Speaking Stata: Correlation with confidence, or Fisher's z revisited

Ronald Aylmer Fisher suggested transforming correlations by using the inverse hyperbolic tangent, or atanh function, a device often called Fisher's z transformation. This article reviews that function and its inverse, the hyperbolic tangent, or tanh function, with discussions of their definitions and behavior, their use in statistical inference with correlations, and how to apply them in Stata. Examples show the use of Stata and Mata in calculator style. New commands corrci and corrcii are also presented for correlation confidence intervals. The results of using bootstrapping to produce confidence intervals for correlations are also compared. Various historical comments are sprinkled throughout.