A Limited Information Estimator for Dynamic Factor Models

Structural equation modeling (SEM) is an increasingly popular method for examining multivariate time series data. As in cross-sectional data analysis, structural misspecification of time series models is inevitable, and further complicated by the fact that errors occur in both the time series and measurement components of the model. In this article, we introduce a new limited information estimator and local fit diagnostic for dynamic factor models within the SEM framework. We demonstrate the implementation of this estimator and examine its performance under both correct and incorrect model specifications via a small simulation study. The estimates from this estimator are compared to those from the most common system-wide estimators and are found to be more robust to the structural misspecifications considered.     

Some local models for correlation experiments

This paper constructs two classes of models for the quantum correlation experiments used to test the Bell-type inequalities, synchronization models and prism models. Both classes employ deterministic hidden variables, satisfy the causal requirements of physical locality, and yield precisely the quantum mechanical statistics. In the synchronization models, the joint probabilities, for each emission, do not factor in the manner of stochastic independence, showing that such factorizability is not required for locality. In the prism models the observables are not random variables over a common space; hence these models throw into question the entire random variables idiom of the literature. Both classes of models appear to be testable.Work on this paper was supported, in part, by National Science Foundation Grant SES 79-25917.     

A Sandwich-Type Standard Error Estimator of SEM Models with Multivariate Time Series

Structural equation models are increasingly used as a modeling tool for multivariate time series data in the social and behavioral sciences. Standard error estimators of SEM models, originally developed for independent data, require modifications to accommodate the fact that time series data are inherently dependent. In this article, we extend a sandwich-type standard error estimator of independent data to multivariate time series data. One required element of this estimator is the asymptotic covariance matrix of concurrent and lagged correlations among manifest variables, whose closed-form expression has not been presented in the literature. The performance of the adapted sandwich-type standard error estimator is evaluated using a simulation study and further illustrated using an empirical example.     

Regime-Switching Bivariate Dual Change Score Model

Mixture structural equation model with regime switching (MSEM-RS) provides one possible way of representing over-time heterogeneities in dynamic processes by allowing a system to manifest qualitatively or quantitatively distinct change processes conditional on the latent “regime” the system is in at a particular time point. Unlike standard mixture structural equation models such as growth mixture models, MSEM-RS allows individuals to transition between latent classes over time. This class of models, often referred to as regime-switching models in the time series and econometric applications, can be specified as regime-switching mixture structural equation models when the number of repeated measures involved is not large. We illustrate the empirical utility of such models using one special case—a regime-switching bivariate dual change score model in which two growth processes are allowed to manifest regime-dependent coupling relations with one another. The proposed model is illustrated using a set of longitudinal reading and arithmetic performance data from the Early Childhood Longitudinal Study, Kindergarten Class of 1998–99 study (ECLS-K; U.S. Department of Education, National Center for Education Statistics, 2010).     

Analysis of Variance Models with Stochastic Group Weights

The analysis of variance (ANOVA) is still one of the most widely used statistical methods in the social sciences. This article is about stochastic group weights in ANOVA models – a neglected aspect in the literature. Stochastic group weights are present whenever the experimenter does not determine the exact group sizes before conducting the experiment. We show that classic ANOVA tests based on estimated marginal means can have an inflated type I error rate when stochastic group weights are not taken into account, even in randomized experiments. We propose two new ways to incorporate stochastic group weights in the tests of average effects – one based on the general linear model and one based on multigroup structural equation models (SEMs). We show in simulation studies that our methods have nominal type I error rates in experiments with stochastic group weights while classic approaches show an inflated type I error rate. The SEM approach can additionally deal with heteroscedastic residual variances and latent variables. An easy-to-use software package with graphical user interface is provided.     

Some contributions to efficient statistics in structural models: Specification and estimation of moment structures

Current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments (e.g., means and covariances), and to distribution theory based on multivariate normality. In psychometrics the multinormality assumption is often incorrect, so that statistical tests on parameters, or model goodness of fit, will frequently be incorrect as well. It is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary. Structural representations are developed for generalizations of the Bentler-Weeks, Jöreskog-Keesling-Wiley, and factor analytic models. Some asymptotically distribution-free efficient estimators for such arbitrary structural models are developed. Limited information estimators are obtained as well. The special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail. The argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory, which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds. Many open research areas are described.     

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.     

A general structural equation model with dichotomous,ordered categorical,and continuous latent variable indicators

A structural equation model is proposed with a generalized measurement part, allowing for dichotomous and ordered categorical variables (indicators) in addition to continuous ones. A computationally feasible three-stage estimator is proposed for any combination of observed variable types. This approach provides large-sample chi-square tests of fit and standard errors of estimates for situations not previously covered. Two multiple-indicator modeling examples are given. One is a simultaneous analysis of two groups with a structural equation model underlying skewed Likert variables. The second is a longitudinal model with a structural model for multivariate probit regressions.This research was supported by Grant No. 81-IJ-CX-0015 from the National Institute of Justice, by Grant No. DA 01070 from the U.S. Public Health Service, and by Grant No. SES-8312583 from the National Science Foundation. I thank Julie Honig for drawing the figures. Requests for reprints should be sent to Bengt Muthén, Graduate School of Education, University of California, Los Angeles, California 90024.     

