On the likelihood ratio test for the number of factors in exploratory factor analysis

Structural Equation Modeling

Volumn 14, Issue 3, 2007, Pages 505-526

  Hayashi, Kentaro ^a   Bentler, Peter M ^b   Yuan, Ke Hai ^c  

^a UNIVERSITY OF HAWAII   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF NOTRE DAME   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

MULTIVARIANT ANALYSIS; PROBABILITY DISTRIBUTIONS; STATISTICAL TESTS;

CHI-SQUARE DISTRIBUTION; EXPLORATORY FACTOR ANALYSIS; LIKELIHOOD RATIO TESTS; MODEL PARAMETERS; NUMBER OF FACTORS; RANK DEFICIENCY;

FACTOR ANALYSIS;

EID: 34548060110     PISSN: 10705511     EISSN: None     Source Type: Journal    
DOI: 10.1080/10705510701301891     Document Type: Article

References (57)

1. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.), The second international symposium on information theory. Budapest: Akadémiai Kiado.
2. Akaike, H. (1987). Factor analysis and AIC. Psychometrika, 52, 317-332.
3. Amemiya, Y., & Anderson, T. W. (1990). Asymptotic chi-square tests for a large class of factor analysis models. The Annals of Statistics, 18, 1453-1463.
4. Bartholomew, D. J., & Knott, M. (1999). Latent variable models and factor analysis (2nd ed.). London: Arnold.
5. Bartlett, M. S. (1950). Tests of significance in factor analysis. British Journal of Psychology (Statistics Section), 3, 77-85.
6. Bartlett, M. S. (1951). The effect of standardisation on an approximation in factor analysis. Biometrika, 38, 337-344.
7. Beauducel, A. (2001). On the generalizability of factors: The influence of changing contexts of variables on different methods of factor extraction. Methods of Psychological Research Online, 6, 69-96.
8. Bekker, P. A., Merckens, A., & Wansbeek, T. J. (1994). Identification, equivalent models, and computer algebra. New York: Academic.
9. Bentler, P. M., & Yuan, K.-H. (1996). Test of linear trend in eigenvalues of a covariance matrix. British Journal of Mathematical and Statistical Psychology, 49, 299-312.
10. Bentler, P. M., & Yuan, K.-H. (1998). Tests for linear trend in the smallest eigenvalues of the correlation matrix. Psychometrika, 63, 131-144.
11. Bentler, P. M., & Yuan, K.-H. (1999). Structural equation modeling with small samples: Test statistics. Multivariate Behavioral Research, 34, 181-197.
12. Bozdogan, H. (Ed.). (1994). Proceedings of the first US/Japan conference on the frontiers of statistical modeling: An informational approach. Boston: Kluwer.
13. Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136-162). Beverly Hills, CA: Sage.
14. Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Pacific Grove, CA: Duxbury.
15. Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1, 245-276.
16. Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. J. (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4, 272-299.
17. Fava, J. L., & Velicer, W. F. (1992). The effects of underextraction in factor and component analysis. Educational and Psychological Measurement, 56, 907-929.
18. Ferguson, T. S. (1996). A course in large sample theory. New York: Chapman & Hall.
19. Fisher, F. M. (1966). The identification problem in economics. New York: McGraw-Hill.
20. Geweke, J. F., & Singleton, K. J. (1980). Interpreting the likelihood ratio statistic in factor models when sample size is small. Journal of the American Statistical Association, 75, 33-137.
21. Gorsuch, R. L. (1983). Factor analysis (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates.
22. Gorsuch, R. L. (2003). Factor analysis. In I. B. Weiner, J. A. Schinka, & W. F. Velicer (Eds.), Handbook of psychology: Vol. 2. Research methods in psychology (pp. 143-164). Hoboken, NJ: Wiley.
23. Guttman, L. (1954). Some necessary conditions for common factor analysis. Psychometrika, 19, 149-162.
24. Hakstian, A. R., Rogers, W. T., & Cattell, R. B. (1982). The behavior of number-of-factor rules with simulated data. Multivariate Behavioral Research, 17, 193-219.
25. Harris, M. L., & Harris, C. W. (1971). A factor analytic interpretation strategy. Educational and Psychological Measurement, 31, 589-606.
26. Harville, D. A. (1997). Matrix algebra from a statistician's perspective. New York: Springer.
27. Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179-185.
28. Hoyle, R. H., & Duvall, J. L. (2004). Determining the number of factors in exploratory and confirmatory factor analysis. In D. Kaplan (Ed.), The Sage handbook of quantitative methodology for the social sciences (pp. 301-315). Thousand Oaks, CA: Sage.
