Partial and Semipartial Correlation

Working With Residuals

Questions

• Give a concrete example (names of vbls, context) where it makes sense to compute a partial correlation. Why a partial rather than semipartial?

• Why is the squared semipartial always less than or equal to the squared partial?

• Give a concrete example where it makes sense to compute a semipartial correlation. Why semi rather than partial?

• Why is regression more closely related to semipartials than partials?

• How could you use ordinary regression to compute 3rd order partials?

Partial Correlation

• People differ in many ways. When one difference is correlated with an outcome, cannot be sure the correlation is not spurious.

• Would like to hold third variables constant, but cannot manipulate.

• Can use statistical control.• Statistical control is based on residuals. If we

regress X2 on X1 and take residuals of X2, this part of X2 will be uncorrelated with X1, so anything X2 resids correlate with will not be explained by X1.

Example of Partials

Person SAT-V HSGPA FGPA PFGPA E1 E21 500 3.0 2.8 2.86 -0.01 -0.062 550 3.2 3.0 3.05 -0.02 -0.053 450 2.8 2.8 2.67 0.01 0.134 400 2.5 2.2 2.48 -0.08 -0.285 600 3.2 3.3 3.24 -0.24 0.066 650 3.8 3.3 3.43 0.15 -0.137 700 3.9 3.5 3.61 0.03 -0.128 550 3.8 3.7 3.05 0.58 0.659 650 3.5 3.4 3.43 -0.15 -0.0310 550 3.1 2.9 3.05 -0.12 -0.15

Use SAT to predict grades (HS & College Fresh)

(HS) (F)HS=.8557+.0043*SAT; F=.9563+.0038*SAT.

R2 for HS = .76; R2 for F = .62 (fictional data).

Example Partials (2)

  SAT-V

HSGPA

FGPA

P E1(HS)

E2 (F)

SAT-V 1          

HSGPA .87 1        

FGPA .81 .92 1      

P 1.00 .87 .81 1    

E1 .00 .50 .45 .00 1  

E2 .00 .37 .58 .00 .74 1

There are 2 sets of predicted values; one for each GPA, however, they correlate 1.0 with each other, so only 1 is presented.

A partial correlation; the correlation between the residuals of the two GPAs. The correlation between HS GPA and FGPA holding SAT constant.

Note that P and SAT are perfectly correlated. P & SAT do not correlate with E1 or E2 (residuals).

High correlations

The Meaning of Partials

• The partial is the result of holding constant a third variable via residuals.

• It estimates what we would get if everyone had same value of 3rd variable, e.g., corr b/t 2 GPAs if all in sample have SAT of 500.

• Some examples of partials? Control for SES, prior experience, what else?

Computing Partials from CorrelationsAlthough you compute partials via residuals, sometimes it is handy to compute them with correlations. Also looking at the formulas is (could be?) informative.

Notation. The partial correlation is r12.3 where variable 3 is being partialed from the correlation between 1 and 2. In our example, 74.)2)(1().)((3.12 EESATVFGPAHSGPA rrr

223

213

2313123.12

11 rr

rrrr

74.81.187.1

)81)(.87(.92.223.12

r

The partial correlation can be a little or a lot bigger or smaller than the original.

The Order of a Partial

• If you partial 1 vbl out of a correlation, the resulting partial is called a first order partial correlation.

• If you partial 2 vbls out of a correlation, the resulting partial is called a second order partial correlation. Can have 3rd, 4th, etc., order partials.

• Unpartialed (raw) correlations are called zero order correlations because nothing is partialed out.

• Can use regression to find residuals and compute partial correlations from the residuals, e.g. for r12.34, regress 1 and 2 on both 3 and 4, then compute correlation between 2 sets of residuals.

Partials from Multiple Correlation

We can compute squared partial correlations from various R2 values.

rR R

R12 32 1 23

21 32

1 321.

. .

.

223.1R is the R2 from the regression in which 1 is the DV and 2 and 3 are the Ivs.

22.

22.

212.2

2.1 1 Y

YYY R

RRr

Alternative (possibly friendlier) notation.

Squared Partials from R2 - Venn Diagrams 2

2.

22.

212.2

2.1 1 Y

YYY R

RRr

Y

X1 X2

UY:X1 UY:X2

Shared Y

Shared X

Y

X1X2

R y.122

Y

X1X2

R y.12 R y.2-2

R y.21 -2 2

Here we want the partial correlationBetween Y and X1 holding X2 constant.

1.

2.

3.4. Y

X1X2

Exercise – Find a Partial

1 2 3

1 ANX 1

2 Fam History

.20 1

3 DOC Visit

.35 .15 1

What is the correlation between trait anxiety and the number of doctor visits controlling for family medical history?

Find a partial1 2 3

1 ANX 1

2 Fam History

.20 1

3 DOC Visit

.35 .15 1

232

212

3212132.13

11 rr

rrrr

33.15.12.1

)15)(.2(.35.222.13

r

Semipartial CorrelationWith partial correlation, we find the correlation between X and Y holding Z constant for both X and Y. Sometimes, we want to hold Z constant for just X or just Y. Instead of holding constant for both, hold for only one, therefore it’s a semipartial correlation instead of a partial. With a semipartial, we find the residuals of X on Z or Y on Z but the other is the original, raw variable. Correlate one raw with one residual.

