Estimation of factors using higher-order multi-cumulants in weak factor models

Higher-order exploratory factor analysis of the Reynolds Intellectual Assessment Scales with a referred sample

ObjectiveThe main aim was to perform a systematic literature review of studies investigating the factor structure of the Eating Disorder Examination‐Questionnaire (EDE‐Q), a widely used measure of eating pathology. Secondary aims were to summarize the quality of reporting of latent variable (factor) analyses in these studies and review support for different factor solutions.MethodLiterature was identified through Scopus, Medline, PsycInfo, and ProQuest databases published up to February 23, 2022 and outreach via an international listserv. All studies published in English reporting factor analysis of the EDE‐Q were included with few restrictions. Sixty studies including 63,389 participants met inclusion criteria.ResultsThe originally proposed four‐factor solution received little empirical support, although few alternative models have been robustly evaluated. Items assessing shape and weight concerns frequently coalesce in factor solutions, suggesting that these constructs are closely related. Investigations of brief versions of the EDE‐Q have produced more consistent findings, suggesting that these measures, particularly a seven‐item version, might be useful alternatives to the full version. Quality of studies was reasonable, with important methodological elements of factor analysis often reported.DiscussionThe findings are of relevance to practitioners and researchers, suggesting that the “original” factor structure of the EDE‐Q should be reconsidered and that use of a seven‐item version is to be encouraged.Public SignificanceSelf‐report questionnaires are widely used in the assessment of disordered eating. The current study found that there is little consensus about the structure of a common measure of eating psychopathology. There is more consistent support for a brief, seven‐item, version assessing dietary restraint, body dissatisfaction, and overvaluation of weight and shape.

Summary We discuss how confirmatory factor analysis results should be used to examine potential higher-order constructs and advocate that researchers present five types of evidence, which are as follows: (1) the ability of the higher-order model to reproduce the observed covariation among manifest variables; (2) the ability of the higher-order model to reproduce the observed covariation among manifest variables better than more parsimonious alternative models—and no less well than less parsimonious alternative models; (3) the ability of the higher-order model to reproduce the observed covariation among lower-order factors; (4) the ability of the higher-order factor to explain variation in lower-order factors; and (5) the ability of the higher-order factor to explain variation in manifest variables. We illustrate how this type of evidence could be presented with a worked example and contrast our recommendations with the manner in which higher-order confirmatory factor analysis has been used in the organizational sciences over the past 25 years to support claims regarding higher-order constructs such as core self-evaluations and transformational leadership. Our review shows that a substantial proportion of the 44 examined articles failed to present enough evidence to allow readers to understand the size and importance of higher-order factors. Copyright © 2015 John Wiley & Sons, Ltd.

This study assessed the construct validity of the School Analysis Model (SAM) Instructional Staff Questionnaire. Construct validation was necessary for several reasons. First, it has not been possible to obtain evidence of the latent factor structure of this key component of the School Analysis Model (SAM). A factor analysis using data collected with the questionnaire was conducted to assess and identify the underlying factor structure of the instrument. Second, there is no evidence that the constructs measured by the SAM are associated with attributes of school performance further empirical analysis was done to determine if latent constructs contained within the SAM Instructional Staff Questionnaire accounted for a significant proportion of variance in school effectiveness beyond that accounted for by the control variables. The eight-factor solution of the SISQ was found to be the best representation of the data based on factor loadings, scale alpha reliability estimates, conceptual cohesiveness, and number of items retained. Correlation analyses were conducted to assess the relationship between the SISQ latent factors and the control variables. Findings indicated a significant inverse relationship was found to exist between a school's SPS and poverty. Additionally, an inverse relationship was found to exist between a school's SPS and the size of a school. Several of the latent factors exhibited a relationship to the control variables as well as to other latent factors. Hierarchical multiple regression analysis was conducted in order to determine whether a combination of the latent SISQ factors account for a significant proportion of variance in school effectiveness, as measured by the school SPS. Model 1 indicated that the control variables explained approximately 56% of the variance in SPS. Model 2 indicated that the SISQ latent factors increased the proportion of variance explained by 11%. The results of this study indicated that the SISQ scales did not account for a significant proportion of the variance in SPS scores and therefore, there is substantial room for improvement in the SISQ as a measurement instrument. Results suggest that construct validation should be of primary concern in the development of measures used to evaluate and guide school improvement efforts.

