Recovering Substantive Factor Loadings in the Presence of Acquiescence Bias: A Comparison of Three Approaches

Researchers are often advised to write balanced scales (containing an equal number of positively and negatively worded items) when measuring psychological attributes. This practice is recommended to control for acquiescence bias (ACQ). However, little advice has been given on what to do with such data if the researcher subsequently wants to evaluate a 1-factor model for the scale. This article compares 3 approaches for dealing with the presence of ACQ bias, which make different assumptions: an ipsatization approach based on the work of Chan and Bentler (CB; 1993), a confirmatory factor analysis (CFA) approach that includes an ACQ factor with equal loadings (Billiet & McClendon, 2000; Mirowsky & Ross, 1991), and an exploratory factor analysis (EFA) approach with a target rotation (Ferrando, Lorenzo-Seva, & Chico, 2003). We also examine the “do nothing” approach which fits the 1-factor model to the data ignoring the presence of ACQ bias. Our main findings are that the CFA method performs best overall and that it is robust to the violation of its assumptions, the EFA and the CB approaches work well when their assumptions are strictly met, and the “do nothing” approach can be surprisingly robust when the ACQ factor is not very strong.     

Ordinary Least Squares Estimation of Parameters in Exploratory Factor Analysis With Ordinal Data

Exploratory factor analysis (EFA) is often conducted with ordinal data (e.g., items with 5-point responses) in the social and behavioral sciences. These ordinal variables are often treated as if they were continuous in practice. An alternative strategy is to assume that a normally distributed continuous variable underlies each ordinal variable. The EFA model is specified for these underlying continuous variables rather than the observed ordinal variables. Although these underlying continuous variables are not observed directly, their correlations can be estimated from the ordinal variables. These correlations are referred to as polychoric correlations. This article is concerned with ordinary least squares (OLS) estimation of parameters in EFA with polychoric correlations. Standard errors and confidence intervals for rotated factor loadings and factor correlations are presented. OLS estimates and the associated standard error estimates and confidence intervals are illustrated using personality trait ratings from 228 college students. Statistical properties of the proposed procedure are explored using a Monte Carlo study. The empirical illustration and the Monte Carlo study showed that (a) OLS estimation of EFA is feasible with large models, (b) point estimates of rotated factor loadings are unbiased, (c) point estimates of factor correlations are slightly negatively biased with small samples, and (d) standard error estimates and confidence intervals perform satisfactorily at moderately large samples.     

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.     

Asymptotic bias of normal-distribution-based maximum likelihood estimates of moderation effects with data missing at random

Moderation analysis is useful for addressing interesting research questions in social sciences and behavioural research. In practice, moderated multiple regression (MMR) models have been most widely used. However, missing data pose a challenge, mainly because the interaction term is a product of two or more variables and thus is a non-linear function of the involved variables. Normal-distribution-based maximum likelihood (NML) has been proposed and applied for estimating MMR models with incomplete data. When data are missing completely at random, moderation effect estimates are consistent. However, simulation results have found that when data in the predictor are missing at random (MAR), NML can yield inaccurate estimates of moderation effects when the moderation effects are non-null. Simulation studies are subject to the limitation of confounding systematic bias with sampling errors. Thus, the purpose of this paper is to analytically derive asymptotic bias of NML estimates of moderation effects with MAR data. Results show that when the moderation effect is zero, there is no asymptotic bias in moderation effect estimates with either normal or non-normal data. When the moderation effect is non-zero, however, asymptotic bias may exist and is determined by factors such as the moderation effect size, missing-data proportion, and type of missingness dependence. Our analytical results suggest that researchers should apply NML to MMR models with caution when missing data exist. Suggestions are given regarding moderation analysis with missing data.     

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.     

Investigation of Multiple Imputation in Low-Quality Questionnaire Data

The performance of multiple imputation in questionnaire data has been studied in various simulation studies. However, in practice, questionnaire data are usually more complex than simulated data. For example, items may be counterindicative or may have unacceptably low factor loadings on every subscale, or completely missing subscales may complicate computations. In this article, it was studied how well multiple imputation recovered the results of several psychometrically important statistics in a data set with such properties. Analysis of this data set revealed that multiple imputation was able to recover the results of these analyses well. Also, a simulation study showed that multiple imputation produced small bias in these statistics for simulated data sets with the same properties.     

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.     

