Multiphase mixed-effects models for repeated measures data

Behavior that develops in phases may exhibit distinctively different rates of change in one time period than in others. In this article, a mixed-effects model for a response that displays identifiable regimes is reviewed. An interesting component of the model is the change point. In substantive terms, the change point is the time when development switches from one phase to another. In a mixed-effects model, the change point can be a random coefficient. This possibility allows individuals to make the transition from one phase to another at different ages or after different lengths of time in treatment. Two examples are reviewed in detail, both of which can be estimated with software that is widely available.     

A mixed-effects expectancy-valence model for the Iowa gambling task

The Iowa gambling task (IGT; Bechara, Damasio, Damasio, & Anderson, 1994) was developed to simulate real-life decision making under uncertainty. The task has been widely used to examine possible neurocognitive deficits in normal and clinical populations. Busemeyer and Stout (2002) proposed the expectancy-valence (EV) model to explicitly account for individual participants’ repeated choices in the IGT. Parameters of the EV model presumably measure different psychological processes that underlie performance on the task, and their values may be used to differentiate individuals across different populations. In the present article, the EV model is extended to include both fixed effects and subject-specific random effects. The mixed-effects EV model fits the nested structure of observations in the IGT naturally and provides a unified procedure for parameter estimation and comparisons among groups of populations. We illustrate the utility of the mixed-effects approach with an analysis of gender differences using a real data set. A simulation study was conducted to verify the advantages of this approach.     

A three-stage estimation procedure for structural equation models with polytomous variables

This paper is concerned with the analysis of structural equation models with polytomous variables. A computationally efficient three-stage estimator of the thresholds and the covariance structure parameters, based on partition maximum likelihood and generalized least squares estimation, is proposed. An example is presented to illustrate the method.This research was supported in part by a research grant DA01070 from the U.S. Public Health Service. The production assistance of Julie Speckart is gratefully acknowledged.     

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.     

Fitting correlated residual error structures in nonlinear mixed-effects models using SAS PROC NLMIXED

Nonlinear mixed-effects (NLME) models remain popular among practitioners for analyzing continuous repeated measures data taken on each of a number of individuals when interest centers on characterizing individual-specific change. Within this framework, variation and correlation among the repeated measurements may be partitioned into interindividual variation and intraindividual variation components. The covariance structure of the residuals are, in many applications, consigned to be independent with homogeneous variances, $ {\sigma}^2{\mathbf{I}}_{n_i} $ , not because it is believed that intraindividual variation adheres to this structure, but because many software programs that estimate parameters of such models are not well-equipped to handle other, possibly more realistic, patterns. In this article, we describe how the programmatic environment within SAS may be utilized to model residual structures for serial correlation and variance heterogeneity. An empirical example is used to illustrate the capabilities of the module.     

A comparison of linear and mixture models for discriminant analysis under nonnormality

Methods for discriminant analysis were compared with respect to classification accuracy under nonnormality through Monte Carlo simulation. The methods compared were linear discriminant analyses based both on raw scores and on ranks; linear logistic discrimination; and mixture discriminant analysis. Linear discriminant analysis and linear logistic discrimination were suboptimal in a number of scenarios with skewed predictors. Linear discriminant analysis based on ranks yielded the highest rates of classification accuracy in only a limited number of situations and did not produce a practically important advantage over competing methods. Mixture discriminant analysis, with a relatively small number of components in each group, attained relatively high rates of classification accuracy and was most useful for conditions in which skewed predictors had relatively small values of kurtosis.     

An elicitation method for multiple linear regression models

This paper describes a method of quantifying subjective opinion about a normal linear regression model. Opinion about the regression coefficients and experimental error is elicited and modeled by a multivariate probability distribution (a Bayesian conjugate prior distribution). The distribution model is richly parameterized and various assessment tasks are used to estimate its parameters. These tasks include the revision of opinion in the light of hypothetical data, the assessment of credible intervals, and a task commonly performed in cue-weighting experiments. A new assessment task is also introduced. In addition, implementation of the method in an interactive computer program is described and the method is illustrated with a practical example.     

