요인 개수 결정을 위한 평행분석의 정확성 평가 = Evaluating the Accuracy of Parallel Analysis for Determining the Number of Common Factors

다국어 초록 (Multilingual 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.

국문 초록 (Abstract)

평행분석은 표본 자료의 고윳값과 무선 자료의 고윳값을 비교하여 요인의 개수를 추정하는 방법이다. 이 방법은 고윳값에 근거하여 요인의 개수를 추정하는 다른 절차들(카이저 방법, 스크리 검사)보다 이론적으로 더 타당한 근거를 갖고 있고 경험적으로도 요인의 개수를 더 정확히 추정하는 것으로 평가 받는다. 그러나 평행분석의 이론적 근거의 타당성에 대해서도 여러 비판이 제기되어 왔고 이에 따른 다양한 수정 절차들도 제안되었다. 또한 평행분석이 상대적으로 낮은 정확성을 나타내는 조건들에 대한 우려도 있어 왔다. 본 연구는 고윳값과 평행분석이 요인의 개수를 추정하는 데 사용될 수 있는 이론적 근거와 한계를 검토하고 이를 바탕으로 평행분석의 정확성이 낮게 나타날 수 있는 조건들을 구체화하였다. 또한 모의실험을 통해 평행분석의 정확성에 영향을 줄 수 있는 다양한 요인들의 효과를 검토하고 평행분석의 사용에 주의를 요하는 조건들을 확인하였다. 모의실험의 결과는 평행분석의 요인수 추정 정확률이 요인상관의 크기, 요인부하량의 크기, 요인의 개수, 요인당 변수의 개수에 따라 크게 영향을 받으며 정확률이 낮은 조건에서 표본크기의 영향이 매우 크다는 것을 보였다. 또한 요인부하량이 낮은 변수들로 구성된 약한 요인이 포함된 경우 요인수 추정의 정확률이 크게 낮아짐을 확인하였다. 전반적으로 상관행렬에 대한 평행분석(PA-PCA)보다 축소상관행렬에 대한 평행분석(PA-PAF)의 정확률이 높았으며 특히 요인상관이 높은 경우에는 PA-PAF가, 요인상관이 낮은 경우에는 PA-PCA가 높은 정확률을 보이는 경향이 확인되었다. 마지막으로 모의실험의 결과를 기초로 요인상관의 크기, 요인부하량의 크기, 요인의 개수의 조합으로 구성되는 다양한 조건에서 평행분석이 90% 이상의 정확률을 제공하기 위해 요구되는 표본크기를 제안하였다.

참고문헌 (Reference)

1 이순묵, "탐색적 요인분석: 어떻게 달라지나?" 한국심리학회 35 (35): 217-255, 2016

2 장승민, "리커트 척도 개발을 위한 탐색적 요인분석의 사용" 한국임상심리학회 34 (34): 1079-1100, 2015

3 Green, S. B., "Type I and II error rates and overall accuracy of the revised parallel analysis method for determining the number of factors" 73 : 428-457, 2014

4 Cattell, R. B., "The scree test for the number of factors" 1 : 245-276, 1966

5 Fava, J. L., "The effects of overextraction on factor and component analysis" 27 (27): 387-415, 1992

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

7 Kaiser, H. F., "The application of electronic computers to factor analysis" 20 (20): 141-151, 1960

8 Cochran, W. G., "The analysis of variances when experimental errors follow the Poisson or binomial laws" 11 : 335-347, 1940

9 Bartlett, M. S., "Tests of significance in factor analysis. British Journal of Psychology" 3 : 77-85, 1950

10 Steiger, J. H., "Statistically based tests for the number of common factors" 1980

11 Cohen, "Statistical power analysis for the behavioral sciences" Erlbaum 1988

12 Guttman, L., "Some necessary conditions for common-factor analysis" 19 : 149-161, 1954

13 MacCallum, R. C., "Sample size in factor analysis" 4 : 84-99, 1999

14 O’Connor, B. P., "SPSS and SAS programs for determining the number of components using parallel analysis and Velicer’s MAP test" 32 : 396-402, 2000

15 Preacher, K. J., "Repairing Tom Swifts’s electric factor analysis machine" 2 (2): 13-43, 2003

16 Buja, A., "Remarks on parallel analysis" 27 (27): 509-540, 1992

17 R Core Team, "R: A Language And Environment For Statistical Computing" R Foundation for Statistical Computing

18 Nunnally, J. C., "Psychometric theory" McGraw-Hill 1994

19 Velicer, W. F., "Problems and solutions in human assessment: Honoring Douglas N. Jackson at seventy" Kluwer Academic/Plenum 41-71, 2000

20 MacCallum, R. C., "Power analysis and determination of sample size for covariance structure modeling" 1 : 130-149, 1996

21 Humphreys, L. G., "Note on a criterion for the number of common factors" 29 (29): 571-578, 1969

22 Mulaik, S. A., "Foundations of factor analysis" Cahpman & Hall 2010

23 Hayton, J. C., "Factor retention decisions in exploratory factor analysis : A tutorial on parallel analysis" 7 : 191-205, 2004

24 Floyd, F. J., "Factor analysis in the development and refinement of clinical assessment instruments" 7 (7): 286-, 1995

25 Reise, S. P., "Factor analysis and scale revision" 12 : 287-297, 2000

26 Dinno, A., "Exploring the sensitivity of Horn's parallel analysis to the distributional form of random data" 44 (44): 362-388, 2009

27 Crawford, A. V., "Evaluation of parallel analysis methods for determining the number of factors" 70 (70): 885-901, 2010

28 Fabrigar, L. R., "Evaluating the use of exploratory factor analysis in psychological research" 4 : 272-299, 1999

29 Timmerman, M. E., "Dimensionality assessment of ordered polytomous items with parallel analysis" 16 (16): 209-220, 2011

30 Velicer, W. F., "Determining the number of components from the matrix of partial correlations" 41 : 321-327, 1976

31 Hu, L., "Cutoff criteria for fit indexes in covariance structure analysis : Conventional criteria versus new alternatives" 6 (6): 1-55, 1999

32 Zwick, W. R., "Comparison of five rules for determining the number of components to retain" 99 (99): 432-442, 1986

33 Bentler, P. M., "Comparative fit indexes in structural models" 107 (107): 238-, 1990

34 Preacher, K. J., "Choosing the optimal number of factors in exploratory factor analysis : A model selection perspective" 48 (48): 28-56, 2013

35 Jaeger, T. F., "Categorical data analysis : Away from ANOVAs(transformation or not)and towards logit mixed models" 59 (59): 434-446, 2008

36 Glorfeld, L. W., "An improvement on Horn’s parallel analysis methodology for selecting the correct number of factors to retain" 55 : 377-393, 1995

37 Tucker, L. R., "A reliability coefficient for maximum likelihood factor analysis" 38 (38): 1-10, 1973

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

39 Green, S. B., "A proposed solution to the problem with using completely random data to assess the number of factors with parallel analysis" 72 : 357-374, 2012