David A. Kenny

August 29, 2011

**Structural Models with Latent
Variables**

Go to:

Introduction

Measurement Model

Structural Model

Example

Problems with the Wald Test

Williams and Hazer Approach to Measurement Error

Standardization

Missing Data

Causal models with latent variables represent a mix of path analysis and confirmatory factor analysis which have been called a

- definition
- the mapping of measures onto theoretical constructs
- constituent parts
- loadings of the measures on the theoretical constructs
- error variances
- error covariances (correlated errors)
- 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 variables)
- Before the structural model is interpreted, it must first be established that the measurement model fits.

· 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

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__

A

I → B: 0.415 4.653 <.001 23.999 <.001

The Problem (see the Gonzalez & Griffin,

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 (Wald tests).

Modification indices..

Solution:

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

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

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 **

Rubin and Little
typology:

1. The data are missing at random (MCAR).

2. Data missing due a variable in the data set which is not missing (MAR).

3. 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

Could be important with MAR.