Restricted accessResearch articleFirst published online 2001-06
Modeling the Observation-to-Variable Ratio Necessary for Determining the Number of Factors by the Standard Error Scree Procedure Using Logistic Regression
Logistic regression was used for modeling the observation-to-variable (n/v) ratio required for the standard error scree (SEscree) procedure to correctly identify the number of factors in simulated data. The correlation matrices were generated to possess known characteristics: number of factors (f), number of variables (v), sample size (n), magnitude of pattern coefficients (p), and degree of interfactor correlations (r). The results indicated that under all conditions, the n/v ratio required for the SEscree procedure to correctly identify the true number of factors with high probability exceeded the minimum of 5:1 recommended in some of the related literature. This study demonstrated the ability of the logistic regression to simplify summarizing and reporting findings from simulation studies that involve a large number of conditions.
Get full access to this article
View all access options for this article.
References
1.
Arrindell, W. A. , & van der Ende, J. (1985). An empirical test of the utility of the observation-to-variables ratio in factor and component analysis. Applied Psychological Measurement, 9, 15-178.
2.
Barrett, P. T. , & Kline, P. (1981). The observation to variable ratio in factor analysis. Personality Study and Group Behavior, 1, 23-33.
3.
Bentler, P. M. (1989). EQS structural equations program manual. Los Angeles: BMDP Statistical Software.
4.
Bentler, P. M. , & Kano, Y. (1990). On the equivalence of factors and components. Multivariate Behavioral Research, 25, 67-74.
5.
Bollen, K. A. (1989). Structural equations with latent variables. New York: John Wiley.
6.
Boomsma, A. (1982). The robustness of LISREL against small sample sizes in factor analysis models. In K. G. Jöreskog & H. Wold (Eds.), Systems under indirect observation (Part 1, pp. 149-173). Amsterdam, the Netherlands: North Holland.
7.
Cattell, R. B. (1952). Factor analysis: An introduction and manual for the psychologist and the social scientist. New York: Harper & Brothers.
8.
Cattell, R. B. (1978). The scientific use of factor analysis in behavioral sciences. New York: Plenum.
9.
Cliff, N. (1987). Analyzing multivariate data. San Diego, CA: Harcourt Brace Jovanovich.
10.
Cochran, W. G. (1977). Sampling techniques (3rd ed.). New York: John Wiley.
11.
Cohen, J. (1988). Statistical power analysis for behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
12.
Comrey, A. L. (1978). Common methodological problems in factor analytic studies. Journal of Counseling and Clinical Psychology, 46, 648-659.
13.
Comrey, A. L. , & Lee, H. B. (1992). A first course in factor analysis (2nd ed.). New York: Academic Press.
14.
Crawford, A. L. (1975). Determining the number of interpretable factors. Psychological Bulletin, 82, 226-237.
15.
Fava, J. L. , & Velicer, W. F. (1996). The effects of underextraction in factor and component analyses. Educational and Psychological Measurement, 56, 907-929.
16.
Gorsuch, R. L. (1983). Factor analysis (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
17.
Green, S. B. (1991). How many subjects does it take to do regression analysis?Multivariate Behavioral Research, 26, 499-510.
18.
Guadagnoli, E. , & Velicer, W. F. (1988). Stability of component patterns: A simulation study. Psychological Bulletin, 103, 265-275.
19.
Guertain, W. H. , & Bailey, J. P. (1970). Introduction to modern factor analysis. Ann Arbor, MI: Edwards Brothers.
20.
Hakstian, A. R. , & Muller, V. J. (1973). Some notes on the number of factors problem. Multivariate Behavioral Research, 8, 461-475.
21.
Hakstian, R. A. , Rogers, T., & Cattell, R. (1982). The behavior of number-of-factors rules with simulated data. Multivariate Behavioral Research, 17, 193-219.
22.
Hays, W. L. (1988). Statistics (4th ed.). New York: Holt, Rinehart & Winston.
23.
Horn, J. L. (1965). A rationale and test of the number of factors in factor analysis. Psychometrika, 30, 179-185.
24.
Horn, J. L. , & Engstrom, R. (1979). Cattell’s scree test in relation to Bartlett’s chi-square test and other observations on the number of factors problem. Multivariate Behavioral Research, 14, 283-300.
25.
Howell, D. C. (1987). Statistical methods for psychology (2nd ed.). Boston: Duxbury.
26.
Huberty, C. J. (1994). Applied discriminant analysis. New York: John Wiley.
27.
Jaeger, R. M. (1984). Sampling in education and the social sciences. New York: Longman.
28.
Kaiser, H. F. , & Dickman, K. (1962). Sample and population score matrices and sample correlation matrices from arbitrary population correlation matrices. Psychometrika, 27, 179-182.