Confirmatory factor analysis is a statistical technique used when investigating the structure of multivariate data. Each of a set of n observed variables is represented as a linear combination of m ( m < n ) unobserved latent variables or factors, plus an independent error term. The n × n covariance matrix corresponding to the n ‐dimensional observed variables contains the individual variable variances on the diagonal, and the covariances for all pairs of the individual observed variables in the corresponding off‐diagonal positions. A matrix equation, called the covariance equation , relates this n × n covariance matrix to the matrix whose n rows correspond to the coefficients of the m latent factors referred to as factor loadings , and the covariance matrices of the unobserved latent factors and the errors. Confirmatory factor analysis is used to test whether a hypothesized structure is appropriate for multivariate data. The hypothesized structure constrains the matrices appearing in the covariance equation. Individual covariances among the latent factors or among the error terms can be assumed equal or set to zero. Likewise, selected variances (diagonal entries) may be presumed to be equal within each of these matrices. Also, selected factor loadings may be set to zero. A random sample of multivariate observations is used to estimate the corresponding sample covariance matrix with and without the constraints imposed by the hypothesized structure. The “constraint free” covariance matrix is the matrix containing the typical sample descriptive statistics. Maximum likelihood estimation is typically used to determine estimates, with the constraints imposed, of the covariance matrices of the latent factors and errors, and the matrix of factor loadings. A statistical test is conducted to determine whether the hypothesized structure fits the data by comparing the sample covariance matrix to its counterpart produced from the covariance equation using the matrices estimated by maximum likelihood (constraints imposed).

Introduction This article studies the economic determinants of risk premiums on insurer stocks. Using a life insurance stock index, a property-liability insurance stock index, and some financial and real estate variables, we address the following question: How do changing economic conditions affect the risk premiums on insurance stocks? It is important for insurers to understand the determinants of equity risk premiums since risk premiums affect not only their investment decisions but also their financing decisions. As is well known in the corporate finance literature, the weighted average cost of capital is a weighted average of the cost of debt and the cost of equity. The higher the equity risk premium, the higher the required rate of return on equity; thus, the higher the weighted average cost of capital. Higher equity risk premiums should lead to reductions in the promised minimum rates of return paid to some universal/variable policyholders since certain profit levels must be maintained in order to ensure risk-adjusted returns to shareholders. The variation of risk premiums is also of interest to regulators because it contains information about market perceptions of insurer risk and cost of capital. Thus, in states where policy premiums are regulated, regulators should allow for higher policy premium increases if there is substantial increase in the equity risk premium. Our study finds substantial variation in risk premiums on insurance stocks that is predictably based on a small set of economic variables. We also find that the risk premiums (expected excess returns) on insurer stocks and equity real estate investment trusts (REITs) have behaved in similar fashion. Preliminary evidence indicates that insurers have been perceived by the market to have increased their real estate risk exposure in the 1980s due to a turbulent real estate market, despite the fact that their actual holdings, as a percentage of assets, remained relatively unchanged during the period. We also find that the time variation in risk premiums could be explained by the changing price of risk of one or two systematic factors. This study employs a multifactor latent-variable model widely used in the finance literature to study the risk premium of market indices. The risk premium of insurance stocks has not been examined using this method. This methodology, which is designed to capture the movement in expected excess returns due to a changing economic environment, is appropriate for our purpose because it allows for time-varying risk premiums. Specifically, it provides a concise framework to study the co-movement of insurance stocks and real estate market returns. The next two sections outline the asset pricing framework and estimation procedure. They are followed by a description of our data set and an empirical study of the time variation in risk premiums on insurance stocks. A final section summarizes the results. The Asset Pricing Framework Using the multifactor latent-variable model of Campbell (1987), Campbell and Hamao (1992), and Ferson (1989), we begin by assuming that asset returns are generated by the following K-factor model: |Mathematical Expression Omitted~ where |Mathematical Expression Omitted~ is the return on asset i held from time t to time t+1, in excess of the Treasury bill rate. |Mathematical Expression Omitted~ is the expected excess return on asset i, conditional on information known to investors at the end of time period t. The unexpected return on asset i equals the sum of K factor realizations |Mathematical Expression Omitted~ times their betas or factor loadings ||Beta~.sub.ik~, plus an idiosyncratic error |Mathematical Expression Omitted~. We assume that |Mathematical Expression Omitted~, and |Mathematical Expression Omitted~. Here, the conditional expected excess return, |Mathematical Expression Omitted~, is allowed to vary over time according to investors' conditional information set at time t. …

