Partial and Semipartial Correlation

1
Partial and Semipartial Correlation

• Working With Residuals

2
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?

3
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.

4
Example of Partials
Use SAT to predict grades (HS College Fresh)
(HS) (F)
HS.8557.0043SAT F.9563.0038SAT.
R2 for HS .76 R2 for F .62 (fictional data).
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Example Partials (2)
There are 2 sets of predicted values one for
each GPA, however, they correlate 1.0 with each
other, so only 1 is presented.
High correlations
Note that P and SAT are perfectly correlated. P
SAT do not correlate with E1 or E2 (residuals).
A partial correlation the correlation between
the residuals of the two GPAs. The correlation
between HS GPA and FGPA holding SAT constant.
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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 for SES,
  prior experience, what else?

7
Computing Partials from Correlations
Although 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,
The partial correlation can be a 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 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.

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Partials from Multiple Correlation
We can compute squared partial correlations from
various R2 values.
is the R2 from the regression in which 1 is
the DV and 2 and 3 are the Ivs.
Alternative (possibly friendlier) notation.
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Squared Partials from R2 - Venn Diagrams
Here we want the partial correlation Between Y
and X1 holding X2 constant.
2.
1.
3.
4.
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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?
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Find a partial
1 2 3
1 ANX 1
2 Fam History .20 1
3 DOC Visit .35 .15 1
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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
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.
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Semipartials from Correlations
Partial
Semipartial
Note that r1(2.3) means the semipartial
correlation between variables 1 and 2 where 3 is
partialled only from 2. In our example
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.
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Partial vs. Semipartial
Partial
Semipartial
Why is the squared partial larger than the
squared semipartial? 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 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.

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Semipartial and Regression 2
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 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

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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 than partial?

21
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.)

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Suppressor Effects (2)
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.
23
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.

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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?

25
Exercise Find a Semipartial
What is the correlation between Y and X1 holding
X2 constant only for X1?
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Find a Semipartial
The correlation of X1 with Y after controlling
for X2 (from X1 only) is rather small.
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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.