David A. Kenny
August 29, 2011Structural Models with Latent Variables

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Measurement Model

Structural Model


Problems with the Wald Test

Williams and Hazer Approach to Measurement Error


Missing Data


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.

Measurement Model

Structural Model

·        definition

o   the causal and correlational links between theoretical variables

  • constituent parts
    • paths
    • variances of the exogenous variables
    • covariances between exogenous variables
    • variances of the disturbances of endogenous variables
    • covariances between disturbances
    • 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 more details)

Ajzen & Madden (Ajzen, I., & Madden, T. J. (1986). Prediction of goal-directed 'font-size:13.5pt;font-family:"Arial","sans-serif"'> 


          Measurement Model

                    Two indicators of Intention, Attitude, and Social Norms

One indicator of Behavior which is assumed to have no measurement error.

          Structural Equations

                    Intention = Attitude + Social Norms +U

                    Behavior = Intention + V


          Measurement Model

                    3 loadings (one for Intention, Attitude and Norms)

                    6 error variances

                    4 variances of factors

                    6 covariances between factors (Behavior is considered a factor)

          Structural Model

                    3 paths

                    1 covariance between exogenous

                    2 exogenous variances

                    2 disturbance variances

Degrees of Freedom

          Measurement Model

                    knowns: (8)(7)/2 = 28

                    unknowns (parameters): 19

                    df: 9

          Structural Model

                    knowns: (5)(4)/2 = 10

                              There are just 4 “variables” in the structural model.

                    unknowns: 8

                    df: 2

          Total Model

                    df: 2 + 9 = 11

Pure Structural Model

CFA Model

Just-Identified Structural Model

Test 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

Test the Structural Model

Specified Paths

                   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  

B:           0.415         4.653       <.001      23.999  <.001

Problems with Testing of Parameters Using the Wald Test

The Problem (see the Gonzalez & Griffin, Psychological Methods, 2001)

                    Markers                A I (CR)

A1, I1                    4.116

A2, I1                    4.028

A1, I2                    3.736

A2, I2                    3.670

Result: 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:

Chi square and the df.

All standard fit indices.

Standardized loadings and paths.

Standardized residuals.
          R squared.

What changes:

Unstandardized loadings and paths.

Critical ratios.

Modification indices.

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 Error


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.


          Fewer variables

          Usually smaller standard errors for the paths

          Easier to estimate

                    No Heywood cases

                    Fewer convergence issues


          No test of the measurement model

          Assumes the measurement model is correct

          Not so traditional and so may meet editorial objections

Standardization (being revised)
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)

Missing Data (being revised)

Rubin and Little typology

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).

Strategies for handing for missing data – given MAR and MCAR

          Data Deletion

Pairwise Deletion – can be problematic

                    Listwise Deletion

                              May result in the loss of too many cases

                              Sample biased: means, variances, & covariances

                    Imputation – Substitute a Value

                              Traditionally the mean



                                        Maximum Likelihood

                    Multiple Imputation (not used often in SEM)

Full Information Maximum Likelihood (FIML)

By far the most common approach to missing data in SEM

                              Creates Multiple Groups

                              Do not get the following output in Amos

                                        Variance-Covariance Matrix of the Measures

                                        Modification Indices

                                        Standardized Residuals

                                        Some Measures of Fit

                              How to handle an auxiliary variables, i.e., variables in the dataset but not in the model.

                                        Could be important with MAR.

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