Fisher transformation

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In statistics, the Fisher transformation (or Fisher z-transformation) of a Pearson correlation coefficient is its inverse hyperbolic tangent (artanh). When the sample correlation coefficient r is near 1 or -1, its distribution is highly skewed, which makes it difficult to estimate confidence intervals and apply tests of significance for the population correlation coefficient ρ. The Fisher transformation solves this problem by yielding a variable whose distribution is approximately normally distributed, with a variance that is stable over different values of r. Given a set of N bivariate sample pairs (Xi, Yi), i = 1, ..., N, the sample correlation coefficient r is given by Here cov ⁡ ( X , Y ) {\displaystyle \operatorname {cov} (X,Y)} stands for the covariance between the variables X {\displaystyle X} and Y {\displaystyle Y} and σ {\displaystyle \sigma } stands for the standard deviation of the respective variable. Fisher's z-transformation of r is defined as where "ln" is the natural logarithm function and "artanh" is the inverse hyperbolic tangent function. If (X, Y) has a bivariate normal distribution with correlation ρ and the pairs (Xi, Yi) are independent and identically distributed, then z is approximately normally distributed with mean and a standard deviation which does not depend on the value of the correlation ρ (i.e., this is a Variance-stabilizing transformation) and which is given by where N is the sample size, and ρ is the true correlation coefficient. This transformation, and its inverse can be used to construct a large-sample confidence interval for r using standard normal theory and derivations. See also application to partial correlation. Hotelling gives a concise derivation of the Fisher transformation. To derive the Fisher transformation, one starts by considering an arbitrary increasing, twice-differentiable function of r {\displaystyle r} , say G ( r ) {\displaystyle G(r)} .