\name{transf}
\alias{transf.rtoz}
\alias{transf.ztor}
\alias{transf.logit}
\alias{transf.ilogit}
\alias{transf.arcsin}
\alias{transf.iarcsin}
\alias{transf.pft}
\alias{transf.ipft}
\alias{transf.ipft.hm}
\alias{transf.isqrt}
\alias{transf.irft}
\alias{transf.iirft}
\alias{transf.ztor.int}
\alias{transf.exp.int}
\alias{transf.ilogit.int}
\title{Transformation Functions}
\description{
	A set of transformation functions useful for meta-analyses.
}
\usage{
transf.rtoz(xi, \dots)
transf.ztor(xi, \dots)
transf.logit(xi, \dots)
transf.ilogit(xi, \dots)
transf.arcsin(xi, \dots)
transf.iarcsin(xi, \dots)
transf.pft(xi, ni, \dots)
transf.ipft(xi, ni, \dots)
transf.ipft.hm(xi, targs, \dots)
transf.isqrt(xi, \dots)
transf.irft(xi, ti, \dots)
transf.iirft(xi, ti, \dots)
transf.ztor.int(xi, targs, \dots)
transf.exp.int(xi, targs, \dots)
transf.ilogit.int(xi, targs, \dots)
}
\arguments{
	\item{xi}{a vector of values to be transformed.}
	\item{ni}{a vector of sample sizes.}
	\item{ti}{a vector of person-times at risk.}
	\item{targs}{a list with additional arguments for the transformation function. See \sQuote{Details}.}
	\item{\dots}{other arguments.}
}
\details{
	The following transformation functions are currently implemented:
	\itemize{
	\item \code{transf.rtoz}: Fisher's r-to-z transformation.
	\item \code{transf.ztor}: inverse of the Fisher's r-to-z transformation.
	\item \code{transf.logit}: logit (log odds) transformation.
	\item \code{transf.ilogit}: inverse of the logit transformation.
	\item \code{transf.arcsin}: arcsine transformation.
	\item \code{transf.iarcsin}: inverse of the arcsine transformation.
	\item \code{transf.pft}: Freeman-Tukey (double arcsine) transformation for proportions. See Freeman & Tukey (1950). The \code{xi} argument is used to specify the proportions and the \code{ni} argument the corresponding sample sizes.
	\item \code{transf.ipft}: inverse of the Freeman-Tukey (double arcsine) transformation for proportions. See Miller (1978).
	\item \code{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 \code{targs} argument (the list element should be called \code{ni}).
	\item \code{transf.isqrt}: inverse of the square-root transformation (i.e., function to square a number).
	\item \code{transf.irft}: Freeman-Tukey transformation for incidence rates. See Freeman & Tukey (1950). The \code{xi} argument is used to specify the incidence rates and the \code{ti} argument the corresponding person-time at risk.
	\item \code{transf.iirft}: inverse of the Freeman-Tukey transformation for incidence rates.
	\item \code{transf.ztor.int}: integral transformation method for the z-to-r transformation.
	\item \code{transf.exp.int}: integral transformation method for the exponential transformation.
	\item \code{transf.ilogit.int}: integral transformation method for the inverse of the logit transformation.
	}
	The integral transformation method for a transformation function \eqn{h(z)} integrates \eqn{h(z) f(z)} over \eqn{z} using the limits \code{targs$lower} and \code{targs$upper}, where \eqn{f(z)} is the density of a normal distribution with mean equal to \code{xi} and variance equal to \code{targs$tau2}. An example is provided below.
}
\value{
	A vector with the transformed values.
}
\author{
	Wolfgang Viechtbauer \email{wvb@metafor-project.org} \cr
	project homepage: \url{http://www.metafor-project.org/} \cr
	author homepage: \url{http://www.wvbauer.com/}
}
\references{
	Fisher, R. A. (1921). On the \dQuote{probable error} of a coefficient of correlation deduced from a small sample. \emph{Metron}, \bold{1}, 1--32.

	Freeman, M. F. & Tukey, J. W. (1950). Transformations related to the angular and the square root. \emph{Annals of Mathematical Statistics}, \bold{21}, 607--611.

	Miller, J. J. (1978). The inverse of the Freeman-Tukey double arcsine transformation. \emph{American Statistician}, \bold{32}, 138.

	Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. \emph{Journal of Statistical Software}, \bold{36}(3), 1--48. \url{http://www.jstatsoft.org/v36/i03/}.
}
\examples{
### load BCG vaccine data
data(dat.bcg)

### meta-analysis of the log relative risks using a random-effects model
res <- rma(ai=tpos, bi=tneg, ci=cpos, di=cneg, 
           measure="RR", data=dat.bcg, method="REML")

### average relative risk with 95\% CI
predict(res, transf=exp)

### average relative risk with 95\% CI using the integral transformation
predict(res, transf=transf.exp.int, targs=list(tau2=res$tau2, lower=-4, upper=4))
}
\keyword{manip}