Statistical Control Techniques: Partial and Semipartial Correlation

Statistical Control Techniques: Partial and Semipartial Correlation

®
Statistical Control Techniques:
Partial and Semipartial Correlation
Chapter 7
Chapter 7
Slide 1 of 37
Today’s Lecture
®
Overview
● Today’s Lecture
Statistical Control
●
Statistical control techniques.
●
Partial correlation.
●
Semipartial correlation.
Partial Correlation
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 2 of 37
Statistical Control
®
Overview
Statistical Control
● Visualizing
Control
● Example Data Set
●
Statistical control can be equated to controlling for variability.
●
In experimental research (with human subjects), this is often
accomplished by random assignment to a set of
experimenter controlled conditions.
●
In quasi-experimental research random assignment cannot
be accomplished, although experimental conditions may
exist.
●
In nonexperimental or observational research, the
“experimenter” does not control anything...but can observe
many things.
●
To obtain control when none is present, statistics can be
useful.
Partial Correlation
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 3 of 37
Venn Diagrams
®
Overview
One way to visualize the concepts described in this lecture is
to imagine the variability of a variable as being represented by
a circle.
Statistical Control
● Visualizing
Control
● Example Data Set
Partial Correlation
Measurement
Errors
Semipartial
Correlation
Y
X1
X2
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 4 of 37
Venn Diagrams
®
Using X1 to predict Y :
Overview
Statistical Control
● Visualizing
Control
● Example Data Set
Partial Correlation
Measurement
Errors
Y
X1
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 5 of 37
Venn Diagrams
®
Using X1 and X2 to predict Y :
Overview
Statistical Control
● Visualizing
Control
● Example Data Set
Partial Correlation
Measurement
Errors
Semipartial
Correlation
X1
Y
X2
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 6 of 37
Venn Diagrams
®
Using X1 to predict Y , while controlling for the effects of X2 :
Overview
Statistical Control
● Visualizing
Control
● Example Data Set
Partial Correlation
Measurement
Errors
Y
X1
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 7 of 37
Today’s Example Data Set
®
Overview
Statistical Control
● Visualizing
Control
● Example Data Set
From Weisberg (1985, p. 240).
“Property taxes on a house are supposedly dependent on the
current market value of the house. Since houses actually sell
only rarely, the sale price of each house must be estimated
every year when property taxes are set. Regression methods
are sometimes used to make up a prediction function.”
Partial Correlation
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 8 of 37
Erie, Pennsylvania
®
We have data for 27 houses sold in the mid 1970’s in Erie,
Pennsylvania:
Overview
●
Statistical Control
● Visualizing
Control
● Example Data Set
Partial Correlation
Measurement
Errors
Semipartial
Correlation
●
●
●
●
●
●
●
Multiple Partial
Correlation
●
Wrapping Up
●
Chapter 7
X1 : Current taxes (local, school, and county) ÷ 100 (dollars).
X2 : Number of bathrooms.
X3 : Lot size ÷ 1000 (square feet).
X4 : Living space ÷ 1000 (square feet).
X5 : Number of garage spaces.
X6 : Number of rooms.
X7 : Number of bedrooms.
X8 : Age of house (years).
X9 : Number of fireplaces.
Y : Actual sale price ÷ 1000 (dollars).
Slide 9 of 37
Erie, Pennsylvania
Lake Erie
Chapter 7
Slide 10 of 37
Erie, Pennsylvania
Jack
Chapter 7
Slide 11 of 37
Erie, Pennsylvania
Paula
Chapter 7
Slide 12 of 37
Erie, Pennsylvania
®
For our example today, we will limit the size of the data to a
smaller set of variables:
Overview
●
Statistical Control
● Visualizing
Control
● Example Data Set
Partial Correlation
Measurement
Errors
●
●
●
●
X1 : Current taxes (local, school, and county) ÷ 100 (dollars).
X2 : Number of bathrooms.
X3 : Living space ÷ 1000 (square feet).
X4 : Age of house (years).
Y : Actual sale price ÷ 1000 (dollars).
