Factor Retention in Exploratory Factor Analysis With Missing Data

Determining the number of factors in exploratory factor analysis is arguably the most crucial decision a researcher faces when conducting the analysis. While several simulation studies exist that compare various so-called factor retention criteria under different data conditions, little is known about the impact of missing data on this process. Hence, in this study, we evaluated the performance of different factor retention criteria—the Factor Forest, parallel analysis based on a principal component analysis as well as parallel analysis based on the common factor model and the comparison data approach—in combination with different missing data methods, namely an expectation-maximization algorithm called Amelia, predictive mean matching, and random forest imputation within the multiple imputations by chained equations (MICE) framework as well as pairwise deletion with regard to their accuracy in determining the number of factors when data are missing. Data were simulated for different sample sizes, numbers of factors, numbers of manifest variables (indicators), between-factor correlations, missing data mechanisms and proportions of missing values. In the majority of conditions and for all factor retention criteria except the comparison data approach, the missing data mechanism had little impact on the accuracy and pairwise deletion performed comparably well as the more sophisticated imputation methods. In some conditions, especially small-sample cases and when comparison data were used to determine the number of factors, random forest imputation was preferable to other missing data methods, though. Accordingly, depending on data characteristics and the selected factor retention criterion, choosing an appropriate missing data method is crucial to obtain a valid estimate of the number of factors to extract.     

Incorporating Uncertainty Into Parallel Analysis for Choosing the Number of Factors via Bayesian Methods

A number of psychometricians have suggested that parallel analysis (PA) tends to yield more accurate results in determining the number of factors in comparison with other statistical methods. Nevertheless, all too often PA can suggest an incorrect number of factors, particularly in statistically unfavorable conditions (e.g., small sample sizes and low factor loadings). Because of this, researchers have recommended using multiple methods to make judgments about the number of factors to extract. Implicit in this recommendation is that, when the number of factors is chosen based on PA, uncertainty nevertheless exists. We propose a Bayesian parallel analysis (B-PA) method to incorporate the uncertainty with decisions about the number of factors. B-PA yields a probability distribution for the various possible numbers of factors. We implement and compare B-PA with a frequentist approach, revised parallel analysis (R-PA), in the contexts of real and simulated data. Results show that B-PA provides relevant information regarding the uncertainty in determining the number of factors, particularly under conditions with small sample sizes, low factor loadings, and less distinguishable factors. Even if the indicated number of factors with the highest probability is incorrect, B-PA can show a sizable probability of retaining the correct number of factors. Interestingly, when the mode of the distribution of the probabilities associated with different numbers of factors was treated as the number of factors to retain, B-PA was somewhat more accurate than R-PA in a majority of the conditions.     

Extension Procedures for Confirmatory Factor Analysis

We present factor extension procedures for confirmatory factor analysis that provide estimates of the relations of common and unique factors with external variables that do not undergo factor analysis. We present identification strategies that build upon restrictions of the pattern of correlations between unique factors and external variables. The first restriction minimizes the sum of squared correlations between unique factors and external variables. This approach is similar to the traditional factor extension procedure. The second restriction minimizes the complexity of the pattern of external correlations of unique factors. This approach has similarities with the simple structure ideal imposed on most factor rotation strategies. The procedures are illustrated with a real data example that demonstrates their applicability to real-world research questions.     

The Poor Fit of Model Fit for Selecting Number of Factors in Exploratory Factor Analysis for Scale Evaluation

Model fit indices are being increasingly recommended and used to select the number of factors in an exploratory factor analysis. Growing evidence suggests that the recommended cutoff values for common model fit indices are not appropriate for use in an exploratory factor analysis context. A particularly prominent problem in scale evaluation is the ubiquity of correlated residuals and imperfect model specification. Our research focuses on a scale evaluation context and the performance of four standard model fit indices: root mean square error of approximate (RMSEA), standardized root mean square residual (SRMR), comparative fit index (CFI), and Tucker–Lewis index (TLI), and two equivalence test-based model fit indices: RMSEAt and CFIt. We use Monte Carlo simulation to generate and analyze data based on a substantive example using the positive and negative affective schedule (N = 1,000). We systematically vary the number and magnitude of correlated residuals as well as nonspecific misspecification, to evaluate the impact on model fit indices in fitting a two-factor exploratory factor analysis. Our results show that all fit indices, except SRMR, are overly sensitive to correlated residuals and nonspecific error, resulting in solutions that are overfactored. SRMR performed well, consistently selecting the correct number of factors; however, previous research suggests it does not perform well with categorical data. In general, we do not recommend using model fit indices to select number of factors in a scale evaluation framework.     

