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



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Introduction


Measurement Model


Structural Model


Example


Problems with the Wald Test


Williams and Hazer Approach to Measurement Error


Standardization


Missing Data



Introduction


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"'> 


Model


          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


Parameters


          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  


                   I
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


Overview


A variant of correction for attenuation


Single indicator, not multiple indicators


Must know the reliability of each measure


How


          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.


Advantages


          Fewer variables


          Usually smaller standard errors for the paths


          Easier to estimate


                    No Heywood cases


                    Fewer convergence issues


Disadvantages


          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


                              Alternatives


                                        Regression


                                        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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