Exploratory Bi-factor Analysis: The Oblique Case

Bi-factor analysis is a form of confirmatory factor analysis originally introduced by Holzinger and Swineford (Psychometrika 47:41?C54, 1937). The bi-factor model has a general factor, a number of group factors, and an explicit bi-factor structure. Jennrich and Bentler (Psychometrika 76:537?C549, 2011) introduced an exploratory form of bi-factor analysis that does not require one to provide an explicit bi-factor structure a priori. They use exploratory factor analysis and a bifactor rotation criterion designed to produce a rotated loading matrix that has an approximate bi-factor structure. Among other things this can be used as an aid in finding an explicit bi-factor structure for use in a confirmatory bi-factor analysis. They considered only orthogonal rotation. The purpose of this paper is to consider oblique rotation and to compare it to orthogonal rotation. Because there are many more oblique rotations of an initial loading matrix than orthogonal rotations, one expects the oblique results to approximate a bi-factor structure better than orthogonal rotations and this is indeed the case. A surprising result arises when oblique bi-factor rotation methods are applied to ideal data.     

双因子模型：多维构念测量的新视角

双因子模型是一种既有全局因子又有局部因子的模型,近年来有了许多应用。本文讨论了双因子模型和高阶因子模型在数学模型、参数之间的关系,概念上和应用上的差异;概述了双因子模型在信度研究、平衡量表、探索性因子分析和项目反应理论中的应用。作为例子,在Rosenberg自尊量表结构的研究中,通过双因子模型分析了自尊特质效应与项目表述方法效应。     

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。     

探索性结构方程建模(ESEM):EFA和CFA的整合

探索性结构方程建模(ESEM)是在测量模型部分使用了类似于EFA模型的SEM.作为一种高级统计方法,ESEM整合了EFA和CFA两种因子分析方法的功能和优点.通过ESEM,研究者既可以灵活地探索因子结构,又可以系统地验证因子模型,为潜变量的关系分析提供更适宜的测量模型.ESEM已经在某些社科领域的研究中得到应用,是一种值得推介的因子分析方法.ESEM的具体应用问题,例如因子旋转方法的选用、测验信度评价等,仍有待探讨.     

Iteration of Partially Specified Target Matrices: Application to the Bi-Factor Case

The current study proposes a new bi-factor rotation method, Schmid-Leiman with iterative target rotation (SLi), based on the iteration of partially specified target matrices and an initial target constructed from a Schmid-Leiman (SL) orthogonalization. SLi was expected to ameliorate some of the limitations of the previously presented SL bi-factor rotations, SL and SL with target rotation (SLt), when the factor structure either includes cross-loadings, near-zero loadings, or both. A Monte Carlo simulation was carried out to test the performance of SLi, SL, SLt, and the two analytic bi-factor rotations, bi-quartimin and bi-geomin. The results revealed that SLi accurately recovered the bi-factor structures across the majority of the conditions, and generally outperformed the other rotation methods. SLi provided the biggest improvements over SL and SLt when the bi-factor structures contained cross-loadings and pure indicators of the general factor. Additionally, SLi was superior to bi-quartimin and bi-geomin, which performed inconsistently across the types of factor structures evaluated. No method produced a good recovery of the bi-factor structures when small samples (N = 200) were combined with low factor loadings (0.30–0.50) in the specific factors. Thus, it is recommended that larger samples of at least 500 observations be obtained.     

Testing Models of Psychopathology in Preschool-aged Children Using a Structured Interview-based Assessment

A number of studies have found that broadband internalizing and externalizing factors provide a parsimonious framework for understanding the structure of psychopathology across childhood, adolescence, and adulthood. However, few of these studies have examined psychopathology in young children, and several recent studies have found support for alternative models, including a bi-factor model with common and specific factors. The present study used parents’ (typically mothers’) reports on a diagnostic interview in a community sample of 3-year old children (n?=?541; 53.9 % male) to compare the internalizing-externalizing latent factor model with a bi-factor model. The bi-factor model provided a better fit to the data. To test the concurrent validity of this solution, we examined associations between this model and paternal reports and laboratory observations of child temperament. The internalizing factor was associated with low levels of surgency and high levels of fear; the externalizing factor was associated with high levels of surgency and disinhibition and low levels of effortful control; and the common factor was associated with high levels of surgency and negative affect and low levels of effortful control. These results suggest that psychopathology in preschool-aged children may be explained by a single, common factor influencing nearly all disorders and unique internalizing and externalizing factors. These findings indicate that shared variance across internalizing and externalizing domains is substantial and are consistent with recent suggestions that emotion regulation difficulties may be a common vulnerability for a wide array of psychopathology.     

