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Lois and Rainer,
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You both appear to anticipate that corresponding =
regressions (CR) will be a type of path analysis. =
This is not strictly so. True, we will use path
diagrams to visually portray the sequences =
of causes, but CR is not analytically a type of =
path or structural equation modeling. CR is an =
inductive procedure. Path analysis follows the =
hypothetico-deductive strategy. This will be
more evident later, after CR is explained and =
can be contrasted with path analysis. In this =
posting I would like to propose the core of the =
method of corresponding regressions. =
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The Core of the Method
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First, lets return to points made in an early =
posting that reveal the core of CR. Let's =
create a causal model. Let Y =3D X1 + X2, =
where X1 and X2 are two columns containing =
48 uniformly distributed random numbers, and =
Y is their sums across each row. Thus Y is =
logically dependent on the both X1 and X2
while X1 and X2 are logically independent of =
one another and Y- since they were generated =
independently of one another and of Y. =
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In terms of logical implication, the value of Y =
carries implications concerning the probable =
values of X1 and X2. A high value of Y implies =
the random pairing of high values of X1 and =
X2. A high value of an X variable, however, =
does not imply a high value of Y. A high X1 =
value could have been randomly paired with =
a low X2 value, creating a mid-range value =
of Y. Another way to think of this is that a =
child implies sexually mature adults =
(parents), while sexually mature adults do =
not necessarily imply a child. They may be =
childless.
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In other, words, X1 and X2 are independent =
variables (formal causes) and Y is
their dependent variable (formal effect).
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X1 and X2 are uncorrelated with one another. =
Their conjugation (pairing) is random.
Both X1 and X2 correlate with Y at about .7, =
since they each explain about half of Y. =
Remember that it is the coefficient of determination =
(square of the correlation) that reveals the percent =
of overlap in variables. The square of .7 is about .5., =
i.e 50%.
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Numerical Example
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The following variables were generated by the =
above strategy. I list them so that they are sorted =
by Y,in order to conserve space. When they came =
out of the computer originally, they were in random =
order.
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Lowest Y's The midranges of Y
Part A Part B Part C
X1 X2 Y X1 X2 Y X1 X2 Y
0 1 1 3 5 8 4 6 10 =
0 1 1 2 6 8 8 2 10
0 2 2 0 8 8 5 5 10
2 1 3 4 4 8 8 2 10
0 3 3 4 5 9 5 6 11
1 2 3 3 6 9 9 2 11
3 3 6 7 2 9 9 2 11
2 4 6 6 3 9 2 9 11
0 6 6 3 6 9 9 2 11
1 6 7 1 8 9 9 3 12
5 2 7 4 6 10 4 8 12
6 1 7 5 5 10 7 5 12
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Highest Y's
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Part D
X1 X2 Y
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3 9 12
9 3 12
7 6 13 =
7 6 13
6 7 13
5 9 14
8 6 14
8 7 15
7 8 15
8 7 15
6 9 15 =
9 7 16
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Partitioning by Ranges
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As above, first sort all three columns by Y, so that =
the rows of Y go from least to most in value. Now, =
take the rows of Y, X1 and X2 that contain the =
lowest 12 values of Y (Part A) and the rows of Y, =
X1, and X2 containing the highest 12 values of Y =
(PartD). Concatenate these two matrices and call =
them the extremes by Y (EOY). Partition the
remaining matrix of Y, X1, and X2 values- those =
containing the mid-range of Y values (Parts B & C) =
- and call them the midrange by Y (MOY).
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Our data have been partitioned into two sets, the =
data corresponding to the extreme values of Y =
(EOY) and those corresponding to the midrange of =
Y (MOY).
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Correlate Y, X1, and X2 in EOY:
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X1 X2 Y
X1 1.00 .49 .88
X2 .49 1.00 .84
Y .88 .84 1.00
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Correlate Y, X1, and X2 in MOY:
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X1 X2 Y
X1 1.00 -.88 .59
X2 -.88 1.00 -.13
Y .59 -.13 1.00
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You get very different correlations across the =
two matrices. At the extremes of Y (EOY), =
variables X1 and X2 tend to be positively =
correlated. In the midrange by Y matrix, X1 and =
X2 will tend to be negatively correlated. This is
because at the extremes of Y, the X variables =
are similar to one another. At the mid-range of Y
(MOY), the X values tend to be different and they =
cancel one another out to produce mid-range =
values of Y.
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Thus the correlations between the X (independent) =
variables polarize across the extreme versus =
midrange of Y (the dependent variable), i.e =
=2E49 versus -.88. This is the core of the method of =
corresponding regressions. The polarization only =
occurs when the data are sorted by the dependent =
variable, not by the independent variable(s).
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To show that the polarization does not occur when =
the data are sorted by independent variables, put the =
two matrices back together, to return to our
original data. In this matrix, X1 and X2 will tend to be =
correlated zero. In our example the r was -.05, which =
is close enough. This is not surprising in that both =
X1 and X2 are just random numbers. Now sort
Y, X1, and X2 by X1. Partition Y, X1 and X2 into =
two matrices that correspond to the extreme of X1 =
(EOX1) and the midranges of X1 (MOX1). Find the =
correlations between the Y, X1, and X2 for each =
matrix.
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EOX1 correlations
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X1 X2 Y
X1 1.00 -.08 .82
X2 -.08 1.00 .50
Y .82 .50 1.00
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MOX1 correlations
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X1 X2 Y
X1 1.00 - .01 .58
X2 -.01 1.00 .81
Y .58 .81 1.00
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No polarization of X1 to X2 correlations occurs, =
i.e. -.08 versus -.01. The independent variables =
X1 and X2 will still be correlated approximately =
zero. The same happens when you partition by X2.
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The polarization of the correlation between =
independent variables across the ranges of the =
dependent variable, but not across the ranges =
of the independent variables, is the core of the =
method of corresponding regressions.
The rest is just expressing this fact in the =
algebra of regression analysis. If we can not =
agree on this polarization business, the rest =
will just be only fancy math and will only serve =
to confuse things. It is the CORE of the =
method of corresponding regressions.
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What do ya'll think of this core?
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Bill =
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References
=
Chambers, W. V. (1991). Inferring formal causation =
from corresponding regressions. Jounral of Mind =
and Behavior, 12,1, 49-70.
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Lamiell, James T. (1991). Beware the illusion of =
technique. JMB, 12, 1, 71-7
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Williams, R. N. (1991) Untangling cause, necessity, =
temporality and method: response to Chambers' method =
of correspondinge regressions. JMB, 12,1,77-83.
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Chambers, W. V. (1991). Corresponding regressions,
procedural evidence, and the dialectics of substantive =
theory, metaphysics and methodology. JMB, 12,1, 83-92.
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=
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