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, ...)

Arguments

xi

vector of values to be transformed.

ni

vector of sample sizes.

ti

vector of person-times at risk.

targs

list with additional arguments for the transformation function. See ‘Details’.

...

other arguments.

Details

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.

Value

A vector with the transformed values.

References

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. http://www.jstatsoft.org/v36/i03/.

Examples

### 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)
#> #> pred ci.lb ci.ub cr.lb cr.ub #> 0.4894 0.3441 0.6962 0.1546 1.5490 #>
### 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))
#> #> pred ci.lb ci.ub cr.lb cr.ub #> 0.5724 0.4024 0.8142 0.1809 1.8117 #>