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
(Observed) / Indicators / Items
21
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…2 Factors
Orthogonal or Oblique (แต่�ละองค์ประกอบ มี�-ไมี�มี�ค์วามีสั�มีพั�นธ์ก�น)
Measured variables
(Observed) / Indicators / Items
1
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
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: CFA
… 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.
Exploratory versus Confirmatory Strategies
Exploratory versus Confirmatory Strategies
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
Confirmatory Factor Analysis: CFA
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
The Factor Loading or the Structure/Pattern Coefficient
Latent Construct Unmeasured
variables
Errors or Uniqueness
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
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 Measurement Model (Path Model)
X10
10,3
An Example of Confirmatory Factor Analysis (CFA)
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
Parameter estimated = 6
X1 X2 X3X1 1X2 4 2X3 5 6 3
Variance & Covariance Matrix
(No. of Var, Cov) = (No. of Parameter Estimated)Just-identified Model
Parameter estimated = 8
X1 X2 X3 X4X1 1X2 5 2X3 6 7 3X4 8 9 10 4
Overidentified Model
Variance & Covariance Matrix
(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
Stage 4: 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.
• 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.
Stage 4: 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.
• 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)
Effects of Method Bias: Multi-traits Mono-Method (Bias)
Effects of Method Bias: Multi-traits Mono-Method (Bias)
Let’s try
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