1. 9: Basic Confidence and Prediction Intervals
Meta-analysis in R with Metafor
9: Basic Confidence and Prediction Intervals
Confidence intervals for the REVC & I-squared, Prediction or Credibility intervals for the mean

2. Confidence Intervals for the Variance
We typically find confidence intervals only for the mean, but
REVC is very important in meta-analysis
REVC is often poorly estimated. Therefore
Important to know the magnitude of uncertainty about it
I-squared is an estimate of the proportion of variability due to between studies variance. This is analogous to R-squared for true between studies variance, that is, it tells us the proportion of variance attributable to random effects.
How much uncertainty do we have about this estimate?

3. Prediction or Credibility Intervals
If the REVC is greater than zero, the ‘true’ or underlying or infinite-sample effect sizes have a distribution, even after accounting for sampling error.
The Prediction or Credibility Interval creates a range of values expected to contain a percentage (usually 95 percent) of the ‘true’ effect sizes (assuming they are Normally distributed).
Metafor will compute prediction/credibility intervals for you. Metafor uses z (1.96) for the computation of the prediction interval. If you want the Higgins version that uses t instead of z (to account for imprecision in the estimate of the REVC with small numbers of studies), you must compute it from information contained in the Metafor output. I’ve written code you can use.

4. Higgins’ Prediction Intervals
Makes sense if random effects.
M is the random effects mean (summary effect).
The value of t is from the t table with your alpha and df equal to (k-2) where k is the number of independent effect sizes (studies).
T-square is the REVC, and the variance of the mean is the squared standard error of the RE summary effect (squared SEM).
Metafor replaces the t(.975, df) with z=1.96 for the 95 percent PI.
Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis (p. 129, equation 17.7). Chichester, UK: Wiley.

5. R code: McNatt data (Pygmalion studies)
McDaniel data (predictive validity of the interview for job performance)