Presentation of cfa 13 by Mahfoudh Mgammal

Confirmatory Factor Analysis Overview

Prof.. Dr. AllaEldin Hassan Kassam

ADVANCED QUANTITATIVE ANALYSIS BDMR8043

Mahfoudh Hussein Hussein Mgammal

Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall. 13-2

• 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.

Confirmatory Factor Analysis Overview


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.

Confirmatory Factor Analysis Defined


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.

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.


• 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.



Correlational

indicator variables

Errors associated

Lx1---Lx8R between the latent constructs

and the respective measured

items


Confirmatory Factor Analysis Stages

Stage 1: Defining Individual Constructs Stage 2: Developing the Overall Measurement ModelStage 3: Designing a Study to Produce Empirical ResultsStage 4: Assessing the Measurement Model Validity

Stage 5: Specifying the Structural ModelStage 6: Assessing Structural Model Validity

Note: CFA involves stages 1 – 4 above. SEM is stages 5 and 6.


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.


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.


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.


Within-construct error

covariance

Between-construct error

covariance

covariance among error terms


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.


Stage 2: A Congeneric Measurement Model

Compensation

X1 X2 X3 X4

e1 e2 e3 e4

Lx1

Lx 2

Lx 3

Lx 4

Teamwork

X5 X6 X7 X8

e5 e6 e7 e8

Lx 5 L 6 Lx 7Lx 8

Each measured variable is related to exactly one construct.


X5 X6 X7 X8

δ5 δ6 δ7 δ8

λx5,2 λx6,2 λx7,2

λx8,2

X1 X2 X3 X4

λx1,1 λx2,1 λx3,1

λx4,1

λx3,2λx5,1

δ1 δ2 δ3 δ4

Ф21

θδ 2,1 θδ 7,4

Figure 11.2 A Measurement Model with Hypothesized Cross-Loadings and Correlated Error Variance

Each measured variable is not related to exactly one construct – errors are not independent.

Stage 2: A Measurement Model that is Not Congeneric

Compensation Teamwork


Under-ideruifiedThe 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-IdentifiedUsing 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



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.



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.


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.

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.

• 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.)

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.

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.

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)

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 )

Problems in Estimationmost 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.

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.

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.

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)

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.

Construct ValidityConstruct 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.

• 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.

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.

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:

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.

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. HBAT3- 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)

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)

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Presentation of cfa 13 by Mahfoudh Mgammal