加入惩罚因子的电商平台协同过滤推荐算法

为提高推荐算法挖掘数据长尾信息的能力,降低推荐结果流行度,使推荐结果更多样,在传统协同过滤推荐算法基础上,分别将热门项目与活跃用户的惩罚因子引入相似性计算中,依据准确度、覆盖率、流行度等评价标准,在上海某电商平台销售数据集上进行比较,并通过多组实验验证不同参数对推荐算法的影响。结果显示,加入惩罚因子后基于用户的协同过滤推荐算法在N值取10、K值取3时,流行度为3.97,比传统方法降低了7.31%;加入惩罚因子后基于项目的协同过滤推荐算法在N值取10、K值取3时,准确率为7.65%,比传统方法提高了5.25%。由此证明加入惩罚因子的协同过滤推荐算法在保持算法准确率的同时,可在一定程度上降低推荐结果流行度。     

A Comparison of CFA,ESEM, and BSEM in Test Structure Analysis

Minor cross-loadings on non-targeted factors are often found in psychological or other instruments. Forcing them to zero in confirmatory factor analyses (CFA) leads to biased estimates and distorted structures. Alternatively, exploratory structural equation modeling (ESEM) and Bayesian structural equation modeling (BSEM) have been proposed. In this research, we compared the performance of the traditional independent-clusters-confirmatory-factor-analysis (ICM-CFA), the nonstandard CFA, ESEM with the Geomin- or Target-rotations, and BSEMs with different cross-loading priors (correct; small- or large-variance priors with zero mean) using simulated data with cross-loadings. Four factors were considered: the number of factors, the size of factor correlations, the cross-loading mean, and the loading variance. Results indicated that ICM-CFA performed the worst. ESEMs were generally superior to CFAs but inferior to BSEM with correct priors that provided the precise estimation. BSEM with large- or small-variance priors performed similarly while the prior mean for cross-loadings was more important than the prior variance.     

Should Regularization Replace Simple Structure Rotation in Exploratory Factor Analysis?

Exploratory factor analysis (EFA) is an important tool when the measurement structure of psychological constructs is uncertain. Typically, factor rotation is applied to obtain interpretable results resembling a simple structure. However, an overwhelming multitude of rotation techniques is available of which none is unequivocally superior. Recently, regularization has been suggested as an alternative to factor rotation. In two simulation studies, we addressed the question if regularized EFA is a suitable alternative for rotated EFA. We compared their performance in recovering predefined factor loading patterns with varying amounts of cross-loadings. Elastic net regularized EFA yielded estimates comparable to rotated EFA. For complex loading patterns, both rotated and regularized EFA tended to underestimate cross-loadings and inflate factor correlations, but regularized EFA was able to recover loading patterns as long as a subset of items followed a simple structure. We conclude that regularization is a suitable alternative to factor rotation for psychometric applications.     

Exploratory Factor Analysis (EFA) Programs in R

We provide a brief overview of two R packages that can conduct exploratory factor analysis (EFA): psych and EFAutilities. After introducing EFA and the exemplar data used in this paper we discuss best practices for EFA. Next, we describe the approaches used in the two packages for EFA. During this explanation, we provide sample code and discuss the usage and results of two empirical datasets. Finally, we highlight the similarities and distinctions of each package on modeling EFA.     

师范生择业效能感问卷的初步编制

为了编制师范生择业效能感问卷,并检验所编制问卷的信度和效度,采用探索性因素分析初步探究理论结构,用验证性因素分析验证理论结构的合理性和正确性。验证结果为:探索性因素分析确定该问卷包括5个因子,全问卷的内部一致性系数为0.96,各分问卷的信度均在0.85以上,各分问卷全问卷得分之间的相关系数在0.859~0.918之间。从而得出结论:编制的师范生择业效能感问卷共包括5个因子,量表具有较好的信效度。     

