Too Many Factors in Factor Analysis?

Guilford (April 1941) reports that "a factor analysis of the ten sub-tests of the Seashore test of pitch discrimination [of 0.5 to 30 cycles per second differences] revealed that more than one ability is involved. One factor, which accounted for the greater share of the variances, had loadings that decreased systematically with increasing difficulty [from 30 to 0.5 cps differences]. A second factor had strongest loadings among the more difficult items with frequency differences of 2 to 5 cps. A third had strongest loadings at differences of 5 to 12 cps. No explanation for the three factors is apparent, but the hypothesis is accepted that they represent distinct abilities. In tests so homogeneous as to content and form, where a single common factor might well have been expected, the appearance of additional common factors emphasizes the importance of considering the difficulty level of test items, both in the attempt to interpret new factors and in the practice of testing. The same kind of item may measure different abilities according as it is easy or difficult for the individuals to whom it is applied" (p.67).

Can there really be three distinct hearing discrimination abilities within such a narrow range of frequency differences? Guilford, an expert on psychophysics, cannot explain how that could be. He allows his data analysis to control his thinking. Yet he gives us a clue: "the importance of considering the difficulty level of test items.. in the attempt to interpret new factors."

Let us follow this clue to Ferguson (October 1941), a paper with no direct reference to Guilford: "In general, the greater the number of degrees of difficulty among the items in a test or among the tests in a battery, the higher the rank of the matrix of inter-correlations; that is differences in difficulty are represented in the factorial configuration as additional factors" (p. 323).

Is this the reality underlying Guilford's three factors? Not three abilities, but three item difficulty levels pertaining to one ability? In attempting to discover the meaning of the three factors of pitch discrimination, appeal to Ferguson's explanation of general statistical effects is more convincing than to Guilford's hypothesis of an inexplicable local psychophysical manifestation.

Ferguson reports the predicament that factor analysis identifies difficulty levels with different factors. Rasch analysis avoids this predicament by constructing one latent variable that spans all difficulty levels. Since the analyst's motivation is to span the data set with one meaning, rather than to stratify it, Rasch analysis is the method of choice.

Trevor G Bond 1994 RMT 8:1 p. 347
School of Education
James Cook University of North Queensland

Ferguson GA. 1941. The factorial interpretation of test difficulty. Psychometrika 6:5 323-329.

Guilford JP. 1941. The difficulty of a test and its factor composition. Psychometrika 6:2 67-77.

Note: A further confusing aspect of Factor Analysis is the analyst's choice of diagonalization, maximization, rotation and obliqueness. These can make highly-correlated content strands, which from the Rasch perspective are part of the dimension, appear to the casual reader to be orthogonal dimensions. Also small perturbations in the data can be made to appear to be large factors. If the factor loadings are plotted against item difficulty, it can be seen which factors may be due to nodes in the item distributions. To identify which factors are distribution-related, simulate data with the estimated Rasch measures as generators, and then apply EFA in order to obtain a baseline against which to interpret the empirically-based EFA. Principal Components Analysis of Residuals avoids many of the problems associated with Common Factor Analysis.

Too Many Factors?

"Therefore, one might expect the emergence of only one factor when a factor analysis would be performed on all newly defined subsets [of unidimensional items]. However, factor analysis of the newly defined subsets yielded two factors. Further inspection of the factor plot showed that the emergence of a second factor could be considered as an artefact due to the skewness of the subset scores."
Van der Ven, A.H.G.S., & Ellis, J.L. (2000). A Rasch Analysis of Raven's Standard Progressive Matrices. Personality and Individual Differences, 29 (1), 45-64.

reported in RMT 16:1

Duncan in his 1984 book "Notes on Social Measurement" has a good discussion on the limitations of factor analysis. In particular how inter-item correlations and item loadings are affected by item difficulty. Thus if a pool of items containing both easy and difficult items is factor analyzed, it will produce 2 factors, even if all the items are on exactly the same topic.
Scott Bayley

Duncan O.D. (1984). Notes on social measurement: Historical and critical. New York: Russell Sage Foundation.

For more information,
The Impact of Rasch Item Difficulty on Confirmatory Factor Analysis , S.V. Aryadoust … Rasch Measurement Transactions, 2009, 23:2 p. 1207
Confirmatory factor analysis vs. Rasch approaches: Differences and Measurement Implications, M.T. Ewing, T. Salzberger, R.R. Sinkovics … Rasch Measurement Transactions, 2009, 23:1 p. 1194-5
Conventional factor analysis vs. Rasch residual factor analysis, Wright, B.D. … 2000, 14:2 p. 753.
Rasch Analysis First or Factor Analysis First? Linacre J.M. … 1998, 11:4 p. 603.
Factor analysis and Rasch analysis, Schumacker RE, Linacre JM. … 1996, 9:4 p.470
Too many factors in Factor Analysis? Bond TG. … 1994, 8:1 p.347
Comparing factor analysis and Rasch measurement, Wright BD. … 1994, 8:1 p.350
Factor analysis vs. Rasch analysis of items, Wright BD. … 5:1 p.134

Too many factors in Factor Analysis? Bond TG. … Rasch Measurement Transactions, 1994, 8:1 p.347

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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