Stochastic search item selection for factor analytic models

In this paper we implement a Markov chain Monte Carlo algorithm based on the stochastic search variable selection method of George and McCulloch (1993) for identifying promising subsets of manifest variables (items) for factor analysis models. The suggested algorithm is constructed by embedding in the usual factor analysis model a normal mixture prior for the model loadings with latent indicators used to identify not only which manifest variables should be included in the model but also how each manifest variable is associated with each factor. We further extend the suggested algorithm to allow for factor selection. We also develop a detailed procedure for the specification of the prior parameters values based on the practical significance of factor loadings using ideas from the original work of George and McCulloch (1993). A straightforward Gibbs sampler is used to simulate from the joint posterior distribution of all unknown parameters and the subset of variables with the highest posterior probability is selected. The proposed method is illustrated using real and simulated data sets.     

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。     

A Unified Framework for the Comparison of Treatments with Ordinal Responses

Different latent variable models have been used to analyze ordinal categorical data which can be conceptualized as manifestations of an unobserved continuous variable. In this paper, we propose a unified framework based on a general latent variable model for the comparison of treatments with ordinal responses. The latent variable model is built upon the location-scale family and is rich enough to include many important existing models for analyzing ordinal categorical variables, including the proportional odds model, the ordered probit-type model, and the proportional hazards model. A flexible estimation procedure is proposed for the identification and estimation of the general latent variable model, which allows for the location and scale parameters to be freely estimated. The framework advances the existing methods by enabling many other popular models for analyzing continuous variables to be used to analyze ordinal categorical data, thus allowing for important statistical inferences such as location and/or dispersion comparisons among treatments to be conveniently drawn. Analysis on real data sets is used to illustrate the proposed methods.     

Effects of divided attention in the word-fragment completion task with unique and multiple solutions

A study with 114 young adults investigated the correlations of intelligence factors and working-memory capacity with reaction time (RT) tasks. Within two sets of four-choice RT tasks, stimulus–response compatibility was varied over three levels: compatible, incompatible, and arbitrary mappings. Two satisfactory measurement models for the RTs could be established: A general factor model without constraints on the loadings and a nested model with two correlated factors, distinguishing compatible from arbitrary mappings, with constraints on the loadings. Structural models additionally including factors for working memory and intelligence showed that the nested model with correlated factors is superior in fit. Working-memory capacity and fluid intelligence were correlated strongly with the nested factor for the RT tasks with arbitrary mappings, and less with the general RT factor. The results support the hypothesis that working memory is needed to maintain arbitrary bindings between stimulus representations and response representations, and this could explain the correlation of working-memory capacity with speed in choice RT tasks.     

Quantifying the Strength of General Factors in Psychopathology: A Comparison of CFA with Maximum Likelihood Estimation,BSEM, and ESEM/EFA Bifactor Approaches

Whether or not importance should be placed on an all-encompassing general factor of psychopathology (or p factor) in classifying, researching, diagnosing, and treating psychiatric disorders depends (among other issues) on the extent to which comorbidity is symptom-general rather than staying largely within the confines of narrower transdiagnostic factors such as internalizing and externalizing. In this study, we compared three methods of estimating p factor strength. We compared omega hierarchical and explained common variance calculated from confirmatory factor analysis (CFA) bifactor models with maximum likelihood (ML) estimation, from exploratory structural equation modeling/exploratory factor analysis models with a bifactor rotation, and from Bayesian structural equation modeling (BSEM) bifactor models. Our simulation results suggested that BSEM with small variance priors on secondary loadings might be the preferred option. However, CFA with ML also performed well provided secondary loadings were modeled. We provide two empirical examples of applying the three methodologies using a normative sample of youth (z-proso, n = 1,286) and a university counseling sample (n = 359).     

Measures of invariance and comparability in factor analysis for fixed variables

New procedures are presented for measuring invariance and matching factors for fixed variables and for fixed or different subjects. Two of these, the coefficient of invariance for factor loadings and the coefficient of factor similarity, utilize factor scores computed from the different sets of factor loadings and one of the original standard score matrices. Another, the coefficient of subject invariance, is obtained by using one of the sets of factor loadings in conjunction with the different standard score matrices. These coefficients are correlations between factor scores of the appropriate matrices. When the best match of factors is desired, rather than degree of resemblance, the method of assignment is proposed.     

Factor copula models for mixed data

We develop factor copula models to analyse the dependence among mixed continuous and discrete responses. Factor copula models are canonical vine copulas that involve both observed and latent variables, hence they allow tail, asymmetric and nonlinear dependence. They can be explained as conditional independence models with latent variables that do not necessarily have an additive latent structure. We focus on important issues of interest to the social data analyst, such as model selection and goodness of fit. Our general methodology is demonstrated with an extensive simulation study and illustrated by reanalysing three mixed response data sets. Our studies suggest that there can be a substantial improvement over the standard factor model for mixed data and make the argument for moving to factor copula models.     

