19 partial and semi

8/2/2019 19 Partial and Semi

1/27

Partial and Semipartial

Correlation

Working With Residuals


2/27

Questions

Give a concrete

example (names of

vbls, context) where it

makes sense to

compute a partialcorrelation. Why a

partial rather than

semipartial?

Why is the squaredsemipartial always less

than or equal to the

squared partial?

Give a concreteexample where itmakes sense tocompute a semipartial

correlation. Why semirather than partial?

Why is regression moreclosely related tosemipartials than

partials? How could you use

ordinary regression tocompute 3rd order

partials?


3/27

Partial Correlation

People differ in many ways. When onedifference 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 notbe explained by X1.


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6/27

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 forSES, prior experience, what else?


7/27

Computing Partials from

CorrelationsAlthough you compute partials via residuals, sometimes itis 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 isbeing partialed from the correlation between 1 and 2. In our

example, 74.)2)(1().)((3.12 !!! EESATVFGPAHSGPA rrr

2

23

2

13

2313123.12

11 rr

rrrr

!

74.81.187.1

)81)(.87(.92.

223.12 !

!r

The partial correlation can bea little or a lot bigger or

smaller than the original.


8/27

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 asecond orderpartial correlation.

Can have 3rd, 4th, etc., order partials.

Unpartialed (raw) correlations are calledzero order

correlationsbecause 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.


9/27

Partials from Multiple

CorrelationWe can compute squared partial correlations from various R2values.

rR R

R12 3

2 1 23

2

1 3

2

1 3

21.

. .

.

!

2

23.1R is the R2 from the regression in

which 1 is the DV and 2 and 3 are

the Ivs.

22.

2

2.

2

12.2

2.1 1 Y

YY

Y R

RRr

!

Alternative (possibly friendlier) notation.


10/27

Squared Partials from R2 -

Venn Diagrams2

2.

2

2.

2

12.2

2.1 1 Y

YY

Y 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 correlation

Between Y and X1 holding X2

constant.

1.

2.

3.4. Y

X1

X2


11/27

Exercise Find a Partial

1 2 3

1 ANX 1

2 FamHistory

.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?


12/27

Find a partial

1 2 3

1 ANX 1

2 Fam

History

.20 1

3 DOC

Visit

.35 .15 1

2

32

2

12

3212132.13

11 rr

rrrr

!

33.15.12.1

)15)(.2(.35.

222.13 !

!r


13/27

Semipartial Correlation

With 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 its a

semipartialcorrelation instead of a partial. With asemipartial, 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.


14/27

Semipartials from

Correlationsr

r r r

r r12 3

12 13 23

13

2

23

21 1

. !

Partial:

Semipartial: rr r r

rand r

r r r

r1 2 312 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 32

92 87 81

1 8137( . )

. (. )(. )

..!

! r2 1 3 2

92 87 81

1 8744( . )

. (. )(. )

..!

!

Agrees with earlier results within rounding error.


15/27

Squared Semipartials from

Multiple Correlations

Partial:

Semipartial:

Squared semipartial is an increment in R2.

Y

X1 X2

UY:X1UY:X2

Shared YShared X

2

2.

2

2.

2

12.2

2.11 Y

YYY

R

RRr

!

2

2.

2

12.

2

)2.1( YYY RRr !

1:

1

1:22.

2

12.

2

)2.1( XUYXUY

RRr YYY !!!

Y

X1

X2


16/27

Partial vs. Semipartial

Partial Semipartial

Y

X1

X2

Y

X1X2

R y.12 R y.2-2

R y.21 -2 2

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


17/27

Regression and Semipartial

Correlation Regression is essentially about semipartials

Each X is residualized on the otherXvariables.

For each X we add to the equation, we ask,What is the unique contribution of this Xabove and beyond the others? Increment inR2 when added last.

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

b is the slope of Y on X, holding the other X

variables constant.


18/27

Semipartial and Regression 2

2

12

12212.1

1 r

rrr YYY

!F

2

12

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.


19/27

Uses of Partial and

Semipartial The partial correlation is most often used

when some third variable z is a plausibleexplanation of the correlation between X andY.

Job characteristics and job sat by NA

Cog ability and grades by SES

The semipartial is most often used when wewant to show that some variable adds

incremental variance in Y above and beyondother X variable

Pilot performance and Cog ability, motor skills

Patient well being and surgery, social support


20/27

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 thanpartial?


21/27

SuppressorEffects

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.)


22/27

SuppressorEffects (2)

Y X1 X2Y 1

X1 .50 1

X2 .00 .50 1

Note that X2 is correlated withX1 but NOT with Y. Will X2

be useful in a regression

equation?

If we solve for beta weights, we find, beta1=.667 andbeta2 = -.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 ris (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.


23/27

SuppressorEffects (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 suppressesthe correlation of Y and X1.

Inspection of correlation matrix not sufficient

to see value of variables.

Looking dumb.

Venn diagram.


24/27

Review

Why is the squared semipartial always

less than or equal to the squared partial?

Why is regression more closelyrelated to semipartials than partials?

How could you use ordinaryregression to compute 3rd orderpartials?


25/27

Exercise Find a Semipartial

Y X1 X2Y 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


26/27

Find a Semipartial

Y X1 X2Y 1

X1 .20 1

X2 .30 .40 1

2

12

1221

)2.1(

1 r

rrrr

yy

y

!

087.40.1

)40)(.30(.20.2

)2.1( !

!

yr

The correlation of X1 with Yafter controlling for X2 (from

X1 only) is rather small.


27/27

ComputerExercise

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 equationafter gender and hassles.

19 partial and semi

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