广州澳门两地青少年价值观的比较研究

以920名广州青少年及665名澳门青少年为被试,通过探索性因素分析和验证性因素分析,编制青少年价值观问卷,并考察两地青少年价值观的特点及其影响因素。结果表明：（1）青少年价值观问卷由社会观、人际观、家庭观、成就观、理想观、尊严观、生活观,及爱情观等八个因素组成;（2）问卷具有较好的信度和效度;（3）家庭观、成就观是两地青少年的主导价值观,但两地青少年在某些价值观维度的得分上存在差异;（4）两地青少年在“影响价值观形成的因素”的看法上存在一定的差异,但都将“家庭的影响”和“个人生活经历的变化”视为最重要的影响因素。     

Exploratory and Confirmatory Factor Analysis of the School Level Environment Questionnaire (SLEQ)

The purpose of the present study was to validate an existing school environment instrument, the School Level Environment Questionnaire (SLEQ). The SLEQ consists of 56 items, with seven items in each of eight scales. One thousand, one hundred and six (1106) teachers in 59 elementary schools in a southwestern USA public school district completed the instrument. An exploratory factor analysis was undertaken for a random sample of half of the completed surveys. Using principal axis factoring with oblique rotation, this analysis suggested that 13 items should be dropped and that the remaining 43 items could best be represented by seven rather than eight factors. A confirmatory factor analysis was run with the other half of the original sample using structural equation modeling. Examination of the fit indices indicated that the model came close to fitting the data, with goodness-of-fit (GOF) coefficients just below recommended levels. A second model was then run with two of the seven factors, with their associated items removed. That left five factors with 35 items. Model fit was improved. A third model was tried, using the same five factors with 35 items but with correlated residuals between some of the items within a factor. This model seemed to fit the data well, with GOF coefficients in recommended ranges. These results led to a refined, more parsimonious version of the SLEQ that was then used in a larger study. Future research is needed to see if this model would fit other samples in different elementary schools and in secondary schools both in the USA and in other countries. This revised version was published online in July 2006 with corrections to the Cover Date.     

大学生功利心理问卷的初步编制

目的:编制大学生功利心理问卷。方法:采用逻辑分析和因素分析相结合的综合法初步编制了大学生功利心理问卷,并对其进行信效度检验。采取随机整群的抽样方法,在预施测和正式施测中各抽取大学生400名。结果:探索性因素分析得到四个因子:利他与利己、利远与利近、道义与名利、理性与欲求。验证性因素分析得到χ2/df=1.96,RMSEA=0.057,IFI、CFI大于0.9。总问卷的内部一致性系数为α=0.723;分半信度为0.803(p<.001)。结论:大学生功利心理问卷的信效度符合心理测量学要求,可在研究中试用。     

非学业自我描述量表（NSDQⅢ）在大学生中试用的信效度研究

目的：对非学业自我描述量表进行修订和信效度检验.方法：以浙江师范大学492名学生为被试,使用SPSS16.0和Amos5.0进行探索性和验证性因素分析.结果表明：（1）经过探索性因素分析,量表包含7个因子,共45个项目,各个项目负荷在0.416~0.810之间,7个二级因子解释了总方差的54.611%;（2）将7个二级因子按Marsh的多层次自我描述量表依托的理论合并为4个一级因子,提高了内部一致性水平,4个因子及总量表内部一致性系数为0.651~0.821.各因子间相关呈中等偏低相关,结构效度良好;（3）通过验证性因素分析,4个一级因子的模型拟合优度指标分别为：χ2/df〈5,RMSEA〈0.05,GFI,NFI,TLI,CFI为0.989~0.997;（4）在体能、与双亲关系和一般自我3个因子上存在性别的显著性差异,男生比女生评价更积极.鉴于以上结果,非学业自我描述量表的信效度水平达到了测量学的要求.     

网络游戏行为的心理因素探析

本研究采用探索性因素分析,对网络游戏行为的心理因素进行探析。分析结果表明,网络游戏的心理因素由成就体验、缓解压力与宣泄、寻求刺激、逃避现实和交往与归属五个维度构成。游戏者的性别和游戏龄在心理因素上无显著差异;大学生在成就体验和逃避现实这两个因素上的得分显著高于高职生。     

On the Exploratory Road to Unraveling Factor Loading Non-invariance: A New Multigroup Rotation Approach

