Estimating confidence intervals for eigenvalues in exploratory factor analysis

Exploratory factor analysis (EFA) has become a common procedure in educational and psychological research. In the course of performing an EFA, researchers often base the decision of how many factors to retain on the eigenvalues for the factors. However, many researchers do not realize that eigenvalues, like all sample statistics, are subject to sampling error, which means that confidence intervals (CIs) can be estimated for each eigenvalue. In the present article, we demonstrate two methods of estimating CIs for eigenvalues: one based on the mathematical properties of the central limit theorem, and the other based on bootstrapping. References to appropriate SAS and SPSS syntax are included. Supplemental materials for this article may be downloaded from http://brm.psychonomic-journals.org/content/supplemental.     

Evaluating Exploratory Factor Analysis Which Initial-Extraction Techniques Provide the Best Factor Fidelity?

Attacks by communication scholars on exploratory factor analysis (EFA) have cast doubt on prior findings based on the technique. The present study is one in a series of studies performed to test the ability of EFA to produce results that replicate known dimensions in a data set. It was designed to determine which of 7 initial extraction techniques produce the highest factor fidelity across 5 item distribution shapes and 3 sample sizes. Monte-Carlo-created data sets with known factors, known item distribution shapes, and a 30% error rate were submitted to EFA. Results from an analysis of variance (ANOH) indicate that image analysis reaches perfect factor fidelity with a smaller number of cases regardless of item distribution shape. This and other results reported in this study suggest that3ndings based on EFA should be viewed with cautious optimism and be evaluated according to the findings from this and similar studies.     

Enhancing Interpretability in Factor Analysis by Means of Mathematical Optimization

Abstract

Exploratory Factor Analysis (EFA) is a widely used statistical technique to discover the structure of latent unobserved variables, called factors, from a set of observed variables. EFA exploits the property of rotation invariance of the factor model to enhance factors’ interpretability by building a sparse loading matrix. In this paper, we propose an optimization-based procedure to give meaning to the factors arising in EFA by means of an additional set of variables, called explanatory variables, which may include in particular the set of observed variables. A goodness-of-fit criterion is introduced which quantifies the quality of the interpretation given this way. Our methodology also exploits the rotational invariance of EFA to obtain the best orthogonal rotation of the factors, in terms of the goodness-of-fit, but making them match to some of the explanatory variables, thus going beyond traditional rotation methods. Therefore, our approach allows the analyst to interpret the factors not only in terms of the observed variables, but in terms of a broader set of variables. Our experimental results demonstrate how our approach enhances interpretability in EFA, first in an empirical dataset, concerning volumes of reservoirs in California, and second in a synthetic data example.     

Exploratory Factor Analysis With Small Sample Sizes

Exploratory factor analysis (EFA) is generally regarded as a technique for large sample sizes (N), with N = 50 as a reasonable absolute minimum. This study offers a comprehensive overview of the conditions in which EFA can yield good quality results for N below 50. Simulations were carried out to estimate the minimum required N for different levels of loadings (λ), number of factors (f), and number of variables (p) and to examine the extent to which a small N solution can sustain the presence of small distortions such as interfactor correlations, model error, secondary loadings, unequal loadings, and unequal p/f. Factor recovery was assessed in terms of pattern congruence coefficients, factor score correlations, Heywood cases, and the gap size between eigenvalues. A subsampling study was also conducted on a psychological dataset of individuals who filled in a Big Five Inventory via the Internet. Results showed that when data are well conditioned (i.e., high λ, low f, high p), EFA can yield reliable results for N well below 50, even in the presence of small distortions. Such conditions may be uncommon but should certainly not be ruled out in behavioral research data.     

The Use of Exploratory Factor Analysis and Principal Components Analysis in Communication Research

Exploratory factor analysis is a popular statistical technique used in communication research. Although exploratory factor analysis (EFA) and principal components analysis (PCA) are different techniques, PCA is often employed incorrectly to reveal latent constructs (i.e., factors) of observed variables, which is the purpose of EFA. PCA is more appropriate for reducing measured variables into a smaller set of variables (i.e., components) by keeping as much variance as possible out of the total variance in the measured variables. Furthermore, the popular use of varimax rotation raises some concerns about the relationships among the factors that researchers claim to discover. This paper discusses the distinct purposes of PCA and EFA, using two data sets as examples to highlight the differences in results between these procedures, and also reviews the use of each technique in three major communication journals: Communication Monographs, Human Communication Research, and Communication Research.     