This article presents a short and non-technical introduction to Structural Equation Modeling or SEM. SEM is a powerful technique that can combine complex path models with latent variables (factors). Using SEM, researchers can specify confirmatory factor analysis models, regression models, and complex path models. We present the basic elements of a structural equation model, introduce the estimation technique, which is most often maximum Likelihood (ML), and discuss some problems concerning the assessment and improvement of the model fit, and model extensions to multigroup problems including factor means. Finally, we discuss some of the software, and list useful handbooks and Internet sites. What is Structural Equation Modeling? Structural Equation Modeling, or SEM, is a very general statistical modeling technique, which is widely used in the behavioral sciences. It can be viewed as a combination of factor analysis and regression or path analysis. The interest in SEM is often on theoretical constructs, which are represented by the latent factors. The relationships between the theoretical constructs are represented by regression or path coefficients between the factors. The structural equation model implies a structure for the covariances between the observed variables, which provides the alternative name covariance structure modeling. However, the model can be extended to include means of observed variables or factors in the model, which makes covariance structure modeling a less accurate name. Many researchers will simply think of these models as ‘Lisrel-models,’ which is also less accurate. LISREL is an abbreviation of LInear Structural RELations, and the name used by Joreskog for one of the first and most popular SEM programs. Nowadays structural equation models need not be linear, and the possibilities of SEM extend well beyond the original Lisrel program. Browne (1993), for instance, discusses the possibility to fit nonlinear curves. Structural equation modeling provides a very general and convenient framework for statistical analysis that includes several traditional multivariate procedures, for example factor analysis, regression analysis, discriminant analysis, and canonical correlation, as special cases. Structural equation models are often visualized by a graphical path diagram. The statistical model is usually represented in a set of matrix equations. In the early seventies, when this technique was first introduced in social and behavioral research, the software usually required setups that specify the model in terms of these matrices. Thus, researchers had to distill the matrix representation from the path diagram, and provide the software with a series of matrices for the different sets of 1 Note: The authors thank Alexander Vazsonyi and three anonymous reviewers for their comments on a previous version. We thank Annemarie Meijer for her permission to use the quality of sleep data. Introduction Structural Equation Modeling 2 parameters, such as factor loadings and regression coefficients. A recent development is software that allows the researchers to specify the model directly as a path diagram. This works well with simple problems, but may get tedious with more complicated models. For that reason, current SEM software still supports the commandor matrix-style model specifications too. This review provides a brief and non-technical review of the basic issues involved in SEM, including issues of estimation, model fit, and statistical assumptions. We include a list of available software, introductory books, and useful Internet resources. Examples of SEM-Models In this section, we set the stage by discussing examples of a confirmatory factor analysis, regression analysis, and a general structural equation model with latent variables. Structural equation modeling has its roots in path analysis, which was invented by the geneticist Sewall Wright (Wright, 1921). It is still customary to start a SEM analysis by drawing a path diagram. A path diagram consists of boxes and circles, which are connected by arrows. In Wright’s notation, observed (or measured) variables are represented by a rectangle or square box, and latent (or unmeasured) factors by a circle or ellipse. Single headed arrows or ‘paths’ are used to define causal relationships in the model, with the variable at the tail of the arrow causing the variable at the point. Double headed arrows indicate covariances or correlations, without a causal interpretation. Statistically, the single headed arrows or paths represent regression coefficients, and double-headed arrows covariances. Extensions of this notation have been developed to represent variances and means (cf. McArdle, 1996). The first example in Figure 1 is a representation of a confirmatory factor analysis model, with six observed variables and

More than 1000 respondents in Sweden (2013) and the US (2014) were asked to report their subjective opinions and attitudes about situations that caused them regret, concern, worry, and anxiety. US respondents self-identified as Black. Although exploratory factor analyses extracted many latent factors from the 80 questions, a common latent inner factor was extracted from five questions that examined key psychological phenomena: worry at the present time, bothersome concerns in the present, regret for the past, anxiety about the future, and unpleasant experience in the past. Confirmatory factor analyses and structural equation modeling of the latent variables (SEM/LV) provided convincing evidence of the existence of a common latent inner factor in both countries. Because each of the five key phenomena reflected concerns involving the self, the common latent inner factor was labeled “Being unable to detach from concerns involving the self.” Then, the same latent inner factor was also confirmed in SEM/LV of combinations of data from Swedish and US Black respondents, and from respondents in a previous study (Japanese, and US respondents who identified as White; Hayase, 2016). Women, younger people, and people with lower levels of education were less able to detach from concerns involving the self than men, older people, and people with higher levels of education. Confirmatory factor analyses by SEM/LV provided additional evidence of the existence of a common latent inner factor for the five phenomena, worry, bothersome concerns, regret, anxiety, and unpleasant experience. Psychological and philosophical implications of the latent inner factor with regard to genuine happiness were discussed.

