Structural Coefficients in Recursive Models/Evils of Standardization
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Page 1
Structural Coefficients in Recursive Models/ Evils of Standardization
R
ECURSIVE DEFINED
.
A model is said to be recursive if all the causal linkages run “one way”,
that is, no two variables are reciprocally related in such a way that each affects and depends on
the other, and no variable feeds back upon itself through any indirect concatenation of causal
linkages. The model we have been looking at is recursive:
X1
X2
X3
X4
u
v
w
The model would be nonrecursive if, for example, X4 also affected X3, i.e. the causation ran in
both directions. Non
recursive models are much more difficult to work with, and we’ll discuss
them later in the course.
R
ECURSIVE MODELS WITH STANDARDIZED VARIABLES
.
We have been examining a 4variable
recursive model in which variables were standardized.
The b’s in such models are referred to as
the
path coefficients.
The advantages of standardized variables are:
1.
Certain algebraic steps are simplified
2.
Sewell Wright’s rule for expressing correlations in terms of path coefficients can be
applied
without modification
3.
Continuity is maintained with the earlier literature on path analysis and causal models in
Sociology
4.
It shows how an investigator whose data are only available in the form of a correlation matrix
can, nevertheless, make use of a clearly specified model in interpreting those correlations.
Nevertheless, standardization should generally be avoided. Standardization tends to
obscure the distinction between the structural coefficients of the model and the several variances
and covariances that describe the joint distribution of the variables in a certain population. To
illustrate this, we will first see what happens when variables are not standardized.
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Structural Coefficients in Recursive Models/Evils of Standardization
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Page 2
R
ECURSIVE MODELS WITH UNSTANDARDIZED VARIABLES
.
We will continue to assume that al
l X’s
have a mean of 0. (The only thing this affects is the intercepts.) We will not assume that
variances all = 1. Under these conditions,
E(X
i
2
) = E(X
i
X
i
) = V(X
i
) =
σ
i
2
E(X
i
X
j
) = Cov(X
i
X
j
) = σ
ij
To get the normal equations, we proceed as before: Multiply each structural equation by the
predetermined variables and then take expectations. In addition, to get the variances, we multiply
the structural equation by the DV of the equation and take expectations. Hence,
(1)
For X2, the structural equation is
X
X
u
2
21
1
The only predetermined variable is X1. Hence, if we multiply both sides of the above equation by
X1 and then take expectations, we get the normal equation
2
1
21
21
1
2
1
21
2
1
)
(
)
(
)
(
u
X
E
X
E
X
X
E
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 Spring '11
 RichardWilliams
 Normal Distribution, Regression Analysis, Standard Deviation, Variance, X1, structural coefﬁcients

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