Regression among factor scores

Structural equation models with latent variables are sometimes estimated using an intuitive three-step approach, here denoted factor score regression. Consider a structural equation model composed of an explanatory latent variable and a response latent variable related by a structural parameter of scientific interest. In this simple example estimation of the structural parameter proceeds as follows: First, common factor models areseparately estimated for each latent variable. Second, factor scores areseparately assigned to each latent variable, based on the estimates. Third, ordinary linear regression analysis is performed among the factor scores producing an estimate for the structural parameter. We investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables. It is demonstrated that the conventional approach to factor score regression performs very badly. Revised factor score regression, using Regression factor scores for the explanatory latent variables and Bartlett scores for the response latent variables, produces consistent estimators for all parameters.     

Describing the elephant: Structure and function in multivariate data

There is a unity underlying the diversity of models for the analysis of multivariate data. Essentially, they constitute a family models, most generally nonlinear, for structural/functional relations between variables drawn from a behavior domain.     

Classical test theory based and factor analytic structural models for analyzing change: A note on differences and similarities

This note is concerned with differences and similarities between structural models for analyzing change, which are conceptualized within two different modelling traditions: the one based on the classical test theory, and that within the factor-analytic approach. It is shown that these two possibilities lead to models for studying change, which are indistinguishable when using for data analytic purposes structural modeling programs, such as LISREL, EQS, COSAN, LISCOMP, RAMONA, EzPATH, SAS PROC CALIS. The reason for this data-analytic equivalence of the two conceptually different types of models is the confounding of their differences in the corresponding implied covariance matrix structures.     

The rank of reduced dispersion matrices

Psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements. Psychometricians concentrate on cases in which low rank can be attained, preferably rank one, the Spearman case. Econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one, the Frisch case. In this paper we give an extensive historial discussion of both fields, we prove the two key results in a more satisfactory and uniform way, we point out various small errors and misunderstandings, and we present a methodological comparison of factor analysis and regression on the basis of our results.Financial support by the Netherlands Organization for the Advancement of Pure Research (ZWO) is gratefully acknowledged.     

Classical Test Theory as a first-order Item Response Theory: Application to true-score prediction from a possibly nonparallel test

We give an account of Classical Test Theory (CTT) in terms of the more fundamental ideas of Item Response Theory (IRT). This approach views classical test theory as a very general version of IRT, and the commonly used IRT models as detailed elaborations of CTT for special purposes. We then use this approach to CTT to derive some general results regarding the prediction of the true-score of a test from an observed score on that test as well from an observed score on a different test. This leads us to a new view of linking tests that were not developed to be linked to each other. In addition we propose true-score prediction analogues of the Dorans and Holland measures of the population sensitivity of test linking functions. We illustrate the accuracy of the first-order theory using simulated data from the Rasch model, and illustrate the effect of population differences using a set of real data.This research is collaborative in every respect and the order of authorship is alphabetical. It was begun when both authors were on the faculty of the Graduate School of Education at the University of California, Berkeley.We would like to thank both Neil Dorans, Skip Livingston and two anonymous referees for many suggestions that have greatly improved this paper.     

Generalized multilevel structural equation modeling

A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent variables. The response model generalizes GLMMs to incorporate factor structures in addition to random intercepts and coefficients. As in GLMMs, the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower-level units in the higher-level units and missing data. A wide range of response processes can be modeled including ordered and unordered categorical responses, counts, and responses of mixed types. The structural model is similar to the structural part of a SEM except that it may include latent and observed variables varying at different levels. For example, unit-level latent variables (factors or random coefficients) can be regressed on cluster-level latent variables. Special cases of this framework are explored and data from the British Social Attitudes Survey are used for illustration. Maximum likelihood estimation and empirical Bayes latent score prediction within the GLLAMM framework can be performed using adaptive quadrature in gllamm, a freely available program running in Stata.gllamm can be downloaded from http://www.gllamm.org. The paper was written while Sophia Rabe-Hesketh was employed at and Anders Skrondal was visiting the Department of Biostatistics and Computing, Institute of Psychiatry, King's College London.     