Development and validation of the Causes of Re-Injury Worry Questionnaire

Re-injury worry is an important construct in competitive sport that may influence performance and increase the risk of re-injury. However, there are currently no available instruments to measure the causes of re-injury worry. The purpose of this study was to develop the Causes of Re-Injury Worry Questionnaire (CR-IWQ). The study was conducted in three independent research phases to investigate the following: (a) the content relevance, (b) the factor structure and the factorial validity, (c) the concurrent validity, (d) the discriminant validity, and (e) the test-retest reliability (intraclass correlation coefficients; ICC), and the internal consistency of the instrument. Exploratory factor analysis (EFA) was chosen to examine the factor structure of the CR-IWQ. Confirmatory factor analysis (CFA) was used to examine further the factorial validity of the instrument. A number of valid constructs were used to assess the concurrent and discriminant validity of the CR-IWQ. The reliability of the new instrument was examined using Pearson r (ICC) and Cronbach α. Three hundred and seventy athletes with an acute musculoskeletal sport injury in the last year participated in the study. EFA revealed a 12-item model, representing two factors ("Re-injury worry due to rehabilitation" and "Re-injury worry due to opponent's ability"). CFA supported the two-factor model of the CR-IWQ. The concurrent and discriminant validity of the CR-IWQ was confirmed by examining correlations between the CR-IWQ with other constructs. The ICCs and the Cronbach α indices of the CR-IWQ were acceptable. We have demonstrated that the CR-IWQ is a good psychometric instrument that can be used for clinical and research purposes.     

Semi-sparse PCA

It is well known that the classical exploratory factor analysis (EFA) of data with more observations than variables has several types of indeterminacy. We study the factor indeterminacy and show some new aspects of this problem by considering EFA as a specific data matrix decomposition. We adopt a new approach to the EFA estimation and achieve a new characterization of the factor indeterminacy problem. A new alternative model is proposed, which gives determinate factors and can be seen as a semi-sparse principal component analysis (PCA). An alternating algorithm is developed, where in each step a Procrustes problem is solved. It is demonstrated that the new model/algorithm can act as a specific sparse PCA and as a low-rank-plus-sparse matrix decomposition. Numerical examples with several large data sets illustrate the versatility of the new model, and the performance and behaviour of its algorithmic implementation.

     

Two Tracks of Thought: A Structural Model of the Test for Creative Thinking-Drawing Production (TCT-DP)

The Test for Creative Thinking-Drawing Production (TCT-DP, Urban & Jellen, 1986) is one of the most used instruments for the assessment of creative potential. Few studies exist regarding its factorial structure, and all of them were limited to using an exploratory approach. The aim of this research was to assess the factorial structure of the TCT-DP (Form A) for undergraduate and postgraduate Portuguese students. Two studies were performed, where Study 1 (967 participants) consisted of an exploratory factor analysis (EFA) that yielded a 2-factor solution and Study 2 (920 participants) consisted of a confirmatory factor analysis (CFA) used to test the fit and compare the suitability of the 2 factorial solutions with alternative models. The 2-factor model exhibited good and acceptable indices of fit. The factors represented Innovativeness and Adaptiveness. This model was called Two Tracks of Thought (TTT). Its structure suggests the importance of both non-conventional and conventional thinking for the creative process.     

Exploratory Factor Analysis With Small Samples and Missing Data

Exploratory factor analysis (EFA) is an extremely popular method for determining the underlying factor structure for a set of variables. Due to its exploratory nature, EFA is notorious for being conducted with small sample sizes, and recent reviews of psychological research have reported that between 40% and 60% of applied studies have 200 or fewer observations. Recent methodological studies have addressed small size requirements for EFA models; however, these models have only considered complete data, which are the exception rather than the rule in psychology. Furthermore, the extant literature on missing data techniques with small samples is scant, and nearly all existing studies focus on topics that are not of primary interest to EFA models. Therefore, this article presents a simulation to assess the performance of various missing data techniques for EFA models with both small samples and missing data. Results show that deletion methods do not extract the proper number of factors and estimate the factor loadings with severe bias, even when data are missing completely at random. Predictive mean matching is the best method overall when considering extracting the correct number of factors and estimating factor loadings without bias, although 2-stage estimation was a close second.     

