22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9 2,1 3,1 3,2 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

Confirmatory Factor Analysis

Sakesan Tongkhambanchong, Ph.D.(Applied Behavioral Sciences Research)

2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9

2,1

3,1

3,2

2,11,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

Exploratory Factor Analysis: EFA

… it is exploratory in the sense that researchers adopt the inductive strategy of determining the factor structure empirically. (bottom-up Strategy)

… researcher allow the statistical procedure to examine the correlations between the variables and to generate a factor structure based on those relationships.

…from the perspective of the researchers at the start of the analysis, any variable may be associated with any component or factor.

Exploratory vs. Confirmatory Strategies

Exploratory Factor Analysis: EFA

… in EFA the researcher has little or no knowledge about the factor structure regarding:1. The number of factors or dimensions of the

constructs.2. Whether these dimensions are orthogonal or

oblique.3. The number of indicators of each factor.4. Which indicators represent which factor.

… there is very little theory that can be used for answering the questions. The researcher may collect data and explore or search for a factor structure or theory which can explain the correlation among the indicators.

Exploratory vs. Confirmatory Strategies

Measured variables

(Observed) / Indicators / Items

2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9

The Factor Loading or the Structure/Pattern Coefficient

Factor structure / Component / Dimensions / Unmeasured

variables

An Exploratory Factor Model (EFA)

Errors or Uniqueness

May be…3 Factors

Orthogonal or Oblique (แต่�ละองค์ประกอบ มี�-ไมี�มี�ค์วามีสั�มีพั�นธ์ก�น)

Measured variables


21

X1 X2 X3 X4 X5 X6 X7 X8 X9



variables



May be…2 Factors

Orthogonal or Oblique (แต่�ละองค์ประกอบ มี�-ไมี�มี�ค์วามีสั�มีพั�นธ์ก�น)

Measured variables


1

X1 X2 X3 X4 X5 X6 X7 X8 X9



variables



Or…may be…1 Factors

An Exploratory Factor Analytic Model (Based on Covariance)

A1

A2

A3A

B1

B2

B3B

A4

B4

B5

B6

A1

A2

A3

A5

A6

A7

A

A4

A8

A9

A10

A1

A2

A3

A

B2

B3

B4

B

B1

C1

C2

C3

C

One-Factor Model

Two-Factor Model

Three-Factor Model

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

Confirmatory Factor Analysis: CFA

… Confirmatory factor analysis, by contrast, requires researchers to use a deductive strategy. (Top-down Approach)

… within this strategy, the factors and the variables that are held to represent them are postulated at the beginning of the procedure rather than emerging from the analysis.

… the statistical procedure is then performed to determine how well this hypothesized theoretical structure fits the empirical data.

Exploratory versus Confirmatory Strategies


… Confirmatory factor analysis, assumes that the structure is known or hypothesized a priori.

Ex. Psychological Construct X is hypothesized as a general factor with three subdimensions or subfactors. Each of these subdimensions is measured by its respective 3-indicators.

The indicators are measures of one and only one factor. The complete factor structure along with the respective indicators and the nature of the pattern loadings is specified a priori.

…The objective is to empirically verify or confirm the factor structure.



Psychological Construct X with 3 subdimensions or subfactors

2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9

2,1

3,1

3,2

2,11,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

Objectives of Confirmatory Factor Analysis


Given the sample covariance matrix, to estimate the parameters of the hypothesized factor model.

To determine the fit of the hypothesized factor model. That is, how close is the estimated covariance matrix: , to the sample covariance matrix: S ?

Measured variables (Observed) /

Indicators / Items

2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9


Latent Construct Unmeasured

variables


A Confirmatory Factor Analytic Model (CFA)-Based on Theory

2,1

3,1

3,2

Some Errors are correlated

Some Factors are correlated/ Some Factors are not correlated

2,11,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

An Example of Confirmatory Factor Analysis (CFA)

2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9

2,1

3,1

3,2

2,11,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

Hypothesized Model of Justice Model

X10

10,3

Some factors are correlated Some factors are not correlated

Uniqueness or Error terms are not Independent (correlated)

Confirmatory Factor Analysis StagesStage 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

*Note: CFA involves stages 1 – 4 above.

Stage 5: Specifying the Structural Model

Stage 6: Assessing Structural Model Validity

SEM is stages 1-4 and 5, 6.

A Seven steps process for Analyzing CFA

Develop a Theoretically Based Model

Construct a Path Diagram

(Factor Model)

Convert the Path Diagram

Choose the Input Matrix type

Correlation matrix

Covariance matrix

Research Design Issue

Assess the Identification of

the Model

Evaluate model Estimates

Evaluate model Goodness-of-fit

Model Interpretation

Model Modification

Final Model


2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9

2,1

3,1

3,2

2,11,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

Hypothesized Measurement Model (Path Model)

X10

10,3


2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9

2,1 = 0.52

3,1 =

0.71

3,2 =

0.47

Last Trimming Model of Justice Model

X10

CR = .782VE = .473

CR = .600VE = .449

CR = .823VE = .540

.62 .71 .72

.68

.79

.92

.67 .70 .75 .81

.616 .496 .482 .538 .370 .160 .550 .510 .440 .340

.36

.35

.13

-.11

Result of Analysis with LISREL program

Measured variables (Observed) /

Indicators / Items

2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9

First order Factor

Errors or Uniquenesses

Alternative Model: Second-order CFA Model

2,13,1 3,1

Some Errors are correlated

Some Factors are correlated/ Some Factors are not correlated

2,11,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

1 Second order Factor

Analysis of CFA with LISREL

Sakesan Tongkhambanchong, Ph.D.(Applied Behavioral Sciences Research)

2 31

X1 X2 X3 X4 X5 X6 X7 X8 X9

2,1

3,1

3,2

2,11,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3

• 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.

