Confirmatory Factor Analysis Fit Statistics

Nicola Ritter, M.Ed.

EPSY 643: Multivariate Methods

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

Take Away Points

1. Researchers should consult several fit statistics when evaluating model fit.

2. There are similarities and differences between all fit statistics.

3. Sample size impacts the chi-square statistic.

4. There are numerous fit statistics.

5. Fit statistics are estimated using a covariance matrix.

5. Fit statistics are estimated using a covariance matrix.

• Analyze matrix of associations (i.e. covariance matrix)

• Recall: Pattern coefficients are the weightsPVxF PFxV

’ = RVxV

+

RVxX - RVxV+ = RVxV

- Rodrigo Jimenez

Factor pattern coefficients to Fit Evaluation

• If PVxF perfectly reproduces RVxV+ then,

1) RVxV- = 011...01c

0r1…0rc

AND

2)RVxV+ = RVxX

No information or variance left in the residual matrix

4. There are numerous fit statistics.

Most Common Fit Statistics

1. Χ² statistical significance test2. Normed fit index (NFI; Bentler & Bonnett, 1980)

3. Comparative fit index (CFI; Bentler, 1990)

4. Root mean-square error of approximation (RMSEA; Steiger & Lind, 1980)

Chi-squared statistical significance test

• Compares sample matrix and reproduced matrix

H0: RVxV = RVxV+

H0: COVVxV = COVVxV+

• Here we do not want to reject the null hypothesis (i.e., not statistically significant) for models that we like.

Degrees of Freedom in Χ² test

• Function of the number of measured variables (n) and number of estimated parameters

dfTOTAL = n (n+1) / 2

• Suppose: 6 variables

dfTOTAL = 6 (6+1) / 2 = 21

• If 6 factor pattern coefficients, 6 error variances, and 1 factor covariance are estimated then:

dfMODEL = dfTOTAL - # of estimated parameters

dfMODEL = 21 - 13 = 8

3. Sample size impacts the chi-square statistic.

Limitation of Χ² test• Biased when:

– MLE is used

– Multivariate normality assumption is not met• NOTE: Satorra & Bentler (1994) propose a correction

• Influenced by sample size, not useful in evaluating the fit of a single model– Demonstrate in AMOS

• Location of Fit Statistics in Output

• Change in Χ² and pcalc

• No change in parameters and fit statistics

Comparison with Varying Sample Sizes

Table 1.

n=1000 n=2000 n=2969

Χ² 16.915 57.799 97.398

df 8 8 8

pcalc 0.0310064488 0.0000000013 0.0000000000

Total # of parameters 21 21 21

Toal # of estimated parameters 13 13 13

NFI (≥ 0.95 -> reasonable fit) 0.997 0.994 0.993

CFI (≥ 0.95 -> reasonable fit) 0.998 0.995 0.994

RMSEA (≤ 0.06 -> reasonable fit) 0.033 0.056 0.061

Strength of Χ² test

• Helpful when comparing nested models

Model A

Model B1

1

1

1

1

1

Normed Fit Index (NFI; Bentler & Bonnett, 1980)

• Compares Χ²TESTED MODEL to Χ²BASELINE MODEL

• Assumes measured variables are uncorrelated.

• Min: 0 Max: 1.0• NFI ≥ 0.95 -> reasonable fit

"Bad Model" Good ModelBaseline model "Ideal Model"

Comparative Fit Index (CFI; Bentler, 1990)

• Compares Χ²TESTED MODEL to Χ²BASELINE MODEL

• Assumes noncentral Χ² distribution• Min: 0 Max: 1.0• CFI ≥ 0.95 -> reasonable fit

"Bad Model" Good ModelBaseline model "Ideal Model"

Root-mean-square error of approximation (RMSEA; Steiger & Lind, 1980)

• Compares sample COV matrix and population COV matrix

• Assumes measured variables are uncorrelated. (Bentler & Bonett, 1980)

• When:

Sample COV matrix = population COV matrix

RMSEA = 0• RMSEA ≤ 0.06 -> reasonable fit

Strength of RMSEA

• Relatively minimal influence by sample size

• Not overly influenced by estimation methods

• Sensitive to model misspecification (Fan, Thompson, & Wang, 1999)

2. There are similarities and differences between all fit statistics.

Table 2      NFI CFI RMSEA

NFI

Compares Χ²TESTED MODEL to Χ²BASELINE MODEL

Assumes measured variables are uncorrelated.

CFI

Assumes noncentral Χ² distribution

RMSEA

Compares sample COV matrix and population COV matrix

1. Researchers should consult several fit statistics when evaluating model fit.

• Fit indices were developed with different rationales.

• No single index will meet all our expectations for an ideal index

(Fan, Thompson, & Wang, 1999)

Take Away Points

1. Researchers should consult several fit statistics when evaluating model fit.

2. There are similarities and differences between all fit statistics.

3. Sample size impacts the chi-square statistic.

4. There are numerous fit statistics.

5. Fit statistics are estimated using a covariance matrix.

References

Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238-246.

Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88, 588-606.

Fan. X., Thompson, B.. & Wang. L. (1999). Effects of sample size, estimation methods, and model specification on structural equation modeling fit indices. Structural Equation Modeling. 6, 56-83.

Satorra, A. & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structural analysis. In A. von Eye & C. C. Clogg (Eds.), Latent variable analysis: Applications for developmental research (pp. 399-419). Thousand Oaks, CA: Sage.

Steiger, J. H. & Lind, J. C. (1980, June). Statistically based test for the number of common factors. Paper presented at the annual meeting of the Psychometric Society, Iowa City, IA.

Sun, J. (2005). Assessing goodness of fit in confirmatory factor analysis. Measurement and Evaluation in Counseling and Development, 37, 240- 256.

Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. Washington, DC: American Psychological Association. (International Standard Book Number: 1-59147-093-5)