Structural Equation Modeling of Multivariate Time Series

The covariance structure of a vector autoregressive process with moving average residuals (VARMA) is derived. It differs from other available expressions for the covariance function of a stationary VARMA process and is compatible with current structural equation methodology. Structural equation modeling programs, such as LISREL, may therefore be employed to fit the model.

Particular attention is given to assumptions concerning the process before the first observation. An application to a repeated time series is used to demonstrate the effect of these assumptions on the structure of the reproduced covariance matrix.     

SEM Based CARMA Time Series Modeling for Arbitrary N

This article explains in detail the state space specification and estimation of first and higher-order autoregressive moving-average models in continuous time (CARMA) in an extended structural equation modeling (SEM) context for N = 1 as well as N > 1. To illustrate the approach, simulations will be presented in which a single panel model (T = 41 time points) is estimated for a sample of N = 1,000 individuals as well as for samples of N = 100 and N = 50 individuals, followed by estimating 100 separate models for each of the one-hundred N = 1 cases in the N = 100 sample. Furthermore, we will demonstrate how to test the difference between the full panel model and each N = 1 model by means of a subject-group-reproducibility test. Finally, the proposed analyses will be applied in an empirical example, in which the relationships between mood at work and mood at home are studied in a sample of N = 55 women. All analyses are carried out by ctsem, an R-package for continuous time modeling, interfacing to OpenMx.     

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.     

A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models

Group-level variance estimates of zero often arise when fitting multilevel or hierarchical linear models, especially when the number of groups is small. For situations where zero variances are implausible a priori, we propose a maximum penalized likelihood approach to avoid such boundary estimates. This approach is equivalent to estimating variance parameters by their posterior mode, given a weakly informative prior distribution. By choosing the penalty from the log-gamma family with shape parameter greater than 1, we ensure that the estimated variance will be positive. We suggest a default log-gamma(2,λ) penalty with λ→0, which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero, thus remaining consistent with the data while being nondegenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior. Our default method provides better estimates of model parameters and standard errors than the maximum likelihood or the restricted maximum likelihood estimators. The log-gamma family can also be used to convey substantive prior information. In either case—pure penalization or prior information—our recommended procedure gives nondegenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.     

Standard Errors of Multivariate Range-Corrected Validities

The multivariate rather than the univariate range correction is used for estimating unrestricted applicant population validities in many military test validity studies but not uniformly. A Monte Carlo approach compared the standard errors of range-corrected validities under various experimental conditions adhering to the assumptions underlying correction accuracy. The multivariate corrected validities had smaller standard errors than both the univariate-corrected validities and the unrestricted validities. We conclude that using the univariate correction could fail to reveal the most valid selection instrument and that the multivariate correction should be used when scores for relevant predictors are available for the unrestricted population.     

A Polychoric Instrumental Variable (PIV) Estimator for Structural Equation Models with Categorical Variables

This paper presents a new polychoric instrumental variable (PIV) estimator to use in structural equation models (SEMs) with categorical observed variables. The PIV estimator is a generalization of Bollen’s (Psychometrika 61:109–121, 1996) 2SLS/IV estimator for continuous variables to categorical endogenous variables. We derive the PIV estimator and its asymptotic standard errors for the regression coefficients in the latent variable and measurement models. We also provide an estimator of the variance and covariance parameters of the model, asymptotic standard errors for these, and test statistics of overall model fit. We examine this estimator via an empirical study and also via a small simulation study. Our results illustrate the greater robustness of the PIV estimator to structural misspecifications than the system-wide estimators that are commonly applied in SEMs. Kenneth Bollen gratefully acknowledges support from NSF SES 0617276, NIDA 1-RO1-DA13148-01, and DA013148-05A2. Albert Maydeu-Olivares was supported by the Department of Universities, Research and Information Society (DURSI) of the Catalan Government, and by grant BSO2003-08507 from the Spanish Ministry of Science and Technology. We thank Sharon Christ, John Hipp, and Shawn Bauldry for research assistance. The comments of the members of the Carolina Structural Equation Modeling (CSEM) group are greatly appreciated. An earlier version of this paper under a different title was presented by K. Bollen at the Psychometric Society Meetings, June, 2002, Chapel Hill, North Carolina.     

Bayesian Analysis of Multivariate Probit Models with Surrogate Outcome Data

A new class of parametric models that generalize the multivariate probit model and the errors-in-variables model is developed to model and analyze ordinal data. A general model structure is assumed to accommodate the information that is obtained via surrogate variables. A hybrid Gibbs sampler is developed to estimate the model parameters. To obtain a rapidly converged algorithm, the parameter expansion technique is applied to the correlation structure of the multivariate probit models. The proposed model and method of analysis are demonstrated with real data examples and simulation studies.     

