Modeling Individual Damped Linear Oscillator Processes with Differential Equations: Using Surrogate Data Analysis to Estimate the Smoothing Parameter

Among the many methods available for modeling intraindividual time series, differential equation modeling has several advantages that make it promising for applications to psychological data. One interesting differential equation model is that of the damped linear oscillator (DLO), which can be used to model variables that have a tendency to fluctuate around some typical, or equilibrium, value. Methods available for fitting the damped linear oscillator model using differential equation modeling can yield biased parameter estimates when applied to univariate time series. The degree of this bias depends on a smoothing–like parameter, which balances the need for increasing smoothing to minimize error variance but not smoothing so much as to obscure change of interest. This article explores a technique that uses surrogate data analysis to select such a parameter, thereby producing approximately unbiased parameter estimates. Furthermore the smoothing parameter, which is usually researcher-selected, is produced in an automated manner so as to reduce the experience required by researchers to apply these methods. Focus is placed on the damped linear model; however, similar issues are expected with other differential equation models and other techniques in which parameter estimates depend on a smoothing parameter. An example using affect data from the Notre Dame Longitudinal Study of Aging (2004) is presented, which contrasts the use of a single smoothing parameter for all individuals versus use of a smoothing parameter for each individual.     

Representing Sudden Shifts in Intensive Dyadic Interaction Data Using Differential Equation Models with Regime Switching

A growing number of social scientists have turned to differential equations as a tool for capturing the dynamic interdependence among a system of variables. Current tools for fitting differential equation models do not provide a straightforward mechanism for diagnosing evidence for qualitative shifts in dynamics, nor do they provide ways of identifying the timing and possible determinants of such shifts. In this paper, we discuss regime-switching differential equation models, a novel modeling framework for representing abrupt changes in a system of differential equation models. Estimation was performed by combining the Kim filter (Kim and Nelson State-space models with regime switching: classical and Gibbs-sampling approaches with applications, MIT Press, Cambridge, 1999) and a numerical differential equation solver that can handle both ordinary and stochastic differential equations. The proposed approach was motivated by the need to represent discrete shifts in the movement dynamics of \(n= 29\) mother–infant dyads during the Strange Situation Procedure (SSP), a behavioral assessment where the infant is separated from and reunited with the mother twice. We illustrate the utility of a novel regime-switching differential equation model in representing children’s tendency to exhibit shifts between the goal of staying close to their mothers and intermittent interest in moving away from their mothers to explore the room during the SSP. Results from empirical model fitting were supplemented with a Monte Carlo simulation study to evaluate the use of information criterion measures to diagnose sudden shifts in dynamics.     

Latent State-Trait Modeling of Satisfaction with Life Scale: An Item-Level Analysis Using Dutch Panel Data

Journal of Happiness Studies - This study sought to examine the stability and change of the 5 items of the Satisfaction With Life Scale over several years. The multi-trait-multi-state model was...     

Modeling Incomplete Longitudinal Substance Use Data Using Latent Variable Growth Curve Methodology

Longitudinal data sets typically suffer from attrition and other forms of missing data. When this common problem occurs, several researchers have demonstrated that correct maximum likelihood estimation with missing data can be obtained under mild assumptions concerning the missing data mechanism. With reasonable substantive theory, a mixture of cross-sectional and longitudinal methods developed within multiple-group structural equation modeling can provide a strong basis for inference about developmental change. Using an approach to the analysis of missing data, the present study investigated developmental trends in adolescent (N = 759) alcohol, marijuana, and cigarette use across a 5-year period using multiple-group latent growth modeling. An associative model revealed that common developmental trends existed for all three substances. Age and gender were included in the model as predictors of initial status and developmental change. Findings discuss the utility of latent variable structural equation modeling techniques and missing data approaches in the study of developmental change.     

