Psychology 282
Lecture #8 Outline
Semipartial and Partial Correlations
Overall strength of association in MLR with 2 IVs is
measured by squared multiple correlation:
2
Y .12
R
rY21 + rY22 − 2rY 1rY 2 r12
=
1 − r122
This is the proportion of variance in Y accounted for
by its linear relationship with X1 and X2.
Consider the problem of determining the contribution
of each of the two IVs in accounting for variance in Y.
This information would help us understand the
relative importance of each IV.
As we shall see, there are different ways to measure
these contributions statistically, and these measures
have different meanings.
2
Representation of relationship among variables in
form of Venn diagram:
Y
e
b
a
c
d
X2
X1
Squared correlations as represented in diagram:
r122 = c + d
rY21 = a + c
rY22 = b + c
RY2.12 = a + b + c
Now consider how to define the contribution of one
IV; start with X1.
What is the unique contribution of X1?
Could say that unique contribution of X1 is
represented by area a in diagram.
3
This area is called the squared semipartial
2
correlation of X1 with Y: sr1
2
In terms of the diagram: sr1 = a
Computing this value:
2
R
Squared multiple correlation: Y .12 = a + b + c
2
Variance accounted for by X2: rY 2 = b + c
Variance accounted for uniquely by X1 would then be
found by:
sr12 = RY2.12 − rY22
That is, variance accounted for by X1 above and
beyond that accounted for by X2 is obtained by taking
total variance accounted for by X1 and X2 and
subtracting that portion accounted for by X2.
Important concept: This value measures the increase,
or increment, in R2 when we add X1 as an IV to a
regression model that already contains X2 as an IV.
Note the conceptual and statistical difference
between the simple contribution of X1 to explaining
2
r
the variance of Y (defined by Y 1 ) and the unique
contribution of X1 to explaining the variance of Y
2
(defined by sr1 ).
4
Now consider applying same rationale to determine
the unique contribution of X2:
Diagram: Area b
Defines squared semipartial correlation of X2 with Y:
sr22 = b
Computing this value:
2
Squared multiple correlation: RY .12 = a + b + c
2
Variance accounted for by X1: rY 1 = a + c
Variance accounted for uniquely by X2 would then be
found by:
sr22 = RY2.12 − rY21
That is, variance accounted for by X2 above and
beyond that accounted for by X1 is obtained by taking
total variance accounted for by X1 and X2 and
subtracting that portion accounted for by X1.
Important concept: This value measures the increase,
or increment, in R2 when we add X2 as an IV to a
regression model that already contains X1 as an IV.
In applications of MLR it is useful to obtain these sr2
values for each IV. They are interpreted as
representing the unique contribution of each IV in
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accounting for the variance in Y. Equivalently, they
represent the increment in variance accounted for by
that IV when it is added to the model.
2
2
2
Important: Note that sr1 + sr2 ≠ RY .12
We cannot add the unique contributions of the
separate IVs and obtain the total variance accounted
for. The total variance accounted includes a portion
that is accounted for “redundantly” by the two IVs
due to their correlation with each other.
That redundant portion is represented by area c in the
diagram. It would be tempting to try to define area c
2
2
2
formally as RY .12 − sr1 − sr2 . However, this is not
correct because this computation can produce a
negative value in some situations. Thus, the
redundant contribution of the IVs cannot be
calculated.
Note also that if r12 is high, the unique contribution of
each IV will be small. That is, the squared
semipartial correlations will be small.
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There is another way to define the contribution of
each separate IV. Look at the diagram again:
Y
e
b
a
c
d
X1
X2
The first approach focused on defining a proportion
of the total variance in Y that is accounted for by each
IV; this led us to focus on areas a and b as a
proportion of the full Y circle.
Considering first X1, note that we could define its
contribution in terms of a proportion of that part of
the Y variance that is not accounted for by X2.
The Y variance not accounted for by X2 is represented
by area a+e.
The portion of that area that is explained by X1 is
represented by area a.
Taking a proportion, we can define the contribution
of X1 as a/(a+e).
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This quantity is called a squared partial correlation:
pr12 =
a
a+e
Consider how this value could be computed.
As noted earlier, area a is equivalent to the squared
semipartial correlation for X1 with Y:
sr12 = RY2.12 − rY22
Area (a+e) would be the variance in Y not explained
2
(
1
−
r
by X2, which could be calculated as:
Y2)
Combining these two values yields:
RY2.12 − rY22
pr =
1 − rY22
2
1
This value tells us the following: Of the variance in
Y that is not explained by X2, what proportion is
explained by X1.
Interpretation in terms of increments in R2 is as
follows: This value tells us the ratio of the increment
in R2 associated with X1 divided by the maximum
increment in R2 that could have been achieved.
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Note that pr1 ≥ sr1 . These values will be equal only
2
when rY 2 = 0 .
2
2
Now we can define the same squared partial
correlation to represent the contribution of X2: the
proportion of that part of the Y variance that is not
accounted for by X1.
The Y variance not accounted for by X1 is represented
by area b+e.
The portion of that area that is explained by X2 is
represented by area b.
Taking a proportion, we can define the contribution
of X2 as b/(b+e).
This quantity is called a squared partial correlation:
pr22 =
b
b+e
Consider how this value could be computed.