Confirmatory Factor Analysis and Profile Analysis via Multidimensional Scaling

This paper describes the Confirmatory Factor Analysis (CFA) parameterization of the Profile Analysis via Multidimensional Scaling (PAMS) model to demonstrate validation of profile pattern hypotheses derived from multidimensional scaling (MDS). Profile Analysis via Multidimensional Scaling (PAMS) is an exploratory method for identifying major profiles in a multi-subtest test battery. Major profile patterns are represented as dimensions extracted from a MDS analysis. PAMS represents an individual observed score as a linear combination of dimensions where the dimensions are the most typical profile patterns present in a population. While the PAMS approach was initially developed for exploratory purposes, its results can later be confirmed in a different sample by CFA. Since CFA is often used to verify results from an exploratory factor analysis, the present paper makes the connection between a factor model and the PAMS model, and then illustrates CFA with a simulated example (that was generated by the PAMS model) and at the same time with a real example. The real example demonstrates confirmation of PAMS exploratory results by using a different sample. Fit indexes can be used to indicate whether the CFA reparameterization as a confirmatory approach works for the PAMS exploratory results.     

Full-information item bi-factor analysis

A plausibles-factor solution for many types of psychological and educational tests is one that exhibits a general factor ands − 1 group or method related factors. The bi-factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of thes − 1 group factors. This paper derives a bi-factor item-response model for binary response data. In marginal maximum likelihood estimation of item parameters, the bi-factor restriction leads to a major simplification of likelihood equations and (a) permits analysis of models with large numbers of group factors; (b) permits conditional dependence within identified subsets of items; and (c) provides more parsimonious factor solutions than an unrestricted full-information item factor analysis in some cases. Supported by the Cognitive Science Program, Office of Naval Research, Under grant #N00014-89-J-1104. We would like to thank Darrell Bock for several helpful suggestions.     

Independent component analysis as a rotation method: A very different solution to Thurstone's box problem

In this paper we consider the well‐known Thurstone box problem in exploratory factor analysis. Initial loadings and components are extracted using principal component analysis. Rotating the components towards independence rather than rotating the loadings towards simplicity allows one to accurately recover the dimensions of each box and also produce simple loadings. It is shown how this may be done using an appropriate rotation criterion and a general rotation algorithm. Methods from independent component analysis are used, and this paper may be viewed as an introduction to independent component analysis from the perspective of factor analysis.     

The Ok-Religious Attitude Scale (Islam): introducing an instrument originated in Turkish for international use

This study reports the content and psychometric properties of the Ok Religious Attitude Scale (in an Islamic tradition). Among two samples of university students (N = 934 and 388), high alpha coefficients were recorded (ranges between .81 and .91). Both exploratory and confirmatory factor analyses confirm that the scale with its four subscales (cognitive, emotional, behavioural and relational) form an ideal (first-order) or acceptable (higher-order) model. The scale revealed powerful criterion validity through its comparison with adapted versions of Francis Scale of Attitude Towards Christianity [FSAC] and Intrinsic Religiosity scale. In its final form the scale can be commended as a reliable, valid and viable instrument to be used in social science research.     

基于双因子模型的测验总分和维度分的合成方法

双因子模型可以同时包含一个全局因子和多个局部因子,在描述多维测验结构时有其独特优势,近些年应用越来越广泛。文章基于双因子模型,提出了4种合成总分和维度分的方法,分别是:原始分法,加和法,全局题目加权加和法和局部题目加权加和法,并采用模拟的方法,在样本量、测验长度、维度间相关变化的条件下考察了这些方法与传统多维IRT方法的表现。最后,通过实证研究对结果进行了验证。结果显示:(1)全局加权加和法和局部加权加和法,尤其是局部加权加和法合成的总分和维度分与真值最接近、信度最高。(2)在维度间相关较高,测验长度较长的条件下,局部加权加和法的结果较好,部分条件下甚至优于多维IRT法。(3)仅有局部加权加和法合成的维度分能够反应维度间真实的相关关系。     

An empirical application of confirmatory factor analysis to the multitrait-multimethod matrix

Though most social psychologists are aware of the use of factor analysis as an exploratory method to uncover latent dimensions, factor analysis can also be used in a confirmatory mode to test specific hypotheses. A confirmatory solution is unique and cannot be rotated. One possible use of confirmatory factor analysis is with the multitrait-multimethod matrix. Jaccard, Weber, and Lundmark's (1975) multitrait-multimethod matrix of two traits and four methods is analyzed using confirmatory factor analysis. Given the very small sample size the analysis is primarily illustrative. A simple two factor model with errors of measurement correlated across measures using the same method satisfactorily fits the data. Both discriminant and convergent validity are high and none of the methods has higher reliability or less correlated measurement error. The reliability of measures estimated by maximum likelihood factor analysis is lower than the test-retest reliability since method variance is subtracted out. Confirmatory factor analysis is recommended over the Campbell-Fiske criteria.     