The SIMCLAS Model: Simultaneous Analysis of Coupled Binary Data Matrices with Noise Heterogeneity Between and Within Data Blocks

In many research domains different pieces of information are collected regarding the same set of objects. Each piece of information constitutes a data block, and all these (coupled) blocks have the object mode in common. When analyzing such data, an important aim is to obtain an overall picture of the structure underlying the whole set of coupled data blocks. A further challenge consists of accounting for the differences in information value that exist between and within (i.e., between the objects of a single block) data blocks. To tackle these issues, analysis techniques may be useful in which all available pieces of information are integrated and in which at the same time noise heterogeneity is taken into account. For the case of binary coupled data, however, only methods exist that go for a simultaneous analysis of all data blocks but that do not account for noise heterogeneity. Therefore, in this paper, the SIMCLAS model, being a Hierarchical Classes model for the simultaneous analysis of coupled binary two-way matrices, is presented. In this model, noise heterogeneity between and within the data blocks is accounted for by downweighting entries from noisy blocks/objects within a block. In a simulation study it is shown that (1) the SIMCLAS technique recovers the underlying structure of coupled data to a very large extent, and (2) the SIMCLAS technique outperforms a Hierarchical Classes technique in which all entries contribute equally to the analysis (i.e., noise homogeneity within and between blocks). The latter is also demonstrated in an application of both techniques to empirical data on categorization of semantic concepts.     

Robust Structural Equation Modeling with Missing Data and Auxiliary Variables

The paper develops a two-stage robust procedure for structural equation modeling (SEM) and an R package rsem to facilitate the use of the procedure by applied researchers. In the first stage, M-estimates of the saturated mean vector and covariance matrix of all variables are obtained. Those corresponding to the substantive variables are then fitted to the structural model in the second stage. A sandwich-type covariance matrix is used to obtain consistent standard errors (SE) of the structural parameter estimates. Rescaled, adjusted as well as corrected and F-statistics are proposed for overall model evaluation. Using R and EQS, the R package rsem combines the two stages and generates all the test statistics and consistent SEs. Following the robust analysis, multiple model fit indices and standardized solutions are provided in the corresponding output of EQS. An example with open/closed book examination data illustrates the proper use of the package. The method is further applied to the analysis of a data set from the National Longitudinal Survey of Youth 1997 cohort, and results show that the developed procedure not only gives a better endorsement of the substantive models but also yields estimates with uniformly smaller standard errors than the normal-distribution-based maximum likelihood.     

An Idiographic Approach to Estimating Models of Dyadic Interactions with Differential Equations

We present an idiographic approach to modeling dyadic interactions using differential equations. Using data representing daily affect ratings from romantic relationships, we examined several models conceptualizing different types of dyadic interactions. We fitted each model to each of the dyads and the resulting AICc values were used to classify the most likely configuration of interaction for each dyad. Additionally, the AICc from the different models were used in parameter averaging across models. Averaged parameters were used in models involving predictors of relationship dynamics, as indexed by these parameters, as well as models wherein the parameters predicted distal outcomes of the dyads such as relationship satisfaction and status. Results indicated that, within our sample, the most likely interaction style was that of independence, without evidence of emotional interrelations between the two individuals in the couple. Attachment-related avoidance and anxiety showed significant relations with model parameters, such that ideal levels of affect for males were negatively influenced by higher levels of avoidance from their partner while their own levels of anxiety had positive effects on their levels of dyadic coregulation. For females coregulation was negatively influenced by both time in the relationship and their partner’s level of avoidance. Analysis involving distal outcomes showed modest influences from the individual’s level of ideal affect.     

Modeling and Testing Differential Item Functioning in Unidimensional Binary Item Response Models with a Single Continuous Covariate: A Functional Data Analysis Approach

Differential item functioning (DIF), referring to between-group variation in item characteristics above and beyond the group-level disparity in the latent variable of interest, has long been regarded as an important item-level diagnostic. The presence of DIF impairs the fit of the single-group item response model being used, and calls for either model modification or item deletion in practice, depending on the mode of analysis. Methods for testing DIF with continuous covariates, rather than categorical grouping variables, have been developed; however, they are restrictive in parametric forms, and thus are not sufficiently flexible to describe complex interaction among latent variables and covariates. In the current study, we formulate the probability of endorsing each test item as a general bivariate function of a unidimensional latent trait and a single covariate, which is then approximated by a two-dimensional smoothing spline. The accuracy and precision of the proposed procedure is evaluated via Monte Carlo simulations. If anchor items are available, we proposed an extended model that simultaneously estimates item characteristic functions (ICFs) for anchor items, ICFs conditional on the covariate for non-anchor items, and the latent variable density conditional on the covariate—all using regression splines. A permutation DIF test is developed, and its performance is compared to the conventional parametric approach in a simulation study. We also illustrate the proposed semiparametric DIF testing procedure with an empirical example.     

