Identifying between‐person and within‐person factors to enhance understanding of observed score profiles

This study aims to reparameterize ordinary factors into between‐ and within‐person factor effects and utilize an array of the within‐person factor loadings as a latent profile which encapsulates all score responses of individuals in a population. To illustrate, the Woodcock–Johnson III (WJ‐III) tests of cognitive abilities were analysed and one between‐ and two within‐person factors were identified. The scoring patterns of individuals in the WJ‐III sample were interpreted according to the within‐person factor patterns. Regression analyses were performed to examine how much the within‐person factors accounted for the person scoring patterns and criterion variables. Finally, the importance and applications of the between‐ and within‐person factors are discussed.     

Oddity, schizotypy/dissociation, and personality

The construct of Oddity was examined in relation to the dimensions of normal personality variation and a dimension of schizotypy and dissociation. In 2 studies involving samples of community adults (N = 409) and college students (N = 378), Oddity—as operationalized in terms of perceived strangeness or eccentricity—was found to be moderately related to a Schizotypy/Dissociation factor and also to factors of normal personality variation, particularly Openness to Experience. The modest loading of Oddity on the Schizotypy/Dissociation factor, along with the somewhat stronger projection of Oddity within the space of normal personality dimensions, indicates that the Schizotypy/Dissociation factor should not be interpreted as a dimension of Oddity. The interpretation of the Schizotypy/Dissociation factor is discussed, as are the implications of these results for proposed dimensional models of personality disorders.     

Identifying Misfitting Achievement Estimates in Performance Assessments: An Illustration Using Rasch and Mokken Scale Analyses

Researchers apply individual person fit analyses as a procedure for checking model-data fit for individual test-takers. When a test-taker misfits, it means that the inferences from their test score regarding what they know and can do may not be accurate. One problem in applying individual person fit procedures in practice is the question of how much misfit it takes to make the test score an untrustworthy estimate of achievement. In this paper, we argue that if a person’s responses generally follow a monotonic pattern, the resulting test score is “good enough” to be interpreted and used. We present an approach that applies statistical procedures from the Rasch and Mokken measurement perspectives to examine individual person fit based on this good enough criterion in real data from a performance assessment. We discuss how these perspectives may facilitate thinking about applying individual person fit procedures in practice.     

Identifying criterion-related patterns of predictor scores using multiple regression

Along with examples involving vocational interests and mathematics achievement, the authors describe a multiple regression based, pattern recognition procedure that can be used to identify a pattern of predictor scores associated with high scores on a criterion variable. This pattern is called the criterion pattern. After the criterion pattern has been identified, a second regression procedure can be used to estimate the proportion of variation attributable to the criterion pattern. Cross-validation can then be used to estimate the variation attributable to a criterion pattern derived from regression weights estimated in another sample. Finally, issues of criterion pattern invariance and interpretation are discussed.     

Application of the concept of simple structure to Alexander's data

A battery of 20 tests originally analyzed by Alexander (1) was reworked according to the principle of simple structure. His results were sustained in general. Both analyses yielded five factors in the first-order domain. Of these, three factors in the re-analysis (v,X andF) have almost exactly the same loadings as the corresponding factors in the original work, and were interpreted in the same way. The loading pattern of a fourth factor,Z, left uninterpreted in the original study, happened to be more clear in the re-analysis, and an interpretation was attempted. It appears to be a factor of perceptual synthesis, and seems to play an important role in intellectual processes. A fifth factor, not present in Alexander's results, appeared in the new analysis: the reasoning factor, involved in inductive and deductive thinking. All four cognitive factors are related to a general factor that can be thought of as representing abstraction and education of relations and correlates, these processes being, therefore, the essential feature underlying intellectual behavior, at least in that sector surveyed by the tests of the present battery.The analysis of the data was done in the Psychometric Laboratory of The University of Chicago, under a fellowship from The University of Madrid (Junta de Relaciones Culturales). The author wishes to express his gratitude to Professor L. L. Thurstone, whose advice as a scientist and kindness as a friend, have been the principal stimulus for this work.     