29.
Kirk, R. E. (1982). Experimental design: Procedures for the behavioral sciences (2nd ed.). Belmont, CA: Brooks/Cole.
30.
Kish, L. (1975). Representation, randomization, and control. In H. M. Blalock, A. Aganbegian, F. M. Borodkin, R. Roudon, & V. Capecchi (Eds.), Quantitative sociology: International perspectives on mathematical and statistical modeling (pp. 261-284). New York: Academic Press.
31.
Kraemer, H. C. , & Thiemann, S. (1987). How many subjects? Statistical power analysis in research. Newbury Park, CA: Sage.
32.
Kunce, T. J. , Cook, W. D., & Miller, E. D. (1975). Random variables and correlational overkill. Educational and Psychological Measurement, 35, 529-534.
33.
Lambert, Z. V. , Wildt, A. R., & Durand, R. M. (1990). Assessing sampling variation relative to number-of-factors criteria. Educational and Psychological Measurement, 50, 33-48.
34.
MacCallum, R. (1983). A comparison of factor analysis programs in SPSS, BMDP, and SAS. Psychometrika, 48, 223-231.
35.
Marascuilo, L. , & Levin, J. R. (1983). Multivariate statistics in the social sciences. Monterey, CA: Brooks/Cole.
36.
Marsh, H. W. , Hau, K.-T., Balla, J. R., & Grayson, D. (1998). Is more ever too much? The number of indicators per factor in confirmatory factor analysis. Multivariate Behavioral Research, 33, 181-220.
37.
Nasser, F. (1997). The performance of regression-based variations of the scree procedure for determining the number of common factors. Dissertation Abstracts International, 58-08, A3003-A3003.
38.
Nunnally, J. C. (1978). Psychometric theory (2nd ed.). New York: McGraw-Hill.
39.
Pearson, E. S. , & Hartley, H. O. (1951). Charts of power function for analysis of variance tests derived from non-central F distribution. Biometrika, 38, 112-130.
40.
Pedhazur, J. E. , & Pedhazur-Schmelkin, L. (1991). Measurement, design, and analysis: An integrated approach. Hillsdale, NJ: Lawrence Erlbaum.
41.
Rotton, J. , & Schönemann, P. H. (1978). Power tables for analysis of variance. Educational and Psychological Measurement, 38, 213-229.
42.
Rummel, R. J. (1970). Applied factor analysis. Evanston, IL: Northwestern University Press.
43.
SAS Institute . (1989). SAS/IML software: Usage and reference (Version 6, 1st ed.). Cary, NC: Author.
44.
Tanaka, S. T. (1987). “How big is big enough?” Sample size and goodness of fit in structural equation models with latent variables. Child Development, 58, 134-146.
45.
Thompson, B. , & Daniel, L. G. (1996). Factor analytic evidence for the construct validity of scores: An historical overview and some guidelines. Educational and Psychological Measurement, 56, 213-224.
46.
Tiku, M. L. (1967). Tables of power of the F test. Journal of American Statistical Association, 62, 525-539.
47.
Velicer, W. F. (1976). Determining the number of components from matrix of partial correlations. Psychometrika, 41, 321-326.
48.
Velicer, W. F. , & Fava, J. L. (1998). Effects of variable and subject sampling on factor pattern recovery. Psychological Methods, 3, 231-251.
49.
Velicer, W. F. , & Jackson, D. N. (1990). Component analysis versus common factor analysis: Some further observations. Multivariate Behavioral Research, 25, 97-114.
50.
Velicer, W. F. , Peacock, A., & Jackson, D. N. (1982). A comparison of component and factor patterns: A Monte Carlo approach. Multivariate Behavioral Research, 17, 371-388.
51.
Winer, B. J. (1971). Statistical principles in experimental design (2nd ed.). New York: McGraw-Hill.
52.
Wood, J. M. , Tataryn, D. J., & Gorsuch, R. L. (1996). Effects of under-and overextraction on principal axis factor analysis with varimax rotation. Psychological Methods, 1, 354-365.
53.
Zoski, K. W. , & Jurs, S. (1993). Using multiple regression to determine the number of factors to retain in factor analysis. Journal of Multiple Linear Regression View Points, 20, 5-9.
54.
Zoski, K. , & Jurs, S. (1996). An objective counterpart to visual test for factor analysis: The standard error scree. Educational and Psychological Measurement, 56, 443-451.
55.
Zwick, W. R. , & Velicer, W. F. (1982). Factors influencing four rules for determining the number of components to retain. Multivariate Behavioral Research, 17, 253-269.
56.
Zwick, W. R. , & Velicer, W. F. (1986). A comparison of five rules for determining the number of factors to retain. Psychological Bulletin, 99, 432-442.
