I^2 (total heterogeneity / total variability): 0.00% Res <- rma (yi, vi, method = 'EE', data =dat, weighted =FALSE ) The unweighted mean can be obtained with: For more details, see the question How is the Freeman-Tukey transformation of proportions and incidence rates computed? under the FAQ section.Īs described by Miller (1978), we can aggregate the transformed values, either by computing an unweighted or a weighted mean (with inverse-variance weights). Whether one includes the multiplicative constant or not is irrelevant, as long as one uses the correct equation for the sampling variance. Therefore, the transformed values given in the table by Miller are twice as large as the ones given above. The escalc() function includes the multiplicative constant $1/2$, while Miller (1978) leaves this out. Note: One can find two different definitions of the Freeman-Tukey transformation in the literature that differ only by the multiplicative constant $1/2$. The yi values are the Freeman-Tukey (double arcsine) transformed proportions, while the vi values are the corresponding sampling variances. The data used by Miller (1978) to illustrate the transformation and its inversion can be re-created with:ĭat <- ame (xi = c ( 3, 6, 10, 1 ), ni = c ( 11, 17, 21, 6 ) )ĭat <- escalc (measure = 'PFT', xi =xi, ni =ni, data =dat )

The corresponding back-transformation equation was derived by Miller (1978). A transformation that works particularly well for normalizing and variance-stabilizing the sampling distribution of proportions is the Freeman-Tukey (double arcsine) transformation (Freeman & Tukey, 1950). While the sampling distribution of a logit transformed proportion is indeed better approximated by a normal distribution, the equation used to compute the corresponding sampling variance can still be quite inaccurate, especially when sample sizes are small. An often recommended transformation in this context is the logit transformation (i.e., the log odds). When meta-analyzing proportions, it is usually advantageous to first transform the proportions into a measure that has better statistical properties (i.e., a sampling distribution that is closer to a normal distribution and whose sampling variance can be better approximated).

Arcsine transformation in r