Using SAS PROC CALIS to fit Level-1 error covariance structures of latent growth models

In the present article, we demonstrates the use of SAS PROC CALIS to fit various types of Level-1 error covariance structures of latent growth models (LGM). Advantages of the SEM approach, on which PROC CALIS is based, include the capabilities of modeling the change over time for latent constructs, measured by multiple indicators; embedding LGM into a larger latent variable model; incorporating measurement models for latent predictors; and better assessing model fit and the flexibility in specifying error covariance structures. The strength of PROC CALIS is always accompanied with technical coding work, which needs to be specifically addressed. We provide a tutorial on the SAS syntax for modeling the growth of a manifest variable and the growth of a latent construct, focusing the documentation on the specification of Level-1 error covariance structures. Illustrations are conducted with the data generated from two given latent growth models. The coding provided is helpful when the growth model has been well determined and the Level-1 error covariance structure is to be identified.     

纳入式分类分析法在潜在剖面模型的后续多元回归中的应用

纳入式分类分析法能克服传统的分类分析法对后续一元回归模型参数的低估,发挥潜在类别模型的后续分析简化变量间交互作用的功能。本文进一步将纳入式分类分析法拓展至潜在剖面模型后续的多元统计分析中。通过蒙特卡洛模拟实验,比较各种纳入变量的方法思路与后续分析模型在四种常见的多元回归模型中参数估计的表现。结果发现,纳入式分类分析法所需纳入的变量取决于后续分析中与因变量、潜类别变量的关系,且后续分析使用含交互作用的模型更为稳健。     

The construct of psychophysiological reactivity: Statistical and psychometric issues

The purpose of this paper is to review major statistical and psychometric issues impacting the study of psychophysiological reactivity and discuss their implications for applied developmental researchers. We first cover traditional approaches such as the observed difference score (DS) and the observed residual score (RS), including a review of classic and recent research on their reliability and validity from two related bodies of work: the measurement of change and the Law of Initial Values. Second, we review several types of latent variable modeling in this context: latent difference score (LDS) models, latent residual score (LRS) models, latent state-trait (LST) models, and latent growth curve (LGC) models. Finally, we provide broad guidelines for applied researchers broken down by key stages of a psychophysiological project: study planning, data analysis, and reporting of results. Our recommendations highlight the need for (1) increased attention to the ubiquitous nature of measurement error in observed variables and the importance of employing latent variable models when possible, and (2) increased specification of theories relating to the construct of reactivity, especially in regards to the distinction between baseline arousal and change over time in broader systems of variables.     

形成性测量模型：结构方程模型的新视角

形成性测量模型(Formative Model, FM)是指标变异导致潜变量变异的模型, 反映性测量模型(Reflective Model, RM)是潜变量变异导致指标变异的模型。FM在模型界定、识别和估计、信效度评价以及模型应用等方面均与RM存在极大的不同。模型界定错误会使参数估计发生偏差, 影响统计结论的有效性, 应当审慎考虑指标和潜变量之间的关系, 选择恰当的测量模型。进一步揭示两者的区别和误用带来的偏差, 完善FM的识别和估计、信效度评价方法、对变量含义的解释以及高阶FM的理论解释和模型估计是未来的研究方向。     

Stochastic EM for estimating the parameters of a multilevel IRT model

An item response theory (IRT) model is used as a measurement error model for the dependent variable of a multilevel model. The dependent variable is latent but can be measured indirectly by using tests or questionnaires. The advantage of using latent scores as dependent variables of a multilevel model is that it offers the possibility of modelling response variation and measurement error and separating the influence of item difficulty and ability level. The two‐parameter normal ogive model is used for the IRT model. It is shown that the stochastic EM algorithm can be used to estimate the parameters which are close to the maximum likelihood estimates. This algorithm is easily implemented. The estimation procedure will be compared to an implementation of the Gibbs sampler in a Bayesian framework. Examples using real data are given.     