预测视角下双因子模型与高阶模型的模拟比较

双因子模型和高阶因子模型,作为既有全局因子又有局部因子的两个竞争模型,在研究中得到了广泛应用。本文采用Monte Carlo模拟方法,在模型拟合比较的基础上,比较了效标分别为外显变量和内潜变量时,两个模型在各种负荷水平下预测准确度的差异。结果发现,两种模型在拟合效果方面无显著差异;但在预测效度方面,当效标为显变量时,两个模型的结构系数估计值皆为无偏估计;而效标为潜变量时,高阶因子模型表现优于双因子模型:高阶因子模型的结构系数为无偏估计,双因子模型的结构系数估计值则在50%左右的情况下存在偏差。     

Recovery of weak factor loadings in confirmatory factor analysis under conditions of model misspecification

This article presents the results of two Monte Carlo simulation studies of the recovery of weak factor loadings, in the context of confirmatory factor analysis, for models that do not exactly hold in the population. This issue has not been examined in previous research. Model error was introduced using a procedure that allows for specifying a covariance structure with a specified discrepancy in the population. The effects of sample size, estimation method (maximum likelihood vs. unweighted least squares), and factor correlation were also considered. The first simulation study examined recovery for models correctly specified with the known number of factors, and the second investigated recovery for models incorrectly specified by underfactoring. The results showed that recovery was not affected by model discrepancy for the correctly specified models but was affected for the incorrectly specified models. Recovery improved in both studies when factors were correlated, and unweighted least squares performed better than maximum likelihood in recovering the weak factor loadings.     

大五人格背后的高级人格因素：社会关系模型

人格结构研究中的一种观点认为人格特质具有层级结构,其中大五人格结构是一种流行的人格结构模型,近年来有研究关注大五人格背后是否存在更高等级的人格.采用全循环设计,用等级评定和团队内排序方法,在80个4～6人小团队中收集成员及成员之间在五个人格特质上的自我人格知觉和对成员的人格知觉,共获得四种来源的大五人格数据,根据社会关系模型分解出目标效应.对每一种来源的数据进行探索性因素分析和跨来源联合因素分析,将四种来源数据视为四个指标进行结构方程建模,结果显示,两高级因素相关模型对数据有较高的拟合程度,两个高级人格因素对关系质量有显著且不同效应量的预测作用.结果支持大五人格背后存在更高级的人格因素.最后,讨论高级人格因素模型对理解人格结构、进行人格评估的意义;未来研究可从发展角度采用多评定者、多方法进行.     

新世纪20年国内心理统计方法研究回顾

新世纪头20年, 国内心理学11本专业期刊一共发表了213篇统计方法研究论文。研究范围主要包括以下10类(按论文篇数排序)：结构方程模型、测验信度、中介效应、效应量与检验力、纵向研究、调节效应、探索性因子分析、潜在类别模型、共同方法偏差和多层线性模型。对各类做了简单的回顾与梳理。结果发现, 国内心理统计方法研究的广度和深度都不断增加, 研究热点在相互融合中共同发展; 但综述类论文比例较大, 原创性研究论文比例有待提高, 研究力量也有待加强。     

Methods for Mediation Analysis with Missing Data

Despite wide applications of both mediation models and missing data techniques, formal discussion of mediation analysis with missing data is still rare. We introduce and compare four approaches to dealing with missing data in mediation analysis including listwise deletion, pairwise deletion, multiple imputation (MI), and a two-stage maximum likelihood (TS-ML) method. An R package bmem is developed to implement the four methods for mediation analysis with missing data in the structural equation modeling framework, and two real examples are used to illustrate the application of the four methods. The four methods are evaluated and compared under MCAR, MAR, and MNAR missing data mechanisms through simulation studies. Both MI and TS-ML perform well for MCAR and MAR data regardless of the inclusion of auxiliary variables and for AV-MNAR data with auxiliary variables. Although listwise deletion and pairwise deletion have low power and large parameter estimation bias in many studied conditions, they may provide useful information for exploring missing mechanisms.     

Selecting polychoric instrumental variables in confirmatory factor analysis: An alternative specification test and effects of instrumental variables

The polychoric instrumental variable (PIV) approach is a recently proposed method to fit a confirmatory factor analysis model with ordinal data. In this paper, we first examine the small-sample properties of the specification tests for testing the validity of instrumental variables (IVs). Second, we investigate the effects of using different numbers of IVs. Our results show that specification tests derived for continuous data are extremely oversized at all sample sizes when applied to ordinal variables. Possible modifications for ordinal data are proposed in the present study. Simulation results show that the modified specification tests with all available IVs are able to detect model misspecification. In terms of estimation accuracy, the PIV approach where the IVs outnumber the endogenous variables by one produces a lower bias but a higher variation than the PIV approach with more IVs for correctly specified factor loadings at small samples.     

Determining the number of factors to retain in an exploratory factor analysis using comparison data of known factorial structure

Exploratory factor analysis (EFA) is used routinely in the development and validation of assessment instruments. One of the most significant challenges when one is performing EFA is determining how many factors to retain. Parallel analysis (PA) is an effective stopping rule that compares the eigenvalues of randomly generated data with those for the actual data. PA takes into account sampling error, and at present it is widely considered the best available method. We introduce a variant of PA that goes even further by reproducing the observed correlation matrix rather than generating random data. Comparison data (CD) with known factorial structure are first generated using 1 factor, and then the number of factors is increased until the reproduction of the observed eigenvalues fails to improve significantly. We evaluated the performance of PA, CD with known factorial structure, and 7 other techniques in a simulation study spanning a wide range of challenging data conditions. In terms of accuracy and robustness across data conditions, the CD technique outperformed all other methods, including a nontrivial superiority to PA. We provide program code to implement the CD technique, which requires no more specialized knowledge or skills than performing PA.     

