Interpretation problems of the partial correlation with nonnormally distributed variables

The partial correlation is a commonly used measure for assessing the bivariate correlation of two quantitative variables after eliminating the influence of one or more other variables. The partial correlation is generally interpreted as the correlation that would result if the variables to be eliminated were fixed (not allowed to vary and influence the other variables), which is referred to in the statistical literature as conditional correlation. The present paper demonstrates, by means of theoretical derivations and practical examples, that when the assumption of multivariate normality is violated (e.g., as a result of nonlinear relationships among the variables investigated) the usual interpretation of the partial correlation coefficient will be basically incorrect. In extreme cases the value of the partial correlation coefficient may be strongly positive, close to 1, whereas the conditional correlation may have a large negative value. To solve this problem the paper suggests to partial out a certain function (in most cases the square) of the variables whose effects are to be eliminated if nonlinear relationships are likely to occur.