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。     

A multivariate reduced-rank growth curve model with unbalanced data

A multivariate reduced-rank growth curve model is proposed that extends the univariate reducedrank growth curve model to the multivariate case, in which several response variables are measured over multiple time points. The proposed model allows us to investigate the relationships among a number of response variables in a more parsimonious way than the traditional growth curve model. In addition, the method is more flexible than the traditional growth curve model. For example, response variables do not have to be measured at the same time points, nor the same number of time points. It is also possible to apply various kinds of basis function matrices with different ranks across response variables. It is not necessary to specify an entire set of basis functions in advance. Examples are given for illustration.The work reported in this paper was supported by Grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the second author. We thank Jennifer Stephan for her helpful comments on an earlier version of this paper. We also thank Patrick Curran and Terry Duncan for kindly letting us use the NLSY and substance use data, respectively. The substance use data were provided by Grant DA09548 from the National Institute on Drug Abuse.     

Four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences

A class of four simultaneous component models for the exploratory analysis of multivariate time series collected from more than one subject simultaneously is discussed. In each of the models, the multivariate time series of each subject is decomposed into a few series of component scores and a loading matrix. The component scores series reveal the latent data structure in the course of time. The interpretation of the components is based on the loading matrix. The simultaneous component models model not only intraindividual variability, but interindividual variability as well. The four models can be ordered hierarchically from weakly to severely constrained, thus allowing for big to small interindividual differences in the model. The use of the models is illustrated by an empirical example.This research has been made possible by funding from the Netherlands Organization of Scientific Research (NWO) to the first author. The authors are obliged to Tom A.B. Snijders, Jos M.F. ten Berge and three anonymous reviewers for comments on an earlier version of this paper, and to Kim Shifren for providing us with her data set, which was collected at Syracuse University.     

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.     

Cointegration methodology for psychological researchers: An introduction to the analysis of dynamic process systems

Longitudinal data analysis focused on internal characteristics of a single time series has attracted increasing interest among psychologists. The systemic psychological perspective suggests, however, that many long-term phenomena are mutually interconnected, forming a dynamic system. Hence, only multivariate methods can handle such human dynamics appropriately. Unlike the majority of time series methodologies, the cointegration approach allows interdependencies of integrated (i.e., extremely unstable) processes to be modelled. This advantage results from the fact that cointegrated series are connected by stationary long-run equilibrium relationships. Vector error-correction models are frequently used representations of cointegrated systems. They capture both this equilibrium and compensation mechanisms in the case of short-term deviations due to developmental changes. Thus, the past disequilibrium serves as explanatory variable in the dynamic behaviour of current variables. Employing empirical data from cognitive psychology, psychosomatics, and marital interaction research, this paper describes how to apply cointegration methods to dynamic process systems and how to interpret the parameters under investigation from a psychological perspective.     

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).     

Dynamic GSCA (Generalized Structured Component Analysis) with Applications to the Analysis of Effective Connectivity in Functional Neuroimaging Data

We propose a new method of structural equation modeling (SEM) for longitudinal and time series data, named Dynamic GSCA (Generalized Structured Component Analysis). The proposed method extends the original GSCA by incorporating a multivariate autoregressive model to account for the dynamic nature of data taken over time. Dynamic GSCA also incorporates direct and modulating effects of input variables on specific latent variables and on connections between latent variables, respectively. An alternating least square (ALS) algorithm is developed for parameter estimation. An improved bootstrap method called a modified moving block bootstrap method is used to assess reliability of parameter estimates, which deals with time dependence between consecutive observations effectively. We analyze synthetic and real data to illustrate the feasibility of the proposed method.     

A Unified Approach to Power Calculation and Sample Size Determination for Random Regression Models

The underlying statistical models for multiple regression analysis are typically attributed to two types of modeling: fixed and random. The procedures for calculating power and sample size under the fixed regression models are well known. However, the literature on random regression models is limited and has been confined to the case of all variables having a joint multivariate normal distribution. This paper presents a unified approach to determining power and sample size for random regression models with arbitrary distribution configurations for explanatory variables. Numerical examples are provided to illustrate the usefulness of the proposed method and Monte Carlo simulation studies are also conducted to assess the accuracy. The results show that the proposed method performs well for various model specifications and explanatory variable distributions. The author would like to thank the editor, the associate editor, and the referees for drawing attention to pertinent references that led to improved presentation. This research was partially supported by National Science Council grant NSC-94-2118-M-009-004.     

Growth curve models for indistinguishable dyads using multilevel modeling and structural equation modeling: the case of adolescent twins' conflict with their mothers

Growth modeling is a useful tool for studying change over time, and it is becoming increasingly popular with developmental researchers. There is a considerable methodological literature surrounding growth modeling for individuals; however, far less attention has been focused on growth models for pairs of related individuals (i.e., dyads). In this article, the authors consider dyadic growth models for those cases where there are no relevant variables that can empirically distinguish between dyad members (e.g., same-sex twins or best friends). The authors describe how researchers can estimate growth models for indistinguishable dyads using both multilevel modeling and structural equation modeling. Although both approaches can be used to estimate the same underlying models, the authors focus on practical similarities and differences between the two approaches. They illustrate modeling issues using an overtime study of adolescent twins' conflict with their mothers, a substantively important topic given the enduring interest in parent-child relationships during adolescence.     

Studying Multivariate Change Using Multilevel Models and Latent Curve Models

In longitudinal research investigators often measure multiple variables at multiple points in time and are interested in investigating individual differences in patterns of change on those variables. In the vast majority of applications, researchers focus on studying change in one variable at a time. In this article we consider methods for studying relations1.1ips between patterns of change on different variables. We show how the multilevel modeling framework, which is often used to study univariate change, can be extended to the multivariate case to yield estimates of covariances of parameters representing aspects of change on different variables. We illustrate this approach using data from a study of physiological response to marital conflict in older married couples, showing a substantial correlation between rate of linear change on different stress-related hormones during conflict. We also consider how similar issues can be studied using extensions of latent curve models to the multivariate case, and we show how such models are related to multivariate multilevel models.