Evaluating the Accuracy of Parallel Analysis for Determining the Number of Common Factors

& (2017). Evaluating the Accuracy of Parallel Analysis for Determining the Number of Common Factors. Korean Journal of Psychology: General, 36(4), 441-475, https://doi.org/10.22257/kjp.2017.09.36.3.441

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Abstract

Parallel analysis is a method of estimating the number of factors by comparing the eigenvalues of sample data with the eigenvalues of random data. This method is considered to be theoretically more valid and empirically more accurate in estimating the number of factors than other methods, such as Kaiser method and scree test, that estimate the number of factors based on the eigenvalues. However, several criticisms have been raised about the validity of the rationale for parallel analysis and various modifications have been proposed. There have also been concerns about the conditions under which parallel analysis shows relatively low accuracy. The current study examined the rationale and limitations of the use of eigenvalues and parallel analysis to estimate the number of factors, and based on this, we specified the conditions under which the accuracy of parallel analysis may be low. We also examined, through a simulation, the effects of various factors that may affect the accuracy of parallel analysis and confirmed the conditions where cautions are needed when applying parallel analysis. The results of the simulation show that the accuracy of estimating the number of factors in the parallel analysis is greatly influenced by the size of factor correlations, the magnitude of factor loadings, the number of factors, and the number of variables per factor. In addition, we confirmed that the accuracy of the parallel analysis is significantly lower when a factor model includes a weak factor with low factor loadings. Overall, the accuracy of the parallel analysis for the reduced correlation matrix (PA-PAF) was higher than the parallel analysis for the correlation matrix (PA-PCA), which in particular, PA-PAF showed high accuracy when factor correlations were high, and PA-PCA showed high accuracy when factor correlations were low. Based on the results of the simulation analyses, we proposed sample sizes required for parallel analysis to provide accuracy of 90% or higher under conditions with different levels of factor correlation, factor loading, and the number of factors.

keywords: parallel analysis, number of factors, eigenvalues, reduced correlation matrix, sample size, 평행분석, 요인의 개수, 고윳값, 축소상관행렬, 표본크기

Reference

이순묵, 윤창영, 이민형, 정선호 (2016). 탐색적요인분석: 어떻게 달라지나? 한국심리학회지: 일반, 35(1), 217-255.

장승민 (2015). 리커트 척도개발을 위한 탐색적 요인분석의 활용, 한국심리학회지: 임상, 34(4), 1079-1100.

Bartlett, M. S. (1950). Tests of significance in factor analysis. British Journal of Psychology, Statistical Section, 3, 77-85.

Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107(2), 238.

Buja, A., & Eyuboglu, N. (1992). Remarks on parallel analysis. Multivariate Behavioral Research, 27(4), 509-540.

Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1, 245-276.

Cochran, W. G. (1940). The analysis of variances when experimental errors follow the Poisson or binomial laws. The Annals of Mathematical Statistics, 11, 335-347.

Cohen. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ:Erlbaum.

Crawford, A. V., Green, S. B., Levy, R., Lo, W. J., Scott, L., Svetina, D., & Thompson, M. S.(2010). Evaluation of parallel analysis methods for determining the number of factors. Educational and Psychological Measurement, 70(6), 885-901.

10.

Dinno, A. (2009). Exploring the sensitivity of Horn's parallel analysis to the distributional form of random data. Multivariate Behavioral Research, 44(3), 362-388.

11.

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.

12.

Fava, J. L., & Velicer, W. F. (1992). The effects of overextraction on factor and component analysis. Multivariate Behavioral Research, 27(3), 387-415.

13.

Floyd, F. J., & Widaman, K. F. (1995). Factor analysis in the development and refinement of clinical assessment instruments. Psychological Assessment, 7(3), 286.

14.

Glorfeld, L. W. (1995). An improvement on Horn’s parallel analysis methodology for selecting the correct number of factors to retain. Educational and Psychological Measurement, 55, 377-393.

15.

Green, S. B., Levy, R., Thompson, M. S., Lu, M., & Lo, W.-J. (2012). A proposed solution to the problem with using completely random data to assess the number of factors with parallel analysis. Educational and Psychological Measurement, 72, 357-374.

16.

Green, S. B., Thompson, M. S., Levy, R., & Lo, W.-J. (2014). Type I and II error rates and overall accuracy of the revised parallel analysis method for determining the number of factors. Educational and Psychological Measurement, 73, 428-457.

17.

Guttman, L. (1954). Some necessary conditions for common-factor analysis. Psychometrika, 19, 149-161.

18.

Hayton, J. C., Allen, D. G., & Scarpello, V.(2004). Factor retention decisions in exploratory factor analysis: A tutorial on parallel analysis. Organizational Research Methods, 7, 191-205.

19.

Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179-185.

20.

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis:Conventional criteria versus new alternatives. Structural Equation Modeling: A Multidisciplinary Journal, 6(1), 1-55

21.

Humphreys, L. G., & Ilgen, D. R. (1969). Note on a criterion for the number of common factors. Educational and Psychological Measurement, 29(3), 571-578.

22.

Jaeger, T. F. (2008). Categorical data analysis:Away from ANOVAs (transformation or not)and towards logit mixed models, Journal of Memory and Language, 59(4), 434-446.

23.

Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20(1), 141-151.

24.

MacCallum, R. C., Browne, M., & Sugawara, H.,(1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1, 130-149.

25.

MacCallum, R. C., Widaman, K. F., & Zhang, S., & Hong, S. (1999). Sample size in factor analysis. Psychological Methods, 4, 84-99.

26.

Mulaik, S. A. (2010). Foundations of factor analysis (2nd ed.). Boca Raton, FL: Cahpman & Hall.

27.

Nunnally, J. C., & Bernstein, I. H. (1994). Psychometric theory (3rd ed.). New York:McGraw-Hill.

28.

O’Connor, B. P. (2000). SPSS and SAS programs for determining the number of components using parallel analysis and Velicer’s MAP test. Behavior Research Methods, Instrumentation, and Computers, 32, 396-402.

29.

Preacher, K. J., & MacCallum, R. C. (2003). Repairing Tom Swifts’s electric factor analysis machine. Understanding Statistics, 2(1), 13-43.

30.

Preacher, K. J., Zhang, G., Kim, C., & Mels, G.(2013). Choosing the optimal number of factors in exploratory factor analysis: A model selection perspective. Multivariate Behavioral Research, 48(1), 28-56.

31.

R Core Team (2017). R: A Language And Environment For Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.

32.

Reise, S. P., Waller, N. G., & Comrey, A. L.(2000). Factor analysis and scale revision. Psychological Assessment, 12, 287-297.

33.

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

34.

Timmerman, M. E. & Lorenzo-Seva U. (2011). Dimensionality assessment of ordered polytomous items with parallel analysis. Psychological Methods, 16(2), 209-220.

35.

Tucker, L. R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38(1), 1-10.

36.

Turner, N. E. (1998). The effect of common variance and structure pattern on random data eigenvalues: Implications for the accuracy of parallel analysis. Educational and Psychological Measurement, 58(4), 541-568.

37.

Velicer, W. F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321-327.

38.

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 E. Helmes (Ed.), Problems and solutions in human assessment: Honoring Douglas N. Jackson at seventy. (pp. 41-71). New York, NY: Kluwer Academic/Plenum.

39.

Zwick, W. R. & Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99(3), 432-442.

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Article Contents

Vol.36 No.4

Evaluating the Accuracy of Parallel Analysis for Determining the Number of Common Factors

Abstract

Reference

Korean Journal of Psychology: General