A set of transformation functions useful for meta-analyses.
transf.rtoz(xi, …)
transf.ztor(xi, …)
transf.logit(xi, …)
transf.ilogit(xi, …)
transf.arcsin(xi, …)
transf.iarcsin(xi, …)
transf.pft(xi, ni, …)
transf.ipft(xi, ni, …)
transf.ipft.hm(xi, targs, …)
transf.isqrt(xi, …)
transf.irft(xi, ti, …)
transf.iirft(xi, ti, …)
transf.ahw(xi, …)
transf.iahw(xi, …)
transf.abt(xi, …)
transf.iabt(xi, …)
transf.ztor.int(xi, targs, …)
transf.exp.int(xi, targs, …)
transf.ilogit.int(xi, targs, …)vector of values to be transformed.
vector of sample sizes.
vector of person-times at risk.
list with additional arguments for the transformation function. See ‘Details’.
other arguments.
A vector with the transformed values.
The following transformation functions are currently implemented:
transf.rtoz: Fisher's r-to-z transformation for correlations.
transf.ztor: inverse of the Fisher's r-to-z transformation.
transf.logit: logit (log odds) transformation for proportions.
transf.ilogit: inverse of the logit transformation.
transf.arcsin: arcsine square root transformation for proportions.
transf.iarcsin: inverse of the arcsine transformation.
transf.pft: Freeman-Tukey (double arcsine) transformation for proportions. See Freeman & Tukey (1950). The xi argument is used to specify the proportions and the ni argument the corresponding sample sizes.
transf.ipft: inverse of the Freeman-Tukey (double arcsine) transformation for proportions. See Miller (1978).
transf.ipft.hm: inverse of the Freeman-Tukey (double arcsine) transformation for proportions using the harmonic mean of the sample sizes for the back-transformation. See Miller (1978). The sample sizes are specified via the targs argument (the list element should be called ni).
transf.isqrt: inverse of the square root transformation (i.e., function to square a number).
transf.irft: Freeman-Tukey transformation for incidence rates. See Freeman & Tukey (1950). The xi argument is used to specify the incidence rates and the ti argument the corresponding person-times at risk.
transf.iirft: inverse of the Freeman-Tukey transformation for incidence rates.
transf.ahw: Transformation of coefficient alpha as suggested by Hakstian & Whalen (1976).
transf.iahw: Inverse of the transformation of coefficient alpha as suggested by Hakstian & Whalen (1976).
transf.abt: Transformation of coefficient alpha as suggested by Bonett (2002).
transf.iabt: Inverse of the transformation of coefficient alpha as suggested by Bonett (2002).
transf.ztor.int: integral transformation method for the z-to-r transformation.
transf.exp.int: integral transformation method for the exponential transformation.
transf.ilogit.int: integral transformation method for the inverse of the logit transformation.
The integral transformation method for a transformation function \(h(z)\) integrates \(h(z) f(z)\) over \(z\) using the limits targs$lower and targs$upper, where \(f(z)\) is the density of a normal distribution with mean equal to xi and variance equal to targs$tau2. An example is provided below.
Bonett, D. G. (2002). Sample size requirements for testing and estimating coefficient alpha. Journal of Educational and Behavioral Statistics, 27, 335--340.
Fisher, R. A. (1921). On the “probable error” of a coefficient of correlation deduced from a small sample. Metron, 1, 1--32.
Freeman, M. F., & Tukey, J. W. (1950). Transformations related to the angular and the square root. Annals of Mathematical Statistics, 21, 607--611.
Hakstian, A. R., & Whalen, T. E. (1976). A k-sample significance test for independent alpha coefficients. Psychometrika, 41, 219--231.
Miller, J. J. (1978). The inverse of the Freeman-Tukey double arcsine transformation. American Statistician, 32, 138.
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1--48. https://www.jstatsoft.org/v036/i03.
# NOT RUN {
### meta-analysis of the log risk ratios using a random-effects model
res <- rma(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)
### average risk ratio with 95% CI (but technically, this provides an
### estimate of the median risk ratio, not the mean risk ratio!)
predict(res, transf=exp)
### average risk ratio with 95% CI using the integral transformation
predict(res, transf=transf.exp.int, targs=list(tau2=res$tau2, lower=-4, upper=4))
# }
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