## 16.2 Random-Effects Model

For power analyses under the random-effects model, the formula to calculate the variance of my true mean effect looks slightly different:

$V_{\delta}^*=\frac{V_Y+\tau^2}{k}$

We see that again, $$tau^2$$ has to be included to take the between-study heterogeneity into account (see Chapter 4.2 for more details). However, i do not know the between-study heterogeneity of my analysis before i perform it, so what value should i assume?

According to Hedges and Pigott (Hedges and Pigott 2004), the follwing formulae may be used to calculate the power in the random-effect model assuming small, moderate or high heterogeneity:

Small heterogeneity:

$V_{\delta}^*=1.33\times\frac{V_Y}{k}$

Moderate heterogeneity:

$V_{\delta}^*=1.67\times\frac{V_Y}{k}$

Large heterogeneity:

$V_{\delta}^*=2\times\frac{V_Y}{k}$

Again, you don’t have to worry about the statistical details here. We have put the entire calculations into the power.analysis function, which can already introduced before.

I will assume the same parameters used for the fixed-effect model power analysis, but this time i will also have to specify the heterogeneity in the function, which can take the values "low", "moderate" and "high". I will choose "moderate" for this example.

power.analysis(d=0.30,
k=10,
n1=25,
n2=25,
p=0.05,
heterogeneity = "moderate")

The output i get is:

## Random-effects model used (moderate heterogeneity assumed).

## Power:
## [1] 0.7327163

Interestingly, we see that this value is 73%, which is smaller than the value of 91% which was calculated using the fixed-effect model. The value is also below 80%, meaning that i would not have optimal power to find the desired effect of $$d=0.30$$ to be statistically significant if it exists.

This has to do with the larger heterogeneity i assume in this simulation, which decreases the precision of my effect size estimate, and thus increases my need for statistical power.

The graph below visualizes this relationship:

### References

Hedges, Larry V, and Therese D Pigott. 2004. “The Power of Statistical Tests for Moderators in Meta-Analysis.” Psychological Methods 9 (4). American Psychological Association: 426.