Encyclopedia of Sociology

CAUSAL INFERENCE MODELS

ε 1 X 1 X 2 ε 2

ε 4

ε 5

ε 3 X 3 X 4

X 5

Parental

education

Delinquent

acts

Guilt

Self-esteem

Neighborhood

delinquency

Figure 1. Simple Recursive Model

their numerical values to the diagram, and also
evaluate the model in terms of its consistency with
the data. In the model of Figure 1, for instance,
there are no direct arrows between X 2 and X 3 ,
between X 4 and both X 1 and X 2 , and between X 5
and both X 1 and X 3. This means that with controls
for all prior or intervening variables, the respec-
tive partial correlations can be predicted to be
zero, apart from sampling errors. One arrives at
the predictions in equation system 2.

r23.1 = 0 r14.23 = 0 r24.13 = 0

r15.234 = 0 r35.124 = 0

( 2 )

Thus, for each omitted arrow one may write
out a specific ‘‘zero’’ prediction. Where arrows
have been drawn in, it may have been possible to
predict the signs of direct links, and these direc-
tional predictions may also be used to evaluate the
model. Notice a very important property of recur-
sive models. In relating any pair of variables, say,
X 2 and X 3 , one expects to control for antecedent
or intervening variables, but it is not appropriate to
introduce as controls any variables that appear as
subsequent variables in the model (e.g., X 4 or X 5 ).
The simple phrase ‘‘controlling for all relevant
variables’’ should therefore not be construed to
mean variables that are presumed to depend on
both of the variables being studied. In an experi-
mental setup, one would presumably be unable to

carry out such an absurd operation, but in statisti-

cal calculations, which involve pencil-and-paper

controlling only, there is nothing to prevent one

from doing so.

It is unfortunately the case that controls for

dependent variables can sometimes be made inad-

vertently through one’s research design (Blalock

1985). For example, one may select respondents

from a list that is based on a dependent variable

such as committing a particular crime, entering a

given hospital, living in a certain residential area,

or being employed in a particular factory. When-

ever such improper controls are introduced, wheth-

er recognized explicitly or not, our inferences

regarding relationships among causally prior vari-

ables are likely to be incorrect. If, for example, X 1

and X 2 are totally uncorrelated, but one controls

for their common effect, X 3 , then even though r 12

= 0, it will turn out that r12.3 ≠ 0.

Recursive models also provide justifications

for common-sense rules of thumb regarding the

conditions under which it is not necessary to con-

trol for prior or intervening variables. In the mod-

el of Figure 1, for example, it can be shown that

although r24.13 = 0, it would be sufficient to control

for either X 1 or X 3 but not both in order for the

partial to disappear. Similarly, in relating X 3 to X 5 ,

the partial will be reduced to zero if one controls

for either X 2 and X 4 or X 1 and X 4. It is not necessary