Cause (Part II) - Causal Systems

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Cause (Part II) - Causal Systems
Topics
I. The Logic of Multiple Relationships
II. Multiple Correlation
III. Multiple Regression
IV. Path Analysis
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Cause (Part II) - Causal Systems
I. The Logic of Multiple Relationships
One Dependent Variable, Multiple Independent
Variables
NR
Y
X1
R
NR
X2
In this diagram the overlap of any two circles
can be thought of as the r2 between the two
variables. When we add a third variable,
however, we must partial out the redundant
overlap of the additional independent variables.
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Cause (Part II) - Causal Systems
II. Multiple Correlation
NR
Y
X1
R
Y
X2
X1
NR
NR
NR
X2
R2y.x1x2 r2yx1 r2yx2
R2y.x1x2 r2yx1 r2yx2.x1
Notice that when the Independent Variables are
independent of each other, the multiple
correlation coefficient (R2) is simply the sum of
the individual r2, but if the independent
variables are related, R2 is the sum of one zero
order r2 of one plus the partial r2 of the
other(s). This is required to compensate for the
fact that multiple independent variables being
related to each other would be otherwise double
counted in explaining the same portion of the
dependent variable. Partially out this
redundancy solves this problem.
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Cause (Part II) - Causal Systems
II. Multiple Regression
Y a byx1X1 byx2X2
Y
X2
X1
or Standardized
X1
Y Byx1X1 Byx2X2
Y
X2
If we were to translate this into the language of
regression, multiple independent variables, that
are themselves independent of each other would
have their own regression slopes and would simply
appear as an another term added in the regression
equation.
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Multiple Regression
Cause (Part II) - Causal Systems
Y
X1
Y a byx1X1 byx2.x1X2
or Standardized
X2
Y Byx1X1 Byx2.x1X2
X1
Y
X2
Once we assume the Independent Variables are
themselves related with respect to the variance
explained in the Dependent Variable, then we must
distinguish between direct and indirect
predictive effects. We do this using partial
regression coefficients to find these direct
effects. When standardized these B-values are
called Path coefficients or Beta Weights
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Cause (Part II) - Causal Systems
III. Path Analysis The Steps and an Example
1. Input the data
2. Calculate the Correlation Matrix
3. Specify the Path Diagram
4. Enumerate the Equations
5. Solve for the Path Coefficients (Betas)
6. Interpret the Findings
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Path Analysis Steps and Example
Step1 Input the data
Assume you have information from ten respondents
as to their income, education, parents education
and parents income. We would input these ten
cases and four variables into SPSS in the usual
way, as here on the right. In this analysis we
will be trying to explain respondents income
(Y), using the three other independent variables
(X1, X2, X3)
Y DV - income
X3 IV - educ
X2 IV - pedu
X1 IV - pinc
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Path Analysis Steps and Example
Step 2 Calculate the Correlation Matrix
These correlations are calculated in the usual
manner through the analyze, correlate,
bivariate menu clicks.
X1
X2
X3
Y
Notice the zero order correlations of each IV
with the DV. Clearly these IVs must interrelate
as the values of the r2 would sum to an R2
indicating more than 100 of the variance in the
DV which, of course, is impossible.
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Path Analysis Steps and Example
Step 3 Specify the Path Diagram
Therefore, we must specify a model that explains
the relationship among the variables across time
We start with the dependent variable on the right
most side of the diagram and form the independent
variable relationship to the left, indicating
their effect on subsequent variables.
X1
a
e
Y
X3
f
b
d
Y Offsprings income
c
X1 Parents income
X2
X2 Parents education
X3 Offsprings education
Time
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Path Analysis Steps and Example
Step 4 Enumerate the Path Equations
With the diagram specified, we need to articulate
the formulae necessary to find the path
coefficients (arbitrarily indicated here by
letters on each path). Overall correlations
between an independent and the dependent variable
can be separated into its direct effect plus the
sum of its indirect effects.
X1
a
e
X3
Y
b
f
d
1. ryx1 a brx3x1 crx2x1
c
2. ryx2 c brx3x2 arx1x2
X2
3. ryx3 b arx1x3 crx2x3
4. rx3x2 d erx1x2
Click here for solution to two equations in two
unknowns
5. rx3x1 e drx1x2
6. rx1x2 f
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Path Analysis Steps and Example
Step 5 Solve for the Path Coefficients a, b
and c
The easiest way to calculate B is to use the
Regression module in SPSS. By indicating income
as the dependent variable and pinc, pedu and educ
as the independent variables, we can solve for
the Beta Weights or Path Coefficients for each of
the Independent Variables.
These circled numbers correspond to Beta for
paths a, c and b, respectively, in the previous
path diagram.
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Path Analysis Steps and Example
Step 5 Solve for the Path Coefficients d and e
The easiest way to calculate B is to use the
Regression module in SPSS. By indicating
offspring education as the dependent variable and
Parents Inc and Parents Edu as the independent
variables, we can solve for the Beta Weights or
Path Coefficients for each of these Independent
Variables on the DV Offspring Edu.
These circled numbers correspond to Beta for
paths d and e, respectively, in the previous path
diagram.
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Path Analysis Steps and Example
Step 5a Solving for R2
The SPSS Regression module also calculate R2.
According to this statistic, for our data, 50 of
the variation in the respondents income (Y) is
accounted for by the respondents education (X3),
parents education (X2) and parents income (X1)
R2 is calculated by multiplying the Path
Coefficient (Beta) by its respective zero order
correlation and summed across all of the
independent variables (see spreadsheet at right).
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Path Analysis Steps and Example
Checking the Findings
ryx1 a brx3x1 crx2x1
.69 .63 -.21(.75) .31(.68)
e .50
X1
ryx2 c brx3x2 arx1x2
r .69 B .63
.57 .31 -.21(.82) .63(.68)
r .75 B .35
ryx3 b arx1x3 crx2x3
.52 -.21 .63(.75) .31(.82)
r .52 B -.21
Y
X3
r B .68
The values of r and B tells us three things 1)
the value of Beta is the direct effect 2)
dividing Beta by r gives the proportion of direct
effect and 3) the product of Beta and r summed
across each of the variables with direct arrows
into the dependent variable is R2 . The value of
1-R2 is e.
r .82 B .58
r .57 B .31
X2
Time
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Path Analysis Steps and Example
Step 6 Interpret the Findings
Y Offsprings income
X3 Offsprings education
X2 Parents education
X1
e .50
X1 Parents income
.63
Specifying the Path Coefficients (Betas), several
facts are apparent, among which are that Parents
income has the highest percentage of direct
effect (i.e., .63/.69 92 of its correlation is
a direct effect, 8 is an indirect effect).
Moreover, although the overall correlation of
educ with income is positive, the direct effect
of offsprings education, in these data, is
actually negative!
.35
Y
X3
.68
-.21
.58
.31
X2
Time
End
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Exercise - Solving Two Equations in Two Unknowns
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