03a measurement models

7/29/2019 03a Measurement Models

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Measurement Models:

Exploratory and Confirmatory

Factor Analysis

James G. Anderson, Ph.D.Purdue University


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Conceptual Nature of Latent

Variables

Latent variables correspond to some type

of hypothetical construct

Require a specific operational definition

Indicators of the construct need to be

selected

Data from the indicators must beconsistent with certain predictions (e.g.,

moderately correlated with one another)


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Multi-Indicator Approach

A multiple-indicator approach reduces theoverall effect of measurement error of anyindividual observed variable on the accuracyof the results

A distinction is made between observedvariables (indicators) and underlying latentvariables or factors (constructs)

Together the observed variables and thelatent variables make up the measurementmodel


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Principles of Measurement

Reliability is concerned with random error

Validity is concerned with random and

systematic error


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Measurement Reliability

Test-Retest

Alternate Forms

Split-Half/Internal Consistency

Inter-rater Coefficient

0.90 Excellent

0.80 Very Good 0.70 Adequate

0.50 Poor


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Measurement Validity

Content ( (whether an indicators items arerepresentative of the domain of the construct)

Criterion-Related (whether a measure relates to anexternal standard against which it can be evaluated)

Concurrent (when scores on the predictor and criterionare collected at the same time)

Predictive (when scores on the predictor and criterionare collected at different times)

Convergent (items that measure the same constructare correlated with one another)

Discriminant (items that measure different constructsare not correlated highly with one another)


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Types of Measurement Models

Exploratory (EFA)

Confirmatory (CFA)

Multitrait-Multimethod (MTMM) Hierarchical CFA


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An Exploratory Factor Model


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EFA Features

The potential number of factors ranges from

one up to the number of observed variables

All of the observed variables in EFA are

allowed to correlate with every factor An EFA solution usually requires rotation to

make the factors more interpretable.

Rotation changes the correlations between

the factors and the indicators so the pattern

of values is more distinct


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A Confirmatory Factor Model


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CFA Features

The number of factors and the observedvariables (indicators) that load on eachconstruct (factor or latent variable) arespecified in advance of the analysis

Generally indicators load on only oneconstruct (factor)

Each indicator is represented as having two

causes, a single factor that it is suppose tomeasure and all other unique sources ofvariance represented by measurement error


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CFA Features

The measurement error terms are

independent of each other and of the

factors

All associations between factors are

unanalyzed


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EFA vs CFA

The purpose is to determine the number

and nature of latent variables or factors

that account for the variation and

covariation among a set of observedvariables or indicators.

Two types of analysis

Exploratory Factor Analysis

Confirmatory Factor Analysis


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EFA vs CFA

Both types of analysis try to reproduce the

observed relationships among a set of indicators

with a smaller set of latent variables.

EFA is data driven and used to determine thenumber of factors and which observed variables

are indicators of each latent variable.

In EFA all the observed variables are

standardized and the correlation matrix is

analyzed


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EFA vs CFA

CFA is confirmatory. The number offactors and the pattern of indicator factorloadings are specified in advance.

CFA analyzes the variance-covariancematrix of unstandardized variables.

The prespecified factor solution is

evaluated in terms of how well itreproduces the sample covariance matrixof measured variables.


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EFA vs CFA

CFA models fix cross-loadings to zero.

EFA models may involve cross-loadings of

indicators.

In EFA models errors are assumed to be

uncorrelated

In CFA models errors may be correlated.


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EFA Procedures

Decide which indicators to include in the

analysis.

Select the method to establish the factor

model

ML (assumes a multivariate normal

distribution)

Principle Factors (Distribution Free)


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EFA Procedures

Select the appropriate number of factors

Eigenvalues greater than one

Scree test

Goodness of fit of the model

If there is more than one factor, select thetechnique to rotate the initial factor matrix

to simple structure Orthogonal rotation (Varimax) Oblique rotation (e.g., Promax)


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EFA Procedures


Eigenvalues greater than one

Scree test

Goodness of fit of the model

If there is more than one factor, select thetechnique to rotate the initial factor matrix

to simple structure Orthogonal rotation (varimax) Oblique rotation (e.g., oblimin)


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EFA Procedures


Identify which indicators load on each

factor or latent variable

You can calculate factor scores to serve

as latent variables


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Uses of CFA

Evaluation of test instruments

Construct validation

Convergent validity

Discriminant validity

Evaluation of methods effects

Evaluation of measurement invariance Development and testing of the

measurement model for a SEM.


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Advantages of CFA

Test nested models

Test relationships among error variables

or constraints on factor loadings (e.g.,

equality)

Test equivalent measurement models in

two or more groups or at two or more

times.


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Advantages of CFA

The fit of the measurement model can be

determined before estimating the SEM

model.

In SEM models you can establish

relationships among variables adjusting for

measurement error.

CFA can be used to analyze mean

structures.


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CFA Model Identification

Identification pertains to the difference between

the number of estimated model parameters and

the number of pieces of information in the

variance/covariance matrix. Every latent variable needs to have its scale

identified.

Fix one loading of an observed variable on the latent

variable to one

Fix the variance of the latent variable to one


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A Covariance Structure Model


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A Structural Model of the

Dimensions of Teacher Stress Survey of teacher stress, job satisfaction

and career commitment

710 primary school teachers in the U.K.


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Methods

20-Item survey of teacher stress

EFA (N=355)

CFA (N=375) 1-Item overall self-rating of stress

SEM (N=710)


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Table1: An oblique five factor pattern solution (N=170)


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Factors

Factor 1 Workload

Factor 2 Professional Recognition

Factor 3Student Misbehavior Factor 4 - Time/Resource Difficulties

Factor 5 Poor Colleague Relations


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Factor Patterns


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EFA Results

5 Factor solution

4 Items deleted

Fit Statistics: Chi Square = 156.94

df = 70

AGFI = 0.906 RMR = 0.053



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Covariances between exogenous latent traits


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CFA Results

5 Factor solution

2 Items deleted

Fit Statistics: Chi Square = 171.14

df = 70

AGFI = 0.911 RMR = 0.057


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Structural Equation Models

True Null Model - Hypothesizes no significant

covariances among the observed variables

Structural Null Model - Hypothesizes no

significant structural or correlationalrelations among the latent variables

Non-Recursive Model

Mediated Model Regression Model

Non-recursive model


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Non-recursive model


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Regression

Model

Comparison of Fit Indices


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Comparison of Fit Indices


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Results

Two major contributors to teacher stress

Work load

Student Misbehavior

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