Exploratory factor analysis: Current use,methodological developments and recommendations for good practice

Current Psychology - Psychological research often relies on Exploratory Factor Analysis (EFA). As the outcome of the analysis highly depends on the chosen settings, there is a strong need for...     

Exploratory longitudinal factor analysis in multiple populations

For multiple populatios, a longtidinal factor analytic model which is entirely exploratory, that is, no explicit identification constraints, is proposed. Factorial collapse and period/practice effects are allowed. An invariant and/or stationary factor pattern is permitted. This model is formulated stochastically. To implement this model a stagewise EM algorithm is developed. Finally a numerical illustration utilizing Nesselroade and Baltes' data is presented.The authors wish to thank Barbara Mellers and Henry Kaiser for their helpful comments and John Nesselroade for providing us the data for our illustration. This research wwa supported in part by a grant (No. AG03164) from the National Institute on Aging to William Meredith. Details of the derivations and a copy of the PROC MATRIX program are available upon request from the first author.     

Exploratory Factor Analysis: A Users’Guide

In 1978, Comrey wrote a guide to factor analysis in the Journal of Consulting and Clinical Psychology. This paper provides an update of the information given by Comrey in relation to exploratory factor analysis (EFA) for work and organizational psychologists, and particularly those involved with test development, interpretation and validation. In doing so, it offers a user's guide to contemporary methods and available techniques and introduces heuristics for dealing with problems of skew and kurtosis, social desirability response set, and factor naming.     

Exploratory structural equation modeling,bifactor models,and standard confirmatory factor analysis models: Application to the BASC-2 Behavioral and Emotional Screening System Teacher Form

Several psychological assessment instruments are based on the assumption of a general construct that is composed of multiple interrelated domains. Standard confirmatory factor analysis is often not well suited for examining the factor structure of such scales. This study used data from 1885 elementary school students (mean age = 8.77 years, SD = 1.47 years) to examine the factor structure of the Behavioral Assessment System for Children, Second Edition (BASC-2) Behavioral and Emotional Screening System (BESS) Teacher Form that was designed to assess general risk for emotional/behavioral difficulty among children. The modeling sequence included the relatively new exploratory structural equation modeling (ESEM) approach and bifactor models in addition to more standard techniques. Findings revealed that the factor structure of the BASC-2 BESS Teacher Form is multidimensional. Both ESEM and bifactor models showed good fit to the data. Bifactor models were preferred on conceptual grounds. Findings illuminate the hypothesis-generating power of ESEM and suggest that it might not be optimal for instruments designed to assess a predominant general factor underlying the data.     

Comparison of several shortened versions of the EMBU: exploratory and confirmatory factor analyses

The aim of the present study was to analyze the psychometric properties of different versions of the EMBU (Egna Minnen av Barndoms Uppfostran – My memories of upbringing) in a non-clinical sample of adolescents. The factor structure of the questionnaire was studied by means of more adequate exploratory and confirmatory factor procedures in order to ascertain which version achieves a better fit to data. The 64-item version of the EMBU was administered to a non-clinical sample of 832 adolescents. Exploratory and Confirmatory factor analyses were carried out for successive versions of 64, 37, 24 and 23 items. The theoretical four-factor structure of the 64-item version is not replicated in a factor analysis through principal axis extraction with direct oblimin rotation. The "Favoring Subject" factor was not found. 37 items present loadings equal to or larger than 0.30 for the "Rejection", "Emotional Warmth" and "Overprotection" factors. A re-analysis of these items shows a clear three-factor structure. When Arrindell et al .'s (1999 ) 23-item version was analysed, the three expected factors were also found. Confirmatory factor analyses allow us to reduce the 36-item version to a shorter one of 24 items. This last version achieves the best fit to data, including the 23-item version. The new 24-item version improves the construct validity of the EMBU and presents similar reliability coefficients.     

