1. ADVANCED QUANTITATIVE ANALYSIS
BDMR8043
Confirmatory Factor
Analysis Overview
Prof.. Dr. AllaEldin Hassan Kassam
Mahfoudh Hussein Hussein Mgammal

2. Confirmatory Factor Analysis
Overview
• What is it?
CFA is a tool that enables us to either "confirm" or
"reject" our preconceived theory.
• Why use it?
CFA is used to provide a confirmatory set of our
measurement theory. A measurement theory specifies
how measured variables logically and systematically
represent constructs involved in a theoretical model.
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3. 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.
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4. 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 < .40).
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.
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5. A Visual Diagram
• Measurement theories often are represented using
visual diagrams called (path diagrams). The path
diagram shows the linkages between specific measured
variables and their associated constructs, along with
the relationships among constructs. "Paths" from the
latent construct to the measured items (loadings) are
based on the measurement theory. When CFA is
applied, only the loadings theoretically linking a
measured item to its corresponding latent factor are
calculated.
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6. • Figure 1 provides a complete specification of the CFA model. The
two latent constructs are Supervisor Support and Work
Environment The X1—X8 represent the measured indicator
variables and the Lx1— Lx8 are the relationships between the
latent constructs and the respective measured items (i.e., factor
loadings).The four items measuring Supervisor Support are
linked to that latent construct, the other four items to the Work
Environment construct The curved arrow between the two
constructs denotes a correlational relationship between them.
Finally, e1— e8 represent the errors associated with each
measured item.
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7. Lx1---Lx8
R between
the latent
constructs
and the
respective
measured
items
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8. 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. SEM is stages 5 and 6.
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9. Stage 1: Defining Individual Constructs
• 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.
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10. Rules of Thumb 13–2
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.
• Content validity should be of primary importance and
judged both qualitatively (e.g., expert’s opinions) and
empirically (e.g., unidimensionality and convergent
validity).
• A pre-test should be used to purify measures prior to
confirmatory testing.
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11. Stage 2: Developing the Overall
Measurement Model
 Unidimensionality – no cross loadings
One type of relationship among a variables that impacts
unidimensionality is when researchers allow a single measured
variable to be caused by more than one construct.
• The researcher is seeking a model that produces a good fit.
When one frees another path in a model to be estimated, the
value of the estimated path can only make the model more
accurate. That is, the difference between the estimated and
observed covariance matrices (∑k — S) is reduced unless the two
variables are completely uncorrected.
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12. Between-
construct error
covariance
Within-
construct error
covariance covariance among error terms
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13. Congeneric measurement models are considered to be
sufficiently constrained to represent good measurement properties . A
congeneric measurement model that meets these requirements is
hypothesized to have construct validity and is consistent with good
measurement practice.
 Items per construct
More items (measured variables or indicators) are not necessarily
better. Even though more items do produce higher reliability estimates
and generalizability more items also require larger sample sizes and
can make it difficult to produce truly unidimensional factors.
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14. Stage 2: A Congeneric
Measurement Model
Teamwork
Compensation
Lx1 Lx 5 L6 Lx 8
Lx 4 Lx 7
Lx 3
Lx 2
X5 X6 X7 X8
X1 X2 X3 X4
e5 e6 e7 e8
e1 e2 e3 e4
Each measured variable is related to exactly one construct.
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15. Stage 2: A Measurement
Model Measurement Model Error Hypothesized Cross-Loadings and
Figure 11.2 A that is Not Congeneric
Correlated
with
Variance
Ф21
Compensation Teamwork
λx3,2
λx5,1
λx1,1 λx4,1 λx5,2 λx8,2
λx2,1 λx3,1 λx6,2 λx7,2
X1 X2 X3 X4 X5 X6 X7 X8
δ1 δ2 δ3 δ4 δ5 δ6 δ7 δ8
θδ 2,1
θδ 7,4
Each measured variable is not related to exactly one construct
– errors are not independent.
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16. Under-ideruified
The covariance matrix would be 2 by 2, consisting of one unique
covariance and the variances of the two variables. Thus, there are
three unique values. A measurement model of this construct would
require, however, that two factor loadings (Lx1 and Lx2) and two
error variances (e1and e2) be estimated. Thus, a unique solution
cannot be found.
Just-Identified
Using the same logic, the three-item indicator is just-dentified. This
means that there are just enough degrees of freedom to estimate
all free parameters. All of the information is used, which means that
the CFA analysis will reproduce the sample covariance matrix
identically. Because of this, just-identified models have perfect fit.
the equation for degrees of freedom and you will see that the
resulting degrees of freedom for a three-item factor would be
zero:[3(3+l)/2|-6=0
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18. The dimensionality of any construct with only one or two items
can only be established relative to other constructs.
When specifying the number of indicators per construct, the
following is recommended:
• Use four indicators whenever possible.
• Having three indicators per construct is acceptable, particularly
when other constructs have more than three.
• Constructs with fewer than three indicators should be avoided.
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20. Rules of Thumb 13–3
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.
• Formative factors are not latent and are not validated as are
conventional reflective factors. As such, they present greater
difficulties with statistical identification and should be used
cautiously.
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21. Formative Constructs
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.
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22. STAGE 3: DESIGNING A STUDY TO
PRODUCE EMPIRICAL RESULTS
In this stage the researcher's measurement
theory will be tested.
We should note that initial data analysis
procedures should first be performed to
identify any problems in the data, including
issues such as data input errors.
In this stage the researcher must make some
key decisions on designing the CFA model.

23. • 1-Measurement Scales in CFA
• CFA models typically contain reflective
indicators measured with an ordinal or better
measurement scale. Meaning Indicators with
ordinal responses of at least four response
categories can be treated as interval, or at least
as if the variables are continuous.
• 2-SEM and Sampling.(Many times CFA requires
the use of multiple samples. Meaning
sample(s) should be drawn to perform the CFA.
Even after CFA results are obtained.)

