David A. Kenny
August 29, 2011Structural Models with Latent
Problems with the Wald Test
Williams and Hazer Approach to
Causal models with latent variables represent
a mix of path analysis and confirmatory
factor analysis which have been called a hybrid model. In essence, the measurement model is first
estimated and the correlations or covariance matrix between constructs or factors then serves as input to estimate the
structural coefficients between constructs or latent variables. In
actuality, both models are simultaneously estimated by a structural equation
modeling program such as AMOS, LISREL, or EQS.
- the mapping of
measures onto theoretical constructs
- constituent parts
- test of specification error:
- Estimate a
structural model that is just-identified or estimate a confirmatory
factor analysis model (no causation, just correlations between the latent
- Before the
structural model is interpreted, it must first be established that the
measurement model fits.
causal and correlational links between theoretical variables
- constituent parts
- variances of
the exogenous variables
between exogenous variables
- variances of
the disturbances of endogenous variables
- covariances between
disturbances and exogenous variables (usually set to zero)
- test of specification error
- compare the
specified structural model to a model in which the structural model is just-identified
Example (go to Respecification webpage for
& Madden (Ajzen, I., & Madden, T. J. (1986). Prediction of
Two indicators of Intention,
Attitude, and Social Norms
indicator of Behavior which is assumed to have no measurement error.
Intention = Attitude +
Social Norms +U
Behavior = Intention + V
3 loadings (one for
Intention, Attitude and Norms)
6 error variances
4 variances of factors
6 covariances between
factors (Behavior is considered a factor)
1 covariance between
2 exogenous variances
2 disturbance variances
knowns: (8)(7)/2 = 28
unknowns (parameters): 19
knowns: (5)(4)/2 = 10
There are just 4
“variables” in the structural model.
2 + 9 = 11
the Measurement Model: CFA (or equivalently a Just-identified Structural Model)
Note that to make the
structural model just-identified paths must be drawn from Attitude and Social
Norms to Behavior
Fit of both models: χ²(11)
= 16.09, p = .065
The Ajzen & Madden model has decent fit
the Structural Model
path estimate CR p χ² diff p
SN → I: -0.033 -0.144 .885 0.019 .892
A → I: 0.973 4.116 <.001 16.111 <001
I → B: 0.415 4.653 <.001 23.999 <.001
Problems with Testing of Parameters Using the
The Problem (see the Gonzalez
& Griffin, Psychological Methods,
Markers A → I (CR)
A1, I1 4.116
A2, I1 4.028
A1, I2 3.736
A2, I2 3.670
Critical ratios depend on the choice of the marker. If you change the marker
variable, some things change in the model and some things say the same.
What stays the same:
square and the df.
standard fit indices.
loadings and paths.
loadings and paths.
Solution: Use chi-square difference test if the test is important as it does
not depend on the choice of marker.
For the above example it is χ²(1) = 16.111 making the “CR” (the square root of chi
square) equal to 4.014.
Williams and Hazer Option to Measurement
A variant of correction for attenuation
Single indicator, not multiple indicators
Must know the reliability of each measure
Structural Model: latent variables
Each latent variable causes its measure
(path fixed to one)
Each measure has an error path (path
fixed to one)
Error variance fixed to
Variance of the measure
times one minus the reliability
Of for standardized data,
one minus the reliability
Need to test paths using chi square difference test as CR appear
to be too conservative.
Usually smaller standard errors for
Easier to estimate
No Heywood cases
Fewer convergence issues
No test of the measurement model
Assumes the measurement model is
so traditional and so may meet editorial objections
Rubin and Little
The data are missing at random (MCAR).
Data missing due a variable in the data set which
is not missing (MAR).
Data missing due to a
variable that is missing or unmeasured (NMAR).
for handing for missing data – given MAR and MCAR
Pairwise Deletion –
can be problematic
May result in the loss of too many cases
Sample biased: means, variances, & covariances
Imputation – Substitute a
Multiple Imputation (not used often in SEM)
Maximum Likelihood (FIML)
By far the most common
approach to missing data in SEM
not get the following output in Amos
Matrix of the Measures
Measures of Fit
to handle an auxiliary variables, i.e., variables in the dataset but not in the model.
Could be important with MAR.
Standardization can occur at many places within the modeling process.
the raw data
the data matrix
the model specification
the transformed model
If the model is not standardized, tests refer to the model not the standardized solution. There are estimation methods based on the assumption that the correlation matrix has been entered (RAMONA), but is rarely used. This procedure does allow the standardization of latent endogenous variables.
When to Standardize
when the units of measurement not very interpretable
desire to compare coefficients with different units of measurement
more experience with betas than b coefficients
When Not to Standardize
units of measurement are meaningful
paths are usually set equal (so in multiple groups analysis one should analyze the covariance matrix) or paths absolute values compared (e.g., dyadic analysis)
Data (being revised)
to the next SEM page.
Go to the main SEM page.