Identification of Facet Models by Means of Factor Rotation: A Simulation Study and Data Analysis of a Test for the Berlin Model of Intelligence Structure

Abstract

We investigated by means of a simulation study how well methods for factor rotation can identify a two-facet simple structure. Samples were generated from orthogonal and oblique two-facet population factor models with 4 (2 factors per facet) to 12 factors (6 factors per facet). Samples drawn from orthogonal populations were submitted to factor analysis with subsequent Varimax, Equamax, Parsimax, Factor Parsimony, Tandem I, Tandem II, Infomax, and McCammon’s minimum entropy rotation. Samples drawn from oblique populations were submitted to factor analysis with subsequent Geomin rotation and a Promax-based Tandem II rotation. As a benchmark, we investigated a target rotation of the sample loadings toward the corresponding faceted population loadings. The three conditions were sample size (n = 400, 1,000), number of factors (q = 4-12), and main loading size (l = .40, .50, .60). For less than six orthogonal factors Infomax and McCammon’s minimum entropy rotation and for six and more factors Tandem II rotation yielded the highest congruence of sample loading matrices with faceted population loading matrices. For six and more oblique factors Geomin rotation and a Promax-based Tandem II rotation yielded the highest congruence with faceted population loadings. Analysis of data of 393 participants that performed a test for the Berlin Model of Intelligence Structure revealed that the faceted structure of this model could be identified by means of a Promax-based Tandem II rotation of task aggregates corresponding to the cross-products of the facets. Implications for the identification of faceted models by means of factor rotation are discussed.

Keywords

factor rotation exploratory factor analysis facet models Berlin Model of Intelligence Structure

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References

Asparouhov

Muthén

(2009). Exploratory structural equation modeling. Structural Equation Modeling, 16(3), 397-438. https://doi.org/10.1080/10705510903008204

Beauducel

Kersting

(2002). Fluid and crystallized intelligence and the Berlin Model of Intelligence Structure (BIS). European Journal of Psychological Assessment, 18(2), 97-112. https://doi.org/10.1027//1015-5759.18.2.97

Bernaards

C. A.

Jennrich

R. I.

(2005). Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis. Educational and Psychological Measurement, 65(5), 676-696. https://doi.org/10.1177/0013164404272507

Box

G. E. P.

Muller

M. E.

(1958). A note on the generation of random normal deviates. Annals of Mathematical Statistics, 29(2), 610-611. https://doi.org/10.1214/aoms/1177706645

Browne

M. W.

(2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36(1), 111-150. https://doi.org/10.1207/S15327906MBR3601_05

Bucik

Neubauer

(1996). Bimodality in the Berlin Model of Intelligence Structure (BIS): A replication study. Personality and Individual Differences, 21(6), 987-1005. https://doi.org/10.1016/S0191-8869(96)00129-8

Campbell

D. T.

Fiske

D. W.

(1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56(2), 81-105. https://doi.org/10.1037/h0046016

Comrey

A. L.

(1967). Tandem criteria for analytic rotation in factor analysis. Psychometrika, 32(2), 277-295. https://doi.org/10.1007/BF02289422

Crawford

C. B.

Ferguson

G. A.

(1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika, 35(3), 321-332. https://doi.org/10.1007/BF02310792

10.

Eid

(2000). A multitrait–multimethod model with minimal assumptions. Psychometrika, 65(2), 241-261. https://doi.org/10.1007/BF02294377

11.

Eid

Lischetzke

Nussbeck

F. W.

Trierweiler

L. I.

(2003). Separating trait effects from trait-specific method effects in multitrait-multimethod models: A multiple indicator CTC(M–1) model. Psychological Methods, 8(1), 38-60. https://doi.org/10.1037/1082-989X.8.1.38

12.

Grayson

Marsh

H. W.

(1994). Identification with deficient rank loading matrices in confirmatory factor analysis: Multitrait-multimethod models. Psychometrika, 59(1), 121-134. https://doi.org/10.1007/BF02294271

13.

Guilford

J. P.

(1967). The nature of human intelligence. McGraw-Hill.

14.

Guilford

J. P.

(1975). Factors and factors of personality. Psychological Bulletin, 82(5), 802-814. https://doi.org/10.1037/h0077101

15.

Guilford

J. P.

(1988). Some changes in the structure-of-intellect model. Educational and Psychological Measurement, 48(1), 1-4. https://doi.org/10.1177/001316448804800102

16.

Guttman

(1954). An outline of some new methodology for social research. Public Opinion Quarterly, 18(4), 395-404. https://doi.org/10.1086/266532

17.

Guttman

Levy

(1991). Two structural laws for intelligence tests. Intelligence, 15(1), 79-103. https://doi.org/10.1016/0160-2896(91)90023-7

18.

Hendrickson

A. E.

White

P. O.

(1964). Promax: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17(1), 65-70. https://doi.org/10.1111/j.2044-8317.1964.tb00244.x

19.

Holzinger

K. J.

Swineford

(1937). The bi-factor method. Psychometrika, 2(1), 41-54. https://doi.org/10.1007/BF02287965

20.

Jäger

A. O.

(1982). Mehrmodale klassifikation von intelligenzleistungen: Experimentell kontrollierte weiterentwicklung eines deskriptiven intelligenzstrukturmodells [Multimodel classification of intelligence performance: Experimentally controlled development of a descriptive model of intelligence structure]. Diagnostica, 23, 195-225.

21.

