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.     

Joint Modeling of Ability and Differential Speed Using Responses and Response Times

With computerized testing, it is possible to record both the responses of test takers to test questions (i.e., items) and the amount of time spent by a test taker in responding to each question. Various models have been proposed that take into account both test-taker ability and working speed, with the many models assuming a constant working speed throughout the test. The constant working speed assumption may be inappropriate for various reasons. For example, a test taker may need to adjust the pace due to time mismanagement, or a test taker who started out working too fast may reduce the working speed to improve accuracy. A model is proposed here that allows for variable working speed. An illustration of the model using the Amsterdam Chess Test data is provided.     

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.     

Using a Response Time–Based Expected A Posteriori Estimator to Control for Differential Speededness in Computerized Adaptive Test

This study investigates using response times (RTs) with item responses in a computerized adaptive test (CAT) setting to enhance item selection and ability estimation and control for differential speededness. Using van der Linden’s hierarchical framework, an extended procedure for joint estimation of ability and speed parameters for use in CAT is developed following van der Linden; this is called the joint expected a posteriori estimator (J-EAP). It is shown that the J-EAP estimate of ability and speededness outperforms the standard maximum likelihood estimator (MLE) of ability and speededness in terms of correlation, root mean square error, and bias. It is further shown that under the maximum information per time unit item selection method (MICT)—a method which uses estimates for ability and speededness directly—using the J-EAP further reduces average examinee time spent and variability in test times between examinees above the resulting gains of this selection algorithm with the MLE while maintaining estimation efficiency. Simulated test results are further corroborated with test parameters derived from a real data example.     

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.     

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...     

Reciprocal Markov Modeling of Feedback Mechanisms Between Emotion and Dietary Choice Using Experience-Sampling Data

With intensively collected longitudinal data, recent advances in the experience-sampling method (ESM) benefit social science empirical research, but also pose important methodological challenges. As traditional statistical models are not generally well equipped to analyze a system of variables that contain feedback loops, this paper proposes the utility of an extended hidden Markov model to model reciprocal the relationship between momentary emotion and eating behavior. This paper revisited an ESM data set (Lu, Huet, &; Dube, 2011 Lu, J., Huet, C., &; Dubé, L. (2011). Emotional reinforcement as a protective factor for healthy eating in home settings. The American Journal of Clinical Nutrition, 94(1), 254–261. http://dx.doi.org/10.3945/ajcn.110.006361[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) that observed 160 participants' food consumption and momentary emotions 6 times per day in 10 days. Focusing on the analyses on feedback loop between mood and meal-healthiness decision, the proposed reciprocal Markov model (RMM) can accommodate both hidden (“general” emotional states: positive vs. negative state) and observed states (meal: healthier, same or less healthy than usual) without presuming independence between observations and smooth trajectories of mood or behavior changes. The results of RMM analyses illustrated the reciprocal chains of meal consumption and mood as well as the effect of contextual factors that moderate the interrelationship between eating and emotion. A simulation experiment that generated data consistent with the empirical study further demonstrated that the procedure is promising in terms of recovering the parameters.     

Modeling Clustered Data with Very Few Clusters

Small-sample inference with clustered data has received increased attention recently in the methodological literature, with several simulation studies being presented on the small-sample behavior of many methods. However, nearly all previous studies focus on a single class of methods (e.g., only multilevel models, only corrections to sandwich estimators), and the differential performance of various methods that can be implemented to accommodate clustered data with very few clusters is largely unknown, potentially due to the rigid disciplinary preferences. Furthermore, a majority of these studies focus on scenarios with 15 or more clusters and feature unrealistically simple data-generation models with very few predictors. This article, motivated by an applied educational psychology cluster randomized trial, presents a simulation study that simultaneously addresses the extreme small sample and differential performance (estimation bias, Type I error rates, and relative power) of 12 methods to account for clustered data with a model that features a more realistic number of predictors. The motivating data are then modeled with each method, and results are compared. Results show that generalized estimating equations perform poorly; the choice of Bayesian prior distributions affects performance; and fixed effect models perform quite well. Limitations and implications for applications are also discussed.     