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 13 of 37
Correlation Matrix
Variable
X1
X2
X3
X4
Y
®
Overview
Statistical Control
● Visualizing
Control
● Example Data Set
Partial Correlation
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
●
●
●
●
●
Wrapping Up
Chapter 7
X1
1.000
0.876
0.832
-0.371
0.915
X2
X3
X4
Y
1.000
0.901
-0.211
0.924
1.000
-0.178
0.929
1.000
-0.310
1.000
X1 : Current taxes (local, school, and county) ÷ 100 (dollars).
X2 : Number of bathrooms.
X3 : Living space ÷ 1000 (square feet).
X4 : Age of house (years).
Y : Actual sale price ÷ 1000 (dollars).
Slide 14 of 37
Partial Correlation
®
Overview
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
A partial correlation is a correlation between two variables from
which the linear relations, or effects, of another variable(s)
have been removed.
r12 − r13 r23
p
r12.3 = p
2
2
1 − r13 1 − r23
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
1
2
3
Wrapping Up
Chapter 7
Slide 15 of 37
Partial Correlation in SPSS
®
●
To perform a partial correlation in SPSS go to
Analyze...Correlation...Partial.
●
In the Variables box, put the variables you would like to get
partial correlations for.
●
In the Controlling For box, put the variables you would like to
control for.
Overview
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Partial correlation of Y (1) (actual sale price ÷ $1000) with
X3 (2) (living space), controlling for X1 (3) (current taxes):
ryx3 − ryx1 rx3 x1
ryx3 .x1 = q
=
p
2
1 − ryx
1 − rx23 x1
1
0.929 − (0.915 × 0.832)
√
√
= 0.748
2
2
1 − 0.915 1 − 0.832
Slide 16 of 37
Higher-Order Partial Correlation
®
Overview
The second-order partial correlation is the correlation between
two variables with the effects of two other variables being
removed:
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
r12.34
r12.3 − r14.3 r24.3
p
= p
2
2
1 − r14.3 1 − r24.3
4
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
1
2
3
Wrapping Up
Chapter 7
Slide 17 of 37
Second-order Partial Correlation
®
●
To perform a partial correlation in SPSS go to
Analyze...Correlation...Partial.
●
In the Variables box, put the variables you would like to get
partial correlations for.
●
In the Controlling For box, put the variables you would like to
control for - if more than one, a higher-order partial is given.
Overview
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 18 of 37
Second-order Partial Correlation
®
Overview
Partial correlation of Y (1) (actual sale price ÷ $1000) with
X3 (2) (living space), controlling for X1 (3) (current taxes) and
X2 (4) (number of bathrooms):
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
r12.3 = ryx3 .x1 = 0.748
r14.3 = ryx2 .x1 = 0.628
r24.3 = rx3 x2 .x1 = 0.641
r12.34
r12.3 − r14.3 r24.3
p
= p
=
2
2
1 − r14.3 1 − r24.3
0.748 − (0.628 × 0.641)
√
√
= 0.578
2
2
1 − 0.628 1 − 0.641
Wrapping Up
Chapter 7
Slide 19 of 37
Partial Correlations from Multiple Correlations
®
As you could probably guess, partial correlations can be found
from regression analyses...
Overview
Squared partial correlation:
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
Measurement
Errors
2
r12.3
2
2
− R1.3
R1.23
=
2
1 − R1.3
●
2
R1.23
is the R2 from a multiple regression with 1 being Y and
2 and 3 being the predictor variables.
●
2
is the R2 from a simple regression with 1 being Y and 2
R1.3
being X - the single predictor variable.
