Terminology and Basics of SEM
Standard Structural Equation Effect Structural Causal = Sum X + Disturbance Variable Coefficient Variable Path Analytic Equation Effect Path Causal = Sum X + Path X Disturbance Variable Coefficient Variable All variables are standardized although sometimes this equation is given without the disturbance standardized and so it does not have a coefficient.
Standardized Variable
Variable whose mean is zero and variance is one.
Latent Variable
A
variable in the model that is not measured.
Exogenous Variable
A variable that is not caused by another variable in the
model. Usually this variable causes one or more variables in the model.
Endogenous Variable
A variable that is caused by one or more variable in the
model. Note that an endogenous variable may also cause another endogenous
variable in the model.
Structural Coefficient
A measure of the amount of change in the effect variable
expected given a one unit change in the causal variable and no change in any
other variable. Although like a regression coefficient, this coefficient
may not be estimable by multiple regression.
Disturbance
The set of unspecified causes of the effect
variable. Analogous to an error or residual in a prediction
equation. Usually each endogenous variable has a disturbance. Usually treated as a latent variable. (Go to an
exception.)
Structural Model
A
set of structural equations.
Path Diagram
A
diagram that pictorially represents a structural model. Curved lines
represent unanalyzed associations. Measured variables are usually
designated by a box and latent variables, including disturbances, are
represented by circles. Covariances or correlations between exogenous
variables and between disturbances are represented by curved lines with
arrowheads at both ends. Paths are represented by straight lines with an
arrowhead pointing toward the effect variable.
Example of a Structural Model
X3 = aX1 + bX2 + U1
X4 = cX1 + dX2 + eX3 + U2
Note that X3 and X4 are endogenous (i.e., caused), X1 and X2 are exogenous (not caused), and U1 and U2 are disturbances. The path diagram for this model is as follows:
Hierarchical Model:
Models without Feedback
A model in which the structural
equations can be ordered such that any variable appearing as an effect in a
given equation does not appear as a cause in any prior equation.
Such a model has no feedback loops. All non-hierarchical models are said
to be non-recursive. However, not all non-recursive models are
non-hierarchical models. The above path model is hierarchical in that if we order the equations, X3 and X4, no cause of X3 later appears as an effect.
Tracing Rule
The correlation between any pair of variables equals the
sum of the products of the paths or correlations from each tracing. A tracing
between two variables is any route in which the same variable is not entered
twice and no variable is entered and left through an arrowhead. This rule
applies only to hierarchical models -- models without feedback. (To see an example of the tracing rule.)
Specification
The
translation of theory, previous research, design, and common sense into a
structural model.
Specification Error
An assumption made in structural model that is
false. So for instance, if a path in a model is set to zero and that path
is not exactly zero, there would be a specification error. It is
reasonable to believe that all models contain specification error. So one
seeks a model with the least specification error.
Identification
A model is said to be identified if there exists a unique
solution for the model's parameters. If there is no unique solution, then
the model is of little value.
Minimum Condition of
Identifiability
The number of known values must equal or
exceed the number of free parameters in the model. All identified models
meet this rule and if the rule is not met the model is not identified. However,
some models that meet this rule are not identified. As will be seen, the model in the path diagram meets this condition.
Known Values
For
the standard specification, the number of known values is the number of covariances or n(n
+ 1)/2 where n is the number of variables. For the path diagram, the number of knowns is (5)(4)/2 or 10.
For the path analytic specification the number of known values is n(n - 1)/2 where n is the number of variables. For the path diagram, the number of knowns is (4)(3)/2 or 6.
Covariance
The covariance between two variables equals the
correlation times the product of the variables' standard deviations. The
covariance of a variable with itself is the variable's variance.
Free Parameters in a Structural Model
Standard specification: paths, covariances between the
exogenous variables, between the disturbances and between exogenous variables
and disturbances, and variances of the exogenous variables and disturbances of
endogenous variables less the number of linear constraints. For the path diagram, the number of unknowns is 10, 5 paths, 1 curved line, 2 exogenous variable, and 2 endogenous variances.
Path analytic specification: paths (not including the disturbance paths) and correlations between the exogenous variables, between the disturbances, and between the exogenous variables and the disturbances less the number of linear constraints. For the path diagram, the number of unknowns is 6, 5 paths and 1 curved line.
Constraints
Setting of a parameter equal to some function of other
parameters. The simplest constraint is to set one parameter equal to
another parameter. Zero constraints are usually not counted. (For
more information on constraints.)
Degrees of Freedom
of a Model
The numbers of knowns minus the number of free
parameters; used in many measures of fit.
Just-identified or
Saturated Model
An identified model in which the number of free parameters exactly equals the number of known values; a model with zero
degrees of freedom. For the path diagram, it turns out that the model is just-identified. Note the number of knowns exactly equals the number of unknowns.
Under-identified
Model
A model for which it is not possible to estimate all of the model's parameters. For some under-identified models, some
parameters are identified.
Over-identified Model
A model for which all the parameters are identified and
for which there are more knowns than free parameters. An over-identified
places constraints on the correlation or covariance matrix.
Model Fit
The ability of an overidentified model to reproduce the
variables' correlation or covariance matrix.
Steps of Structural Equation
Modeling
STEP 1:
SPECIFICATION
Statement of the theoretical model either as a set of
equations or as a diagram.
STEP 2: IDENTIFICATION
The model can in
theory and in practice be estimated with observed data (to learn the general
rules of identification).
STEP 3: ESTIMATION
The model's
parameters are statistically estimated from data. Multiple regression is
one such estimation method, but typically more complicated estimated methods are
used. Generally, a specialized SEM program (e.g. AMOS or LISREL) is used.
STEP 4: MODEL FIT
The estimated model
parameters are used to predict the correlations or covariances between measured
variables and the predicted correlations or covariances are compared to the
observed correlations or covariances (see measures of model fit).
