Download - 19 Partial and Semi
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Partial and Semipartial
Correlation
Working With Residuals
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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?
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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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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?
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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.
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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.
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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.
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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
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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?
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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
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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.
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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.
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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
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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.
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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.
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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.
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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
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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?
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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.)
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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.
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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.
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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?
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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
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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.
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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.