中国大学生艾森克人格问卷测试因素结构之探讨

本文旨在检验艾森克人格理论对中国人的适合性,探明艾森克人格问卷(成人)的结构。经对2311名大学生艾森克人格问卷的作答数据做验证性因素分析,艾森克人格理论未能得到证明。使用全息项目因素分析方法对EPQ进行因素分析,得出神经质、外向性、友善因素、诚实因素、任性因素和严谨因素六个因素。该方法较好地克服了经典线性因素分析高估维度数。低估因素负荷等缺陷。     

Validation of the NEO Personality Inventory and the five-factor model. Can findings from exploratory and confirmatory factor analysis be reconciled?

The aims of the study were (i) to analyse a Norwegian version of the NEO Personality Inventory (NEO-PI), using both exploratory factor analysis (EFA) and confirmatory factor analysis (CFA); (ii) to compare the results of the two factor analytic strategies, both within the present study and across different studies; and (iii) to discuss possible causes of discrepant findings (across factor-analytic methods and across samples). The sample comprised 961 subjects representative of the non-institutionalized Norwegian adult population. Using an EFA strategy, very high coefficients of factor comparability (r=0.93–0.99) across sexes were found. None of the five main domains turned out to be as homogeneous as suggested by the original five-factor model, but most of the deviations from the assumed simple structure were comparable to results from recent American studies. However, none of the revised EFA-based models were supported using CFA methods. Moreover, a large number of modifications were necessary to obtain a model with acceptable fit. It is argued that these discrepant findings can be accounted for, at least in part, by (i) consequences of different model acceptance criteria in the EFA and CFA tradition, (ii) the inherent logical–semantical structure of the NEO-PI, and (iii) consequences of selection effects (factorial invariance problem). © 1997 by John Wiley & Sons, Ltd.     

Analysis of the factor structure of the Sociocultural Attitudes Towards Appearance Questionnaire (SATAQ-3) in Spanish secondary-school students through exploratory structural equation modeling

The aims of the present study were: (1) to assess the factor structure of the SATAQ-3 in Spanish secondary-school students by means of exploratory factor analysis (EFA), confirmatory factor analysis (CFA) and exploratory structural equation modeling (ESEM) models; and (2) to study its invariance by sex and school grade. ESEM is a technique that has been proposed for the analysis of internal structure that overcomes some of the limitations of EFA and CFA. Participants were 1559 boys and girls in grades seventh to tenth. The results support the four-factor solution of the original version, and reveal that the best fit was obtained with ESEM, excluding Item 20 and with correlated uniqueness between reverse-keyed items. Our version shows invariance by sex and grade. The differences between scores of different groups are in the expected direction, and support the validity of the questionnaire. We recommend a version excluding Item 20 and without reverse-keyed items.     

大学生传统人际价值观的探索性与验证性研究

选取传统谚语、俗语中关于人际关系的语句编制成传统人际价值观问卷,通过探索性因子分析,构建传统人际价值观的维度。并通过验证性因子分析检验理论构想的拟合性。探索性因子分析中,一阶因子分析抽取了14个因子,对14个因子进行二阶因子分析又抽取了人际道德、人际亲疏性、人际防御、人际智慧4个主因子。根据这一结果建构了一个含有人际道德、人际亲疏性、人际防御三个主因子的模型。在验证性因子分析中,对模型进行修正,使之分别含有4个、3个、2个次因子,经检验,模型的拟合比较良好。     

Determining the number of factors to retain: Q windows-based FORTRAN-IMSL program for parallel analysis

Parallel analysis (PA; Horn, 1965) is a technique for determining the number of factors to retain in exploratory factor analysis that has been shown to be superior to more widely known methods (Zwick & Velicer, 1986). Despite its merits, PA is not widely used in the psychological literature, probably because the method is unfamiliar and because modern, Windows-compatible software to perform PA is unavailable. We provide a FORTRAN-IMSL program for PA that runs on a PC under Windows; it is interactive and designed to suit the range of problems encountered in most psychological research. Furthermore, we provide sample output from the PA program in the form of tabled values that can be used to verify the program operation; or, they can be used either directly or with interpolation to meet specific needs of the researcher.     

