Pool the results of two treatment arms

This function allows to pool the mean, standard deviation and sample size of two experimental groups of a study. Results of two treatment arms may be pooled to mitigate the risk of a unit-of-analysis error and to avoid "double-counting" in meta-analyses in which studies with more than two experimental groups are included.

Usage

pool.groups(n1, n2, m1, m2, sd1, sd2)

Arguments

n1: Numeric vector or single number. The number of participants in arm 1.
n2: Numeric vector or single number. The number of participants in arm 2.
m1: Numeric vector or single number. The mean in arm 1.
m2: Numeric vector or single number. The mean in arm 2.
sd1: Numeric vector or single number. The standard deviation ($SD$) in arm 1.
sd2: Numeric vector or single number. The standard deviation ($SD$) in arm 2.

Value

Returns a data.frame containing the following columns:

Mpooled: The pooled mean of both groups
SDpooled: The pooled standard deviation of both groups
Npooled: The pooled number of participants of both groups

Details

Many randomized-controlled trials do not only include a single intervention and control group, but compare the effect of two or more interventions to a control group. It might be tempting in such a scenario to simply include all the comparisons between the intervention groups and control within a study into one meta-analysis. Yet, researchers should abstain from this practice, as this would mean that the control group is used twice for the meta-analysis, thus “double-counting” the participants in the control group. This results in a unit-of-analysis error, as the effect sizes are correlated, and thus not independent, but are treated as if they would stem from independent samples.

One way to deal with this is to synthesize the results of the intervention arms to obtain one single comparison to the control group. Despite its practical limitations (sometimes, this would mean synthesizing the results from extremely different types of interventions), this procedure does avoid the unit-of-analysis error problem.

To synthesize the pooled effect size data (pooled mean, standard deviation and $N$), the following formulae are used:

$$N_{pooled}=N_1+N_2$$ $$M_{pooled}=\frac{N_1M_1+N_2M_2}{N_1+N_2}$$ $$SD_{pooled} = \sqrt{\frac{(N_1-1)SD^{2}_{1}+ (N_2-1)SD^{2}_{2}+\frac{N_1N_2}{N_1+N_2}(M^{2}_1+M^{2}_2-2M_1M_2)} {N_1+N_2-1}}$$

What should I do when a study has more than two intervention groups?

If a study has more than two intervention groups you want to synthesize (e.g. four arms, with three distinct intervention arms), you can pool the effect size data for the first two interventions, and then synthesize the pooled data you calculated with the data from the third group.

References

Harrer, M., Cuijpers, P., Furukawa, T.A, & Ebert, D. D. (2019). Doing Meta-Analysis in R: A Hands-on Guide. DOI: 10.5281/zenodo.2551803. Chapter 13.9

Author

Mathias Harrer & David Daniel Ebert

Examples

pool.groups(n1 = 50, n2 = 56,
    m1 = 7.83, m2 = 8.32,
    sd1 = 3.52, sd2 = 2.25)
#>    Mpooled SDpooled Npooled
#> 1 8.088868 2.914511     106