Transcript The Use of Explanations to Increase User Trust in a

Introduction to CFA
SEM – Confirmatory Factor Analysis
LEARNING OBJECTIVES:
Upon completing this chapter, you should be able to do the following:
 Distinguish between exploratory factor analysis and confirmatory factor
analysis.
 Assess the construct validity of a measurement model.
 Know how to represent a measurement model using a path diagram.
 Understand the basic principles of statistical identification and know some
of the primary causes of SEM identification problems.
 Understand the concept of fit as it applies to measurement models and be
able to assess the fit of a confirmatory factor analysis model.
 Know how SEM can be used to compare results between groups. This
includes assessing the cross-validation of a measurement model.
Confirmatory Factor Analysis Overview
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What is it?
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Why use it?
Confirmatory Factor Analysis Defined
Confirmatory Factor Analysis . . . is similar to EFA in
some respects, but philosophically it is quite different.
With CFA, the researcher must specify both the number
of factors that exist within a set of variables and which
factor each variable will load highly on before results
can be computed. So the technique does not assign
variables to factors. Instead the researcher must be
able to make this assignment before any results can be
obtained. SEM is then applied to test the extent to
which a researcher’s a-priori pattern of factor loadings
represents the actual data.
Review of and Contrast with
Exploratory Factor Analysis
EFA (exploratory factor analysis) explores the data and
provides the researcher with information about how many
factors are needed to best represent the data. With EFA, all
measured variables are related to every factor by a factor loading
estimate. Simple structure results when each measured variable
loads highly on only one factor and has smaller loadings on other
factors (i.e., loadings < .4).
The distinctive feature of EFA is that the factors are
derived from statistical results, not from theory, and so they can
only be named after the factor analysis is performed. EFA can be
conducted without knowing how many factors really exist or
which variables belong with which constructs. In this respect,
CFA and EFA are not the same.
CFA and Construct Validity
One of the biggest advantages of CFA/SEM is its ability to
assess the construct validity of a proposed measurement theory.
Construct validity . . . is the extent to which a set of measured
items actually reflect the theoretical latent construct they are
designed to measure.
Construct validity is made up of four important components:
1. Convergent validity – three approaches:
o Factor loadings.
o Variance extracted.
o Reliability.
2. Discriminant validity.
3. Nomological validity.
4. Face validity.
Rules of Thumb
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Construct Validity: Convergent and Discriminant
Validity
Standardized loading estimates should be .5 or
higher, and ideally .7 or higher.
VE should be .5 or greater to suggest adequate
convergent validity.
Construct reliability should be .7 or higher to
indicate adequate convergence or internal
consistency.
VE estimates for two factors also should be greater
than the square of the correlation between the two
factors to provide evidence of discriminant validity.
Confirmatory Factor Analysis Stages
Stage 1: Defining Individual Constructs
Stage 2: Developing the Overall Measurement Model
Stage 3: Designing a Study to Produce Empirical
Results
Stage 4: Assessing the Measurement Model Validity
Stage 5: Specifying the Structural Model
Stage 6: Assessing Structural Model Validity
Note: CFA involves stages 1 – 4 above.
stages 5 and 6.
SEM is
Stage 1: Defining Individual Constructs
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List constructs that will comprise the
measurement model.
Determine if existing scales/constructs are
available or can be modified to test your
measurement model.
If existing scales/constructs are not
available, then develop new scales.
Rules of Thumb
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Defining Individual Constructs
All constructs must display adequate construct validity,
whether they are new scales or scales taken from
previous research. Even previously established scales
should be carefully checked for content validity.
Experts should judge the items’ content for validity in
the early stages of scale development.
o When two items have virtually identical content, one
should be dropped.
o Items upon which the judges cannot agree should be
dropped.
A pre-test should be used to purify measures prior to
confirmatory testing.
Stage 2: Developing the Overall Measurement Model
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Key Issues:
Unidimensionality.
Measurement model.
Items per construct.
o Identification
Reflective vs. formative measurement
models.