A checklist for testing measurement invariance

The analysis of measurement invariance of latent constructs is important in research across groups, or across time. By establishing whether factor loadings, intercepts and residual variances are equivalent in a factor model that measures a latent concept, we can assure that comparisons that are made on the latent variable are valid across groups or time. Establishing measurement invariance involves running a set of increasingly constrained structural equation models, and testing whether differences between these models are significant. This paper provides a step-by-step guide to analysing measurement invariance.     

Growth modeling with nonignorable dropout: alternative analyses of the STAR*D antidepressant trial

This article uses a general latent variable framework to study a series of models for nonignorable missingness due to dropout. Nonignorable missing data modeling acknowledges that missingness may depend not only on covariates and observed outcomes at previous time points as with the standard missing at random assumption, but also on latent variables such as values that would have been observed (missing outcomes), developmental trends (growth factors), and qualitatively different types of development (latent trajectory classes). These alternative predictors of missing data can be explored in a general latent variable framework with the Mplus program. A flexible new model uses an extended pattern-mixture approach where missingness is a function of latent dropout classes in combination with growth mixture modeling. A new selection model not only allows an influence of the outcomes on missingness but allows this influence to vary across classes. Model selection is discussed. The missing data models are applied to longitudinal data from the Sequenced Treatment Alternatives to Relieve Depression (STAR*D) study, the largest antidepressant clinical trial in the United States to date. Despite the importance of this trial, STAR*D growth model analyses using nonignorable missing data techniques have not been explored until now. The STAR*D data are shown to feature distinct trajectory classes, including a low class corresponding to substantial improvement in depression, a minority class with a U-shaped curve corresponding to transient improvement, and a high class corresponding to no improvement. The analyses provide a new way to assess drug efficiency in the presence of dropout.     

Accounting for sensation seeking in route choice behavior with travel time information

The purpose of this paper is to demonstrate that latent variables, with the focus on sensation seeking concepts, incorporated in new technique of route choice modeling, improve our analyzing of route choice behavior with pre-trip travel time information. The application of a hybrid discrete choice model framework integrates a latent variable model and a route choice model by combining their measurement and structural equations. The model is estimated based on data from a laboratory experiment and a field study of a simple network. The results show that certain sensation seeking domains (e.g., thrill and adventure seeking) alongside traditional variables (e.g., travel time information) enrich our understanding and provide more insight into route choice behavior. Furthermore, observed personal variables, such as gender and marital status, may serve as causal indicators to sensation seeking variables.     

Structural Equation Modeling of Social Networks: Specification,Estimation, and Application

Psychologists are interested in whether friends and couples share similar personalities or not. However, no statistical models are readily available to test the association between personalities and social relations in the literature. In this study, we develop a statistical model for analyzing social network data with the latent personality traits as covariates. Because the model contains a measurement model for the latent traits and a structural model for the relationship between the network and latent traits, we discuss it under the general framework of structural equation modeling (SEM). In our model, the structural relation between the latent variable(s) and the outcome variable is no longer linear or generalized linear. To obtain model parameter estimates, we propose to use a two-stage maximum likelihood (ML) procedure. This modeling framework is evaluated through a simulation study under representative conditions that would be found in social network data. Its usefulness is then demonstrated through an empirical application to a college friendship network.     

Using trait–state models to evaluate the longitudinal consistency of global self-esteem from adolescence to adulthood

The present research used a latent variable trait–state model to evaluate the longitudinal consistency of self-esteem during the transition from adolescence to adulthood. Analyses were based on ten administrations of the Rosenberg self-esteem scale (Rosenberg, 1965) spanning the ages of approximately 13–32 for a sample of 451 participants. Results indicated that a completely stable trait factor and an autoregressive trait factor accounted for the majority of the variance in latent self-esteem assessments, whereas state factors accounted for about 16% of the variance in repeated assessments of latent self-esteem. The stability of individual differences in self-esteem increased with age consistent with the cumulative continuity principle of personality development.     