A response Consistency statistic for the millon clinical multiaxial inventory (MCMI)

Response style in objective psychological testing is an important issue in the reliability and validity of tests as well as in the interpretation of test results. The MCMI provides two response-style indices, the validity scale and the weight factor. The present work presents an additional statistic to assess random response in subjects. The Consistency Coefficient is the correlation between the subjects' endorsement of even and odd items across the 20 MCMI scales. The distributions of 500 patient and 500 randomly generated profiles were compared. Good separation between these distributions was found. The subject data were extremely negatively skewed, whereas the randomly generated data were normally distributed. Data are presented that display positive and negative predictive values, as well as sensitivity and specificity across ranges of prevalence and cut score. These data facilitate the identification of subjects who respond to the MCMI in a random manner so that their scores can be interpreted accordingly.     

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.     

Clearing Up Obstructions: An Image Schema Approach to the Concept of ‘Datong’ 大通 in Chapter 6 of the Zhuangzi

In much of modern scholarship, the notion of datong 大通 in Zhuangzi’s famous zuowang 坐忘 (sitting in disregard) passage is often interpreted as either Dao or a mental/spiritual state of an ideal person, a person who has obtained Dao. In either case, however, the association between datong and such interpretation lacks detailed justification resulting from an insufficiently understood relation between datong and its immediately preceding statements. Different from the more common readings, I propose a cognitive approach based on an image schema related to non-obstruction. In order to demonstrate the philosophical importance of this schema, I will first briefly point out the problems of previous scholarship regarding the zuowang passage. Second, I will introduce the image schema related to non-obstruction based on a number of examples taken from the Zhuangzi. Finally, I will apply this methodology and introduce a cognitive experience-based interpretation of the zuowang passage.     

The many null distributions of person fit indices

This paper deals with the situation of an investigator who has collected the scores ofn persons to a set ofk dichotomous items, and wants to investigate whether the answers of all respondents are compatible with the one parameter logistic test model of Rasch. Contrary to the standard analysis of the Rasch model, where all persons are kept in the analysis and badly fittingitems may be removed, this paper studies the alternative model in which a small minority ofpersons has an answer strategy not described by the Rasch model. Such persons are called anomalous or aberrant. From the response vectors consisting ofk symbols each equal to 0 or 1, it is desired to classify each respondent as either anomalous or as conforming to the model. As this model is probabilistic, such a classification will possibly involve false positives and false negatives. Both for the Rasch model and for other item response models, the literature contains several proposals for a person fit index, which expresses for each individual the plausibility that his/her behavior follows the model. The present paper argues that such indices can only provide a satisfactory solution to the classification problem if their statistical distribution is known under the null hypothesis that all persons answer according to the model. This distribution, however, turns out to be rather different for different values of the person's latent trait value. This value will be called ability parameter, although our results are equally valid for Rasch scales measuring other attributes.As the true ability parameter is unknown, one can only use its estimate in order to obtain an estimated person fit value and an estimated null hypothesis distribution. The paper describes three specifications for the latter: assuming that the true ability equals its estimate, integrating across the ability distribution assumed for the population, and conditioning on the total score, which is in the Rasch model the sufficient statistic for the ability parameter.Classification rules for aberrance will be worked out for each of the three specifications. Depending on test length, item parameters and desired accuracy, they are based on the exact distribution, its Monte Carlo estimate and a new and promising approximation based on the moments of the person fit statistic. Results for the likelihood person fit statistic are given in detail, the methods could also be applied to other fit statistics. A comparison of the three specifications results in the recommendation to condition on the total score, as this avoids some problems of interpretation that affect the other two specifications.The authors express their gratitude to the reviewers and to many colleagues for comments on an earlier version.     

随机截距潜在转变分析(RI-LTA)——个案自我转变与个案间差异的分离

传统的潜在转变分析属于单水平分析, 但其同样也可以看作二水平分析。Muthén和Asparouhov就以二水平分析的视角在单水平分析框架内提出了随机截距潜在转变分析(RI-LTA), 其中跨时间点产生的自我转变可以看作在水平1进行分析, 跨时间点不变的个案间差异可以看作在水平2进行分析, 使个案的自我转变和个案间的初始差异分离, 避免了高估保留在初始类别的概率。某研究型大学2016级本科生的追踪调查数据被用于演示使用随机截距潜在转变分析的过程。该方法的最大优势是通过引入随机截距避免了高估保留在本类别的转变概率。未来研究可以运用蒙特卡洛模拟研究探究随机截距潜在转变分析模型的适用性, 也可以用多水平分析的思路为灵感, 探究多水平随机截距潜在转变分析在统计软件中的实现。     