Invariance of the “NEO-PI-R” factor structure across exploratory and confirmatory factor analyses

Confirmatory factor analyses (CFA) carried out in the last decades to test the Five-factor simple structure have generally obtained negative results. McCrae, Zonderman, Costa, Bond, and Paunonen (1996) suggested that CFA limitations were the main cause of such negative results. They also claim that Procrustes Orthogonal rotation method is an adequate procedure to test the replicability of the Five-factor structure. The aim of the present paper is twofold: (A) comparing several Exploratory factor procedures (including the Procrustes rotation) to test what is the most appropriate one to analyse the replicability of the Five-factor structure, and (B) replicating the CFA results of the McCrae et al.’s study with a larger number of subjects. The normative American (N = 1000), and Spanish (2000) standardization samples of the NEO-PI-R, together with an independent university sample (N = 948) were analysed. Results were replicated in the three samples, and suggest that: (1) structural properties of the Five-factor model, as measured by the NEO-PI-R, are invariant irrespective of the Exploratory factor procedure used, and (2) when CFA limitations are surpassed, the Five-factor structure is supported.     

Exploratory behavior and developmental skill acquisition in infants with Down syndrome

Infants learn about objects by exploring them. Typically developing infants actively explore objects through visual, manual, and oral modalities. Attenuated exploratory behavior has been observed in various neurodevelopmental disorders, including Down syndrome (DS), presumably limiting learning options. However, a direct link between exploration and overall developmental functioning has not been characterized. This study used a Latent Profile Analysis framework to examine within-syndrome variability in exploratory behavior in infants with DS and the developmental correlates of different exploratory behavior profiles. Participants were 45 infants with DS (CA = 9.58 months; SD = 3.62) who completed an object exploration activity and the Bayley Scales of Infant Development-III (BSID-III; Bayley, 2006). Exploration behavior was coded for the percentage of time engaged in visual, manual, and oral exploration. Results indicated that a 2-profile solution provided the best model fit for exploratory behavior, yielding profiles that represented either an Active (57.78% of the sample) or a Passive Exploratory (42.22% of the sample) profile. The Active Exploratory profile was associated with significantly higher age equivalent scores on the BSID-III Cognitive, Communication, and Motor domains than the Passive Exploratory profile. Other factors, such as sex and biomedical risk factors, were not associated with exploratory profiles. These findings offer a more nuanced understanding of early within-syndrome heterogeneity in DS, and demonstrate that impoverished early exploratory behavior may serve as an important indicator of increased risk for more pronounced developmental delays in DS.     

Exploratory factor analysis: its role in item analysis

The special characteristics of items-low reliability, confounds by minor, unwanted covariance, and the likelihood of a general factor-and better understanding of factor analysis means that the default procedure of many statistical packages (Little Jiffy) is no longer adequate for exploratory item factor analysis. It produces too many factors and precludes a general factor even when that means the factors extracted are nonreplicable. More appropriate procedures that reduce these problems are presented, along with how to select the sample, sample size required, and how to select items for scales. Proposed scales can be evaluated by their correlations with the factors; a new procedure for doing so eliminates the biased values produced by correlating them with either total or factor scores. The role of exploratory factor analysis relative to cluster analysis and confirmatory factor analysis is noted.     

Optimal matching analysis in career research: A review and some best-practice recommendations

Optimal matching is a method for the analysis of sequential data. It allows researchers to detect patterns in career sequences or occupational trajectories. We first give a brief introduction to the method and review the existing career literature that employs optimal matching. To examine which data properties are required for optimal matching analysis, we conducted Monte Carlo simulations of career sequences with varying parameters for sequence length, sample size and missing items. We find that sequence length is the relevant factor for correct results, while sample size does not substantially affect result quality. Another important finding is that sequences with up to 30% elements missing can be used for optimal matching analysis. We also show which settings for the optimal matching procedure deliver the best results.     