Model modification

An analysis of empirical data often leads to a rejection of a hypothesized model, even if the researcher has spent considerable efforts in including all available information in the formulation of the model. Thus, the researcher must reformulate the model in some way, but in most instances there is, at least theoretically, an overwhelming number of possible actions that could be taken. In this paper a modification index will be discussed which should serve as a guide in the search for a better model. In statistical terms, the index measures how much we will be able to reduce the discrepancy between model and data, as defined by a general fit function, when one parameter is added or freed or when one equality constraint is relaxed. The modification index discussed in this paper is an improvement of the one incorporated in the LISREL V computer program in that it takes into account changes in all the parameters of the model when one particular parameter is freed.The research reported in this paper has been supported by The Swedish Council for Research in the Humanities and Social Sciences under Research Program Multivariate Statistical Analysis, Project Director Karl G Jöreskog.     

Exploratory structural equation modeling,bifactor models,and standard confirmatory factor analysis models: Application to the BASC-2 Behavioral and Emotional Screening System Teacher Form

Several psychological assessment instruments are based on the assumption of a general construct that is composed of multiple interrelated domains. Standard confirmatory factor analysis is often not well suited for examining the factor structure of such scales. This study used data from 1885 elementary school students (mean age = 8.77 years, SD = 1.47 years) to examine the factor structure of the Behavioral Assessment System for Children, Second Edition (BASC-2) Behavioral and Emotional Screening System (BESS) Teacher Form that was designed to assess general risk for emotional/behavioral difficulty among children. The modeling sequence included the relatively new exploratory structural equation modeling (ESEM) approach and bifactor models in addition to more standard techniques. Findings revealed that the factor structure of the BASC-2 BESS Teacher Form is multidimensional. Both ESEM and bifactor models showed good fit to the data. Bifactor models were preferred on conceptual grounds. Findings illuminate the hypothesis-generating power of ESEM and suggest that it might not be optimal for instruments designed to assess a predominant general factor underlying the data.     

Inequalities among lower bounds to reliability: With applications to test construction and factor analysis

A chain of lower-bound inequalities leading to the greatest lower bound to reliability is established for the internal consistency of a composite of unit-weighted components. The chain includes the maximum split-half coefficient, the lowest coefficient consistent with nonimaginary common factors, and the lowest coefficient consistent with nonimaginary common and unique factors. Optimization theory is utilized to determine the conditions that are requisite for the inequalities. Convergence proofs demonstrate that the coefficients can be attained. Rapid algorithms obtain estimates of the coefficients with sample data. The theory yields methods for splitting items into maximally similar sets and for exploratory factor analysis based on a theoretical solution to the communality problem.     

Asymptotic biases in exploratory factor analysis and structural equation modeling

Formulas for the asymptotic biases of the parameter estimates in structural equation models are provided in the case of the Wishart maximum likelihood estimation for normally and nonnormally distributed variables. When multivariate normality is satisfied, considerable simplification is obtained for the models of unstandardized variables. Formulas for the models of standardized variables are also provided. Numerical examples with Monte Carlo simulations in factor analysis show the accuracy of the formulas and suggest the asymptotic robustness of the asymptotic biases with normality assumption against nonnormal data. Some relationships between the asymptotic biases and other asymptotic values are discussed.The author is indebted to the editor and anonymous reviewers for their comments, corrections, and suggestions on this paper, and to Yutaka Kano for discussion on biases.     

On structural models for analyzing change

The recommendation to base the analysis of multi-wave data upon explicit models for change is advocated. Several univariate and multivariate models are described, which emerge from an interaction between the classical test theory and the structural equation modeling approach. The resulting structural models for analyzing change reflect in some of their parameters substantively interesting aspects of intra- and interindividual change in follow-up studies. The models are viewed as an alternative to an ANOVA-based analysis of longitudinal data, and are illustrated on data from a cognitive intervention study of old adults (Bakes et al , 1986). The approach presents a useful means of analyzing change over time, and is applicable for purposes of (latent) growth curve analysis when analysis of variance assumptions are violated (e.g., Schaie & Hertzog, 1982; Morrison, 1976).     

