On exploratory factor analysis: a review of recent evidence, an assessment of current practice, and recommendations for future use.

Exploratory factor analysis (hereafter, factor analysis) is a complex statistical method that is integral to many fields of research. Using factor analysis requires researchers to make several decisions, each of which affects the solutions generated. In this paper, we focus on five major decisions that are made in conducting factor analysis: (i) establishing how large the sample needs to be, (ii) choosing between factor analysis and principal components analysis, (iii) determining the number of factors to retain, (iv) selecting a method of data extraction, and (v) deciding upon the methods of factor rotation. The purpose of this paper is threefold: (i) to review the literature with respect to these five decisions, (ii) to assess current practices in nursing research, and (iii) to offer recommendations for future use. The literature reviews illustrate that factor analysis remains a dynamic field of study, with recent research having practical implications for those who use this statistical method. The assessment was conducted on 54 factor analysis (and principal components analysis) solutions presented in the results sections of 28 papers published in the 2012 volumes of the 10 highest ranked nursing journals, based on their 5-year impact factors. The main findings from the assessment were that researchers commonly used (a) participants-to-items ratios for determining sample sizes (used for 43% of solutions), (b) principal components analysis (61%) rather than factor analysis (39%), (c) the eigenvalues greater than one rule and screen tests to decide upon the numbers of factors/components to retain (61% and 46%, respectively), (d) principal components analysis and unweighted least squares as methods of data extraction (61% and 19%, respectively), and (e) the Varimax method of rotation (44%). In general, well-established, but out-dated, heuristics and practices informed decision making with respect to the performance of factor analysis in nursing studies. Based on the findings from factor analysis research, it seems likely that the use of such methods may have had a material, adverse effect on the solutions generated. We offer recommendations for future practice with respect to each of the five decisions discussed in this paper.

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

[2]  P. Wieringa,et al.  Exploratory Factor Analysis With Small Sample Sizes , 2009, Multivariate behavioral research.

[3]  T. Schmitt.  Current Methodological Considerations in Exploratory and Confirmatory Factor Analysis , 2011 .

[4]  Ira H. Bernstein,et al.  Factoring items and factoring scales are different: Spurious evidence for multidimensionality due to item categorization. , 1989 .

[5]  Kristopher J Preacher,et al.  Sample Size in Factor Analysis: The Role of Model Error , 2001, Multivariate behavioral research.

[6]  D. Woltz,et al.  Relationship between perceived and actual frequency represented by common rating scale labels. , 2012, Psychological assessment.

[7]  W F Velicer,et al.  An Empirical Comparison Of The Similarity Of Principal Component, Image, And Factor Patterns. , 1977, Multivariate behavioral research.

[8]  Daniel J. Mundfrom,et al.  Minimum Sample Size Recommendations for Conducting Factor Analyses , 2005 .

[9]  J J McArdle,et al.  Principles versus Principals of Structural Factor Analyses. , 1990, Multivariate behavioral research.

[10]  J. Hair.  Multivariate data analysis : a global perspective , 2010 .

[11]  R. Gorsuch,et al.  Component analysis versus common factor analysis: A Monte Carlo study. , 1989 .

[12]  Wayne F. Velicer,et al.  The Relation Between Factor Score Estimates, Image Scores, and Principal Component Scores , 1976 .

[13]  J. Tukey,et al.  Multiple-Factor Analysis , 1947 .

[14]  Dimitra Dodou,et al.  Factor recovery by principal axis factoring and maximum likelihood factor analysis as a function of factor pattern and sample size , 2012 .

[15]  Kristopher J Preacher,et al.  Item factor analysis: current approaches and future directions. , 2007, Psychological methods.

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

[17]  G. H. Thomson,et al.  Hotelling's method modified to give Spearman's g. , 1934 .

[18]  Francisco José Abad,et al.  A new look at Horn's parallel analysis with ordinal variables. , 2013, Psychological methods.

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

[20]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[21]  K. Yuan,et al.  A unified approach to exploratory factor analysis with missing data, nonnormal data, and in the presence of outliers , 2002 .

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

[23]  Charles E. Lance,et al.  Statistical and methodological myths and urban legends : doctrine, verity, and fable in the organizational and social sciences , 2009 .

[24]  W F Velicer,et al.  Component Analysis versus Common Factor Analysis: Some issues in Selecting an Appropriate Procedure. , 1990, Multivariate behavioral research.

[25]  Karl G. Jöreskog,et al.  On the statistical treatment of residuals in factor analysis , 1962 .

