Dear friends,
I am conducting a meta-analysis using Fisher's Z as effect size in a general linear mixed model. I am using rma.mv function from metafor package. My question is about what measure I should use as weight in meta-analytic models. Most researchers use variance, calculated as (standard error)^2. However, I read the following passage of chapter 8 (p. 96) from the book of Koricheva et al. 2013:
"Suppose that each study provides an estimate of the effect of interest Ti, and that this estimate has a within-study sampling variance of Si², which is estimated by Si². Thus, the Ti are the effect estimates from each study, and the meta-analysis model describes the combination of these estimates across studies. It is emphasized that Si² and Si² represent the variance of the effect size estimate Ti; this is usually different from the variance of the data in the primary study. For example, if Ti is the estimated sample mean, then Si² is the estimated standard error of this mean (not the sample variance of the data). (Note that the reader may have seen some of this terminology used to represent statistical parameters within primary studies, but we are using the terms to mean something different here.)"
Therefore, I am not sure about to use variance as weight. Should I use standard error instead of variance? Any contributions will be helpful.
Best whishes,
Rafael.
I am confused. Why would you want to use the standard error as a weight in a meta-analysis? Because you are using Fisher's z, I assume you are doing a meta-analysis on correlations. The within-study variance for Fisher's z is approximated very well by 1/(n-3) where n is the sample size for a study. The weighting of a study in a meta-analysis is equal to 1/V where V equals the within-study variance in case of the fixed-effect model and the within-study variance plus the between-study variance in the case of the random-effects model.
Dear Dr. Warren,
Thank you for the reply. I am using rma.mv function of metafor package to conduct a multi-level metanalysis. The function is the following:
rma.mv(zi, sei, mods=~x, random(studyID, speciesID), data)
Are you suggesting to use the inverse of variance (1/vi) instead of standard error (se)? I think values of 1/vi will be higher than the effect size. For instace, if I calculate a variance iqual 0.12, the 1/0.12 will be 8.33. This does not seem correct.
Cheers,
Rafael.
The absolute value of a study's weighting is arbitrary. For each study, you multiply the weighting by the study's effect size. You then sum up all those products and divide through by the sum of all the weightings. The resulting value will be your overall effect size. See the attached Excel file for an example.
Thank you for the contributions, Dr. Warren. Your explanation was very enlightening. I was confused about the definition of standard error and variance used by meta-analysts due to the passage above from the book of Koricheva et al. (2013). In statistic, these terms represent different measures: standard deviation = square root (variance); and standard error = standard deviation/ square root (number of samples). Then, Koricheva et al. (2013) also discussed that variance is the standard error of the mean effect size.
I was using the inverse of standard error as weight in a phylogenetic meta-analysis on correlations, following recommendations of Zaykin (2011). However, I observed that this approach generated different estimations of I^2 compared with the same models using the inverse of variance. Zaykin (2011) defended that the inverse of standard error has the largest power to reduce heterogeneity. In fact, I found lower heterogeneity in models using this approach compared with models using the inverse of variance as weight. However, I am not sure about what measure I should use as weight. Do you recommend to use the inverse of variance instead of the inverse of standard error or my approach is also valid?
Zaykin, DV (2011) Optimally weighted Z-test is a powerful method for combining probabilities in meta-analysis. JOURNAL OF EVOLUTIONARY BIOLOGY, 24:1836–1841.
Thank you for the contribution, Dr. Holzhauer. The terminology of meta-analysis is very confusing when compared to the statistical language. However, we may adapt after a while. I was worried about the correct measure to use as weight. I am using the inverse of standard error based on Zaykin (2011).
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