Encyclopedia of Sociology

CAUSAL INFERENCE MODELS

of underlying or ‘‘true’’ variables, plus additional
factors that may produce combinations of random
measurement errors, which are unrelated to all
other variables in the theoretical system, and sys-
tematic biases that are explainable in causal terms.
Thus, measures of ‘‘true guilt’’ or ‘‘true self-es-
teem’’ will consist of responses, usually to paper-
and-pencil tests, that may be subject to distortions
produced by other variables, including some of
the variables in the causal system. Perhaps distor-
tions in the guilt measure may be a function of
amount of delinquent behavior or parental educa-
tion. Similarly, measures of behaviors are likely to
overestimate or underestimate true frequencies,
with biases dependent on qualities of the observer,
inaccuracies in official records, or perhaps the
ability of the actor to evade detection.

In all such instances, we may be able to con-
struct an ‘‘auxiliary measurement theory’’ (Blalock
1968; Costner 1969) that is itself a causal model
that contains a mixture of measured and unmeasured
variables, the latter of which constitute the ‘‘true’’
or underlying variables of theoretical interest. The
existence of such unmeasured variables, however,
may introduce identification problems by using
more unknowns than can be estimated from one’s
data. If so, the situation will once more be hopeless
empirically. But if one has available several indica-
tors of each of the imperfectly measured con-
structs, and if one is willing to make a sufficient
number of simplifying assumptions strategically
placed within the overall model, estimates may be
obtainable.

Consider the model of Figure 5 (borrowed
from Costner 1969), which contains only two theo-
retical variables of interest, namely, the unmeasured
variables X and Y. Suppose one has two indicators
each for both X and Y and that one is willing to
make the simplifying assumption that X does not
affect either of Y’s indicators, Y 1 and Y 2 , and that Y
does not affect either of X’s indicators, X 1 and X 2.
For the time being ignore the variable W as well as
the two dashed arrows drawn from it to the indica-
tors X 2 and Y 1. Without W, the nonexistence of
other arrows implies that the remaining causes of
the four indicators are assumed to be uncorrelated
with all other variables in the system, so that one
may assume measurement errors to be strict-
ly random.

If one labels the path coefficients (which all

connect measured variables to unmeasured ones)

by the simple letters a, b, c, d, and, e, then with 4(3)/

2 = 6 correlations among the four indicators, there

will be six pieces of empirical information (equa-

tion system 5) with which to estimate the five

unknown path coefficients.

rx 1 x 2 = ab rx 1 y 2 = ace

ry 1 y 2 = de rx 2 y 1 = bcd

rx 1 y 1 = acd rx 2 y 2 = bce

( 6 )

One may now estimate the correlation or path

coefficient c between X and Y by an equation

derived from equation system 6.

c^2 = ––––––abc = –––––––– = ––––––––

(^2) de
(ab)(de)
rx 1 y 1 rx 2 y 2
rx 1 x 2 rx 1 y 2 rx 1 x 2 ry 1 y 2
rx 1 y 2 rx 2 y 1
( 7 )
Also notice that there is an excess equation that
may be used to check on the consistency of the
model with the data, namely the prediction that rx
1 y^1 rx 2 y 2 = rx 1 y 2 rx 2 y 1 = abc
(^2) de.
Suppose next that there is a source of meas-
urement error bias W that is a common cause of
one of X’s indicators (namely X 2 ) and one of Y’s
(namely Y 1 ). Perhaps these two items have similar
wordings based on a social survey, whereas the
remaining two indicators involve very different
kinds of measures. There is now a different expres-
sion for the correlation between X 2 and Y 1 , namely
r x 2 y 1 = bcd + fg. If one were to use this particular
correlation in the estimate of c^2 , without being
aware of the impact of W, one would obtain a
biased estimate. In this instance one would be able
to detect this particular kind of departure from
randomness because the consistency criterion
would no longer be met. That is, (acd)(bce) ≠ (ace)(bcd

• fg). Had W been a common cause of the two
  indicators of either X or Y alone, however, it can be
  seen that one would have been unable to detect
  the bias even though it would have been present.
  Obviously, most of one’s measurement-error
  models will be far more complex than this, with
  several (usually unmeasured) sources of bias, pos-
  sible nonlinearities, and linkages between some of
  the important variables and indicators of other
  variables in the substantive theory. Also, some
  indicators may be taken as causes of the conceptual
  variables, as for example often occurs when one is