Theoretical and methodological analysis of partial knowledge

A theory of partial knowledge is proposed as an explanation of cognitive development, and methods are described for testing the theory. The theory consists of three structure-process pairs, each of which postulates a type of cognitive structure and a developmental process specific to that type. In restricted knowledge, a unitary algorithm is the cognitive structure, and amendment is the developmental process. In variable sampling, a structure of unitary substitutes is paired with a process of selection. In variable integration, modular components are paired with self-monitoring. Methods for testing the theory form a sequence of mathematical models. The first model in the sequence, called a model of double assessment, is described both verbally and mathematically. Other models in the sequence are described verbally with reference to other articles for the formal mathematics. Also described are some nonmathematical methods to be used as sequels to the double assessment model.     

Two-Step Estimation of Models Between Latent Classes and External Variables

We consider models which combine latent class measurement models for categorical latent variables with structural regression models for the relationships between the latent classes and observed explanatory and response variables. We propose a two-step method of estimating such models. In its first step, the measurement model is estimated alone, and in the second step the parameters of this measurement model are held fixed when the structural model is estimated. Simulation studies and applied examples suggest that the two-step method is an attractive alternative to existing one-step and three-step methods. We derive estimated standard errors for the two-step estimates of the structural model which account for the uncertainty from both steps of the estimation, and show how the method can be implemented in existing software for latent variable modelling.     

The theoretical status of latent variables

This article examines the theoretical status of latent variables as used in modern test theory models. First, it is argued that a consistent interpretation of such models requires a realist ontology for latent variables. Second, the relation between latent variables and their indicators is discussed. It is maintained that this relation can be interpreted as a causal one but that in measurement models for interindividual differences the relation does not apply to the level of the individual person. To substantiate intraindividual causal conclusions, one must explicitly represent individual level processes in the measurement model. Several research strategies that may be useful in this respect are discussed, and a typology of constructs is proposed on the basis of this analysis. The need to link individual processes to latent variable models for interindividual differences is emphasized.     

Latent Class Models for Diary Method Data: Parameter Estimation by Local Computations

The increasing use of diary methods calls for the development of appropriate statistical methods. For the resulting panel data, latent Markov models can be used to model both individual differences and temporal dynamics. The computational burden associated with these models can be overcome by exploiting the conditional independence relations implied by the model. This is done by associating a probabilistic model with a directed acyclic graph, and applying transformations to the graph. The structure of the transformed graph provides a factorization of the joint probability function of the manifest and latent variables, which is the basis of a modified and more efficient E-step of the EM algorithm. The usefulness of the approach is illustrated by estimating a latent Markov model involving a large number of measurement occasions and, subsequently, a hierarchical extension of the latent Markov model that allows for transitions at different levels. Furthermore, logistic regression techniques are used to incorporate restrictions on the conditional probabilities and to account for the effect of covariates. Throughout, models are illustrated with an experience sampling methodology study on the course of emotions among anorectic patients. Frank Rijmen was partly supported by the Fund for Scientific Research Flanders (FWO).     

On measurement properties of continuation ratio models

Three classes of polytomous IRT models are distinguished. These classes are the adjacent category models, the cumulative probability models, and the continuation ratio models. So far, the latter class has received relatively little attention. The class of continuation ratio models includes logistic models, such as the sequential model (Tutz, 1990), and nonlogistic models, such as the acceleration model (Samejima, 1995) and the nonparametric sequential model (Hemker, 1996). Four measurement properties are discussed. These are monotone likelihood ratio of the total score, stochastic ordering of the latent trait by the total score, stochastic ordering of the total score by the latent trait, and invariant item ordering. These properties have been investigated previously for the adjacent category models and the cumulative probability models, and for the continuation ratio models this is done here. It is shown that stochastic ordering of the total score by the latent trait is implied by all continuation ratio models, while monotone likelihood ratio of the total score and stochastic ordering on the latent trait by the total score are not implied by any of the continuation ratio models. Only the sequential rating scale model implies the property of invariant item ordering. Also, we present a Venn-diagram showing the relationships between all known polytomous IRT models from all three classes.     