Factor structure of coping: two studies of mothers with high levels of life stress

Exploratory factor analysis (EFA) and confirmatory factor analysis (CFA) were used to investigate the factor structure of coping in mothers with high levels of life stress. In Study 1, EFA of the Coping Orientation to Problems Experienced (C. S. Carver, M. F. Scheier, & J. K. Weintraub, 1989) in a sample of mothers of full-term or very low birth weight 2-year-old children yielded 7 reliable coping factors. Each factor accounted for significant variance in at least 1 of 6 outcomes measuring maternal-child well-being. In Study 2, CFA was used to cross-validate the EFA model on the basis of the responses of mothers of 2-year-old children with prenatal polysubstance exposure. CFA results revealed a moderately good fit, confirming the factor structure in a 2nd, independent sample of mothers with high levels of life stress.     

ADHD and college students: exploratory and confirmatory factor structures with student and parent data

Exploratory factor analyses (EFAs) and confirmatory factor analyses (CFAs) were used to investigate the structure of the Student Report Inventory (SRI) and Parent Report Inventory (PRI) of the College Attention-Deficit/Hyperactivity Disorder (ADHD) Response Evaluation. The sample was composed of 1,080 college students and their parents and was stratified by ethnicity, gender, ability level, age, grade, region of residence, and psychoeducational classification status. Results varied according to the information source (self-report vs. parent). EFA uncovered and CFA confirmed 3 distinct and reliable dimensions for student reports: Inattention, Hyperactivity, and Impulsivity. By contrast, EFA and CFA uncovered a reliable 2-dimension structure for the parent-report data. Factor structures replicated across genders (3 factors for the SRI, and 2 factors for the PRI). Results are discussed in terms of the divergence of structures.     

Factor structure of the Bulimia Test--Revised in college women from four ethnic groups

The factor structure of the Bulimia Test--Revised (BULIT-R) was investigated using confirmatory factor analysis (CFA) and exploratory factor analysis (EFA). The sample consisted of 2,671 female college students (African American, Asian American, Caucasian American, and Latino American). Reliability coefficients were excellent across groups. African Americans scored significantly lower on the BULIT-R than Caucasian Americans. Across groups, CFA and EFA results suggest a six-factor solution is most appropriate. Consistent across groups were factors representing bingeing, body image, purging, and extreme weight loss behaviors, while few differences were observed across groups. These findings suggest that the measure is reliable and valid for use with diverse ethnic groups. Future research should focus on culturally salient psychological correlates of disordered eating in diverse ethnic groups.     

Dimensionality assessment of ordered polytomous items with parallel analysis

Parallel analysis (PA) is an often-recommended approach for assessment of the dimensionality of a variable set. PA is known in different variants, which may yield different dimensionality indications. In this article, the authors considered the most appropriate PA procedure to assess the number of common factors underlying ordered polytomously scored variables. They proposed minimum rank factor analysis (MRFA) as an extraction method, rather than the currently applied principal component analysis (PCA) and principal axes factoring. A simulation study, based on data with major and minor factors, showed that all procedures consistently point at the number of major common factors. A polychoric-based PA slightly outperformed a Pearson-based PA, but convergence problems may hamper its empirical application. In empirical practice, PA-MRFA with a 95% threshold based on polychoric correlations or, in case of nonconvergence, Pearson correlations with mean thresholds appear to be a good choice for identification of the number of common factors. PA-MRFA is a common-factor-based method and performed best in the simulation experiment. PA based on PCA with a 95% threshold is second best, as this method showed good performances in the empirically relevant conditions of the simulation experiment.     

A Comparative Investigation of Rotation Criteria Within Exploratory Factor Analysis

Exploratory factor analysis (EFA) is a commonly used statistical technique for examining the relationships between variables (e.g., items) and the factors (e.g., latent traits) they depict. There are several decisions that must be made when using EFA, with one of the more important being choice of the rotation criterion. This selection can be arduous given the numerous rotation criteria available and the lack of research/literature that compares their function and utility. Historically, researchers have chosen rotation criteria based on whether or not factors are correlated and have failed to consider other important aspects of their data. This study reviews several rotation criteria, demonstrates how they may perform with different factor pattern structures, and highlights for researchers subtle but important differences between each rotation criterion. The choice of rotation criterion is critical to ensure researchers make informed decisions as to when different rotation criteria may or may not be appropriate. The results suggest that depending on the rotation criterion selected and the complexity of the factor pattern matrix, the interpretation of the interfactor correlations and factor pattern loadings can vary substantially. Implications and future directions are discussed.