Modeling fMRI Data: Challenges and Opportunities

We offer an introduction to the five papers that make up this special section. These papers deal with a range of the methodological challenges that face researchers analyzing fMRI data—the spatial, multilevel, and longitudinal nature of the data, the sources of noise, and so on. The papers all provide analyses of data collected by a multi-site consortium, the Function Biomedical Informatics Research Network. Due to the sheer volume of data, univariate procedures are often applied, which leads to a multiple comparisons problem (since the data are necessarily multivariate). The papers in this section include interesting applications, such as a state-space model applied to these data, and conclude with a reflection on basic measurement problems in fMRI. All in all, they provide a good overview of the challenges that fMRI data present to the standard psychometric toolbox, but also to the opportunities they offer for new psychometric modeling.     

Finite Mixtures for Simultaneously Modeling Differential Effects and Nonnormal Distributions

Regression mixture models have been increasingly applied in the social and behavioral sciences as a method for identifying differential effects of predictors on outcomes. Although the typical specification of this approach is sensitive to violations of distributional assumptions, alternative methods for capturing the number of differential effects have been shown to be robust. Yet, there is still a need to better describe differential effects that exist when using regression mixture models. This study tests a new approach that uses sets of classes (called differential effects sets) to simultaneously model differential effects and account for nonnormal error distributions. Monte Carlo simulations are used to examine the performance of the approach. The number of classes needed to represent departures from normality is shown to be dependent on the degree of skew. The use of differential effects sets reduced bias in parameter estimates. Applied analyses demonstrated the implementation of the approach for describing differential effects of parental health problems on adolescent body mass index using differential effects sets approach. Findings support the usefulness of the approach, which overcomes the limitations of previous approaches for handling nonnormal errors.     

Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes

We examine emotion self-regulation and coregulation in romantic couples using daily self-reports of positive and negative affect. We fit these data using a damped linear oscillator model specified as a latent differential equation to investigate affect dynamics at the individual level and coupled influences for the 2 partners in each couple. Results indicate an absence of damping of relationship-specific affect within individuals in the sample. When both positive and negative affect are modeled at the individual level, the influence of positive affect is greater than that of negative affect. At the dyad level, the findings indicate coupled influences in both positive and negative affect between partners. With regard to positive affect, females are sensitive to their partners' overall displacement from average as well as their rate of change; males are sensitive only to their partners' displacement from average. For negative affect both partners are sensitive to each other's displacement from average, yet there are no coupled influences for rates of change in this dimension. We interpret the influence of the parameters on the system by examining the expected behavior of the system as a function of varying parameter values.     

Item Response Modeling of Paired Comparison and Ranking Data

The comparative format used in ranking and paired comparisons tasks can significantly reduce the impact of uniform response biases typically associated with rating scales. Thurstone's (1927, 1931) model provides a powerful framework for modeling comparative data such as paired comparisons and rankings. Although Thurstonian models are generally presented as scaling models, that is, stimuli-centered models, they can also be used as person-centered models. In this article, we discuss how Thurstone's model for comparative data can be formulated as item response theory models so that respondents' scores on underlying dimensions can be estimated. Item parameters and latent trait scores can be readily estimated using a widely used statistical modeling program. Simulation studies show that item characteristic curves can be accurately estimated with as few as 200 observations and that latent trait scores can be recovered to a high precision. Empirical examples are given to illustrate how the model may be applied in practice and to recommend guidelines for designing ranking and paired comparisons tasks in the future.     

Identifying a Reordering of Rows and Columns for Multiple Proximity Matrices Using Multiobjective Programming

This paper is concerned with a problem where K (n×n) proximity matrices are available for a set of n objects. The goal is to identify a single permutation of the n objects that provides an adequate structural fit, as measured by an appropriate index, for each of the K matrices. A multiobjective programming approach for this problem, which seeks to optimize a weighted function of the K indices, is proposed, and illustrative examples are provided using a set of proximity matrices from the psychological literature. These examples show that, by solving the multiobjective programming model under different weighting schemes, the quantitative analyst can uncover information about the relationships among the matrices and often identify one or more permutations that provide good to excellent index values for all matrices.     

Building Play Skills Using Video Modeling and Matrix Training

Journal of Behavioral Education - Children with autism spectrum disorder often lack fundamental play skills, which can aid development with social, language, and imitation skills (Boutot et al....     