Testing a bi-factor model to disentangle general and specific factors of motivation in self-determination theory

We examined the utility of the bi-factor model for disentangling general motivation and specific motivations (i.e., amotivation, external, introjected, identified, and intrinsic regulations) in relation to goal progress and physical activity (PA). Participants (N = 186 undergraduate students; M_age = 19.26 years) completed assessments of motivation and PA at Time 1. Four weeks later, PA and goal progress were assessed at Time 2. Results indicated that the exploratory bi-factor model specifying motivational regulations as the specific factors and general motivation as the general factor was a good fit to the data. Results of the structural equation model indicated that identified and intrinsic regulations and general motivation predicted concurrent PA at Time 1. A novel finding was that controlling for concurrent PA at Time 1, general motivation emerged as the only predictor of Time 2 goal progress and PA. Results highlight the importance of examining general motivation in addition to quality of motivation in tandem because general motivation emerged as the sole significant longitudinal predictor of PA outcomes.     

大学生网络成瘾量表（修订版）的多元概化分析*

用多元概化理论考察大学生网络成瘾量表在大学生群体中应用的测量学性能。以随机测量模式的概化设计,针对1200名在校大学生进行问卷调查。结果显示双因子结构的相关程度在0.92以上,五因子结构的相关程度均在0.76~0.97间;整体量表的概化系数和可靠性指数均达到了0.94以上,而双因子结构各因子在0.90左右,五因子结构各因子均在0.74~0.85间。所以,整体量表及各因子在大学生群体中应用的信效度较高,可用作常模和标准参照测验;无论双因子还是五因子结构,CIAS-R各因子在分值比和项目数上,设计非常合理和完善。     

Some comparisons of the multiple-factor and the bi-factor methods of analysis

Bi-factor and multiple-factor analyses of the same data are compared in two respects. First, two criteria are suggested for determining when the factorization is adequate. This problem being more acute for the centroid method than for the bi-factor method, the latter is used primarily for comparison only. It is shown also that the omission from the simple structure of entries smaller than .10 yields a pattern which is a poorer fit to the original correlations than is the bi-factor pattern. Second, the second-order general factor obtained from the intercorrelations of the primaries is found to be highly correlated with the general factor of the bi-factor pattern.     

Exploratory and confirmatory factor analyses of the Asian American Racism-Related Stress Inventory

The authors conducted exploratory and confirmatory factor analyses of the Asian American Racism-Related Stress Inventory (AARRSI) to further examine the underlying factor structure in a total sample of 1,273 Asian American participants. In the first step of analysis, an exploratory factor analysis with 651 participants yielded a 13-item two-factor solution to the data. In the second step, a confirmatory factor analysis with 622 participants supported both the 13-item two-factor model and the original 29-item three-factor model in the cross-validation sample and generational and ethnicity analyses. The two-factor and three-factor models produced internal consistency estimates ranging from .81 to .95. In addition, the authors examined convergent and criterion related evidence for 13-item and 29-item versions of the AARRSI. Given its brief nature and generally good fit across generational status and ethnicity, the authors suggest that the 13-item AARRSI might be advantageous for research and assessment endeavors.     

Mental toughness among high school students: a test of its multidimensionality and nomological validity with academic achievement and preference for difficult tasks

Recent studies posit that mental toughness is a relevant construct for predicting achievement outcomes in academic settings. It is a multidimensional construct that encapsulates psychological resources that facilitate consistent performance despite stressors and challenges. However, recent evidence has called into question its multidimensional aspect. The first purpose of this study was to verify, using a bi-factor model, if mental toughness can be operationalized by (a) multiple dimensions, (b) a general factor, or (c) both a general factor and multiple dimensions. The second goal was to test the nomological validity of the construct. Specifically, we verified whether the specific factors predict, beyond the general factor, academic achievement and preference for difficult tasks. Using a correlational cross-sectional design in which 515 high school students (58.8% girls; M age = 15.68; SD = 1.05) were asked to complete a questionnaire, we found that mental toughness is best conceptualized by a general factor. More specifically, most loadings are higher on the mental toughness general factor than on the specific dimensions. Furthermore, the mental toughness general factor predicts better school achievement and preference for difficult tasks than the specific factors. The results are discussed in terms of their implications for theory and practice.     

青少年学生公益心内隐观研究

公益心是个体在公益行为中表现出来的比较稳定的人格特征。以435名青少年为被试，采用内隐观调查法搜集了具有公益心特征的形容词或短语，通过分类与合并找到了能代表具有公益心特征的80个形容词；然后，通过489名青少年对这些形容词或短语的评定得分，采用探索性因素分析得到了青少年公益心的四个因子，即同情友爱、环境意识、集体意识、热情自信；最后，对另外351名青少年的形容词问卷得分进行了验证性因素分析，结果表明青少年公益心四因素模型的拟合度较好，从而确认了青少年公益心的四因素结构模型。