we highly recommend that you also use this measure in you meta-analysis.

In meta‘s metabin and metacont function, Hedges’ g is automatically calculated for each study if we set sm=“SMD”. If you use the metgen function, however, you should calculate Hedges’ g for each study yourself first.

To calculate the effect sizes, we will use Daniel Lüdecke’s extremely helpful esc package (Lüdecke 2018). So, please install this package first using the install.packages("esc") command, and then load it in you library.

library(esc)



Here’s an overview of all calculators covered in this guide

  1. Calculating Hedges’ g from the Mean and SD
  2. Calculating Hedges’ g from a regression coefficient
  3. Calculating an Odd’s Ratio from Chi-square
  4. Calculating Hedges’ g from a one-way ANOVA
  5. Calculating Hedges’ g from the Mean and SE
  6. Calculating Hedges’ g from a correlation
  7. Calculating Hedges’ g from an independent t-test
  8. Calculating Hedges’ g from Cohen’s d
  9. Calculating effect sizes for studies with multiple comparisons



References

Borenstein, Michael, Larry V Hedges, Julian PT Higgins, and Hannah R Rothstein. 2011. Introduction to Meta-Analysis. John Wiley & Sons.

Cohen, Jacob. 1988. “Statistical Power Analysis for the Behavioral Sciences. 1988, Hillsdale, Nj: L.” Lawrence Earlbaum Associates 2.

Hedges, Larry V. 1981. “Distribution Theory for Glass’s Estimator of Effect Size and Related Estimators.” Journal of Educational Statistics 6 (2). Sage Publications Sage CA: Thousand Oaks, CA: 107–28.

Hedges, L, and Ingram Olkin. 1985. “Statistical Models for Meta-Analysis.” New York: Academic Press.

Lüdecke, Daniel. 2018. Effect Size Computation for Meta Analysis. https://CRAN.R-project.org/package=esc.

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