Exploratory Bifactor Analysis: The Schmid-Leiman Orthogonalization and Jennrich-Bentler Analytic Rotations

Analytic bifactor rotations have been recently developed and made generally available, but they are not well understood. The Jennrich-Bentler analytic bifactor rotations (bi-quartimin and bi-geomin) are an alternative to, and arguably an improvement upon, the less technically sophisticated Schmid-Leiman orthogonalization. We review the technical details that underlie the Schmid-Leiman and Jennrich-Bentler bifactor rotations, using simulated data structures to illustrate important features and limitations. For the Schmid-Leiman, we review the problem of inaccurate parameter estimates caused by the linear dependencies, sometimes called “proportionality constraints,” that are required to expand a p correlated factors solution into a (p + 1) (bi)factor space. We also review the complexities involved when the data depart from perfect cluster structure (e.g., item cross-loading on group factors). For the Jennrich-Bentler rotations, we describe problems in parameter estimation caused by departures from perfect cluster structure. In addition, we illustrate the related problems of (a) solutions that are not invariant under different starting values (i.e., local minima problems) and (b) group factors collapsing onto the general factor. Recommendations are made for substantive researchers including examining all local minima and applying multiple exploratory techniques in an effort to identify an accurate model.     

Heterogeneous factor analysis models: A bayesian approach

Multilevel factor analysis models are widely used in the social sciences to account for heterogeneity in mean structures. In this paper we extend previous work on multilevel models to account for general forms of heterogeneity in confirmatory factor analysis models. We specify various models of mean and covariance heterogeneity in confirmatory factor analysis and develop Markov Chain Monte Carlo (MCMC) procedures to perform Bayesian inference, model checking, and model comparison.We test our methodology using synthetic data and data from a consumption emotion study. The results from synthetic data show that our Bayesian model perform well in recovering the true parameters and selecting the appropriate model. More importantly, the results clearly illustrate the consequences of ignoring heterogeneity. Specifically, we find that ignoring heterogeneity can lead to sign reversals of the factor covariances, inflation of factor variances and underappreciation of uncertainty in parameter estimates. The results from the emotion study show that subjects vary both in means and covariances. Thus traditional psychometric methods cannot fully capture the heterogeneity in our data.     

Bayesian factor analysis for multilevel binary observations

Multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs. We develop practical simulation based procedures for Bayesian inference of multilevel binary factor analysis models. We illustrate how Markov Chain Monte Carlo procedures such as Gibbs sampling and Metropolis-Hastings methods can be used to perform Bayesian inference, model checking and model comparison without the need for multidimensional numerical integration. We illustrate the proposed estimation methods using three simulation studies and an application involving student's achievement results in different areas of mathematics. The authors thank Ian Westbury, University of Illinois at Urbana Champaign for kindly providing the SIMS data for the application.     

探索性因子分析及其在应用中存在的主要问题

探索性因子分析的发展非常迅速,已成为教育与社会心理学领域中最常用的统计方法之一。本文全面介绍了探索性因子分析的基本原理,阐述了其发生的机制及基本过程,对其在教育、心理领域应用中存在的问题进行了总结,并针对应用中样本容量和观测变量数目不够、因子求解方法的误用、因子数目的确定标准及因子旋转中存在的问题、因子值缺乏重复验证性、研究结果呈现形式不规范、过于依赖SPSS、缺乏主动性等问题提出了一些相应的建议。     

Higher-order exploratory factor analysis of the Reynolds Intellectual Assessment Scales with a referred sample

The factor structure of the Reynolds Intellectual Assessment Scales (RIAS; [Reynolds, C.R., & Kamphaus, R.W. (2003). Reynolds Intellectual Assessment Scales. Lutz, FL: Psychological Assessment Resources, Inc.]) was investigated with a large (N = 1163) independent sample of referred students (ages 6-18). More rigorous factor extraction criteria (viz., Horn's parallel analysis (HPA); [Horn, J.L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179-185.], and Minimum Average Partial (MAP) analysis; [Velicer, W.F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321-327.]), in addition to those used in RIAS development, were investigated. Exploratory factor analyses using both orthogonal and oblique rotations and higher-order exploratory factor analyses using the Schmid and Leiman [Schmid, J., and Leiman, J.M. (1957). The development of hierarchical factor solutions. Psychometrika, 22, 53-61.] procedure were conducted. All factor extraction criteria indicated extraction of only one factor. Oblique rotations resulted in different results than orthogonal rotations, and higher-order factor analysis indicated the largest amount of variance was accounted for by the general intelligence factor. The proposed three-factor solution was not supported. Implications for the use of the RIAS with similarly referred students are discussed.     