29. Jöreskog, K. G. (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 32, 443-482.
30. Jöreskog, K. G. (1977). Factor analysis by least-squares and maximum-likelihood methods. In K. Enslein, A. Ralston, & H. S. Wilf (Eds.), Statistical methods for digital computers (pp. 125-153). New York: Wiley.
31. Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20, 141-151.
32. Keeling, K. B. (2000). A regression equation for determining the dimensionality of data. Multivariate Behavioral Research, 35, 457-468.
33. Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. Annals of Mathematical Statistics, 22, 79-86.
34. Lawley, D. N., & Maxwell, A. E. (1971). Factor analysis as a statistical method. New York: Elsevier.
35. Linn, R. L. (1968). A Monte Carlo approach to the number of factors problem. Psychometrika, 33, 37-72.
36. MacCallum, R. C., Widaman, K. F., Preacher, K. J., & Hong, S. (2001). Sample size in factor analysis: The role of model error. Multivariate Behavioral Research, 36, 611-637.
37. Magnus, J. R., & Neudecker, H. (1999). Matrix differential calculus with applications in statistics and econometrics (rev. ed.). New York: Wiley.
38. Preacher, K. J., & MacCallum, R. C. (2003). Repairing Tom Swift's electric factor analysis machine. Understanding Statistics, 2, 13-43.
39. Rao, C. R. (1973). Linear statistical inference and its applications (2nd ed.). New York: Wiley.
40. Rothenberg, T. J. (1971). Identification in parametric models. Econometrica, 39, 577-592.
41. Satorra, A., & Bentler, P. M. (1988). Scaling corrections for chi-square statistics in covariance structure analysis. Proceedings of the American Statistical Association, 308-313.
42. Schönemann, P. H., & Wang, M. M. (1972). Some new results on factor indeterminacy. Psychometrika, 37, 61-91.
43. Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461-464.
44. Sen, P. K., & Singer, J. M. (1993). Large sample methods in statistics: An introduction with applications. New York: Chapman & Hall.
45. Shapiro, A. (1986). Asymptotic theory of overparameterized structural models. Journal of the American Statistical Association, 81, 142-149.
46. Shapiro, A., & Browne, M. W (1983). On the investigation of local identifiability: A counterexample. Psychometrika, 48, 303-304.
47. Steiger, J. H. (1990). Structural model evaluation and modification: An interval estimation approach. Multivariate Behavioral Research, 25, 173-180.
48. Steiger, J. H., & Lind, J. C. (1980, May). Statistically-based tests for the number of common factors. Paper presented at the annual spring meeting of the Psychometric Society, Iowa City, IA.
49. Takane, Y., van der Heijden, P. G. M., & Browne, M. W (2003). On likelihood ratio tests for dimensionality selection. In T. Higuchi, Y Iba, & M. Ishiguro (Eds.), Proceedings of science of modeling: The 30th anniversary meeting of the information criterion (AIC) (pp. 348-349). Tokyo: The Institute of Statistical Mathematics.
50. Tucker, L. R., Koopman, R. F., & Linn, R. L. (1969). Evaluation of factor analytic research procedures by means of simulated correlation matrices. Psychometrika, 34, 421-459.
51. Velicer, W. F., Eaton, C. A., & Fava, J. L. (2000). Construct explication through factor or component analysis: A review and evaluation of alternative procedures for determining the number of factors or components. In R. D. Goffin & E. Helmes (Eds.), Problems and solutions in human assessment: A festschrift to Douglas Jackson at seventy (pp. 47-71). Norwell, MA: Kluwer.
52. Wilks, S. S. (1938). The large sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 9, 60-62.
53. Yanai, H., Shigemasu, K., Maekawa, S., & Ichikawa, M. (1990). Factor analysis: Its theory and methods. Tokyo: Asakura-shoten.
54. Yuan, K.-H., & Bentler, P. M. (2000). Three likelihood-based methods for mean and covariance structure analysis with nonnormal missing data. Sociological Methodology, 30, 165-200.
55. Yuan, K.-H., & Hayashi, K. (2003). Bootstrap approach to inference and power analysis based on three statistics for covariance structure models. British Journal of Mathematical and Statistical Psychology, 56, 93-110.
56. Yuan, K.-H., Marshall, L. L., & Bentler, P. M. (2002). A unified approach to exploratory factor analysis with missing data, nonnormal data, and in the presence of outliers. Psychometrika, 67, 95-122.
57. Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99, 432-442.