Encyclopedia of Sociology

CAUSAL INFERENCE MODELS

μ 1

ρ 21

ρ 43

ρ 31

ρ 52

ρ 54

X 1 X 2 μ 2

μ 5

μ 3 X 3 X 4 μ 4

X 5

Figure 2. Model of Figure 1, with Path Coefficients and Standardized Variables

to control for all three simultaneously. More gen-
erally, a number of simplifications become possi-
ble, depending on the patterning of omitted ar-
rows, and these simplifications can be used to
justify the omission of certain variables if these
cannot be measured. If, for example, one could
not measure X 3 , one could draw in a direct arrow
from X 1 to X 4 without altering the remainder of
the model. Without such an explicit causal model
in front of us, however, the omission of variables
must be justified on completely ad hoc grounds.
The important point is that pragmatic reasons for
such omissions should not be accepted without
theoretical justifications.

PATH ANALYSIS AND AN EXAMPLE

Sewall Wright (1934, 1960) introduced a form of
causal modeling long before it became fashion-
able among sociologists. Wright, a population ge-
neticist, worked in terms of standardized variables
with unit variances and zero means. Expressing
any given equation in terms of what he referred to
as path coefficients, which in recursive modeling
are equivalent to beta weights, Wright was able to
derive a simple formula for decomposing the cor-
relation between any pair of variables xi and xj. The
equation for any given variable can be written as xi
= pi1x 1 +pi2x 2 +...+pikxk+ui, where the pij represent
standardized regression coefficients and where

the lower-case x’s refer to the standardized vari-

ables. One may then multiply both sides of the

equation by xj, the variable that is to be correlated

with xi. Therefore, xixj = pi1x 1 xj + pi2x 2 xj +...+ pikxkxj +

uixj. Summing over all cases and dividing by the

number of cases N, one has the results in equation

system 3.

rij = –––– = pil ––––+pi2 ––––+...+pik––––+––––

NNN NN

∑xixj ∑xlxj ∑x 2 xj ∑xkxj ∑uixj

= pilrlj + pi2r2j +...+ pikrkj + 0 = pk∑ ikrkj (^3 )

The expression in equation system 3 enables

one to decompose or partition any total correla-

tion into a sum of terms, each of which consists of a

path coefficient multiplied by a correlation coeffi-

cient, which itself may be decomposed in a similar

way. In Wright’s notation the path coefficients are

written without any dots that indicate control

variables but are indeed merely the (partial) re-

gression coefficients for the standardized vari-

ables. Any given path coefficient, say p 54 , can be

interpreted as the change that would be imparted

in the dependent variable x 5 , in its standard devia-

tion units, if the other variable x 4 were to change

by one of its standard deviation units, with the

remaining explicitly included independent vari-

ables (here x 1 , x 2 , and x 3 ) all held constant. In

working with standardized variables one is able to

simplify these expressions owing to the fact that rij