Mathematical coupling of data: correction of a common error for linear calculations.

The need frequently arises in the scientific environment to investigate the relationship between quantities that are calculated from a common set of directly measured variables. However, the presence of error in the common set of measured variables distorts the relationship among the calculated quantities and can lead to incorrect conclusions. This article presents a method of correcting for such distortions in the Pearson correlation coefficient and in the linear regression coefficient for linear calculations involving two measured variables. The errors considered may be either independent of, or proportional to, the value of the variable being measured. Tests to determine whether these popular coefficients have values significantly different from zero are presented. An example from the physiology literature is presented to illustrate these techniques.