The Latent Structure of Spatial Skills and Mathematics: A Replication of the Two-Factor Model

In a previous study, Mix et al. (2016) reported that spatial skill and mathematics were composed of 2 highly correlated, domain-specific factors, with a few cross-domain loadings. The overall structure was consistent across grade (kindergarten, 3rd grade, 6th grade), but the cross-domain loadings varied with age. The present study sought to replicate these patterns. Using the data from Wave 1 (n = 854) and data collected from a 2nd sample of kindergarten (n = 251), 3rd-grade (n = 247), and 6th-grade students (n = 241) with the same measures as in Wave 1, we carried out a multigroup confirmatory factor analysis to compare the 2 waves. We also completed several analyses of the Wave 2 data alone. The overall pattern obtained in Wave 1—2 highly correlated domain-specific factors—was clearly replicated in Wave 2. However, more subtle effects involving cross-domain loading were only partially replicated and generally appear fragile and context-specific. In Wave 2, we also included 2 new measures (i.e., proportion matching and fraction identification) that were analyzed in a separate model. Including these new measures did not change the overall pattern of factors and domain-specific factor loadings but did alter some of the cross-domain loadings.     

Race, social class and ability patterns on the WISC-R

The national standardization sample of whites and blacks on the Wechsler Intelligence Scale for Children—Revised (WISC-R) was the basis for a detailed analysis of the psychometric nature of racial and social class differences on the original 13 subscales of the WISC-R. The profiles of subtest scores of whites and blacks were compared directly and also after the racial groups were statistically equated on Full Scale IQ (FSIQ). Under the latter condition, the races differ only very slightly, although significantly, on some of the subtests, in ways generally contrary to popular expectations. The profile of white-black differences on the WISC-R subtests is markedly different, and negatively correlated with, the profiles of social class differences within each racial group, indicating that the pattern of racial differences is not explainable in terms of the difference in the average socio-economic status (SES) of blacks and whites. A Schmid-Leiman orthogonalized hierarchical factor analysis yields virtually identical factor structures and highly congruent factor loadings on the subtests for whites and blacks. Analysis of factor scores shows that by far the largest proportion of the variance between races is attributable to the general factor (g) common to all the subtests, whereas the group factors (verbal, performance and memory) contribute only minutely to the interracial variance. Hence the white-black differences on the diverse subtests of the WISC-R, and in the Full Scale IQ, are interpreted primarily as a difference in Spearman's g, rather than as differences in the more specific factors peculiar to particular content, knowledge, acquired skills or type of test. However, some slight but significant differences in patterns of ability also occur that are independent of g.     

A new iterative method for correcting erroneous communality estimates in factor analysis

A new method for correcting erroneous communality estimates is applicable to any completed orthogonal factor solution. It seeks, by direct correction of factor loadings, to make the residuals conform to the chance error criteria of zero mean and zero skewness for each row separately. Two numerical examples, with one and two factors, respectively, are presented. The method can be used as a short cut for Dwyer's extension in adding variables to a matrix. It can also be used as a short cut in cross-validation factor studies. Successful use on problems with many variables and numerous factors is claimed. Factors can be made oblique,after correction, if desired.     

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).     

An exploration of genetic g: Hierarchical factor analysis of cognitive data from the western reserve twin project

Hierarchical factor analyses involving Schmid-Leiman transformations (Schmid & Leiman, 1957) were conducted on specific cognitive abilities data collected in a sample of 148 identical (MZ) and 135 same-sex fraternal (DZ) twin pairs. Two main questions were addressed: First, are genetic influences on specific cognitive abilities simply a reflection of their g loading, or are different sets of genes affecting separate abilities? Second, to the extent that specific cognitive abilities are affected by common genetic variance, how similar is the common genetic factor to a phenotypic factor reflecting g? Model fitting results suggest that genetic influences on specific abilities are a reflection of both general intelligence and genetic influences specific to separate abilities and that loadings on the common genetic factor are more highly correlated with phenotypic g loadings than are common environmental factor loadings.     