Stage 2: A Measurement Model (and SEM)
A SEM diagram commonly has certain standard elements: latents are
ellipses, indicators are rectangles, error and residual terms are circles, singleheaded arrows are causal relations (note causality goes from a latent to its
indicators), and double-headed arrows are correlations between indicators or
between exogenous latents. Path coefficient values may be placed on the
arrows from latents to indicators, or from one latent to another, or from an
error term to an indicator, or from a residual term to a latent.
Each endogenous variable (the one 'Dependent variable' in the model below)
has an error term, sometimes called a disturbance term or residual error, not
to be confused with indicator error, e, associated with each indicator variable.
Measurement
Model
Rules of Thumb
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Developing the Overall Measurement Model
In standard CFA applications testing a measurement theory,
within and between error covariance terms should be fixed
at zero and not estimated.
In standard CFA applications testing a measurement theory,
all measured variables should be free to load only on one
construct.
Latent constructs should be indicated by at least three
measured variables, preferably four or more. In other
words, latent factors should be statistically identified.
Rules of Thumb
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Developing the Overall Measurement Model
Formative factors are not latent and are not validated as are
conventional reflective factors. Internal consistency and reliability
are not important. The variables that make up a formative factor
should explain the largest portion of variation in the formative
construct itself and should relate highly to other constructs that are
conceptually related (minimum correlation of .5):
o Formative factors present greater difficulties with statistical
identification.
o Additional variables or constructs must be included along with a
formative construct in order to achieve an over-identified model.
o A formative factor should be represented by the entire
population of items that form it. Therefore, items should not be
dropped because of a low loading.
o With reflective models, any item that is not expected to correlate
highly with the other indicators of a factor should be deleted.
Rules of Thumb
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Designing a Study to Provide Empirical Results
The ‘scale’ of a latent construct can be set by either:
o Fixing one loading and setting its value to 1, or
o Fixing the construct variance and setting its value to 1.
Congeneric, reflective measurement models in which all
constructs have at least three item indicators should be
statistically identified.
The researcher should check for errors in the specification
of the measurement model when identification problems
are indicated.
Models with large samples (more than 300) that adhere to
the three indicator rule generally do not produce Heywood
cases.
Stage 3: Designing a Study to Produce
Empirical Results
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Key Issues:
Measurement scales in CFA.
SEM/CFA and sampling.
Specifying the model:
o Which indicators belong to each construct?
o Setting the scale to “1” for one indicator on
each construct.
Issues in identification.
Problems in estimation:
o Heywood cases.
o Illogical standardized parameters.
Identification
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Recognizing Identification Problems:
Very large standard errors.
Inability to invert the information matrix (no solution
can be found).
Wildly unreasonable estimates including negative error
variances.
Unstable parameter estimates.
Stage 4: Assessing Measurement Validity
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Key Issues:
Assessing fit:
o GOF.
o Construct validity.
Diagnosing problems:
o Path estimates.
o Standardized residuals.
o Modification indices.
o Specification search.
Rules of Thumb
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Assessing Measurement Model Validity
Loading estimates can be statistically significant but still be too
low to qualify as a good item (standardized loadings below |.5|).
In CFA, items with low loadings become candidates for deletion.
Completely standardized loadings above +1.0 or below -1.0 are
out of the feasible range and can be an important indicator of
some problem with the data.
Typically, standardized residuals less than |2.5| do not suggest a
problem.
o Standardized residuals greater than |4.0| suggest a
potentially unacceptable degree of error that may call for the
deletion of an offending item.
o Standardized residuals between |2.5| and |4.0| deserve some
attention, but may not suggest any changes to the model if no
other problems are associated with those two items.
Rules of Thumb
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Assessing Measurement Model Validity
The researcher should use the modification indices only as a
guideline for model improvements of those relationships
that can theoretically be justified.
Specification searches based on purely empirical grounds
are discouraged because they are inconsistent with the
theoretical basis of CFA and SEM.
CFA results suggesting more than minor modification should
be re-evaluated with a new data set. For instance, if more
than two out of every 15 measured variables are deleted,
then the modifications can not be considered minor.
CFA Learning Checkpoint
1. What is the difference between EFA and CFA?
2. Describe the four stages of CFA.
3. What is the difference between reflective and
formative measurement models?
4. What is “statistical identification” and how can
it be avoided?
5. How do you decide if CFA is successful?
The End