Meta-analysis using linear mixed models

Psychologists often use special computer programs to perform meta-analysis. Until recently, this had been necessary because standard statistical packages did not provide procedures for such analysis. This paper introduces linear mixed models as a framework for meta-analysis in psychological research, using a popular general purpose statisticalpackage, SAS. The approach is illustrated with three examples, usingsas peoc mixed.     

Equivalent linear logistic test models

This paper is about the Linear Logistic Test Model (LLTM). We demonstrate that there are infinitely many equivalent ways to specify a model. An implication is that there may well be many ways to change the specification of a given LLTM and achieve the same improvement in model fit. To illustrate this phenomenon, we analyze a real data set using a Lagrange multiplier test for the specification of the model. This Lagrange multiplier test is similar to the modification index used in structural equation modeling.     

Statistical inference in generalized linear mixed models: A review

We present a review of statistical inference in generalized linear mixed models (GLMMs). GLMMs are an extension of generalized linear models and are suitable for the analysis of non‐normal data with a clustered structure. A GLMM contains parameters common to all clusters (fixed regression effects and variance components) and cluster‐specific parameters. The latter parameters are assumed to be randomly drawn from a population distribution. The parameters of this population distribution (the variance components) have to be estimated together with the fixed effects. We focus on the case in which the cluster‐specific parameters are normally distributed. The cluster‐specific effects are integrated out of the likelihood so that the fixed effects and variance components can be estimated. Unfortunately, the integral over the cluster‐specific effects is intractable for most GLMMs with a normal mixing distribution. Within a classical statistical framework, we distinguish between two broad classes of methods to handle this intractable integral: methods that rely on a numerical approximation to the integral and methods that use an analytical approximation to the integrand. Finally, we present an overview of available methods for testing hypotheses about the parameters of GLMMs.     

Behavioral evidence for a differential modulation of semantic processing and lexical production by aging: a full linear mixed-effects modeling approach

The effect of normal aging on lexical production and semantic processing was evaluated in 72 healthy participants. Four tasks were used, picture naming (PN), picture categorization (PC), numerical judgment (NJ), and color judgment (CJ). The dependence of reaction time (RT) and correct responses with age was accounted by mixed-effects models. Participants underwent neuropsychological testing for verbal, executive, and memory functions. The RTs increase significantly with age for all tasks. After parceling out the non-specific cognitive decline, as reflected by the NJ task, the RT for the PN task decreases with age. Behavioral data were interpreted in relation with neuropsychological scores. Our results suggest that (a) naming becomes more automatic and semantic processing slightly more difficult with age, and (b) a non-specific general slowdown of cognitive processing occurs with age. Lexical production remained unaltered, based on compensatory automatic processes. This study also suggests a possible slowdown of semantic processing, even in normal aging.     

A comparison of statistical selection strategies for univariate and bivariate log‐linear models

In this study, eight statistical selection strategies were evaluated for selecting the parameterizations of log‐linear models used to model the distributions of psychometric tests. The selection strategies included significance tests based on four chi‐squared statistics (likelihood ratio, Pearson, Freeman–Tukey, and Cressie–Read) and four additional strategies (Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), and a measure attributed to Goodman). The strategies were evaluated in simulations for different log‐linear models of univariate and bivariate test‐score distributions and two sample sizes. Results showed that all eight selection strategies were most accurate for the largest sample size considered. For univariate distributions, the AIC selection strategy was especially accurate for selecting the correct parameterization of a complex log‐linear model and the likelihood ratio chi‐squared selection strategy was the most accurate strategy for selecting the correct parameterization of a relatively simple log‐linear model. For bivariate distributions, the likelihood ratio chi‐squared, Freeman–Tukey chi‐squared, BIC, and CAIC selection strategies had similarly high selection accuracies.     