Evidence for a general factor model of ADHD in adults

Objective: To examine factor structures of Diagnostic and Statistical Manual of Mental Disorders (4th ed.) symptoms of ADHD in adults. Method: Two sets of models were tested: (a) models with inattention and hyperactivity/impulsivity as separate but correlated latent constructs and (b) hierarchical general factor models with a general factor for all symptoms and separate specific factors for inattention and hyperactivity/impulsivity. Participants were 751 adults with ADHD. Two models with correlated factors and two general factor models of ADHD symptoms were tested. Results: The general factor model provided a better fit of the data than the correlated models. The general factor model with one general and three (inattention, motoric, and verbal hyperactivity/impulsivity) specific factors best accounted for the adults' symptoms. Conclusion: These results suggest a unitary component to ADHD symptoms as well as dimensional specific factors. The replication of a general factor in adults suggests continuity of symptom presentation from childhood into adulthood. Clinical implications are discussed.     

Some relationships between factors and components

The asymptotic correlations between the estimates of factor and component loadings are obtained for the exploratory factor analysis model with the assumption of a multivariate normal distribution for manifest variables. The asymptotic correlations are derived for the cases of unstandardized and standardized manifest variables with orthogonal and oblique rotations. Based on the above results, the asymptotic standard errors for estimated correlations between factors and components are derived. Further, the asymptotic standard error of the mean squared canonical correlation for factors and components, which is an overall index for the closeness of factors and components, is derived. The results of a Monte Carlo simulation are presented to show the usefulness of the asymptotic results in the data with a finite sample size.The author is indebted to anonymous referees for their comments, corrections and suggestions which have led to the improvement of this article.     

The Factor Structure of ADHD – Different Models,Analyses and Informants in a Bifactor Framework

The methodological approach of exploratory structural equation modelling (ESEM) has only been applied once to the construct of Attention-deficit/hyperactivity disorder (ADHD). We decided to compare bifactor models based on confirmatory factor analyses (Bi-CFA) and exploratory equation modeling (Bi-ESEM) only, as there is a growing support of a bifactor structure of ADHD. To examine the factorial validity of the construct we compared three possible bifactor models. One model with two specific factors (inattention and hyperactivity/impulsivity), another model with three specific factors (inattention, hyperactivity and impulsivity) and an alternative, incomplete model with one general ADHD and two specific factors (inattention and impulsivity). We used parent- (N = 1386; Age: M = 11.70, SD = 3.18; Sex: 74.5 % male) and teacher-ratings (N = 110; Age: M = 11.27, SD = 3.04; Sex: 77.5 % male) from clinically referred children between the age of 6 and 18. The results indicate that both methods lead to equally good model fit and for both informants the reliable variance of the specific factor hyperactivity is almost completely explained by the general factor. However, in the teacher condition cross-loadings seem to be of particular importance. Across both methods and informants covariation among ADHD symptom items can be in most part attributed to a general ADHD factor as well as to three (inattention, hyperactivity and impulsivity) or two (inattention and impulsivity) weakly defined specific factors. Further research regarding associations between the specific factors of ADHD and other disorders (e.g. conduct disorder) should be conducted.     

项目表述效应对人格测验信度和效标效度的影响——以核心自我评价量表(CSES)为例

项目表述效应是指由项目表述方式的差异引起的与测量内容无关的系统变异,项目表述效应模型的统计本质是一种双因子模型。本研究以核心自我评价量表(CSES)为例,探讨项目表述效应对人格测验信效度的影响。采用核心自我评价量表、生活满意度量表和积极情感消极情感量表对340名“蚁族”进行测查。结果表明,CSES在核心自我评价特质以外,还存在一个反向题项目表述效应因子;忽视项目表述效应对CSES的同质性信度和效标关联效度有重要影响：高估CSES的同质性信度,低估核心自我评价与生活满意度、积极情感的正相关,高估核心自我评价与消极情感的负相关。     

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.     

A general class of latent variable models for ordinal manifest variables with covariate effects on the manifest and latent variables

Previous work on a general class of multidimensional latent variable models for analysing ordinal manifest variables is extended here to allow for direct covariate effects on the manifest ordinal variables and covariate effects on the latent variables. A full maximum likelihood estimation method is used to estimate all the model parameters simultaneously. Goodness‐of‐fit statistics and standard errors are discussed. Two examples from the 1996 British Social Attitudes Survey are used to illustrate the methodology.     

