## Exploratory factor analysis: New directions (21)

*Chair: Joost Van Ginkel, Thursday 23rd July, 11.30 - 12.50, Dirac Room, Fisher Building.*

**Shu-chuan Kao** and Yeow Meng Thum, *Pearson VUE, USA*. Characterizing the dimensionality of a set of random variables. (230)

**Joost van Ginkel** and Pieter M. Kroonenberg, *Faculty of Social and Behavioural Sciences, Leiden University, The Netherlands*. Using procrustes analysis to combine the results from principal components analysis in multiple imputation. (074)

**David Flora** and E.M. Romero Escobar, *Department of Psychology, York University, Toronto, Canada*. Effect of minor model error on confirmatory factor analysis of ordinal variables with polychoric correlations. (035)

**Steffen Unkel **and Nickolay T. Trendafilov, *Department of Mathematics and Statistics, Open University, UK.* Exploratory factor analysis of data matrices with more variables than observations. (001)

ABSTRACTS

**Characterizing the dimensionality of a set of random variables**. (230)

*Shu-chuan Kao and Yeow Meng Thum*

Tools for assessing dimensionality have long depended on the relations among the successive eigenvalues obtained from an eigen decomposition, such as eigenvalue greater than 1.0 rule (Kaiser, 1960), skree plot test (Cattell, 1966), parallel analysis method (Horn, 1965), and the ratio difference index (Roznowski, Tucker, & Humphreys, 1991). This paper proposes an approach representing the dimensionality of a correlation matrix with a continuous function of eigenvalues and the determinant of the underlying correlation matrix. For example, given positive semi-definite correlation matrices of the same order, it is possible to differentiate and compare various correlation matrices using the slope of eigenvalues and the determinant: the former reflects the magnitude and pattern of the latent factors, and the latter has a functional relationship with its eigenvalues. The level of dimensionality can be mapped onto an arbitrary scale and the dimensional structure of latent factors can be evaluated accordingly. The subspace plots for two and three dimensional cases are provided by simulation, and the results reflect how the concentration of subspaces shifts on the proposed scale. With the proposed method, we argue that the dimensionality of a set of correlation matrices can be better communicated when their comparison are desired. *References: Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1, 245-276. Horn, J. L. (1965). A rational and test for the number of factors in factor analysis. Psychometrika, 30(179-185). Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20, 141-151. Roznowski, M., Tucker, L. R., & Humphreys, L. G. (1991). Three approaches to determining the dimensionality of binary items. Applied Psychological Measurement, 15(2), 109-127. *

**Using procrustes analysis to combine the results from principal components analysis in multiple imputation**. (074)

*Joost van Ginkel and Pieter M. Kroonenberg*

Multiple imputation is one of the most highly recommended procedures for dealing with missing data. Several rules for combining the results from statistical analyses in multiple imputation are available. However, to date, very few authors have investigated procedures for combining the results from principal component analyses after multiple imputation. In this presentation, Procrustes analysis is proposed for aligning the component loadings of the principal component analyses on the multiply imputed data, and its centroid solution is put forward as the final estimate for the component loadings

**Effect of minor model error on confirmatory factor analysis of ordinal variables with polychoric correlations**. (035)

*David Flora and E.M. Romero Escobar*

A popular limited-information approach to factor analyzing item-level ordinal variables involves fitting the model to a polychoric correlation matrix using a least-squares estimator. Simulation studies have reported promising findings for this polychoric approach when perfectly specified models are estimated, but no study has explicitly focused on its finite sample properties in the presence of minor model error. The perspective taken is that no model is ever perfect, and instead a small degree of model-data misfit at the level of the population (“approximation discrepancy”) should be tolerated for a model to be deemed useful. We conducted a comprehensive simulation study to examine the finite-sample properties of fit statistics and parameter estimates for confirmatory factor analysis models fitted to ordinal variables using the polychoric approach. In one condition, population root mean-squared error of approximation (RMSEA) = 0, indicating perfect specification, and in another population RMSEA = .05, indicating minor model error. In the no model error condition, results followed expected patterns. In the model error condition, the sampling distributions of model fit statistics showed complex patterns, primarily as a function of factor loading strength and the number of categories for the observed variables, whereas parameter estimates were not strongly affected.

**Exploratory factor analysis of data matrices with more variables than observations**. (001)

*Steffen Unkel and Nickolay T. Trendafilov*

The standard fitting problem in Exploratory Factor Analysis (EFA) is to find estimates for the factor loadings matrix and the matrix of unique factor variances which give the best fit to the sample correlation matrix with respect to some goodness-of-fit criterion. If the number of variables exceeds the number of available observations, the sample correlation matrix is singular. Then, the most common factor extraction methods such as maximum-likelihood factor analysis or generalized least squares factor analysis cannot be applied. In this talk, a novel approach for fitting the EFA model is presented. The EFA model is considered as a specific data matrix decomposition with fixed unknown matrix parameters. Fitting the EFA model directly to the data matrix yield simultaneous solutions for both loadings and factor scores. A new algorithm is introduced for the least squares estimation of all EFA model unknowns. As in principal component analysis, the new method is based on the efficient numerical procedure for singular value decomposition of data matrices. The developed algorithm is illustrated with Thurstone’s 26-variable box data and a real high-dimensional data set.