In our example, we found the correlation between E1 (HSGPA) and FGPA to be .45. This is the semipartial correlation between HSGPA and FGPA holding SAT constant for HSGPA only.

Semipartials from Correlations

rr r r

r r12 3

12 13 23

132

2321 1

.

Partial:

Semipartial: rr r r

rand r

r r r

r1 2 3

12 13 23

232 2 1 3

12 13 23

1321 1

( . ) ( . )

Note that r1(2.3) means the semipartial correlation between variables 1 and 2 where 3 is partialled only from 2. In our example:

r1 2 3 2

92 87 81

1 8137( . )

. (. )(. )

..

r2 1 3 2

92 87 81

1 8744( . )

. (. )(. )

..

Agrees with earlier results within rounding error.

Squared Semipartials from Multiple Correlations

Partial:

Semipartial:

Squared semipartial is an increment in R2.

Y

X1 X2

UY:X1 UY:X2

Shared Y

Shared X

22.

22.

212.2

2.1 1 Y

YYY R

RRr

22.

212.

2)2.1( YYY RRr

1:1

1:22.

212.

2)2.1( XUY

XUYRRr YYY

Y

X1X2

Partial vs. Semipartial

Partial Semipartial

Y

X1X2

Y

X1X2

R y.12 R y.2-2

R y.21 -2 2

Why is the squared partial larger than the squared semipartial? Look at the respective areas for Y.

Regression and Semipartial Correlation• Regression is essentially about semipartials• Each X is residualized on the other X variables.• For each X we add to the equation, we ask,

“What is the unique contribution of this X above and beyond the others?” Increment in R2 when added last.

• We do NOT residualize Y, just X.• Semipartial because X is residualized but Y is

not.• b is the slope of Y on X, holding the other X

variables constant.

Semipartial and Regression 2

212

12212.1 1 r

rrr YYY

212

1221)2.1(

1 r

rrrr YYY

Standardized regression coefficient

Semipartial correlation

The difference is the square root in the denominator. The regression coefficient can exceed 1.0 in absolute value; the correlation cannot.

Uses of Partial and Semipartial• The partial correlation is most often used when

some third variable z is a plausible explanation of the correlation between X and Y.– Job characteristics and job sat by NA– Cog ability and grades by SES

• The semipartial is most often used when we want to show that some variable adds incremental variance in Y above and beyond other X variable– Pilot performance and Cog ability, motor skills– Patient well being and surgery, social support

Review

• Give a concrete example (names of vbls, context) where it makes sense to compute a partial correlation. Why a partial rather than semipartial?

• Give a concrete example where it makes sense to compute a semipartial correlation. Why semi rather than partial?

Suppressor Effects

• Hard to understand, but– Inspection of r not enough to tell value– Need to know to avoid looking dumb– Show problems with Venn diagrams

• Think of observed variable as composite of different stuff, e.g., satisfaction with car (price, prestige, etc.)

Suppressor Effects (2)  Y X1 X2

Y 1    

X1 .50 1  

X2 .00 .50 1

Note that X2 is correlated with X1 but NOT with Y. Will X2 be useful in a regression equation?

If we solve for beta weights, we find, beta1=.667 and beta2 = -.333. Notice that the beta weight for the first is actually larger than r (.50), and the second has become negative. Can also happen that r is (usually slightly) positive and beta is negative. This is a suppressor effect. Always inspect your correlations along with your regression weights to see if this is happening.

What does it mean that beta2 is negative? Sometimes people forget that there are other X variables in the equation. “The results mean that we should feed people more to get them to lose weight.”

Suppressor Effects (3)

• Can also happen in path analysis, CSM.• Explanation – X2 is a measure of prediction

error in X1. If we subtract X2, will have a more useful measure of X1. X2 ‘suppresses’ the correlation of Y and X1.

• Inspection of correlation matrix not sufficient to see value of variables.

• Looking dumb.• Venn diagram.

Review

• Why is the squared semipartial always less than or equal to the squared partial?

•Why is regression more closely related to semipartials than partials?

•How could you use ordinary regression to compute 3rd order partials?

Exercise – Find a Semipartial  Y X1 X2

Y 1    

X1 .20 1  

X2 .30 .40 1

What is the correlation between Y and X1 holding X2 constant only for X1?

?)2.1( yr

Find a Semipartial  Y X1 X2

Y 1    

X1 .20 1  

X2 .30 .40 1

212

1221)2.1(

1 r

rrrr yy

y

087.40.1

)40)(.30(.20.2)2.1(

yr

The correlation of X1 with Y after controlling for X2 (from X1 only) is rather small.

Computer Exercise

• Go to labs and download 2IV Example.• Find the partial correlation between hassles

and well being holding gender and anger constant (2nd order partial).

• Find the squared semipartial for anger when well being is the DV and gender and hassles are the other IVs, that is, find the increment in R-square when anger is added to the equation after gender and hassles.