David
A. Kenny

August 15,
2011

**Path Analysis**** **

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

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.

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

Standardized variables

Paths: beta weights from the regression equation

Disturbance Path: 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

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. In some cases, the model should not be trimmed if either the goal is to compare the paths from one model to another or if the goal is determine the absolute size of the paths.

a) test of the individual paths: standard

b) test of all of the paths in a given equation

**(N - p - 1)(R _{2}^{2} - R_{1}^{2}) **

________________

k(1 - R_{2}^{2})

**where N
is the overall sample size, p is the number of deleted plus specified paths, k
is the number of deleted paths, R _{1}^{2} is the multiple correlation squared (not adjusted) from the
equation with only the specified paths, and R_{2}^{2} is the multiple correlation squared (not adjusted) from
the equation with the specified and deleted paths. **

c) 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, MPLUS, or LISREL.

**Correlation = Direct Effect +
Indirect Effects + Spuriousness **

Correlation between an endogenous variable and an exogenous variable:

**Correlation = Direct Effect +
Indirect Effects + Unspecified Covariance **

**Example
**

The coefficients in this model can be estimated by multiple regression:
For the Intention equation, it disturbance, U_{1} is uncorrelated with
both Attitude and Social Norms and for the Behavior equation, it disturbance, U_{2} is uncorrelated with both Intention.

**There are two deleted paths model: One from Attitude to
Behavior and the other from Social Norms to Behavior. If these two paths
are added to the model, it would be saturated. The following three steps
are suggested:
**

**STEP ONE: TEST OF DELETED PATHS
**

Because this is a fairly standard model, we probably would not trim. If we did and say found that path b, the path from Social Norms to Intention, was not needed, we re-estimate the model with that path dropped and then report the resulting coefficients.