Analysis of Interactions and Nonlinear Effects with Missing Data: A Factored Regression Modeling Approach Using Maximum Likelihood Estimation

Abstract

When estimating multiple regression models with incomplete predictor variables, it is necessary to specify a joint distribution for the predictor variables. A convenient assumption is that this distribution is a multivariate normal distribution, which is also the default in many statistical software packages. This distribution will in general be misspecified if predictors with missing data have nonlinear effects (e.g., x²) or are included in interaction terms (e.g., x·z). In the present article, we introduce a factored regression modeling approach for estimating regression models with missing data that is based on maximum likelihood estimation. In this approach, the model likelihood is factorized into a part that is due to the model of interest and a part that is due to the model for the incomplete predictors. In three simulation studies, we showed that the factored regression modeling approach produced valid estimates of interaction and nonlinear effects in regression models with missing values on categorical or continuous predictor variables under a broad range of conditions. We developed the R package mdmb, which facilitates a user-friendly application of the factored regression modeling approach, and present a real-data example that illustrates the flexibility of the software.     

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.     

运用基因表达式编程的自陈量表数据建模

探讨基因表达式编程对自陈量表测量数据的建模方法。运用威廉斯创造力测验和认知需求量表获得400位中学生的测量分数,通过数据清洗,保留383个被试的分数作为建模的数据集。运用哈曼单因素检验方法没有发现共同方法偏差。采用均匀设计方法对基因表达式编程中的5个参数进行优化配置,在测试拟合度最大的试验条件下,找到了测试误差最小的模型。比较基因表达式编程和BP神经网络、支持向量回归机、多元线性回归、二次多项式回归所建模型的预测精度。研究表明,基因表达式编程能用于自陈量表测量数据的建模,该模型比传统方法所建的模型具有更高的预测精度,而且模型是稳健的。     

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.     

Testing Group Mean Differences of Latent Variables in Multilevel Data Using Multiple-Group Multilevel CFA and Multilevel MIMIC Modeling

Considering that group comparisons are common in social science, we examined two latent group mean testing methods when groups of interest were either at the between or within level of multilevel data: multiple-group multilevel confirmatory factor analysis (MG ML CFA) and multilevel multiple-indicators multiple-causes modeling (ML MIMIC). The performance of these methods were investigated through three Monte Carlo studies. In Studies 1 and 2, either factor variances or residual variances were manipulated to be heterogeneous between groups. In Study 3, which focused on within-level multiple-group analysis, six different model specifications were considered depending on how to model the intra-class group correlation (i.e., correlation between random effect factors for groups within cluster). The results of simulations generally supported the adequacy of MG ML CFA and ML MIMIC for multiple-group analysis with multilevel data. The two methods did not show any notable difference in the latent group mean testing across three studies. Finally, a demonstration with real data and guidelines in selecting an appropriate approach to multilevel multiple-group analysis are provided.     

A Method of Correcting Estimation Failure in Latent Differential Equations with Comparisons to Kalman Filtering

Abstract

Studies have used the latent differential equation (LDE) model to estimate the parameters of damped oscillation in various phenomena, but it has been shown that correct, non-zero parameter estimates are only obtained when the latent series exhibits little or no process noise. Consequently, LDEs are limited to modeling deterministic processes with measurement error rather than those with random behavior in the true latent state. The reasons for these limitations are considered, and a piecewise deterministic approximation (PDA) algorithm is proposed to treat process noise outliers as functional discontinuities and obtain correct estimates of the damping parameter. Comprehensive, random-effects simulations were used to compare results with those obtained using a state-space model (SSM) based on the Kalman filter. The LDE with the PDA algorithm (LDEPDA) successfully recovered the simulated damping parameter under a variety of conditions when process noise was present in the latent state. The LDEPDA had greater precision and accuracy than the SSM when estimating parameters from data with sparse jump discontinuities, but worse performance for diffusion processes overall. All three methods were applied to a sample of postural sway data. The basic LDE estimated zero damping, while the LDEPDA and SSM estimated moderate to high damping. The SSM estimated the smallest standard errors for both frequency and damping parameter estimates.     

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.