Part Correlation

The part correlation coefficient ry(1·2), which is also called the semipartial correlation coefficient by Hays (1972), differs from the partial correlation coefficient ry1·2 in the following respect: The influence of the control variable X2 is removed from X1 but not from Y.

The three steps in partial correlation analysis are now reduced to two. In the first step, the residual scores X1 − X^1 from a bivariate regression analysis of X1 on X2 are calculated. In a second and last step, we calculate the zero-order correlation coefficient between these residual scores X1 − X^1 and the original Y scores. This, then, is the part correlation coefficient ry(1·2). For higher-order coefficients, such as ry(1·23), it holds in the same way that the influence of the control variables X2 and X3 is removed from X1 but not from Y. An example of ry(1·2) can be the correlation between the extracurricular activity of students (variable X1) and their popularity (variable Y), controlling for their scholastic achievement (X2). To clarify the mechanism of part correlation, wegive a simple (fictitious) sample of five students with the following scores on the three variables: 1, 3, 2, 6, 4 for Y ; 0, 1, 3, 6, 8 for X1; and 4, 4, 1, 2, 0 for X2. The part correlation coefficient ry(1·2) is now calculated as the zero order correlation between the X1 − X^1 scores −0.70, 0.30, −2.53, 2.08, 0.86 and the original Y scores 1, 3, 2, 6, 4, which yields ry(1·2) = 0.83. This is then the correlation between popularity and extracurricular activity, in which the latter is cleared of the influence of scholastic achievement.

When dealing with considerations of content, part correlation finds little application. One reason why part correlation could be preferred to partial correlation in concrete empirical research is that the variance of Ywould almost disappear when removing the influence of the control variable X2 from it, so that there would be little left to explain. In such a case, the control variable would, of course, be a very special one because it [Page 789]would share its variance almost entirely with one of the considered variables.

An example of a more appropriate use of the part correlation coefficient can be found in Colson's (1977, p. 216) research of the influence of the mean age (A) on the crude birth rate (B) in a sample of countries. But as the latter is not independent of the age structure and vice versa, because the mean age A contains a fertility component, a problem of contamination arose. This problem can be solved in two ways if we include the total (period) fertility rate as a variable, which is measured as the total of the age-specific fertility rates (F) and is independent of the age structure. A first procedure is the calculation of the correlation coefficient between A and F (i.e., the association between the mean age and age-free fertility). A second procedure, which was chosen by the author, is the calculation of the part correlation coefficient rB(A·F) between crude birth rates (measured as B) and the fertility-free age structure (residual scores of A after removing the influence of F).

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