PEARSON,KARL AND THE CORRELATION CURVE

We present an extension of the Galton-Pearson correlation coefficient to cases of nonlinear heteroscedastic regression. This extension, the correlation curve, addresses what Karl Pearson called in 1905 'the question of generalising correlation'. The correlation curve measures the local variance explained by regression and thus provides a local measure of association in examples where the strength of relationship between a response variable Y and a covariate X differs for different values of the covariate. As an illustration of the use of the correlation curve, we explore a data set first analyzed by Pearson in 1905 as part of his attempts to generalise correlation beyond the linear model. We refine Pearson's analysis to obtain estimates of the correlation curve, and show that Pearson came close to defining the correlation curve in 1905.