多变量追踪研究的模型整合与拓展：考察往复式影响与增长趋势

追踪研究当中, 交叉滞后模型可以探究多变量之间往复式影响, 潜增长模型可以探究个体增长趋势。对两类模型进行整合, 例如同时关注往复式影响与个体增长趋势, 同时可以定义测量误差、随机截距等变异成分, 衍生出随机截距交叉滞后模型、特质-状态-误差模型、自回归潜增长模型、结构化残差潜增长模型等。以交叉滞后模型和潜增长模型分别作为基础模型, 从个体间/个体内变异分解的角度对上述各类模型梳理, 整合出此类模型的分析框架, 并拓展建立“因子结构化潜增长模型(factor latent curve model with structured reciprocals)”作为统合框架。通过实证研究(早期儿童的追踪研究-幼儿园版, ECLS-K), 建立21049名儿童的阅读和数学能力的往复式影响与增长趋势。研究发现, 分离了稳定特质的模型拟合最优。研究也对模型建模思路和模型选择提供了建议。     

When constraints interact: a caution about reference variables,identification constraints,and scale dependencies in structural equation modeling

In traditional approaches to structural equations modeling, variances of latent endogenous variables cannot be specified or constrained directly and, consequently, are not identified, unless certain precautions are taken. The usual method for achieving identification has been to fix one factor loading for each endogenous latent variable at unity. An alternative approach is to fix variances using newer constrained estimation algorithms. This article examines the philosophy behind such constraints and shows how their appropriate use is neither as straightforward nor as noncontroversial as portrayed in textbooks and computer manuals. The constraints on latent variable variances can interact with other model constraints to interfere with the testing of certain kinds of hypotheses and can yield incorrect standardized solutions with some popular software.     

Development and validation of a Rasch-derived CES-D short form

The current study presents a Rasch-derived short form of the Center for Epidemiologic Studies-Depression scale (CES-D) for use as a depression screening tool in the general population. In contrast to short forms developed with reliance on classical measurement techniques, those developed using techniques based on item response theory produce a measure that offers true interval scaling, provide enhanced information about responders with extreme scores, and expand understanding of the underlying latent structure. Cross-validation of the Rasch-derived CES-D short form supported its utility and structural validity across samples. Tests of structural validity using latent variable modeling methodology indicated that a hierarchical, single-factor model of depression had the best fit for the original full form and the Rasch-derived short form of the CES-D. This finding challenges depression researchers and theorists to reconsider the interfactor relationships in the study and assessment of depression.     

Polytomous Latent Scales for the Investigation of the Ordering of Items

We propose three latent scales within the framework of nonparametric item response theory for polytomously scored items. Latent scales are models that imply an invariant item ordering, meaning that the order of the items is the same for each measurement value on the latent scale. This ordering property may be important in, for example, intelligence testing and person-fit analysis. We derive observable properties of the three latent scales that can each be used to investigate in real data whether the particular model adequately describes the data. We also propose a methodology for analyzing test data in an effort to find support for a latent scale, and we use two real-data examples to illustrate the practical use of this methodology.     

Single and Multiple Ability Estimation in the SEM Framework: A Noninformative Bayesian Estimation Approach

Latent variable models with many categorical items and multiple latent constructs result in many dimensions of numerical integration, and the traditional frequentist estimation approach, such as maximum likelihood (ML), tends to fail due to model complexity. In such cases, Bayesian estimation with diffuse priors can be used as a viable alternative to ML estimation. This study compares the performance of Bayesian estimation with ML estimation in estimating single or multiple ability factors across 2 types of measurement models in the structural equation modeling framework: a multidimensional item response theory (MIRT) model and a multiple-indicator multiple-cause (MIMIC) model. A Monte Carlo simulation study demonstrates that Bayesian estimation with diffuse priors, under various conditions, produces results quite comparable with ML estimation in the single- and multilevel MIRT and MIMIC models. Additionally, an empirical example utilizing the Multistate Bar Examination is provided to compare the practical utility of the MIRT and MIMIC models. Structural relationships among the ability factors, covariates, and a binary outcome variable are investigated through the single- and multilevel measurement models. The article concludes with a summary of the relative advantages of Bayesian estimation over ML estimation in MIRT and MIMIC models and suggests strategies for implementing these methods.