# 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**

- Calculating Hedges’
*g*from the Mean and SD - Calculating Hedges’
*g*from a regression coefficient - Calculating an Odd’s Ratio from
*Chi-square* - Calculating Hedges’
*g*from a one-way ANOVA - Calculating Hedges’
*g*from the Mean and SE - Calculating Hedges’
*g*from a correlation - Calculating Hedges’
*g*from an independent t-test - Calculating Hedges’
*g*from Cohen’s*d* - 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.