As noted earlier, area b is equivalent to the squared
semipartial correlation for X2 with Y:
sr22 = RY2.12 − rY21
9
Area (b+e) would be the variance in Y not explained
2
by X1, which could be calculated as: (1 − rY 1 )
Combining these two values yields:
RY2.12 − rY21
pr =
1 − rY21
2
2
This value tells us the following: Of the variance in
Y that is not explained by X1, what proportion is
explained by X2.
These pr2 values can be obtained for each IV in the
regression model.
Interpretation in terms of increments in R2 would be
as follows: The pr2 value for a particular IV would
indicate the ratio of the increment in R2 that is
associated with that variable divided by the
maximum possible increment that could have been
achieved.
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Defining semipartial and partial correlations by
“residualizing”
In the developments presented above we defined
squared semipartial and squared partial correlations
in terms of portions of variance in Y accounted for, or
in terms of increments in R2.
Another way to define these values is based on the
concept of residualizing.
Based on residualizing, here is how we would define
the semipartial correlation of X1 with Y:
First, suppose we were to conduct a SLR analysis
treating X1 as a dependent variable and X2 as an
independent variable, ignoring Y.
Xˆ 1 = B0 + B X1 X 2 X 2
Residuals:
e X1 = X 1 − Xˆ 1
These residuals represent that part of X1 that is not
accounted for by X2. (X1 has been residualized by
removing the effect of X2 from it. Another way to
say this is that the effect of X2 on X1 has been
partialed out.)
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Now suppose we compute the correlation between
these residuals and variable Y: reX 1Y
This correlation is the semipartial correlation of X1
with Y, removing the effect of X2 from X1.
That is, sr1 = re X 1Y
By writing out the full expression for reX 1Y , then
simplifying, one can show that
sr1 = re X 1Y =
rY 1 − rY 2 r12
1 − r122
Squaring this value then would produce the squared
semipartial correlation defined earlier.
To define sr2 by residualizing, one would follow
exactly the same procedure just shown, but reverse
the roles of X1 and X2:
SLR predicting X2 from X1:
Residuals:
Xˆ 2 = B0 + BX 2 X 1 X 1
eX 2 = X 2 − Xˆ 2
These residuals represent that part of X2 that is not
accounted for by X1.
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Obtain correlation between these residuals and Y:
sr2 = re X 2Y =
rY 2 − rY 1r12
1 − r122
This is the semipartial correlation of X2 with Y,
partialing X1 from X2. Squaring this value produces
the squared semipartial as presented earlier.
We can also use the concept and method of
residualizing to define a partial correlation. This is
an extension of the procedure just described in that
for a partial correlation we partial out the effect of
one predictor from the other predictor and also from
the dependent variable.
To define the partial correlation of X1 with Y,
partialing out X2:
First partial X2 from X1:
SLR:
Residuals:
Xˆ 1 = B0 + B X1 X 2 X 2
e = X − Xˆ
X1
1
1
These residuals represent that part of X1 that is not
accounted for by X2.
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Then partial X2 from Y:
Yˆ = B0 + BYX X 2
SLR:
2
Residuals:
eY = Y − Yˆ
These residuals represent that part of Y that is not
accounted for by X2.
The partial correlation is then obtained by correlating
the two sets of residuals:
pr1 = re X 1eY
By writing out this correlation algebraically and then
simplifying, it can be shown that
pr1 = re X 1eY =
rY 1 − rY 2 r12
1 − rY22 1 − r122
This is the partial correlation of X1 with Y, partialing
out the effect of X2.
Squaring this value produces the squared partial
correlation covered earlier, and the corresponding
interpretation.
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To define the partial correlation for X2 with Y,
partialing out X1, we follow exactly the same
approach but reverse the roles of X1 and X2.
First partial X1 from X2:
SLR:
Residuals:
Xˆ 2 = B0 + BX 1 X 2 X 1
e = X − Xˆ
X2
2
2
These residuals represent that part of X2 that is not
accounted for by X1.
Then partial X1 from Y:
Yˆ = B0 + BYX X 1
SLR:
1
Residuals:
eY = Y − Yˆ
These residuals represent that part of Y that is not
accounted for by X1.
The partial correlation is then obtained by correlating
the two sets of residuals:
pr2 = re X 2eY
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By writing out this correlation algebraically and the
simplifying, it can be shown that
pr2 = re X 2eY =
rY 2 − rY 1r12
1 − rY21 1 − r122
This is the partial correlation of X2 with Y, partialing
out the effect of X1.
The concept and methods of residualizing are
important in correlational methods. They can be used
in many situations when one wishes to statistically
control for effects of some variables on other
variables.
A relationship among the various partial coefficients
We use the term “partial coefficients” to refer to any
of the various partial regression coefficients and
correlations we have defined (including raw and
standardized regression coefficients, and semipartial
and partial correlations).
Consider the partial coefficients associated with X1.
If we look back at our definitional formulas for the
standardized regression weight (β1), the semipartial
correlation (sr1), and the partial correlation (pr1), we
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can see that all of these formulas have the same
numerator: (rY1 – rY2r12).
This observation has an important implication: If any
one of these values is zero, then they all must be zero.
This relationship extends to the raw score regression
weight (B1), since there is a simple multiplicative
relationship between B1 and β1.
Thus, in considering any single IV, if any of the
partial coefficients representing the effect of that IV
are zero, then all are zero. This fact has useful
consequences in the context of testing hypotheses
about partial coefficients, which will be covered
when we study inferences in multiple regression.