Multilevel Factor Analysis and Structural Equation Modeling of Daily Diary Coping Data: Modeling Trait and State Variation

This study used multilevel modeling of daily diary data to model within-person (state) and between-person (trait) components of coping variables. This application included the introduction of multilevel factor analysis (MFA) and a comparison of the predictive ability of these trait/state factors. Daily diary data were collected on a large (n = 366) multiethnic sample over the course of 5 days. Intraclass correlation coefficient for the derived factors suggested approximately equal amounts of variability in coping usage at the state and trait levels. MFAs showed that Problem-Focused Coping and Social Support emerged as stable factors at both the within-person and between-person levels. Other factors (Minimization, Emotional Rumination, Avoidance, Distraction) were specific to the within-person or between-person levels but not both. Multilevel structural equation modeling (MSEM) showed that the prediction of daily positive and negative affect differed as a function of outcome and level of coping factor. The Discussion section focuses primarily on a conceptual and methodological understanding of modeling state and trait coping using daily diary data with MFA and MSEM to examine covariation among coping variables and predicting outcomes of interest.     

Proactive Coping as a Mediator Between Resources and Outcomes: A Structural Equations Modeling Analysis

Proactive coping is a recent construct in the field of positive psychology. It refers to the efforts aimed at building general forces to facilitate the path toward challenging goals and personal growth. The current study adopted a theoretical model of proactive coping as a mediator of resources with positive and negative outcomes. Our aim was to examine two separate models with satisfaction with life and depression as the proposed outcomes of proactive coping. The proposed models presumed direct relationships between self-efficacy, optimism and social support with life satisfaction or depression, as well as indirect paths through proactive coping. We administered a set of questionnaires that consisted of demographic data, the Proactive Coping Scale, the General Self-efficacy Scale, the Life Orientation Test-Revised, the Perceived Social Support Scale, the Beck Depression Inventory, and the Satisfaction with Life Scale. Data were obtained from a sample of high school graduates (N?=?485, age 17 to 19, 174 males and 311 females). Our analyses confirmed presumed direct and indirect relationships between resources and outcomes. The results showed that the data fit well in the model with life satisfaction as an outcome, while the data did not fit well in the model with depression as an outcome. A subsequent model was formed in which satisfaction with life was used as a full mediator between proactive coping and depression; this model provided a good fit to the data. Theoretical implications, as well as suggestions for future research, are discussed.     

Personality and Emotional Intelligence in Older Adults: A Predictive Model Based on Structural Equations Modeling

Recent research on emotions in aging points to emotional intelligence (EI) as a factor that plays an important role in this process, and different conceptualizations of EI show that this construct is closely linked to personality in the general population. The main purpose of this study was to find out whether findings obtained in the general population indicating a predictive relationship between personality and EI are also confirmed during the aging process. A sample of 233 healthy older subjects between 60 and 90?years old was used. Participants answered two self-report scales on EI and personality, respectively. Structural equation modeling was used to test the predictive role of personality in EI. Personality was found to be a predictor of EI in older people, and the weight of the prediction was significant in all the dimensions of the big five personality factors, except the dimension of neuroticism, which is known to vary greatly during the aging process. These results indicate that personality influences EI differently in the older adult population, compared to the general population. This is a relevant finding that should be examined further in order to better understand the influence of personality on positive emotional development in this population.     

SEM with Missing Data and Unknown Population Distributions Using Two-Stage ML: Theory and Its Application

This article provides the theory and application of the 2-stage maximum likelihood (ML) procedure for structural equation modeling (SEM) with missing data. The validity of this procedure does not require the assumption of a normally distributed population. When the population is normally distributed and all missing data are missing at random (MAR), the direct ML procedure is nearly optimal for SEM with missing data. When missing data mechanisms are unknown, including auxiliary variables in the analysis will make the missing data mechanism more likely to be MAR. It is much easier to include auxiliary variables in the 2-stage ML than in the direct ML. Based on most recent developments for missing data with an unknown population distribution, the article first provides the least technical material on why the normal distribution-based ML generates consistent parameter estimates when the missing data mechanism is MAR. The article also provides sufficient conditions for the 2-stage ML to be a valid statistical procedure in the general case. For the application of the 2-stage ML, an SAS IML program is given to perform the first-stage analysis and EQS codes are provided to perform the second-stage analysis. An example with open- and closed-book examination data is used to illustrate the application of the provided programs. One aim is for quantitative graduate students/applied psychometricians to understand the technical details for missing data analysis. Another aim is for applied researchers to use the method properly.