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 20 of 37
Partial Correlations from Multiple Correlations
®
Partial correlation of Y (1) (actual sale price ÷ $1000) with
X3 (2) (living space), controlling for X1 (3) (current taxes):
Overview
2
R1.23
= 0.928
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
Measurement
Errors
2
R1.3
= 0.838
2
r12.3
2
2
R1.23
− R1.3
0.928 − 0.838
=
=
= 0.865
2
1 − R1.3
1 − 0.838
q
√
2
r12.3 = r12.3 = 0.865 = 0.748
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 21 of 37
Partial Correlations from Multiple Correlations
®
Alternatively:
Overview
2
r12.3
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
Partial correlation of Y (1) (actual sale price ÷ $1000) with
X3 (2) (living space), controlling for X1 (3) (current taxes):
2
R2.13
= 0.865
2
R2.3
= 0.693
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
2
2
R2.13
− R2.3
=
2
1 − R2.3
2
r12.3
2
2
R2.13
− R2.3
0.865 − 0.693
=
= 0.865
=
2
1 − R2.3
1 − 0.693
q
√
2
r12.3 = r12.3 = 0.865 = 0.748
Wrapping Up
Chapter 7
Slide 22 of 37
Higher-Order Partial Correlations
®
Overview
Second-order squared partial correlation:
2
r12.34
2
2
R1.234
− R1.34
=
2
1 − R1.34
2
r12.34
2
2
R2.134
− R2.34
=
2
1 − R2.34
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
Measurement
Errors
Semipartial
Correlation
Third-order squared partial correlation:
2
r12.345
2
2
R1.2345
− R1.345
=
2
1 − R1.345
2
r12.345
2
2
R2.1345
− R2.345
=
2
1 − R2.345
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 23 of 37
Higher-Order Partial Correlations
®
Overview
Partial correlation of Y (1) (actual sale price ÷ $1000) with
X3 (2) (living space), controlling for X1 (3) (current taxes) and
X2 (4) (number of bathrooms):
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
Measurement
Errors
2
R1.234
= 0.935
2
R1.34
= 0.902
2
r12.34
2
2
R1.234
− R1.34
0.935 − 0.901
=
=
= 0.337
2
1 − R1.34
1 − 0.902
q
√
2
r12.34 = r12.34 = 0.337 = 0.578
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 24 of 37
Partial Correlations in Regression
®
Overview
Statistical Control
Partial Correlation
● Partial Correlation
● Higher-Order
Partial Correlation
● Partial Correlation
From Regression
● Higher Orders
from Regression
● More SPSS
You can obtain partial correlations (and semipartial correlations
- called part correlations by SPSS) from the regression
analysis subroutine:
●
Go to Analyze...Regression...Linear.
●
Click on Statistics box.
●
Check Part and Partial Correlations.
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 25 of 37
Correction For Attenuation
®
●
Often in behavioral research, variables are not measured
without error (non-perfect reliability).
●
For these times, the correlation between two variables is
dampened by extra noise associated with measurement.
●
To correct for this noise, a so-called correction for attenuation
accounts for the imperfections in measurement.
●
Using these corrected values, one can then build corrected
partial correlations.
Overview
Statistical Control
Partial Correlation
Measurement
Errors
● Correction For
Attenuation
● Corrected Partial
Correlation
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Correction for attenuation:
∗
r12
r12
=√ √
r11 r22
Slide 26 of 37
Corrected Partial Correlation
®
Overview
Statistical Control
Partial Correlation
Measurement
Errors
● Correction For
Attenuation
● Corrected Partial
Correlation
Recall the formula for the first-order partial correlation:
r12.3
r12 − r13 r23
p
= p
2
2
1 − r13 1 − r23
First-order partial correlation correction for attenuation:
r33 r12 − r13 r23
∗
p
r12.3
= p
2
2
r11 r33 − r13 r22 r33 − r23
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 27 of 37
Semipartial Correlation
®
●
Semipartial correlation removes the effects of additional
variables from one of the variables under study (typically X).
Overview
First-order semipartial correlation:
Statistical Control
r1(2.3)
Partial Correlation
Measurement
Errors
Semipartial
Correlation
● Semipartial
Correlation from
Regression
● Tests of
Significance
● Multiple
Regression
Or...