This paper proposes an improved procedure for factor analysis applied to flow data in terms of solving relevant technical problems, and attempts to systematize it (see Fig. 5).After confirming the proposed procedure's validity, the utility of higher-order factor analysis is examined, taking as a case study automobile traffic flows in the Keihanshin Metropolitan Area with a complicated connectivity structure. Compared with the orthogonal rotation which results in identifying uncorrelated functional regions, higher-order factor analysis has advantages of being able to extract correlated functional regions and to clarify the hierarchical structure of functional regions.The proposed procedure of factor analysis applied to the Origin-Destination data matrix is as follows: i) as to the form of the O-D data matrix, an asymmetrical matrix including diagonal elements or intra-flows is preferable; ii) as to the input data matrix in extracting the initial factors, the result of factor analysis of a hypothetical O-D data matrix reveals that the cross-product standardized by the sum of squares (ΣXj2=1.0) matrix may be more adequate than the correlation matrix; iii) as to the number-of-factors which is one of the most intractable problems on factor analysis, the number of factors showing the most interpretable factor structure are regarded as the number of common factors to be extracted, based on three criteria of 1) each area's communality value of greater than 0.1, 2) the change of percentage of the accumulated variance explained and 3) existence or not of bipolar factors; iv) in interpreting factors, under the assumption that the high factor loadings may specify groups of destinations receiving trips from common origins, and that the high factor scores specify these (groups of) origins, the internal structure of each functional region (the primate central area-type, the multiprimate central areas-type and the interdependent-type) is identified based on the ranking of the factor scores and their differences from the first ranking factor score.Next, higher-order factor analysis with oblique rotation is applied to the 212×212 O-D data matrix of automobile traffic flows in the Keihanshin Metropolitan Area in 1979. As a result of the analysis, it turned out that the Keihanshin Metropolitan Area consisted of 30 first-order functional regions with the above-mentioned internal structures, and that these first-order functional regions were integrated into 13 relatively independent second-order functional regions.First, as for the first-order functional regions (see Fig. 9), the northern part of Osaka City whose central area is Osaka-Kita Ward, is classified as the primate central area-type, the southern part of Osaka City is classified as the interdependent-type, Kyoto City is classified as the interdependent-type, and Kobe City whose central areas are Kobe-Ikuta and Kobe-Hyogo Wards is classified as the multi-primate central areas-type. They are identified as functional regions corresponding to the three metropolitan areas of Osaka, Kyoto and Kobe respectively.In addition, ‘satellite cities’ of Himeji, Nara and Wakayama located in periphery of the Metropolitan Area are characterized as functional regions classified as the primate central area-type. One set of ‘satellite cities’ surrounding Osaka City such as Higashi-Osaka, Sakai and Tondabayashi, and another set of ‘satellite cities’ such as Toyonaka-Suita-Ibaragi, Hirakata-Neyagawa-Moriguchi-Kadoma and Amagasaki-Nishinomiya, show different internal structures. The former is the primate central area-type, and the latter is the multi-primate central areas-type.Second, as for the second-order functional regions (see Fig. 10), it could be pointed out that they integrate a few neighboring first-order functional regions, and that some boundaries between them correspond to the boundaries of prefectures.

Is the Repeatable Battery for the Assessment of Neuropsychological Status Factor Structure Appropriate for Inpatient Psychiatry? An Exploratory and Higher-Order Analysis

This article reports the use of higher-order factor analysis to examine the underlying dimensions of an instrument designed to measure community acceptance of nurse practitioners/physician's assistants. The instrument consisted of both dichotomous and Likert scale items. Use of factor analysis with dichotomous data is controversial. Higher-order factor analysis provides a potential solution to this dilemma. Following initial factor analysis using maximum likelihood extraction with oblique rotation, the factor correlation matrix was factored to obtain second order factors. The second order factors were interpreted using the factor pattern matrix and examining correlations of individual survey items with the second order factors. This procedure provides additional factorial validity of the underlying uni- and multidimensional concepts related to acceptance of NPs and PAs in rural settings: knowledge, competence, access, and trust.

