A new look at Horn's parallel analysis with ordinal variables.

Previous research evaluating the performance of Horn's parallel analysis (PA) factor retention method with ordinal variables has produced unexpected findings. Specifically, PA with Pearson correlations has performed as well as or better than PA with the more theoretically appropriate polychoric correlations. Seeking to clarify these findings, the current study employed a more comprehensive simulation study that included the systematic manipulation of 7 factors related to the data (sample size, factor loading, number of variables per factor, number of factors, factor correlation, number of response categories, and skewness) as well as 3 factors related to the PA method (type of correlation matrix, extraction method, and eigenvalue percentile). The results from the simulation study show that PA with either Pearson or polychoric correlations is particularly sensitive to the sample size, factor loadings, number of variables per factor, and factor correlations. However, whereas PA with polychorics is relatively robust to the skewness of the ordinal variables, PA with Pearson correlations frequently retains difficulty factors and is generally inaccurate with large levels of skewness. In light of these findings, we recommend the use of PA with polychoric correlations for the dimensionality assessment of ordinal-level data.

[1]  W. Velicer,et al.  The Effects of Overextraction on Factor and Component Analysis. , 1992, Multivariate behavioral research.

[2]  Alexis Dinno,et al.  Implementing Horn's Parallel Analysis for Principal Component Analysis and Factor Analysis , 2009 .

[3]  Alberto Maydeu-Olivares,et al.  Factor Analysis with Ordinal Indicators: A Monte Carlo Study Comparing DWLS and ULS Estimation , 2009 .

[4]  Richard L. Gorsuch,et al.  Effects of under- and overextraction on principal axis factor analysis with varimax rotation. , 1996 .

[5]  J. Horn.  A rationale and test for the number of factors in factor analysis , 1965, Psychometrika.

[6]  Louis W. Glorfeld.  An Improvement on Horn's Parallel Analysis Methodology for Selecting the Correct Number of Factors to Retain , 1995 .

[7]  Emin Babakus,et al.  The Sensitivity of Confirmatory Maximum Likelihood Factor Analysis to Violations of Measurement Scale and Distributional Assumptions , 1987 .

[8]  Wayne F. Velicer,et al.  The Effects of Underextraction in Factor and Component Analyses , 1996 .

[9]  A. Buja,et al.  Remarks on Parallel Analysis. , 1992, Multivariate behavioral research.

[10]  W. Velicer.  Determining the number of components from the matrix of partial correlations , 1976 .

[11]  Donald A. Jackson,et al.  How many principal components? stopping rules for determining the number of non-trivial axes revisited , 2005, Comput. Stat. Data Anal..

[12]  B. Price.  A First Course in Factor Analysis , 1993 .

[13]  S. Reise,et al.  Factor analysis and scale revision. , 2000, Psychological assessment.

[14]  W. Velicer,et al.  Comparison of five rules for determining the number of components to retain. , 1986 .

[15]  Howard B. Lee,et al.  A First Course in Factor Analysis 2nd Ed , 1973 .

[16]  Lloyd G. Humphreys,et al.  Note On a Criterion for the Number of Common Factors , 1969 .

[17]  L. Guttman.  Some necessary conditions for common-factor analysis , 1954 .

[18]  P. Lachenbruch.  Statistical Power Analysis for the Behavioral Sciences (2nd ed.) , 1989 .

[19]  D. Knol,et al.  Empirical Comparison Between Factor Analysis and Multidimensional Item Response Models. , 1991, Multivariate behavioral research.

[20]  R. Henson,et al.  Use of Exploratory Factor Analysis in Published Research , 2006 .

[21]  A. Formann,et al.  Performance of Parallel Analysis in Retrieving Unidimensionality in the Presence of Binary Data , 2009 .

[22]  Chung-Ping Cheng,et al.  Parallel Analysis with Unidimensional Binary Data , 2005 .

[23]  K. Bollen,et al.  Pearson's R and Coarsely Categorized Measures , 1981 .

[24]  M. Timmerman,et al.  Dimensionality assessment of ordered polytomous items with parallel analysis. , 2011, Psychological methods.

[25]  K. Jöreskog,et al.  Factor Analysis of Ordinal Variables: A Comparison of Three Approaches , 2001, Multivariate behavioral research.

[26]  R. Darlington,et al.  Factor Analysis , 2008 .

[27]  André Beauducel,et al.  Problems with parallel analysis in data sets with oblique simple structure , 2001 .

[28]  F. Floyd,et al.  Factor analysis in the development and refinement of clinical assessment instruments. , 1995 .

[29]  K. Dickman,et al.  Factorial Validity of a Rating Instrument , 1960 .

[30]  Pere J. Ferrando,et al.  FACTOR: A computer program to fit the exploratory factor analysis model , 2006, Behavior research methods.

[31]  H. Kaiser.  The Application of Electronic Computers to Factor Analysis , 1960 .

[32]  B P O'Connor,et al.  SPSS and SAS programs for determining the number of components using parallel analysis and Velicer’s MAP test , 2000, Behavior research methods, instruments, & computers : a journal of the Psychonomic Society, Inc.

[33]  Wayne F. Velicer,et al.  Construct Explication through Factor or Component Analysis: A Review and Evaluation of Alternative Procedures for Determining the Number of Factors or Components , 2000 .

[34]  Sun-Joo Cho,et al.  Accuracy of the Parallel Analysis Procedure With Polychoric Correlations , 2009 .

[35]  U. Olsson.  On The Robustness Of Factor Analysis Against Crude Classification Of The Observations. , 1979, Multivariate behavioral research.

[36]  Glenn Gamst,et al.  Applied Multivariate Research: Design and Interpretation , 2005 .

[37]  Samuel B. Green,et al.  Evaluation of Parallel Analysis Methods for Determining the Number of Factors , 2010 .

[38]  Jacob Cohen.  Statistical Power Analysis for the Behavioral Sciences , 1969, The SAGE Encyclopedia of Research Design.

[39]  J. S. Long,et al.  Testing Structural Equation Models , 1993 .

[40]  B. Muthén,et al.  A comparison of some methodologies for the factor analysis of non‐normal Likert variables , 1985 .

[41]  Ulf Olsson,et al.  Maximum likelihood estimation of the polychoric correlation coefficient , 1979 .

[42]  R. Cattell.  The Scree Test For The Number Of Factors. , 1966, Multivariate behavioral research.

[43]  Vicente Ponsoda,et al.  Performance of Velicer’s Minimum Average Partial Factor Retention Method With Categorical Variables , 2011 .

[44]  James C. Hayton,et al.  Factor Retention Decisions in Exploratory Factor Analysis: a Tutorial on Parallel Analysis , 2004 .

[45]  A L Comrey,et al.  Distortions In A Commonly Used Factor Analytic Procedure. , 1979, Multivariate behavioral research.

[46]  D. Flora,et al.  An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. , 2004, Psychological methods.

[47]  K. Widaman.  Common Factor Analysis Versus Principal Component Analysis: Differential Bias in Representing Model Parameters? , 1993, Multivariate behavioral research.

[48]  A. Colman,et al.  Optimal number of response categories in rating scales: reliability, validity, discriminating power, and respondent preferences. , 2000, Acta psychologica.

[49]  Duane T. Wegener,et al.  Evaluating the use of exploratory factor analysis in psychological research. , 1999 .