预测视角下双因子模型与高阶因子模型的一般性模拟比较

高阶因子模型本质上是一种特殊的双因子模型, 应用中却常被当做双因子模型的竞争模型。已有研究以满足比例约束的双因子模型(此时等价于一个高阶因子模型)为真实测量模型产生模拟数据, 比较了用双因子模型和高阶因子模型作为测量模型的预测效果。本文使用不满足比例约束的双因子模型(此时不与任何高阶因子模型等价)为真实测量模型产生模拟数据进行比较, 所得结果与满足比例约束的双因子模型的结果有很大差别, 双因子模型结构系数的相对偏差较小、检验力较高, 但第Ⅰ类错误率略高。结论是, 在比例约束条件成立时可以使用高阶因子模型, 否则, 从统计角度看, 一般情况下使用双因子模型进行预测比较好。     

On the sampling errors of factor loadings

The results of three empirical studies on the sampling fluctuation of centroid factor loadings are reported. The first study is based on data which happened to be available on 8 variables for 700 cases and which were factored to three factors for subsamples. The second study is based on fictitious data for 2500 cases which provided separate analyses on 25 samples for each of three situations: 5 variables, one factor; 5 variables, two factors; and 6 variables, three factors. The third study, based on real data for 9 variables and 7000 cases, involves separate factorization for 35 samples of 200 cases. The three studies agree in showing that the sampling behavior of first centroid factor loadings is much like that of correlation coefficients, whereas the sampling fluctuations for loadings beyond the first are disturbingly large.     

How to handle missing data: A comparison of different approaches

Many researchers face the problem of missing data in longitudinal research. Especially, high risk samples are characterized by missing data which can complicate analyses and the interpretation of results. In the current study, our aim was to find the most optimal and best method to deal with the missing data in a specific study with many missing data on the outcome variable. Therefore, different techniques to handle missing data were evaluated, and a solution to efficiently handle substantial amounts of missing data was provided. A simulation study was conducted to determine the most optimal method to deal with the missing data. Results revealed that multiple imputation (MI) using predictive mean matching was the most optimal method with respect to lowest bias and the smallest confidence interval (CI) while maintaining power. Listwise deletion and last observation carried backward also scored acceptable with respect to bias; however, CIs were much larger and sample size almost halved using these methods. Longitudinal research in high risk samples could benefit from using MI in future research to handle missing data. The paper ends with a checklist for handling missing data.     

基于增长模型的非随机缺失数据处理:选择模型和极大似然方法

对含有非随机缺失数据的潜变量增长模型,为了考察基于不同假设的缺失数据处理方法:极大似然(ML)方法与DiggleKenward选择模型的优劣,通过Monte Carlo模拟研究,比较两种方法对模型中增长参数估计精度及其标准误估计的差异,并考虑样本量、非随机缺失比例和随机缺失比例的影响。结果表明,符合前提假设的Diggle-Kenward选择模型的参数估计精度普遍高于ML方法;对于标准误估计值,ML方法存在一定程度的低估,得到的置信区间覆盖比率也明显低于Diggle-Kenward选择模型。     

Estimation and tests of significance in factor analysis

A distinction is drawn between the method of principal components developed by Hotelling and the common factor analysis discussed in psychological literature both from the point of view of stochastic models involved and problems of statistical inference. The appropriate statistical techniques are briefly reviewed in the first case and detailed in the second. A new method of analysis called the canonical factor analysis, explaining the correlations between rather than the variances of the measurements, is developed. This analysis furnishes one out of a number of possible solutions to the maximum likelihood equations of Lawley. It admits an iterative procedure for estimating the factor loadings and also for constructing the likelihood criterion useful in testing a specified hypothesis on the number of factors and in determining a lower confidence limit to the number of factors.     

概化理论缺失数据方差分量估计

各种心理调查、心理实验中, 数据的缺失随处可见。由于数据缺失, 给概化理论分析非平衡数据的方差分量带来一系列问题。基于概化理论框架下, 运用Matlab 7.0软件, 自编程序模拟产生随机双面交叉设计p×i×r缺失数据, 比较和探讨公式法、REML法、拆分法和MCMC法在估计各个方差分量上的性能优劣。结果表明：(1) MCMC方法估计随机双面交叉设计p×i×r缺失数据方差分量, 较其它3种方法表现出更强的优势; (2) 题目和评分者是缺失数据方差分量估计重要的影响因素。