Premature conclusions about the signal-to-noise ratio in structural equation modeling research: A commentary on Yuan and Fang (2023)

In a recent article published in this journal, Yuan and Fang (British Journal of Mathematical and Statistical Psychology, 2023) suggest comparing structural equation modeling (SEM), also known as covariance-based SEM (CB-SEM), estimated by normal-distribution-based maximum likelihood (NML), to regression analysis with (weighted) composites estimated by least squares (LS) in terms of their signal-to-noise ratio (SNR). They summarize their findings in the statement that “[c]ontrary to the common belief that CB-SEM is the preferred method for the analysis of observational data, this article shows that regression analysis via weighted composites yields parameter estimates with much smaller standard errors, and thus corresponds to greater values of the [SNR].” In our commentary, we show that Yuan and Fang have made several incorrect assumptions and claims. Consequently, we recommend that empirical researchers not base their methodological choice regarding CB-SEM and regression analysis with composites on the findings of Yuan and Fang as these findings are premature and require further research.     

题目因素分析：基于SEM和基于IRT的两类方法

心理和教育测量一般只能达到顺序量表的水平,其测量数据与被测因子间并非简单线性关系。题目因素分析是用来描述测量题目与因子间非线性关系的统计模型。题目因素分析主要有基于结构方程模型和基于项目反应理论两类方法,两类方法之间存在紧密的联系,甚至可以看作是同一模型的两种表现形式。本文详细阐述了该关系,同时对两类方法在参数估计、模型拟合指标、测量一致性检验和支撑软件等方面的特点进行了分析和比较,以便研究者选择最为适合其研究的方法。     

Selecting the number of classes under latent class regression: a factor analytic analogue

Recently, the regression extension of latent class analysis (RLCA) model has received much attention in the field of medical research. The basic RLCA model summarizes shared features of measured multiple indicators as an underlying categorical variable and incorporates the covariate information in modeling both latent class membership and multiple indicators themselves. To reduce complexity and enhance interpretability, one usually fixes the number of classes in a given RLCA. Often, goodness of fit methods comparing various estimated models are used as a criterion to select the number of classes. In this paper, we propose a new method that is based on an analogous method used in factor analysis and does not require repeated fitting. Two ideas with application to many settings other than ours are synthesized in deriving the method: a connection between latent class models and factor analysis, and techniques of covariate marginalization and elimination. A Monte Carlo simulation study is presented to evaluate the behavior of the selection procedure and compare to alternative approaches. Data from a study of how measured visual impairments affect older persons’ functioning are used for illustration.This work was supported by National Institute on Aging (NIA) Program Project P01-AG-10184-03. The author wishes to thank Dr. Karen Bandeen-Roche for her stimulating comments and helpful discussions, and Drs. Gary Rubin and Sheila West for kindly making the Salisbury Eye Evaluation data available.     

A comparative evaluation of factor- and component-based structural equation modelling approaches under (in)correct construct representations

Structural equation modelling (SEM) has evolved into two domains, factor-based and component-based, dependent on whether constructs are statistically represented as common factors or components. The two SEM domains are conceptually distinct, each assuming their own population models with either of the statistical construct proxies, and statistical SEM approaches should be used for estimating models whose construct representations correspond to what they assume. However, SEM approaches have often been evaluated and compared only under population factor models, providing misleading conclusions about their relative performance. This is partly because population component models and their relationships have not been clearly formulated. Also, it is of fundamental importance to examine how robust SEM approaches can be to potential misrepresentation of constructs because researchers may often lack clear theories to determine whether a factor or component is more representative of a given construct. Addressing these issues, this study begins by clarifying several population component models and their relationships and then provides a comprehensive evaluation of four SEM approaches – the maximum likelihood approach and factor score regression for factor-based SEM as well as generalized structured component analysis (GSCA) and partial least squares path modelling (PLSPM) for component-based SEM – under various experimental conditions. We confirm that the factor-based SEM approaches should be preferred for estimating factor models, whereas the component-based SEM approaches should be chosen for component models. Importantly, the component-based approaches are generally more robust to construct misrepresentation than the factor-based ones. Of the component-based approaches, GSCA should be chosen over PLSPM, regardless of whether or not constructs are misrepresented.