Nonlinear submodels of orthogonal linear models

A model containing linear and nonlinear parameters (e. g., a spatial multidimensional scaling model) is viewed as a linear model with free and constrained parameters. Since the rank deficiency of the design matrix for the linear model determines the number of side conditions needed to identify its parameters, the design matrix acts as a guide in identifying the parameters of the nonlinear model. Moreover, if the design matrix and the uniqueness conditions constitute anorthogonal linear model, then the associated error sum of squares may be expressed in a form which separates the free and constrained parameters. This immediately provides least squares estimates of the free parameters, while simplifying the least squares problem for those which are constrained. When the least squares estimates for a nonlinear model are obtained in this way,i.e. by conceptualizing it as a submodel, the final error sum of squares for the nonlinear model will be arestricted minimum whenever the side conditions of the model become real restrictions upon its submodel. In this case the design matrix for the embracing orthogonal model serves as a guide in introducing parameters into the nonlinear model as well as in identifying these parameters. The method of overwriting a nonlinear model with an orthogonal linear model is illustrated with two different spatial analyses of a three-way preference table.     

Fitting linear models to ordinal responses

The distribution of an ordinal response can be modelled as a grouping of an underlying quantitative variable whose mean is a linear function of explanatory variables. Possible distributional assumptions about the underlying quantitative response are compared. An iteratively reweighted least squares algorithm for parameter estimation in these models is described in detail and variances and tests of hypotheses are given. Two data sets are analysed to illustrate the methods.     

Relations between linear models: Implications for the lens model

Brunswik's lens model has been widely used in the modeling and analysis of judgment tasks and the lens model equation has been an important part of most analyses. In such analyses, researchers often have based arguments and interpretations upon the magnitude of various components of the lens model equation. One component is G, an index of agreement between a linear model of the subject's judgments and a linear model of the task environment. This paper addresses the index G and shows that while in principle −1 G 1, the distribution of G is often skewed toward high values so that caution must be used in the interpretation of its magnitude. Moreover, the results reported here apply to the relation between linear models in general.     

A coding procedure for teachers

An observational technique for reliably estimating the per cent of time a student engages in appropriate and inappropriate classroom behavior is described. The regular classroom teacher can utilize the procedure without deviating from regular routine, and the obtained data can serve as a basis for dispensing token reinforcement.     

A double bootstrap method to analyze linear models with autoregressive error terms

A new method for the analysis of linear models that have autoregressive errors is proposed. The approach is not only relevant in the behavioral sciences for analyzing small-sample time-series intervention models, but it is also appropriate for a wide class of small-sample linear model problems in which there is interest in inferential statements regarding all regression parameters and autoregressive parameters in the model. The methodology includes a double application of bootstrap procedures. The 1st application is used to obtain bias-adjusted estimates of the autoregressive parameters. The 2nd application is used to estimate the standard errors of the parameter estimates. Theoretical and Monte Carlo results are presented to demonstrate asymptotic and small-sample properties of the method; examples that illustrate advantages of the new approach over established time-series methods are described.     

A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling

Meta-analysis and structural equation modeling (SEM) are two important statistical methods in the behavioral, social, and medical sciences. They are generally treated as two unrelated topics in the literature. The present article proposes a model to integrate fixed-, random-, and mixed-effects meta-analyses into the SEM framework. By applying an appropriate transformation on the data, studies in a meta-analysis can be analyzed as subjects in a structural equation model. This article also highlights some practical benefits of using the SEM approach to conduct a meta-analysis. Specifically, the SEM-based meta-analysis can be used to handle missing covariates, to quantify the heterogeneity of effect sizes, and to address the heterogeneity of effect sizes with mixture models. Examples are used to illustrate the equivalence between the conventional meta-analysis and the SEM-based meta-analysis. Future directions on and issues related to the SEM-based meta-analysis are discussed.