Exploratory second-order analyses for components and factors

Abstract: Exploratory methods using second‐order components and second‐order common factors were proposed. The second‐order components were obtained from the resolution of the correlation matrix of obliquely rotated first‐order principal components. The standard errors of the estimates of the second‐order component loadings were derived from an augmented information matrix with restrictions for the loadings and associated parameters. The second‐order factor analysis proposed was similar to the classical method in that the factor correlations among the first‐order factors were further resolved by the exploratory method of factor analysis. However, in this paper the second‐order factor loadings were estimated by the generalized least squares using the asymptotic variance‐covariance matrix for the first‐order factor correlations. The asymptotic standard errors for the estimates of the second‐order factor loadings were also derived. A numerical example was presented with simulated results.     

探索性因素分析——最近10年的评述

目的：(1)介绍国外心理统计学界对探索性因素分析中几个重大问题的基本观点;(2)系统地评述过去10年里(1991～2000年)我国心理学研究对这一技术的使用情况;(3)强调运用这一技术时值得注意的一些事项,以期这一技术对我国心理学研究发挥更大的作用。方法：作检索了《心理学报》和《心理科学》在1991～2000年间与探索性因素分析有关的章,对其中以探索性因素分析为主要研究方法的重点章(feature article)进行了编码,统计了频数及百分比分布。结果：我国心理学研究在积极使用探索性因素分析这一先进统计技术时还存在一些问题,主要表现在：(1)在确定因素个数时,倾向于机械地依靠桌个单一方法来作决定,(2)大量使用正交旋转,(3)过于依赖SPSS,(4)对因素分析过程中的重要信息／结果报告不够。结论：探索性因素分析在过去十年得到了广泛的应用,如能吸收国外同行的一些观点,探索性因素分析这一技术在我国心理学研究中必将获得更广泛有效的应用。     

Acculturation and mental health: Current findings and recommendations for future research

Results of existing literature suggest that the relationship between acculturation and mental health is complex. Some studies have found a beneficial association between increased acculturation and mental health, whereas others have found a detrimental association or no relationship at all. We reviewed literature on acculturation and mental health in Hispanics/Latino Americans, Asians/Asian Americans, and other ethnic groups. Results indicate that greater acculturation is associated with increased substance use and abuse. Findings for other disorders and symptoms demonstrate considerable heterogeneity and potential reasons for this variation are provided. Acculturation involves a complex set of processes that appear to have differential impact on mental health outcomes. Critical issues in the measurement of acculturation are discussed, and recommendations for future research are offered.     

The big none: No evidence for a general factor of personality in chimpanzees, orangutans, or rhesus macaques

We examined whether a general factor of personality (GFP) was present in chimpanzees, orangutans, or rhesus macaques. We used confirmatory factor analysis (CFA) to model correlations among first-order factors as arising from a GFP. We then conducted principal axis factor analyses (PFA) of the first-order factors to extract a single higher-order factor and then to extract two oblique higher-order factors. The CFA model fit was poor for chimpanzees and orangutans, but not rhesus macaques. The single higher-order factors extracted via PFA did not resemble the GFP in all three species. The oblique higher-order factors extracted via PFA were only weakly correlated in all three species. These results do not support the existence of a GFP in nonhuman primates.     

Regression estimation and post-stratification in factor analysis

Regression estimation and poststratification are methods used in survey sampling to estimate a population mean, when additional information is available for some auxiliary variables. The extension of these methods to factor analysis is considered. These methods may be used either to improve the efficiency of estimation or to adjust for the effects of nonrandom selection. The estimation procedure may be formulated in a LISREL framework.The author is grateful to the referees for their comments.     

A loss function for alpha factor analysis

Most of the factor solutions can be got by minimizing a corresponding loss function. However, up to now, a loss function for the alpha factor analysis (AFA) has not been known. The present paper establishes such a loss function for the AFA. Some analogies to the maximum likelihood factor analysis are discussed.The author is greatly indebted to Prof. Henry F. Kaiser (University of California, Berkeley) for his kind encouragement. He is also indebted to an anonymous referee ofPsychometrika for having confronted him with the problem in 1977. Financial support by the Wiener Hochschuljubiläumsstiftung is gratefully acknowledged.