[26]  Myrsini Katsikatsou,et al.  Pairwise likelihood estimation for factor analysis models with ordinal data , 2012, Comput. Stat. Data Anal..

[27]  José Muñiz,et al.  Effect of the Number of Response Categories on the Reliability and Validity of Rating Scales , 2008 .

[28]  Godfrey H. Thomson.  THE FACTORIAL ANALYSIS OF ABILITY: I. THE PRESENT POSITION AND THE PROBLEMS CONFRONTING US1 , 1939 .

[29]  Robert C MacCallum,et al.  Recovery of Weak Common Factors by Maximum Likelihood and Ordinary Least Squares Estimation , 2003, Multivariate behavioral research.

[30]  D. Lawley.  VI.—The Estimation of Factor Loadings by the Method of Maximum Likelihood , 1940 .

[31]  Soonmook Lee,et al.  A Review of CEFA Software: Comprehensive Exploratory Factor Analysis Program , 2010 .

[32]  W F Velicer,et al.  An Empirical Comparison of Factor, Image, Component, and Scale Scores. , 1992, Multivariate behavioral research.

[33]  Douglas N. Jackson,et al.  Maximum-likelihood estimation in common factor analysis: A cautionary note , 1980 .

[34]  K. Widaman.  Bias in Pattern Loadings Represented by Common Factor Analysis and Component Analysis. , 1990, Multivariate behavioral research.

[35]  D. Jackson,et al.  Component Analysis versus Common Factor Analysis: Some Further Observations. , 1990, Multivariate behavioral research.

[36]  Allen I. Huffcutt,et al.  A Review and Evaluation of Exploratory Factor Analysis Practices in Organizational Research , 2003 .

[37]  K. Widaman.  Common Factors Versus Components: Principals and Principles, Errors and Misconceptions , 2007 .

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

[39]  Karl G. Jöreskog,et al.  On the estimation of polychoric correlations and their asymptotic covariance matrix , 1994 .

[40]  P. Whittle,et al.  Latent Variables in Socio‐Economic Models , 1978 .

[41]  W F Velicer,et al.  Factors Influencing Four Rules For Determining The Number Of Components To Retain. , 1982, Multivariate behavioral research.

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

[43]  M. S. Bartlett,et al.  A FURTHER NOTE ON TESTS OF SIGNIFICANCE IN FACTOR ANALYSIS , 1951 .

[44]  Kristopher J Preacher,et al.  Repairing Tom Swift's Electric Factor Analysis Machine , 2003 .

[45]  Kristine Y. Hogarty,et al.  The Quality of Factor Solutions in Exploratory Factor Analysis: The Influence of Sample Size, Communality, and Overdetermination , 2005 .

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

[47]  J. O. Neuhaus,et al.  THE QUARTIMAX METHOD , 1954 .

[48]  J. Ruscio,et al.  Determining the number of factors to retain in an exploratory factor analysis using comparison data of known factorial structure. , 2012, Psychological assessment.

[49]  R. MacCallum,et al.  Sample size in factor analysis. , 1999 .

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

[51]  Urbano Lorenzo-Seva,et al.  The Hull Method for Selecting the Number of Common Factors , 2011, Multivariate behavioral research.

[52]  Charles B. Crawford,et al.  Note: Inter-Rater Reliability of Scree Test and Mean Square Ratio Test of Number of Factors , 1979 .

[53]  K. G. Jöreskog,et al.  Efficient estimation in image factor analysis. , 1969 .

[54]  Mário Basto,et al.  An SPSS R-Menu for Ordinal Factor Analysis , 2012 .

[55]  Robert C. MacCallum,et al.  Representing sources of error in the common-factor model: Implications for theory and practice. , 1991 .

[56]  H. Harman.  Modern factor analysis , 1961 .

[57]  Paul Kline,et al.  A Handbook of Test Construction : Introduction to Psychometric Design , 1987 .

[58]  Rocco J. Perla,et al.  Ten Common Misunderstandings, Misconceptions, Persistent Myths and Urban Legends about Likert Scales and Likert Response Formats and their Antidotes , 2007 .

[59]  Ou Lydia Liu,et al.  A modified procedure for parallel analysis of ordered categorical data , 2008, Behavior research methods.

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

[61]  Matthew Courtney,et al.  Determining the Number of Factors to Retain in EFA: Using the SPSS R-Menu v2 0 to Make More Judicious Estimations , 2013 .

[62]  W. Hemmerle,et al.  Robustness of the maximum-likelihood estimation procedure in factor analysis , 1966, Psychometrika.

[63]  S S Stevens,et al.  On the Theory of Scales of Measurement. , 1946, Science.