24. 3-Specifying the Model
• distinction between CFA and EFA
• the researcher does not specify cross
loadings, which fixes the loadings at
zero.
• One unique feature in specifying the
indicators for each construct is the
process of "setting the scale" of a
latent factor.

25. 4-Issues in Identification
• overidentification is the desired state
for CFA and SEM models in general.
• During the estimation process, the most
likely cause of the computer program
"blowing up" or producing meaningless
results is a problem with statistical
identification. As SEM models become
more complex.

26. AVOIDING IDENTIFICATION PROBLEMS
(Several guidelines can help determine the
identification status of a SEM model and assist the
researcher in avoiding identification problems)
• Meeting the Order and Rank
Conditions.(required mathematical properties)
• THREE-INDICATOR RULE.(It is satisfied when all
factors in a congeneric model have at least three
significant indicators)
• RECOGNIZING IDENTIFICATION PROBLEMS(Many
times the software programs will provide some
form of solution)

27. SOURCES AND REMEDIES OF
IDENTIFICATION PROBLEMS
Does the presence of identification problems mean
your model is invalid? Although many times
identification issues arise from common mistakes
in specifying the model and the input data.
• Incorrect Indicator Specification. (4 mistakes e.g.)
• "Setting the Scale" of a Construct.(each construct
must have one value specified)
• Too Few Degrees of Freedom.(Small sample size
(fewer than 200) increases the likelihood of
problems )

28. Problems in Estimation
most SEM programs will complete the estimation
process in spite of these issues.
It then becomes the responsibility of the researcher
to identify the illogical results and correct the
model to obtain acceptable results.
• ILLOGICAL STANDARDIZED PARAMETERS. (when
correlation estimates between constructs exceed
|1.0| or even standardized path coefficients exceed
|1.0|. Meaning there is problem with SEM results.
• HEYWOOD CASES A SEM. (solution that produces
an error variance estimate of less than zero (a
negative error variance) is termed a Heywood case.

29. STAGE 4: ASSESSING MEASUREMENT
MODEL VALIDITY
 Once the measurement model is correctly
specified, a SEM model is estimated to provide
an empirical measure of the relationships
among variables and constructs represented by
the measurement theory.
 The results enable us to compare the theory
against reality as represented by the sample
data.
 we see how well the theory fits the data.

30. a-Assessing Fit
The sample data are represented by a
covanance matrix of measured items, and
the theory is represented by the
proposed measurement model. These
equations enable us to estimate reality
by computing an estimated covariance
matrix based on our theory. Fit compares
the two covariance matrices.

31. b-Path Estimates
One of the most fundamental assessments of construct
validity involves the measurement relationships
between items and constructs
• SIZE OF PATH ESTIMATES AND STATISTICAL
SIGNIFICANCE.
loadings should be at least .5 and ideally .7 or higher
meaning Loadings of this size or larger confirm that the
indicators are strongly related to their associated
constructs and are one indication of construct validity.
• IDENTIFYING PROBLEMS.
means(Loadings also should be examined for offending
estimates as indications of overall problems)

32. C- 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
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.

33. Construct Validity
Construct validity is the extent to which a set of measured items
actually reflects the theoretical latent construct those items are
designed to measure.
1- CONVERGENT VALIDITY.
The items that are indicators of a specific construct should converge
• Factor Loadings.
• At a minimum, all factor loadings should be statistically
significant.(standardized loading estimates should be .5 or
higher, and ideally .7 or higher)
• Average Variance Extracted.
• The Li represents the standardized factor
loading, and i is the number of items.
• AVE estimates for two factors also should be greater than the
square of the correlation between the two factors to provide
evidence of discriminant validity.

34. • Reliability.
• Reliability estimate is that .7 or higher
suggests good reliability. Reliability between
.6 and .7 may be acceptable, provided that
other indicators of a model's construct validity
are good.

35. 2- DISCRIMINANT VALIDITY.
the extant to which a construct is truly distinct from
other construct. (The high discriminant validity provides
evidence that a construct is Unique)
3- NOMOLOGICAL VALIDITY AND FACE VALIDITY
(Constructs also should have face validity and
nomological validity)
• face validity: must be established prior to any
theoretical testing when using FA.
• nomological validity: is then tested by examining
whether the corrections among the constructs in a
measurement theory make sense.

36. D- Model Diagnostics
• the process of testing using CFA provides
additional diagnostic information that may
suggest modifications for either addressing
unresolved problems or improving the
model's test of measurement theory.
• Some areas that can be used to identify
problems with measures as following:

37. 1- STANDARDIZED RESIDUALS:
• Residuals: are the individual differences
between observed covariance terms and the
fitted (estimated) covariance terms.
• The standardized residuals: are simply the raw
residuals divided by the standard error of the
residual.
• Residuals: can be either positive or negative,
depending on whether the estimated
covariance is under or over the corresponding
observed covariance.

38. 2- MODIFICATION INDICES:
(is calculated for every possible relationship that
is not estimated in a model)
(of approximately 4.0 or greater suggest that the
fit could be improved significantly) e.g. HBAT
3- SPECIFICATION SEARCHES:
(is an empirical trial-and-error approach that
uses model diagnostics to suggest changes in
the model)
(SEM programs such as AMOS and LISREL can
perform specification searches automatically)

39. 4- CAVEATS IN MODEL RESPECIFICATION:
• CFA results suggesting more than minor
modification should be reevaluated with
a new data set.
• (e.g., if more than 20% of the measured
variables are deleted, then the
modifications cannot be considered
minor)

40. Thanks a lots for
attention
questions ???