were to measure each of N

8/7/2019 were to measure each of N

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were to measure each of N subjects on each of three variables, X, Y,and Z, and find the following correlations:

X versus Y: rXY = +.50 r2XY = .25

X versus Z: rXZ = +.50 r2XZ = .25

Y versus Z:rYZ = +.50

r2

YZ = .25

For the moment, focus on the value of r2, which in each case (for thisstreamlined hypothetical example) is equal to .25. What this means isthat for each pair of variables²XY, XZ, and YZ²the covariance, orvariance overlap, is 25%. As illustrated in the following diagram, 25% of the variability of X overlaps with variability in Y; 25% of the variabilityof X overlaps with variability in Z; and 25% of the variability of Y alsooverlaps with variability in Z.

Note as well that there is one region where all three of thevariability circles overlap. The meaning of this three-wayoverlap is that a certain amount of the correlation foundbetween any two of the variables is tied in with the

correlation that each of those two has with the third. Thus, of the 25%variance overlap found between X and Y, approximately half (judging bythe naked eye) is tied in with the overlaps that exist between XZ and YZ.Similarly for the 25% overlap between X and Z, where about half is boundup with the overlaps for XY and YZ. And similarly as well for the 25%overlap of YZ, where about half is tied up with the overlaps for XY and XZ.

Partial correlation is a procedure that allows us to measure the region of three-way overlap precisely, and then to remove it from the picture inorder to determine what the correlation between any two of the variableswould be (hypothetically) if they were not each correlated with the thirdvariable. Alternatively, you can say that partial correlation allows us todetermine what the correlation between any two of the variables would be(hypothetically) if the third variable were held constant. The partialcorrelation of X and Y, with the effects of Z removed (or held constant),would be given by the formula

rXY·Z = rXY²(rXZ)(rYZ) sqrt[1²r

2XZ] x sqrt[1²r

2YZ]

which for the present example would work out as

rXY·Z = .50²(.50)(.50) sqrt[1².25] x sqrt[1².25]



rXY·Z = +.33 Hence r

2XY·Z = .11

The same general structure would apply for calculating the partial

correlation of X and Z, with the effects of Y removed:

rXZ·Y = rXZ²(rXY)(rYZ) sqrt[1²r

2XY] x sqrt[1²r

2YZ]

and for for calculating the partial correlation of Y and Z, with the effects

of X removed:

rYZ·X = rYZ²(rXY)(rXZ)

sqrt[1²r2XY] x sqrt[1²r

2XZ]

Here is the apparatus of partial correlation applied to a real-life example.The Wechsler Adult Intelligence Scale (WAIS) is a device often used tomeasure "intelligence" beyond the years of childhood. Among its severalsub-scales are three labeled as C, A, and V. The "C" stands for"comprehension," which chiefly reflects the test-taker's ability tocomprehend the meanings and implications of written passages. The "A"

refers to the test-taker's ability to perform tasks that require arithméticability. And the "V" stands for "vocabulary," which as you might imagineis a measure that increases or decreases in accordance with the breadthof the test-taker's vocabulary within the domain of the language in whichthe test is constructed. The following table shows the correlationstypically found among these three sub-scales.

C versus A: rCA = +.49 r2CA = .24

C versus V: rCV = +.73 r2CV = .53

A versus V: rAV = +.59 r2AV = .35

Here the overlaps are less evenly proportioned, although thelogic is quite the same. Of the 24% variance overlap thatoccurs in the relationship between comprehension andarithmétic ability, a substantial portion reflects the fact that

both of these variables are correlated with vocabulary. If we were to



remove the effects of vocabulary from the relationship between C and A,the resulting partial correlation would be

rCA·V =

rCA²(rCV)(rAV) sqrt[1²r

2CV] x sqrt[1²r

2AV]

rCA·V = .49²(.73)(.59) sqrt[1².53] x sqrt[1².35]

rCA·V = +.11 Hence r

2CA·V = .01

In brief: with the effects of vocabulary removed, the correlation betweencomprehension and arithmétic ability collapses down to hardly anything atall. The practical inference is that if we were to administer the WAIS to asample of subjects who were homogeneous with respect to breadth of vocabulary, the correlation between their scores on the comprehensionand arithmétic sub-scales would prove fairly scant, on the order of r=+.11 and r

2=.01.

In most cases a partial correlation of the general form rXY·Z will turn outsmaller than the original correlation rXY. In those cases where it turns outlarger, the third variable, Z, is typically spoken of as a supressor

variable on the assumption that it is suppressing the larger correlationthat would appear between X and Y if Z were held constant. Suppose, for example, that a rather cranky professor has justadministered an exam in his statistics course, and that for each student inthe course we have measures on each of the following three variables:

X = the amount of effort spent on studying for the exam beforehand Y = the student's score on the exam Z = a measure of the degree to which the professor inspires fear and

trembling in the student

And here are the correlations among the three variables:

X versus Y: rXY = +.20 r2XY = .04

X versus Z: rXZ = +.80 r2XZ = .64

Y versus Z: rYZ = ².40 r2YZ = .16



Now isn't it odd that the correlation between X and Y should end up as ascant rXY=+.20 and r

2XY=.04, indicating a mere 4% covariance between

the degrees of effort that students put into to the exam and the scoresthat they receive on it? Examine the other two correlations, however, and

you will see that it is not so odd after all. The greater the fear andtrembling, the greater the effort that students tend to put into preparingfor the exam; hence rXZ=+.80 and r

2XZ=.64. On the other hand, the

greater the fear and trembling, the less well students tend to do on theexam, as witness rYZ=².40 and r

2YZ=.16. Remove the supressing effects

of fear and trembling from the equation,

rXY·Z = .20²(.80)(².40) sqrt[1².64] x sqrt[1².16] rXY=.20 r

2XY=.04

rXY·Z=.95 r2XY·Z=.90

rXY·Z = +.95

and the correlation between effort and exam score goes from a scantrXY=+.20 to an impressive rXY·Z=+.95. Or alternatively: remove the fearand trembling, and the covariance between effort and exam score goes

from a mere 4% to a very substantial 90% (r2XY·Z=.90). The VassarStats computational site includes a page that will calculate thepartial correlation coefficients for any particular set of threeintercorrelated variables.

End of Subchapter 3a.

Return to Top of Subchapter 3a

Go to Subchapter 3b [Rank-Order Correlation]

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were to measure each of N