Abstract
When Tucker's congruence coefficient is used to assess the similarity of factor interpretations, it is desirable to have a critical congruence level less than unity that can be regarded as indicative of identity of the factors. The literature only reports rules of thumb. The present article repeats and broadens the approach used in the study by Haven and ten Berge (1977). It aims to find a critical congruence level on the basis of judgments of factor similarity by practitioners of factor analysis. Our results suggest that a value in the range .85-.94 corresponds to a fair similarity, while a value higher than .95 implies that the two factors or components compared can be considered equal.
References
Acton, G. S. , Schroeder, D. H. (2001). Sensory discrimination as related to general intelligence. Intelligence, 29, 263– 271Arrindell, W. A. , Bridges, K. R. , Van derEnde, J. , Lawrence, J. S. , Gray, S. L. , Harnish, R. , Rogers, R. , Sanderman, R. (2001). Normative studies with the Scale for Interpersonal Behaviour (SIB): II. US students. A cross-cultural comparison with Dutch data. Behaviour Research & Therapy, 39, 1461– 1479Bentler, P. M. , Bonett, D. G. (1980). Significance tests and goodness-of-fit in the analyses of covariance structures. Psychological Bulletin, 88, 588– 606Broadbooks, W. J. , Elmore, P. B. (1987). A Monte Carlo study of the sampling distribution of the congruence coefficient. Educational & Psychological Measurement, 47, 1– 11Burt, C. (1948). The factorial study of temperament traits. British Journal of Psychology, Statistical Section, 1, 178– 203Cattell, R. B. (1978). Matched determines vs. factor invariance: A reply to Korth. Multivariate Behavioral Research, 13, 431– 448Chan, W. , Ho, R. M. , Leung, K. , Chan, D. K-S. , Yung, Y-F. (1999). An alternative method for evaluating congruence coefficients with Procrustes rotation: A bootstrap procedure. Psychological Methods, 4, 378– 402Chico, E. , Tous, J. M. , Lorenzo-Seva, U. , Vigil-Colet, A. (2003). Spanish adaptation of Dickman's impulsivity inventory: Its relationship to Eysenck's personality questionnaire. Personality & Individual Differences, 35, 1883– 1892Church, A. T. , Burke, P. J. (1994). Exploratory and confirmatory tests of the Big Five and Tellegen's three- and four-dimensional models. Journal of Personality & Social Psychology, 66, 93– 114Davenport, E. C. (1990). Significance testing of congruence coefficients: A good idea?. Educational & Psychological Measurement, 50, 289– 296Ferrando, P. J. , Lorenzo, U. (2000). Unrestricted versus restricted factor analysis of multidimensional test items: Some aspects of the problem and some suggestions. Psicológica, 21, 301– 323Guadagnoli, E. , Velicer, W. F. (1991). A comparison of pattern matching indices. Multivariate Behavioral Research, 26, 323– 343Harman, H. H. (1960). Modern factor analysis . (2nd ed.). Chicago: University of Chicago PressHaven, S. , tenBerge, J. M. F. (1977). Tucker's coefficient of congruence as a measure of factorial invariance: An empirical study. Heymans Bulletin 290 EX. unpublished report by the Department of Psychology, University of GroningenHendriks, A. A. J. , Hofstee, W. K. B. , DeRaad, B. (1999). The Five-Factor Inventory (FFPI). Personality & Individual Differences, 27, 307– 325Horn, J. L. , Wanberg, K. W. , Appel, M. (1973). On the internal structure of the MMPI. Multivariate Behavioral Research, 8, 131– 172Korf, L. , Malan, J. (2002). Threat to ethnic identity: The experience of white Afrikaans speaking participants in postapartheid South Africa. Journal of Social Psychology, 142, 149– 169Korth, B. A. (1973). Analytic and experimental analysis of factor matching methods . Unpublished doctoral dissertation, University of Illinois at Urbana-ChampaignKorth, B. A. (1978). A significance test for congruence coefficients for Cattell's factors matched by scanning. Multivariate Behavioral Research, 13, 419– 430Korth, B. A. , Tucker, L. R. (1975). The distribution of chance congruence coefficients from simulated data. Psychometrika, 40, 361– 372Lorenzo-Seva, U. , Ferrando, P. J. (2003). IMINCE: An unrestricted factor-analysis-based program for assessing measurement invariance. Behavior Research Methods, Instruments, & Computers, 35, 318– 321Lorenzo-Seva, U. , Kiers, H. A. L. , tenBerge, J. M. F. (2002). Techniques for oblique factor rotation of two or more loading matrices to a mixture of simple structure and optimal agreement. British Journal of Mathematical & Statistical Psychology, 55, 337– 360MacCallum, R. C. , Widaman, K. F. , Zhang, S. , Hong, S. (1999). Sample size in factor analysis. Psychological Methods, 4, 84– 99McCrae, R. R. , Zonderman, A. B. , Costa, P. T. , Bond, M. H. , Paunonen, S. V. (1996). Evaluating replicability of factors in the revised NEO personality inventory: Confirmatory factor analysis versus Procrustes rotation. Journal of Personality & Social Psychology, 70, 552– 566Miller, L. T. , Vernon, P. A. (1996). Intelligence, reaction time, and working memory in 4 to 6 year old children. Intelligence, 22, 155– 190Mulaik, S. A. (1972). The foundations of factor analysis . New York: McGraw-HillNesselroade, J. R. , Baltes, P. W. (1970). On a dilemma of comparative factor analysis: A study of factor matching based on random data. Educational & Psychological Measurement, 30, 935– 948Nesselroade, J. R. , Baltes, P. W. , Labouvie, E. W. (1971). Evaluating factor invariance in oblique space: Baseline data generated from random numbers. Multivariate Behavioral Research, 6, 233– 241Paunonen, S. V. (1997). On chance and factor congruence following orthogonal Procrustes rotation. Educational & Psychological Measurement, 57, 33– 59Rodríguez-Fornells, A. , Lorenzo-Seva, U. , Andrés-Pueyo, A. (2001). Psychometric properties of the Spanish adaptation of the Five Factor Personality Inventory. European Journal of Psychological Assessment, 17, 145– 153Schneewind, K. A. , Cattell, R. B. (1970). Zum problem der Faktorenidentifikation: Verteilungen und Vertrauensintervalle von Kongruenzkoefficienten f r Persönlichkeitsfaktoren im Bereich objektiv-analytischer Tests. [On the problem of the factor identification: Distributions and confidence intervals of congruence coefficients for personality factors in the area of objective analytic tests.]. Psychologische Beiträge, 12, 214– 226Steer, R. A. , Rissmiller, D. J. , Beck, A. T. (2000). Use of Beck Depression Inventory II with depressed geriatric inpatients. Behaviour Research & Therapy, 38, 311– 318Tucker, L. R. (1951). A method for synthesis of factor analysis studies (Personnel Research Section Report No. 984). Washington, DC: Department of the ArmyValk, A. , Karu, K. (2001). Ethnic attitudes in relation to ethnic pride and ethnic differentiation. Journal of Social Psychology, 141, 583– 601Van deVijver, F. , Leung, K. (1997). Methods and data analysis for cross-cultural research . London: Sage