This page discusses how to use multiple regression to estimate the
parameters of a structural model.

Key Assumption
For an endogenous variable, its disturbance must be uncorrelated with all of the 
specified causal variables.  So for a model, consider each endogenous variable and 
determine that its disturbance is uncorrelated with each of its causes.

Violation of the Assumptions to Use Multiple Regression to Estimate Structural Coefficients
There are three conditions under which the disturbance is correlated with the
an exogenous variable (thus, eliminating multiple regression as an appropriate 
tool to estimate causal paths):
    a) Spuriousness:  A variable causes both the endogenous variable and 
          one its causal variables and that variable is not included in 
          the model.
    b) Reverse Causation:  The endogenous variable causes, 
          either directly or indirectly, one of its causes.
    c) Measurement Error:  There is measurement error in a causal variable.

Estimation Using Multiple Regression
    Standardized variables
         Paths: beta weights from the regression equation
         Disturbance ath: square root of one minus the
            multiple correlation squared 
         Curved lines between exogenous variables: correlations
         Curved lines between disturbances: partial
            correlations with common causal variables of both       
            endogenous variables partialled
    Unstandardized variables
         Paths: b weights from the regression equation
         disturbance variance: the variance of the endogenous
            variable times one minus the multiple correlation
            squared
         Curved lines between exogenous variables: covariances
         Curved lines between disturbances: partial covariances,
            with causal variables of both endogenous variables      
            partialled

Steps in Testing

     STEP ONE: TEST OF DELETED PATHS
          Respecify the model to make it just-identified. That
          is, add the paths that the model specifies to be zero
          and include them in the model.  Test the paths
          specified to be zero making sure that the specified
          paths are included in the equation, but not tested.
          These tests may be done with a reduced alpha (e.g., .01).
    
    STEP TWO: TEST OF SPECIFIED PATHS
          Retaining the significant paths from the previous step,
          test the paths that were specified to be present in the
          model.  Sometimes these tests are done hierarchically.

    STEP THREE: TRIMMED MODEL
          Re-estimate the model, including (a) the paths that were
          specified to be zero but were significant from step one
          and (b) dropping the paths that were specified but were not
          significant in step two.
          
Types of Tests
    test of the individual paths
        standard t or F test of the coefficient

    F test of all of the paths in a given equation

                          (N - p - 1)(R22 - R12)
                          ---------------------
                               k(1 - R22)
            where
                N   overall sample size
                p   the number of deleted plus specified paths
                k   the number of deleted paths
                R12  multiple correlation squared (not adjusted)
                    from the equation with only the specified
                    paths
                R22  multiple correlation squared (not adjusted)
                    from the equation with the specified and
                    deleted paths

The combined test of all of the paths in the model is usually a chi square 
goodness of fit test from a SEM program such as AMOS, EQS, or LISREL.

Determination of Deleted Paths
    If all of the paths in the model can be estimated by multiple 
regression, the number of deleted paths equals the number of knowns minus 
the number of unknowns or the degrees of freedom of the specified model.  
To make the model justidentified, examine each pair of variables and 
determine pairs of variables that are not linked by a path or a 
correlation (including a correlation between disturbances).  Add a path or 
a correlation between disturbance between each pair of these unlinked pairs.  
The direction of the path is given by theory and the requirement that 
feedback not be introduced.
                                
Types of Links Between Two Variables
    Direct effect:  Either X causes Y, Y causes X, or both.  
    Indirect effect:  The relationship between X and Y is said to be indirect if X
causes Z which in turn causes Y.  (To learn more about indirect effects.)
    Spuriousness:  The relationship between X and Y is said to be spurious 
if Z causes X and Y.
    Unexplained covariation:  Both X and Y are exogenous and so variation 
between them is not explained by the model.

Decomposition of a Correlation

Correlation between two endogenous variables:

   Correlation = Direct Effect + Indirect Effects + Spuriousness

Correlation between an endogenous variable and an exogenous variable:

   Correlation = Direct Effect + Indirect Effects + Unspecified
                                                    Covariance