Multigroup exploratory factor analysis (EFA) has gained popularity to address measurement invariance for two reasons. Firstly, repeatedly respecifying confirmatory factor analysis (CFA) models strongly capitalizes on chance and using EFA as a precursor works better. Secondly, the fixed zero loadings of CFA are often too restrictive. In multigroup EFA, factor loading invariance is rejected if the fit decreases significantly when fixing the loadings to be equal across groups. To locate the precise factor loading non-invariances by means of hypothesis testing, the factors’ rotational freedom needs to be resolved per group. In the literature, a solution exists for identifying optimal rotations for one group or invariant loadings across groups. Building on this, we present multigroup factor rotation (MGFR) for identifying loading non-invariances. Specifically, MGFR rotates group-specific loadings both to simple structure and between-group agreement, while disentangling loading differences from differences in the structural model (i.e., factor (co)variances).     

Factor structure of the WISC‐V in four standardization age groups: Exploratory and hierarchical factor analyses with the 16 primary and secondary subtests

This study examined the factor structure of the Wechsler Intelligence Scale for Children‐Fifth Edition (WISC‐V) with four standardization sample age groups (6–8, 9–11, 12–14, 15–16 years) using exploratory factor analysis (EFA), multiple factor extraction criteria, and hierarchical EFA not included in the WISC‐V Technical and Interpretation Manual. Factor extraction criteria suggested that one to four factors might be sufficient despite the publisher‐promoted, five‐factor solution. Forced extraction of five factors resulted in only one WISC‐V subtest obtaining a salient pattern coefficient on the fifth factor in all four groups, rendering it inadequate. Evidence did not support the publisher's desire to split Perceptual Reasoning into separate Visual Spatial and Fluid Reasoning dimensions. Results indicated that most WISC‐V subtests were properly associated with the four theoretically oriented first‐order factors resembling the WISC‐IV, the g factor accounted for large portions of total and common variance, and the four first‐order group factors accounted for small portions of total and common variance. Results were consistent with EFA of the WISC‐V total standardization sample.     

The development of a small-scale survey instrument of UK teachers to study professional use (and non-use) of and attitudes to social media

This paper documents the creation, implementation and analysis of a survey instrument designed to reveal patterns of use and attitudes towards the value of social media by UK teachers. The study was motivated to discover which teachers use social media professionally, how they use it (both personally and professionally) and attitudes to social media as a professional tool (for their students' and their own professional use). The instrument was created from verbal data from two focus group discussions regarding the use of social media in education. Attitude statements were included verbatim when practical. This instrument was placed online and practising teachers invited to complete it (n?=?216). Exploratory factor analysis and hierarchical clustering identified 9 factors from 54 attitude statements and 5 distinct teacher groups. The rich data allowed each group to be carefully defined, providing potentially invaluable information to school leaders when developing social media projects to recognize and accommodate the full range of teacher concerns and experience. The paper also addresses methodological concerns regarding instrument creation, dealing with missing data and the impact of missing data on subsequent analysis.     

积极人格特质问卷PPTQ在中国大学生中的因素结构研究

探索积极人格特质问卷(PPTQ)在中国大学生中的因素结构。采用积极人格特质问卷(Positive Personality Traits Questionnaire)中文版,对648名大学生施测,对其中一半数据使用PASWStatistics18进行探索性因素分析,另一半数据使用AMOS16.0进行验证性因素分析(CFA)。探索性因素分析得出积极自我意象、外向性和文化认同三因素结构。累计解释率为50.457%,验证性因素分析结果显示:χ2/df=2.230,GFI=0.841。AGFI=0.816,CFI=0.822,RMSEA=0.062,中文版三因素结构在中国大学生人群中较为合理。     

大学生学习适应状况量表的编制

在有关当代大学生学习适应状况研究的基础上,探讨了大学生学习适应状况的主要因素,编制了一套适合当代大学生学习适应状况的量表,对9所院校900名大学生进行施测,对所得数据进行探索性因素分析。结果表明,大学生学习适应状况的主要因素包括:学习方法、学习热情、学习态度、专业兴趣、学习动力、学习环境等六个因素。验证性因素分析结果表明,所提取的6个因素与构想模型拟合较好,测验具有较好的信效度。     

大学生学习情况调查问卷的探索性因素分析——以红河学院为例

以红河学院为例对在校大学生的学习情况进行问卷调查,探索性因素分析结果表明:影响红河学院在校大学生学习情况的主要因素有学习规划和学习方法、学习态度、学习气氛、教师等其他因素。