Encyclopedia of Sociology

CAUSAL INFERENCE MODELS

W

X 1 X 2 Y 1 Y 2

X c Y

a

b

d

e

f g

Figure 5. Simple Measurement-Error Model, with Two Unmeasured Variables X and Y, Two Indicators of Each, and
Source of Bias W

Here, k = 3, so that at least two variables must
be left out of each equation, meaning that their
respective coefficients have been set equal to zero.
One can rather simply check on the necessary
condition by counting arrowheads coming to each
variable. In this instance there can be no more
than two arrows into each variable, whereas in the
case of guilt X 1 there are three. The equation for
X 1 is referred to as being ‘‘underidentified,’’ mean-
ing that the situation is empirically hopeless. The
coefficients simply cannot be estimated by any
empirical means. There are exactly two arrows
coming into delinquency X 3 , and one refers to this
as a situation in which the equation is ‘‘exactly
identified.’’ With only a single arrow coming into
self-esteem X 2 , one has an ‘‘overidentified’’ equa-
tion for which one actually has an excess of empirical
information compared to the number of unknowns
to be estimated. It turns out that overidentified
equations provide criteria for evaluating goodness
of fit, or a test of the model, in much the same way
that, for recursive models, one obtains an empiri-
cal test of a null hypothesis for each causal arrow
that has been deleted.

Since the equation for X 1 is underidentified,
one must either remove one of the arrows, on a
priori grounds, or search for at least one more
predetermined variable that does not belong in
this equation, that is, a predetermined variable
that is assumed not to be a direct cause of level of

guilt. Perhaps school performance can be intro-

duced as Z 3 by making the assumption that Z 3

directly affects both self-esteem and delinquency

but not guilt level. A check of this revised model

indicates that all equations are properly identified,

and one may proceed to estimation. Although

space does not permit a discussion of alternative

estimation methods that enable one to get around

the violated assumption required by ordinary least

squares, there are various computer programs

available to accomplish this task. The simplest

such alternative, two-stage least squares (2SLS),

will ordinarily be adequate for nearly all sociologi-

cal applications and turns out to be less sensitive to

other kinds of specification errors than many of

the more sophisticated alternatives that have been

proposed.

CAUSAL APPROACH TO

MEASUREMENT ERRORS

Finally, brief mention should be made of a grow-

ing body of literature—closely linked to factor

analysis—that has been developed in order to

attach measurement-error models to structural-

equation approaches that presume perfect meas-

urement. The fundamental philosophical starting

point of such models involves the assumption that

in many if not most instances, measurement errors

can be conceived in causal terms. Most often, the

indicator or measured variables are taken as effects