CAUSAL INFERENCE MODELSX (^2) X
4
X 5
X 1 X 3
Father's
occupation
First
job
Occupation
in 1962
Respondent's
education
Father's
education
0.818
0.440
0.859
0.394
0.310
0.516 0.279
0.115
0.224
0.753
0.281
Figure 3. Path Diagram for Blau-Duncan Model
= ∑xixj/N, but one must pay the price of then
having to work with standard deviation units that
may vary across samples or populations. This, in
turn, means that two sets of path coefficients for
different samples, say men and women, cannot
easily be compared since the standard deviations
(say, in income earned) may be different.
In the case of the model of Figure 2, which is
the same causal diagram as Figure 1, but with the
relevant pij inserted, one may write out expres-
sions for each of the rij as shown in equation
system 4.
r 12 = p 21 r 11 = p 21 (1) = p 21
r 13 = p 31 r 11 = p 31
r 23 = p 31 r 12 = p 31 p 21 = r 13 r 12 (or r 23. 1 = 0)
r 14 = p 43 r 13 = p 43 p 31
r 24 = p 43 r 23 = p 43 p 31 p 21
r 34 = p 43 r 33 = p 43
r 15 = p 52 r 21 +p 54 r 41 = p 52 p 21 +p 54 p 43 p 31
r 25 = p 52 r 22 +p 54 r 42 = p 52 p 54 p 43 p 31 p 21
r 35 = p 52 r 23 +p 54 r 43 = p 52 p 31 p 21 + p 54 p 43
r 45 = p 52 r 24 +p 54 r 44 = p 52 p 43 p 31 p 21 + p 54
( 4 )
In decomposing each of the total correlations,
one takes the path coefficients for each of the
arrows coming into the appropriate dependent
variable and multiplies each of these by the total
correlation between the variable at the source of
the arrow and the ‘‘independent’’ variable in which
one is interested. In the case of r 12 , this involves
multiplying p 21 by the correlation of x 1 with itself,
namely r 11 = 1.0. Therefore one obtains the simple
result that r 12 = p 21. Similar results obtain for r 13
and r 34. The decomposition of r 23 , however, re-
sults in the expression r 23 = p 31 r12 = p 31 p 21 = r 12 r 13 ,
which also of course implies that r23.1 = 0.
When one comes to the decomposition of
correlations with x 5 , which has two direct paths
into it, the expressions become more complex but
also demonstrate the heuristic value of path analy-
sis. For example, in the case of r 35 , this total
correlation can be decomposed into two terms,
one representing the indirect effects of x 3 via the
intervening variable x 4 , namely the product p 54 p 43 ,
and the other the spurious association produced
by the common cause x 1 , namely the more com-
plex product p 52 p 31 p 21. In the case of the correla-
tion between x 4 and x 5 one obtains a similar result
except that there is a direct effect term represent-
ed by the single coefficient p 54.
As a numerical substantive example consider
the path model of Figure 3, which represents the
basic model in Blau and Duncan’s classic study,
The American Occupational Structure (1967, p. 17).
Two additional features of the Blau-Duncan mod-
el may be noted. A curved, double-headed arrow
has been drawn between father’s education and
father’s occupation, indicating that the causal paths
between these two exogenous or independent
variables have not been specified. This means that