In investment management, multifactor risk modeling is the most common application of financial modeling. Multifactor risk models, or simply factor models, are linear regressions over a number of variables called factors. Factors can be exogenous variables or abstract variables formed by portfolios. Exogenous factors (or known factors) can be identified from traditional fundamental analysis or economic theory from macroeconomic factors. Abstract factors, also called unidentified or latent factors, can be determined with factor analysis or principal component analysis. Principal component analysis identifies the largest eigenvalues of the variance-covariance matrix or the correlation matrix. The largest eigenvalues correspond to eigenvectors that identify the entire market and sectors that correspond to industry classification. Factor analysis can be used to identify the structure of the latent factors. Keywords: Factor models; Linear factor models; factor loadings; static models; dynamic; normal factor model; Principal components analysis; principal components; factor analysis; expectation maximization

In two studies conducted in the US and Japan in 2012, more than 1000 respondents in each country were asked to report their subjective opinions and attitudes about situations that caused them regret, concern, worry, and anxiety. Although exploratory factor analyses extracted many latent factors from the 80 questions, a common latent inner factor was extracted from five questions that examined key psychological phenomena: worry at the present time, bothersome concerns in the present, regret for the past, anxiety about the future, and unpleasant experience in the past. Confirmatory factor analyses and structural equation modeling of the latent variables (SEM/LV) provided convincing evidence of the existence of the common latent inner factor in both countries. Because each of the five key phenomena reflected concerns involving the self, the common latent inner factor was labeled “Being unable to detach from concerns involving the self.” The same latent inner factor was also confirmed in SEM/LV of the combined US- Japanese data. Women, younger people, and people with lower levels of education were less able to detach from concerns involving the self than were men, older people, and people with higher levels of education. This was true in the samples from both independent (US) and interdependent (Japan) cultures. Psychological and philosophical implications of the latent inner factor were discussed.

The task of ranking the university in international rating systems is urgent. An approach to solving the problem is proposed to ensure the required values of the basic indicators of the university’s activity in the international institutional ranking QS using models developed on the basis of methods of correlation-regression analysis and factor analysis. Estimates of basic indicators and ratings were obtained based on the methods of correlation and regression analysis. A comparative analysis of the results obtained for universities in the reference group Interpretation of the results of factor analysis revealed a set of latent factors that have a significant impact on the baseline indicators. It is shown that measures to achieve the specified indicators must be carried out considering the identified correlation dependences of latent exponential factors and basic indicators, as well as the results of interpretation of the developed factor model. The novelty of the developed proposals lies in the assessment of the significance of latent factors affecting the basic indicators of the analysis of the university’s activities, based on the use of correlation-regression methods and methods of factor analysis. The developed factorial model made it possible to group and structure the obtained data, as well as to reduce the dimension of the problem being solved. The results obtained allow us to solve the problem of identifying latent factors and to substantiate the conditions for achieving the required indicators of the university ranking in the international financial ranking QS.KeywordsCorrelation-regression analysisFactor analysisBasic indicatorsInstitutional rating

ObjectiveThe aim of this study was to assess the factor structure of the English version of the Young Schema Questionnaire Long Form‐3 (YSQ‐L3) using a large clinical sample and smaller non‐clinical population. The items in the YSQ‐L3 were evaluated as to how well they assess the underlying theoretical constructs of schema and schema domains.MethodA primary and a higher order factor (HOF) analysis was undertaken on a large heterogeneous clinical sample (N = 574) and the total sample (N = 838) that included a small non‐clinical population (N = 264).ResultsThe primary factor analysis revealed 20 early maladaptive schemas (EMS). Of the 232 items, 182 loaded above .4 and were retained for the final analysis. The original Emotional Inhibition schema separated into Emotional Constriction and Fear of Losing Control, and Punitiveness likewise separated into Punitiveness (Self) and Punitiveness (Other). The HOF analysis indicated four domains: Emotional Dysregulation, Disconnection, Impaired Autonomy/Underdeveloped Self, and Excessive Responsibility/Overcontrol. These overlap with the domains proposed by Young et al. (2003) but with some differences. The Emotion Dysregulation domain was unique to the current study. The mean item loading for each factor ranged from .52 to .82. The revised scale showed excellent overall internal consistency (a = .91).ConclusionThis is the first study that investigated the psychometric properties of the English version of the YSQ‐L3. The resultant domains fitted with existing literature of meaningful clinical phenomenon such as attachment and emotion dysregulation and their role in maintaining chronic psychological disorders.