[64]  Francisco Pablo Holgado Tello,et al.  Polychoric versus Pearson correlations in exploratory and confirmatory factor analysis of ordinal variables , 2010 .

[65]  Deborah L. Bandalos,et al.  Four Common Misconceptions in Exploratory Factor Analysis , 2008 .

[66]  Daniel A. Sass,et al.  Rotation Criteria and Hypothesis Testing for Exploratory Factor Analysis: Implications for Factor Pattern Loadings and Interfactor Correlations , 2011 .

[67]  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.

[68]  W. Velicer,et al.  Affects of variable and subject sampling on factor pattern recovery. , 1998 .

[69]  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 .

[70]  Daniel A Sass,et al.  Please Scroll down for Article Multivariate Behavioral Research a Comparative Investigation of Rotation Criteria within Exploratory Factor Analysis , 2022 .

[71]  E. Abt.  Understanding statistics 3 , 2010, Evidence-Based Dentistry.

[72]  Leandre R. Fabrigar,et al.  Exploratory Factor Analysis , 2011 .

[73]  Louis Guttman,et al.  Image theory for the structure of quantitative variates , 1953 .

[74]  G. A. Ferguson,et al.  A general rotation criterion and its use in orthogonal rotation , 1970 .

[75]  A. Goldberger,et al.  Factor analysis by generalized least squares , 1972 .

[76]  B. Everitt,et al.  Multivariate Exploratory Data Analysis: A Perspective on Exploratory Factor Analysis. , 1988 .

[77]  Stephen Tchudi,et al.  A Final Comment. , 1988 .

[78]  M. Browne.  Generalized Least Squares Estimators in the Analysis of Covariance Structures. , 1973 .

[79]  James H. Steiger,et al.  On the validity of indeterminate factor scores , 1978 .

[80]  Henry F. Kaiser,et al.  Alpha factor analysis , 1965, Psychometrika.

[81]  M. Browne.  An Overview of Analytic Rotation in Exploratory Factor Analysis , 2001 .

[82]  R. Jennrich,et al.  Rotation for simple loadings , 1966, Psychometrika.

[83]  M. Browne.  A comparison of factor analytic techniques , 1968, Psychometrika.

[84]  C. Spearman.  General intelligence Objectively Determined and Measured , 1904 .

[85]  H. Kaiser.  The varimax criterion for analytic rotation in factor analysis , 1958 .

[86]  Lee T. Ostrom,et al.  Using Likert-Type Scales in the Social Sciences. , 2011 .

[87]  Calyampudi R. Rao.  Estimation and tests of significance in factor analysis , 1955 .

[88]  R. Cattell.  The Scientific Use of Factor Analysis in Behavioral and Life Sciences , 2012 .

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

[90]  C. Gaskin,et al.  Power of mental health nursing research: a statistical analysis of studies in the International Journal of Mental Health Nursing. , 2013, International journal of mental health nursing.

[91]  Robert Cudeck,et al.  Factor analysis at 100 : historical developments and future directions , 2007 .

[92]  Roger Watson,et al.  Use of factor analysis in Journal of Advanced Nursing: literature review. , 2006, Journal of advanced nursing.

[93]  Chen-Yuan Chen,et al.  A simulation study using EFA and CFA programs based the impact of missing data on test dimensionality , 2012, Expert systems with applications.

[94]  R. D. Bock,et al.  Item response theory in a general framework , 2006 .

[95]  H. Kaiser.  A second generation little jiffy , 1970 .

[96]  B. Tabachnick,et al.  Using Multivariate Statistics , 1983 .

[97]  Raymond Hubbard,et al.  An empirical comparison of alternative methods for principal component extraction , 1987 .

[98]  Ulf Jakobsson,et al.  Statistical presentation and analysis of ordinal data in nursing research. , 2004, Scandinavian journal of caring sciences.

[99]  M. Bartlett.  TESTS OF SIGNIFICANCE IN FACTOR ANALYSIS , 1950 .

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

[101]  Guangjian Zhang,et al.  Ordinary Least Squares Estimation of Parameters in Exploratory Factor Analysis With Ordinal Data , 2012, Multivariate behavioral research.

[102]  Y Kano,et al.  On the Equivalence of Factors and Components. , 1990, Multivariate behavioral research.

[103]  Roy Levy,et al.  A Proposed Solution to the Problem With Using Completely Random Data to Assess the Number of Factors With Parallel Analysis , 2012 .

[104]  David J. Bartholomew,et al.  Latent Variable Models and Factor Analysis: A Unified Approach , 2011 .

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

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