Statistical Methods for Psychology

some would not reach it, but the points could be fit reasonably well by this plane. The ver-

tical distances of the points from the plane, the distances , would be the residuals.

Just as in the one-predictor case, the residuals represent the vertical distance of the points

from the best-fitting line (or, in this three-dimensional case, the best-fitting plane).

We can derive one additional insight from this three-dimensional model. The plane we

have been discussing forms some angle (in this case, positive) with the axis (Expend). In

other words, the plane rises from right to left. The slope of that plane relative to is.

Similarly, the slope of the plane with respect to (LogPctSAT) is. The height of the

plane at the point ( , ) 5 (0,0) would be.

15.8 Partial and Semipartial Correlation

Two closely related correlation coefficients involve partialling out, or controlling for, the

effects of one or more other variables. These correlations are the partial and semipartial

correlation coefficients.

Partial Correlation

We have seen that a partial correlation r01.2is the correlation between two variables with

one or more variables partialled out of both Xand Y. More specifically, it is the correlation

between the two sets of residuals formed from the prediction of the original variables by

one or more other variables.

Consider an experimenter who wanted to investigate the relationship between earned

income and success in college. He obtained measures for each variable and ran his correla-

tion, which turned out to be significant. Elated with the results, he harangued his students

with the admonition that if they did not do well in college they were not likely to earn large

salaries. In the back of the class, however, was a bright student who realized that both vari-

ables were (presumably) related to IQ. She argued that people with high IQs tend to do

well in college and also earn good salaries, and that the correlation between income and

college success is an artifact of this relationship.

The simplest way to settle this argument is to calculate the partial correlation between

Income and college Success with IQ partialled out of both variables. Thus, we regress In-

come on IQ and obtain the residuals. These residuals represent the variation in Income that

cannot be attributed to IQ. You might think of this as a “purified” income measure—purified

of the influence of IQ. We next regress Success on IQ and again obtain the residuals, which

here represent the portion of Success that is not attributable to IQ. We can now answer the

important question: Can the variation in Income not explained by (independent of) IQ be

predicted by the variation in Success that is also independent of IQ? The correlation between

these two variables is the partial correlation of Income and Success, partialling out IQ.

The partial correlation coefficient is represented by. The two subscripts to

the left of the dot represent the variables being correlated, and the subscripts to the right of

the dot represent those variables being partialled out of both.

Semipartial Correlation

A type of correlation that will prove exceedingly useful both here and in Chapter 16 is the

semipartial correlation r0(1.2)sometimes called the part^5 correlation. As the name

suggests, a semipartial correlation is the correlation between the criterion and a partialled

r01.23... p

X 1 X 2 b 0

X 2 b 2

X 1 b 1

X 1

(Y 2 YN)

15.8 Partial and Semipartial Correlation 535

(^5) The only text that I have seen using “part correlation” was McNemar (1969) when I was just out of graduate
school. But the name seems to have stuck, and you will find SPSS employing that term.
partial
correlation r01.2
semipartial
correlation r0(1.2)