A Confirmatory Approach to Examining the Factor Structure of the Strengths and Difficulties Questionnaire (SDQ): A Large Scale Cohort Study

The aim of this study was to examine the factor structure of the Strengths and Difficulties Questionnaire (SDQ) using a Structural Confirmatory Factor Analytic approach. The Danish translation of the SDQ was distributed to 71,840 parents and teachers of 5–7 and 10–12-year-old boys and girls from four large scale cohorts. Three theoretical models were examined: 1. a model with five first order factors (i.e., hyperactivity/inattention, conduct, emotional, peer problems and prosocial), 2. a model adding two internalising and externalising second order factors to model 1, and 3. a model adding a total difficulties second order factor to model 1. Model fits were evaluated, multi-group analyses were carried out and average variance extracted (AVE) and composite reliability (CR) estimates were examined. In this general population sample, low risk sample models 1 and 2 showed similar good overall fits. Best model fits were found when two positively worded items were allowed to cross load with the prosocial scale, and cross loadings were allowed for among three sets of indicators. The analyses also revealed that model fits were slightly better for teachers than for parents and better for older children than for younger children. No convincing differences were found between boys and girls. Factor loadings were acceptable for all groups, especially for older children rated by teachers. Some emotional, peer, conduct and prosocial subscale problems were revealed for younger children rated by parents. The analyses revealed more internal consistency for older children rated by teachers than for younger children rated by parents. It is recommended that model 1 comprising five first order factors, or alternatively model 2 with additionally two internalising/externalising second order factors, should be used when employing the SDQ in low risk epidemiological samples.     

A General Factor of Personality (GFP) from two meta-analyses of the Big Five: and

In two studies, we used structural equation models to test the hypothesis that a General Factor of Personality (GFP) occupies the apex of the hierarchy of personality. In Study 1, we found a GFP that explained 45% of the reliable variance in a model that went from the Big Five to the Big Two to the Big One in the 14 studies of inter-scale correlations (N = 4496) assembled by Digman (1997). A higher order factor of Alpha/Stability was defined by Conscientiousness, Emotional Stability, and Agreeableness, with loadings of from 0.61 to 0.70, while Beta/Plasticity was defined by Openness and Extraversion with loadings of 0.55 and 0.77. In turn, the GFP was defined by Alpha and Beta with loadings of 0.67. In Study 2, a GFP explained 44% of the reliable variance in a similar model using data from a published meta-analysis of the Big Five (N = 4000) by Mount, Barrick, Scullen, and Rounds (2005). Strong general factors such as these, based on large data sets with good model fits that cross validate perfectly, are unlikely to be due to artifacts and response sets.     

When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions

A simulation study compared the performance of robust normal theory maximum likelihood (ML) and robust categorical least squares (cat-LS) methodology for estimating confirmatory factor analysis models with ordinal variables. Data were generated from 2 models with 2-7 categories, 4 sample sizes, 2 latent distributions, and 5 patterns of category thresholds. Results revealed that factor loadings and robust standard errors were generally most accurately estimated using cat-LS, especially with fewer than 5 categories; however, factor correlations and model fit were assessed equally well with ML. Cat-LS was found to be more sensitive to sample size and to violations of the assumption of normality of the underlying continuous variables. Normal theory ML was found to be more sensitive to asymmetric category thresholds and was especially biased when estimating large factor loadings. Accordingly, we recommend cat-LS for data sets containing variables with fewer than 5 categories and ML when there are 5 or more categories, sample size is small, and category thresholds are approximately symmetric. With 6-7 categories, results were similar across methods for many conditions; in these cases, either method is acceptable. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Estimating, testing, and comparing specific effects in structural equation models: the phantom model approach

The phantom model approach for estimating, testing, and comparing specific effects within structural equation models (SEMs) is presented. The rationale underlying this novel method consists in representing the specific effect to be assessed as a total effect within a separate latent variable model, the phantom model that is added to the main model. The following favorable features characterize the method: (a) It enables the estimation, testing, and comparison of arbitrary specific effects for recursive and nonrecursive models with latent and manifest variables; (b) it enables the bootstrapping of confidence intervals; and (c) it can be applied with all standard SEM programs permitting latent variables, the specification of equality constraints, and the bootstrapping of total effects. These features along with the fact that no manipulation of matrices and formulas is required make the approach particularly suitable for applied researchers. The method is illustrated by means of 3 examples with real data sets.     

The Recoverability of P-technique Factor Analysis

It seems that just when we are about to lay P–technique factor analysis finally to rest as obsolete because of newer, more sophisticated multivariate time-series models using latent variables—dynamic factor models—it rears its head to inform us that an obituary may be premature. We present the results of some simulations demonstrating that even though it does not explicitly model lagged information, P–technique's ability to recover the parameters of underlying dynamic processes involving lagged relations among the manifest variables is apparently robust and accurate. An empirical example is presented using 103 days of affective mood self-ratings from a young pregnant woman. Implications of the simulation and empirical findings are briefly discussed.     

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.