Examining the Factor Structure of the Self-Compassion Scale in Four Distinct Populations: Is the Use of a Total Scale Score Justified?

This study examined the factor structure of the Self-Compassion Scale (SCS) using a bifactor model, a higher order model, a 6-factor correlated model, a 2-factor correlated model, and a 1-factor model in 4 distinct populations: college undergraduates (N = 222), community adults (N = 1,394), individuals practicing Buddhist meditation (N = 215), and a clinical sample of individuals with a history of recurrent depression (N = 390). The 6-factor correlated model demonstrated the best fit across samples, whereas the 1- and 2-factor models had poor fit. The higher order model also showed relatively poor fit across samples, suggesting it is not representative of the relationship between subscale factors and a general self-compassion factor. The bifactor model, however, had acceptable fit in the student, community, and meditator samples. Although fit was suboptimal in the clinical sample, results suggested an overall self-compassion factor could still be interpreted with some confidence. Moreover, estimates suggested a general self-compassion factor accounted for at least 90% of the reliable variance in SCS scores across samples, and item factor loadings and intercepts were equivalent across samples. Results suggest that a total SCS score can be used as an overall mesure of self-compassion.     

随机截距因子分析模型在控制条目表述效应中的应用

本研究用中文修订版罗森博格自尊量表(RSES-R)考察随机截距因子分析模型在控制条目表述效应时的表现。用RSES-R和过分宣称问卷组成的量表调查621名中学生。结果表明,随机截距模型在建模时,拟合指数良好、因子方差与负荷合理,自尊因子分与RSES-R总分有极高相关,表明该模型能有效分离RSES-R得分的特质与表述效应。分离的表述效应因子分与受测者的自我提升水平具有显著但较弱的相关,表明表述效应与自受测者的社会赞许性有共同的成分。     

Equivalent Dynamic Models

Equivalences of two classes of dynamic models for weakly stationary multivariate time series are discussed: dynamic factor models and autoregressive models. It is shown that exploratory dynamic factor models can be rotated, yielding an infinite set of equivalent solutions for any observed series. It also is shown that dynamic factor models with lagged factor loadings are not equivalent to the currently popular state-space models, and that restriction of attention to the latter type of models may yield invalid results. The known equivalent vector autoregressive model types, standard and structural, are given a new interpretation in which they are conceived of as the extremes of an innovating type of hybrid vector autoregressive models. It is shown that consideration of hybrid models solves many problems, in particular with Granger causality testing.     

An empirical investigation of the influence of social desirability on the factor structure of the Chinese 16PF

This study examined whether social desirability affects the factor structure of the Chinese 16PF. Participants were 4645 Chinese first-year undergraduate and graduate students who completed the Chinese 16PF as part of a mandatory psychological assessment. Based on students’ score on the Impression Management scale (Paulhus, 1998), two groups were formed: (a) the high socially desirable responding group, consisting of students whose IM scores fell within the upper 20% of the distribution, and (b) the low socially desirable responding group, consisting of students whose IM scores fell within the bottom 25% of the distribution. A series of exploratory and confirmatory factor analyses indicated that the factor pattern and factor loadings of the Chinese 16PF were invariant across these two groups.     

Factor Structure of the PTSD Checklist in a Sample of OEF/OIF Veterans Presenting to Primary Care: Specific and Nonspecific Aspects of Dysphoria

Although DSM-IV-TR diagnostic criteria for posttraumatic stress disorder (PTSD) include three primary symptom clusters, recent evidence from confirmatory factor analyses suggest that the latent structure of PTSD is better represented by four factors, which will likely be reflected in the upcoming DSM-5. Given this likely transition from three to four clusters, the present study sought to examine specific and non-specific aspects of dysphoria in the factor structure of PTSD symptoms in a sample of OEF/OIF combat veterans presenting to a Veterans Affairs primary care clinic. PTSD symptoms were assessed using the PCL-M (Weathers et al. 1993). Results from confirmatory factor analyses suggested that a dysphoria factor involving a number of non-specific distress symptoms may be an important part of the PTSD symptom profile. After controlling for variance due to general psychological distress, we further found that factor loadings on the dysphoria factor were attenuated but continued to significantly load onto the factor, suggesting that dysphoria may be a specific part of the PTSD symptom constellation.