Jäger

A. O.

(1984). Intelligenzstrukturforschung: Konkurrierende modelle, neue entwicklungen, perspektiven [Structural research on intelligence: Competing models, new developments, perspectives]. Psychologische Rundschau, 35(1), 21-35.

22.

Jäger

A. O.

Süß

H. M.

Beauducel

(1997). Berliner Intelligenzstruktur-Test. BIS-Test, Form 4 [Test for the Berlin model of intelligence structure, Form 4]. Hogrefe.

23.

Jennrich

R. I.

(2001). A simple general procedure for orthogonal rotation. Psychometrika, 66(2), 289-306. https://doi.org/10.1007/BF02294840

24.

Jennrich

R. I.

Bentler

P. M.

(2011). Exploratory bi-factor analysis. Psychometrika, 76(4), 537-549. https://doi.org/10.1007/S11336-011-9218-4

25.

Jennrich

R. I.

Sampson

P. F.

(1966). Rotation for simple loadings. Psychometrika, 31(3), 313-323. https://doi.org/10.1007/BF02289465

26.

Kaiser

H. F.

(1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23(3), 187-200. https://doi.org/10.1007/BF02289233

27.

MacCallum

R. C.

Roznowski

Necowitz

L. B.

(1992). Model modifications in covariance structure analysis: The problem of capitalizing on chance. Psychological Bulletin, 111(3), 490-504. https://doi.org/10.1037/0033-2909.111.3.490

28.

Marsh

H. W.

Bailey

(1991). Confirmatory factor analyses of multitrait-multimethod data: A comparison of alternative models. Applied Psychological Measurement, 15(1), 47-70. https://doi.org/10.1177/014662169101500106

29.

McCammon

R. B.

(1966). Principal component analysis and its application in large-scale correlation studies. Journal of Geology, 74(5), 721-733. https://doi.org/10.1086/627207

30.

McKeon

J. J.

(1968). Rotation for maximum association between factors and tests [Unpublished manuscript]. George Washington University.

31.

Mulaik

S. A.

Quartetti

D. A.

(1997). First order or higher order general factor? Structural Equation Modeling, 4(3), 193-211. https://doi.org/10.1080/10705519709540071.

32.

Pfister

H.-R.

Jäger

A. O.

(1992). Topografische Analysen zum Berliner Intelligenstrukturmodell [Topographic analyses of the Berlin Intelligence Structure (BIS) model]. Diagnostica, 38, 91-116.

33.

Reise

S. P.

(2012). The rediscovery of bifactor measurement models. Multivariate Behavioral Research, 47(5), 667-696. https://doi.org/10.1080/00273171.2012.715555

34.

Saunders

D. R.

(1962). Trans-varimax: Some properties of the Ratiomax and Equamax criteria for blind orthogonal rotation [Paper presentation]. American Psychological Association, St. Louis, MO, USA.

35.

Schlesinger

I. M.

Guttman

(1969). Smallest space analysis of intelligence and achievement tests. Psychological Bulletin, 71(2), 95-100. https://doi.org/10.1037/h0026868

36.

Schmid

Leiman

J. M.

(1957). The development of hierarchical factor solutions. Psychometrika, 22(1), 53-61. https://doi.org/10.1007/BF02289209

37.

Schmitt

T. A.

Sass

D. A.

(2011). Rotation criteria and hypothesis testing for exploratory factor analysis: Implications for factor pattern loadings and interfactor correlations. Educational and Psychological Measurement, 71(1), 95-113. https://doi.org/10.1177/0013164410387348

38.

Schönemann

P. H.

(1966). A generalized solution of the orthogonal Procrustes problem. Psychometrika, 31(1), 1-10. https://doi.org/10.1007/BF02289451

39.

Shye

(1998). Modern facet theory: Content design and measurement in behavioral research. European Journal of Psychological Assessment, 14(2), 160-171. https://doi.org/10.1027/1015-5759.14.2.160

40.

Süß

H.-M.

Beauducel

(2005). Faceted models of intelligence. In Wilhelm

Engle

(Eds.). Understanding and measuring intelligence (pp. 313-332). Sage.

41.

Süß

H.-M.

Beauducel

(2015). Modeling the construct validity of the Berlin Intelligence Structure Model. Estudos de Psychologia/Pychological Studies, 32(1), 13-25. https://doi.org/10.1590/0103-166X2015000100002

42.

Süß

H. M.

Oberauer

Wittmann

W. W.

Wilhelm

Schulze

(2002). Working-memory capacity explains reasoning ability-and a little bit more. Intelligence, 30(3), 261-288. https://doi.org/10.1016/S0160-2896(01)00100-3

43.

Thurstone

L. L.

(1947). Multiple factor analysis: A development and expansion of vectors of the mind. University of Chicago Press.

44.

Tucker

L. R.

(1951). A method for synthesis of factor analysis studies (Personnel Research Section Report No. 984). Department of the Army.

45.

Velicer

A. F.

Jackson

D. N.

(1990). Component analysis versus common factor analysis: Some issues in selecting an appropriate procedure. Multivariate Behavioral Research, 25(1), 1-28. https://doi.org/10.1207/s15327906mbr2501_1

46.

Weide

A. C.

Beauducel

(2019). Varimax rotation based on gradient projection is a feasible alternative to SPSS. Frontiers in Psychology. Quantitative Psychology and Measurement, 10, Article 645. https://doi.org/10.3389/fpsyg.2019.00645

47.

Yates

(1987). Multivariate exploratory data analysis: A perspective on exploratory factor analysis. State University of New York Press.

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