Autoregressive Generalized Linear Mixed Effect Models with Crossed Random Effects: An Application to Intensive Binary Time Series Eye-Tracking Data

As a method to ascertain person and item effects in psycholinguistics, a generalized linear mixed effect model (GLMM) with crossed random effects has met limitations in handing serial dependence across persons and items. This paper presents an autoregressive GLMM with crossed random effects that accounts for variability in lag effects across persons and items. The model is shown to be applicable to intensive binary time series eye-tracking data when researchers are interested in detecting experimental condition effects while controlling for previous responses. In addition, a simulation study shows that ignoring lag effects can lead to biased estimates and underestimated standard errors for the experimental condition effects.     

Extending Standard Models of ZFC to Models of Nonstandard Set Theories

We study those models of ZFCwhich are embeddable, as the class of all standard sets, in a model of internal set theory >ISTor models of some other nonstandard set theories. This revised version was published online in June 2006 with corrections to the Cover Date.     

Time Series Analysis in Historiometry: A Comment on Simonton

Time series analysis (TSA) is one of a number of new methods of data analysis appropriate for longitudinal data. Simonton (1998) applied TSA to an analysis of the causal relationship between two types of stress and both the physical and mental health of George III. This innovative application demonstrates both the strengths and weaknesses of time series analysis. Time series is applicable to a unique class of problems, can use information about temporal ordering to make statements about causation, and focuses on patterns of change over time, all strengths of the Simonton study. Time series analysis also suffers from a number of weaknesses, including problems with generalization from a single study, difficulty in obtaining appropriate measures, and problems with accurately identifying the correct model to represent the data. While careful attempts are made to minimize these problems, each is present in the Simonton study, although sometimes in a subtle manner. Changes in how the data could be gathered are suggested that might help to solve some of these problems in future studies. Finally, the advantages and disadvantages of employing alternative methods for analyzing multivariate time series data, including dynamic factor analysis, are discussed.     

Spectral Decompositions of Multiple Time Series: A Bayesian Non-parametric Approach

We consider spectral decompositions of multiple time series that arise in studies where the interest lies in assessing the influence of two or more factors. We write the spectral density of each time series as a sum of the spectral densities associated to the different levels of the factors. We then use Whittle’s approximation to the likelihood function and follow a Bayesian non-parametric approach to obtain posterior inference on the spectral densities based on Bernstein–Dirichlet prior distributions. The prior is strategically important as it carries identifiability conditions for the models and allows us to quantify our degree of confidence in such conditions. A Markov chain Monte Carlo (MCMC) algorithm for posterior inference within this class of frequency-domain models is presented. We illustrate the approach by analyzing simulated and real data via spectral one-way and two-way models. In particular, we present an analysis of functional magnetic resonance imaging (fMRI) brain responses measured in individuals who participated in a designed experiment to study pain perception in humans.     

A Defense of Backwards in Time Causation Models in Quantum Mechanics

Synthese - This paper offers a defense of backwards in time causation models in quantum mechanics. Particular attention is given to Cramer's transactional account, which is shown to have the...     

Theories of Truth Which Have No Standard Models

This papers deals with the class of axiomatic theories of truth for semantically closed languages, where the theories do not allow for standard models; i.e., those theories cannot be interpreted as referring to the natural number codes of sentences only (for an overview of axiomatic theories of truth in general, see Halbach[6]). We are going to give new proofs for two well-known results in this area, and we also prove a new theorem on the nonstandardness of a certain theory of truth. The results indicate that the proof strategies for all the theorems on the nonstandardness of such theories are "essentially" of the same kind of structure.     

Stochastic Dynamic Models of Response Time and Accuracy: A Foundational Primer

A large class of statistical decision models for performance in simple information processing tasks can be described by linear, first-order, stochastic differential equations (SDEs), whose solutions are diffusion processes. In such models, the first passage time for the diffusion process through a response criterion determines the time at which an observer makes a decision about the identity of a stimulus. Because the assumptions of many cognitive models lead to SDEs that are time inhomogeneous, classical methods for solving such first passage time problems are usually inapplicable. In contrast, recent integral equation methods often yield solutions to both the one-sided and the two-sided first passage time problems, even in the presence of time inhomogeneity. These methods, which are of particular relevance to the cognitive modeler, are described in detail, together with illustrative applications. Copyright 2000 Academic Press.     

A Latent Growth Curve Modeling Approach to Pooled Interrupted Time Series Analyses

This paper presents a latent variable approach for the estimation of treatment effects within a pooled interrupted time series (ITS) design. Although considered quasi-experimental, the ITS design has been noted as representing one of the strongest alternatives to the randomized experiment, making it highly appropriate for use in documenting the presence of effects that might warrant further evaluation in a large-scale randomized study. Results suggest that the latent variable growth modeling (LGM) is capable of detecting simultaneous differences in both level and slope, and provides tests of significance for these two necessary indicators of an ITS intervention effect. As shown in the analyses, the LGM framework provides a comprehensive and flexible approach to research design and data analysis, making available to a wide audience of researchers an analytical framework for a variety of analyses of growth and developmental processes.