r12 − r13 r23
= p
2
1 − r23
r12 − r13 r23
r2(1.3) = p
2
1 − r13
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 28 of 37
Semipartial Correlation from Regression
®
Overview
Statistical Control
First-order semipartial correlation:
2
2
2
= R1.23
r1(2.3)
− R1.3
Second-order semipartial correlation:
Partial Correlation
Measurement
Errors
Semipartial
Correlation
● Semipartial
Correlation from
Regression
● Tests of
Significance
● Multiple
Regression
2
2
2
r1(2.34)
= R1.234
− R1.34
Third-order semipartial:
2
2
2
r1(2.345)
= R1.2345
− R1.345
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 29 of 37
Semipartial Correlation from Regression
®
Overview
Semipartial correlation of Y (1) (actual sale price ÷ $1000) with
X3 (2) (living space), controlling X3 (2) (living space) for X1 (3)
(current taxes):
Statistical Control
2
R1.23
= 0.928
Partial Correlation
Measurement
Errors
Semipartial
Correlation
● Semipartial
Correlation from
Regression
● Tests of
Significance
● Multiple
Regression
2
R1.3
= 0.838
2
2
2
r1(2.3)
= R1.23
− R1.3
= 0.928 − 0.838 = 0.090
r1(2.3)
q
√
2
= r1(2.3) = 0.090 = 0.300
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 30 of 37
Semipartial Correlation from Regression
®
Overview
Semipartial correlation of Y (1) (actual sale price ÷ $1000) with
X3 (2) (living space), controlling X3 (2) (living space) for X1 (3)
(current taxes) and X2 (4) (number of bathrooms):
Statistical Control
2
R1.234
= 0.935
Partial Correlation
Measurement
Errors
Semipartial
Correlation
● Semipartial
Correlation from
Regression
● Tests of
Significance
● Multiple
Regression
2
R1.34
= 0.902
2
2
2
r1(2.34)
= R1.234
− R1.34
= 0.935 − 0.902 = 0.033
r1(2.3)
q
√
2
= r1(2.3) = 0.090 = 0.182
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 31 of 37
Tests of Significance
®
Overview
Semipartial correlations are found using the formulas
developed for the test of the increment in the proportion of
variance accounted for in regression:
Statistical Control
2
2
Ry.12...k
−
R
/(k1 − k2 )
y.12...k
1
2
F =
,
2
1 − Ry.12...k
/(N
−
k
−
1)
1
1
Partial Correlation
Measurement
Errors
Semipartial
Correlation
● Semipartial
Correlation from
Regression
● Tests of
Significance
● Multiple
Regression
where k1 > k2 , with:
●
●
dfnum = k1 − k2
dfdenom = N − k1 − 1
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 32 of 37
Multiple Regression
®
Overview
Statistical Control
Partial Correlation
You can begin to see how multiple regression uses the statical
concepts from today’s to control the prediction of the Y
variable:
2
2
2
2
2
Ry.1234
= ry1
+ ry(2.1)
+ ry(3.12)
+ ry(3.123)
Measurement
Errors
Semipartial
Correlation
● Semipartial
Correlation from
Regression
● Tests of
Significance
● Multiple
Regression
Multiple Partial
Correlation
Wrapping Up
Chapter 7
Slide 33 of 37
Multiple Partial Correlation
®
Overview
Multiple partial correlation:
2
R1.23(4)
Statistical Control
Partial Correlation
Measurement
Errors
Semipartial
Correlation
Two control variables:
2
R1.23(45)
2
2
R1.234
− R1.4
= p
2
1 − R1.4
2
2
R1.2345
− R1.45
= p
2
1 − R1.45
Multiple Partial
Correlation
● Multiple Partial
Correlation
● Multiple
Semipartial
Correlation
Wrapping Up
Chapter 7
Slide 34 of 37
Multiple Semipartial Correlation
®
Overview
Statistical Control
Multiple semipartial correlation:
2
2
2
= R1.234
R1(23.4)
− R1.4
Two control variables:
Partial Correlation
Measurement
Errors
2
2
2
R1(23.45)
= R1.2345
− R1.45
Semipartial
Correlation
Multiple Partial
Correlation
● Multiple Partial
Correlation
● Multiple
Semipartial
Correlation
Wrapping Up
Chapter 7
Slide 35 of 37
Final Thought
®
●
Often times, experimental
control is not possible.
●
Using statistical
techniques, one can get an
idea of how variables are
related to one another by
removing the effects of
spurious variables
●
Partial and semipartial are statistical techniques to provide
post-hoc control.
Overview
Statistical Control
Partial Correlation
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
● Final Thought
● Next Class
Chapter 7
Slide 36 of 37
Next Time
®
Overview
●
Midterm handed out.
●
Midterm discussion.
Statistical Control
Partial Correlation
Measurement
Errors
Semipartial
Correlation
Multiple Partial
Correlation
Wrapping Up
● Final Thought
● Next Class
Chapter 7
Slide 37 of 37