Stage 1: Defining Individual Constructs

A

B

C

D

E

F

IQ2

1

Hypothesized Measurement Model: Two-Factor model of IQ

IQ1

1

2,1

1,1

3,1

4,2

5,2

6,2

Hypothesized Measurement Model of IQ

Stage 2: Developing the Overall Measurement Model

Key Issues . . .

• Unidimensionality – no cross loadings• Congeneric measurement model – no covariance

between or within construct error variances• Items per construct – identification• Reflective vs. formative measurement models

A

B

C

D

E

F

IQ2

1

CFA Model: Two-Factor model

IQ1

1Congeneric measurement model:

no covariance (correlation) between or within construct

error variances

Unidimensionality: No cross-loading

Reflective measurement models

Congeneric measurement model: Each measured variable is related to exactly one construct

A

B

C

D

E

F

IQ2

1

IQ1

1

CFA Model: Two-Factor model with correlate factor

Cross-loading

covariance

between construct

error variances

Covariance within

construct error

variances

measurement model is Not Congeneric : Each measured variable is not related to exactly one construct /errors are

not independent

1 2

Model Identifications: Underidentified, Just-identified & Over-identified

1

X1 X2

1

X1 X2 X3

1

X1 X2 X3

X4

1 2 3 4

5 6 7 8

3 4

1 2 3

4 5 6

Parameter estimated = 4

X1 X2X1 1X2 3 2

(No. of Var, Cov) < (No. of Parameter Estimated)

Variance & Covariance Matrix

Underidentified Model


X1 X2 X3X1 1X2 4 2X3 5 6 3


(No. of Var, Cov) = (No. of Parameter Estimated)Just-identified Model


X1 X2 X3 X4X1 1X2 5 2X3 6 7 3X4 8 9 10 4

Overidentified Model


(No. of Var, Cov) > (No. of Parameter Estimated)

Stage 2: Developing the Overall Measurement Model

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.

Stage 3: Designing a Study to Produce Empirical Results

• The ‘scale’ of a latent construct can be set by either:

• Fixing one loading and setting its value to 1, or

• Fixing the construct variance and setting its value to 1.

• Congeneric, reflective measurement models in which all constructs have at least three item indicators are statistically identified in models with two or more constructs.

• 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 4: Assessing Measurement Model Validity

• Assessing fit – GOF indices and path estimates (significance and size)

• Construct validity

• Diagnosing problems

• Standardized residuals

• Modification indices (MI)

• Specification searches


• 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.

• Standardized residuals greater than |4.0| suggest a potentially unacceptable degree of error that may call for the deletion of an offending item.

• 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 items.


• The researcher should use the modification indices only as a guideline for model improvements of those relationships that can theoretically be justified.

• CFA results suggesting more than minor modification should be re-evaluated with a new data set (e.g., if more than 20% of the measured variables are deleted, then the modifications can not be considered minor).

A

B

C

D

E

F

IQ2

1

Hypothesized Measurement Model: Two-Factor model of IQ

IQ1

1

2,1

1,1

3,1

4,2

5,2

6,2

Hypothesized Measurement Model of IQ

2,2

1,1

3,3

4,4

5,5

6,6

2,1

Measurement Model of IQ & LISREL Matrix (LX, PH, TD)

LX (NX*NK) (6*2)IQ1 IQ2

A 1,1 1,2B 2,1 2,2C 3,1 3,2D 4,1 4,2E 5,1 5,2F 6,1 6,2

IQ1 IQ2A 0 0B 1 0C 1 0D 0 0E 0 1F 0 1

PA LX

Full Matrix (FU)

PH (NK*NK) (2*2)IQ1 IQ2

IQ1 1,1IQ2 2,2

PH IQ1 IQ2IQ1 1IQ2 0 1

Symetry Matrix (SY)

PA PH

TD (NX*NX) (6*6)A B C D E F

A 1,1B 0 2,2C 0 0 3,3D 0 0 0 4,4E 0 0 0 0 5,5F 0 0 0 0 0 6,6

A B C D E FA 1B 0 1C 0 0 1D 0 0 0 1E 0 0 0 0 1F 0 0 0 0 0 1

Symetry Matrix (SY)

PA TD

Measurement Model of IQ & Data (Input Matrix)

rx y A B C D E FA 1.000B 0.620 1.000C 0.540 0.510 1.000D 0.320 0.380 0.360 1.000E 0.284 0.351 0.336 0.686 1.000F 0.370 0.430 0.405 0.730 0.735 1.000

Correlation Matrix

Fitting Models & Data

Measurement Model of IQ & LISREL Syntax

Result of Analysis (LISREL Path Model)

Result of Analysis (Goodness-of-Fit Index)

Effects of Method Bias: Multi-traits Mono-Method (Bias)



Let’s try

Documents

22 22 33 33 11 11 X1 X2 X3 X4 X5 X6 X7 X8 X9 2,1 3,1 3,2 2,1 1,1 3,1 4,2 5,2 6,2 7,3 8,3 9,3