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--- analyses:miller1978 [2021/12/20 15:38] – Wolfgang Viechtbauer
+++ analyses:miller1978 [2022/06/08 13:50] – Wolfgang Viechtbauer
@@ Line 21: / Line 21: @@
 The ''yi'' values are the Freeman-Tukey (double arcsine) transformed proportions, while the ''vi'' values are the corresponding sampling variances.
-Note that one can find two different definitions of the Freeman-Tukey transformation in the literature that differ only by the multiplicative constant $1/2$. The ''escalc()'' function includes the multiplicative constant $1/2$, while Miller (1978) leaves this out. Therefore, the transformed values given in the table by Miller are twice as large as the ones given above. Whether one includes the multiplicative constant or not is irrelevant, as long as one uses the correct equation for the sampling variance. For more details, see the question [[:faq#how_is_the_freeman-tukey_trans|How is the Freeman-Tukey transformation of proportions and incidence rates computed?]] under the FAQ section.
+**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 ''escalc()'' function includes the multiplicative constant $1/2$, while Miller (1978) leaves this out. Therefore, the transformed values given in the table by Miller are twice as large as the ones given above. Whether one includes the multiplicative constant or not is irrelevant, as long as one uses the correct equation for the sampling variance. For more details, see the question [[:faq#how_is_the_freeman-tukey_trans|How is the Freeman-Tukey transformation of proportions and incidence rates computed?]] under the FAQ section.
 ==== Back-Transformation of Individual Values ====
@@ Line 207: / Line 207: @@
 {{ analyses:miller1978_forest.png?nolink }}
 Now the estimated average and the results from the individual studies are correctly back-transformed.
+==== Note of Caution ====
+It turns out that there are some concerns with bias when back-transforming a pooled Freeman-Tukey (double arcsine) transformed proportion using the back-transformation suggested by Miller (1978) that uses the harmonic mean of the sample sizes in the back-transformation. For details, see Schwarzer et al. (2019). Alternatively, one can use the standard arcsine transformation (''measure="PAS"'' in ''escalc()''), whose back-transformation does not depend on the sample sizes, or switch to a (random-effects) logistic regression model (which can be done with the ''rma.glmm()'' function).
 ==== References ====
-Freeman, M. F., & Tukey, J. W. (1950). Transformations related to the angular and the square root. //Annals of Mathematical Statistics, 21//(4), 607--611.
+Freeman, M. F., & Tukey, J. W. (1950). Transformations related to the angular and the square root. //Annals of Mathematical Statistics, 21//(4), 607--611. https://doi.org/10.1214/aoms/1177729756
+Miller, J. J. (1978). The inverse of the Freeman-Tukey double arcsine transformation. //American Statistician, 32//(4), 138. https://doi.org/10.1080/00031305.1978.10479283
-Miller, J. J. (1978). The inverse of the Freeman-Tukey double arcsine transformation. //American Statistician, 32//(4), 138.
+Schwarzer, G., Chemaitelly, H., Abu-Raddad, L. J. & Rücker, G. (2019). Seriously misleading results using inverse of Freeman-Tukey double arcsine transformation in meta-analysis of single proportions. //Research Synthesis Methods, 10//(3), 476--483. https://doi.org/10.1002/jrsm.1348