A Multilevel AR(1) Model: Allowing for Inter-Individual Differences in Trait-Scores,Inertia, and Innovation Variance

In this article we consider a multilevel first-order autoregressive [AR(1)] model with random intercepts, random autoregression, and random innovation variance (i.e., the level 1 residual variance). Including random innovation variance is an important extension of the multilevel AR(1) model for two reasons. First, between-person differences in innovation variance are important from a substantive point of view, in that they capture differences in sensitivity and/or exposure to unmeasured internal and external factors that influence the process. Second, using simulation methods we show that modeling the innovation variance as fixed across individuals, when it should be modeled as a random effect, leads to biased parameter estimates. Additionally, we use simulation methods to compare maximum likelihood estimation to Bayesian estimation of the multilevel AR(1) model and investigate the trade-off between the number of individuals and the number of time points. We provide an empirical illustration by applying the extended multilevel AR(1) model to daily positive affect ratings from 89 married women over the course of 42 consecutive days.     

Semiparametric Thurstonian Models for Recurrent Choices: A Bayesian Analysis

We develop semiparametric Bayesian Thurstonian models for analyzing repeated choice decisions involving multinomial, multivariate binary or multivariate ordinal data. Our modeling framework has multiple components that together yield considerable flexibility in modeling preference utilities, cross-sectional heterogeneity and parameter-driven dynamics. Each component of our model is specified semiparametrically using Dirichlet process (DP) priors. The utility (latent variable) component of our model allows the alternative-specific utility errors to semiparametrically deviate from a normal distribution. This generates a robust alternative to popular Thurstonian specifications that are based on underlying normally distributed latent variables. Our second component focuses on flexibly modeling cross-sectional heterogeneity. The semiparametric specification allows the heterogeneity distribution to mimic either a finite mixture distribution or a continuous distribution such as the normal, whichever is supported by the data. Thus, special features such as multimodality can be readily incorporated without the need to overtly search for the best heterogeneity specification across a series of models. Finally, we allow for parameter-driven dynamics using a semiparametric state-space approach. This specification adds to the literature on robust Kalman filters. The resulting framework is very general and integrates divergent strands of the literatures on flexible choice models, Bayesian nonparametrics and robust time series specifications. Given this generality, we show how several existing Thurstonian models can be obtained as special forms of our model. We describe Markov chain Monte Carlo methods for the inference of model parameters, report results from two simulation studies and apply the model to consumer choice data from a frequently purchased product category. The results from our simulations and application highlight the benefits of using our semiparametric approach.     

The influence of trend estimation method on forecasting curriculum-based measurement of reading performance

Estimating a trend line through words read correct per minute scores collected across successive weeks is a preferred method to evaluate student response to instruction with curriculum-based measurement of reading (CBM-R). This is due in part, because the slope of that line of best fit is used to predict the trajectory of student performance if the current intervention is maintained. In turn, trend lines should predict future scores with a high degree of accuracy when an intervention is maintained. We evaluated the forecasting accuracy of a trend estimation method currently used in practice (i.e., ordinary least squares), and five alternate methods recently evaluated in CBM-R simulation studies, using actual student data. Results suggest that alternate trend estimation methods predicted future performance with a similar level of accuracy as ordinary least squares trend lines across most conditions, with the exception of slopes estimated via Bayesian analysis. Bayesian trend lines estimated using informed prior distributions yielded noticeably less biased and more precise predictions when applied to short data series relative to all other estimation methods across most conditions. Outcomes from the current study highlight the need to further explore the viability of Bayesian analysis to evaluate individual time series data.     

Single and Multiple Ability Estimation in the SEM Framework: A Noninformative Bayesian Estimation Approach

Latent variable models with many categorical items and multiple latent constructs result in many dimensions of numerical integration, and the traditional frequentist estimation approach, such as maximum likelihood (ML), tends to fail due to model complexity. In such cases, Bayesian estimation with diffuse priors can be used as a viable alternative to ML estimation. This study compares the performance of Bayesian estimation with ML estimation in estimating single or multiple ability factors across 2 types of measurement models in the structural equation modeling framework: a multidimensional item response theory (MIRT) model and a multiple-indicator multiple-cause (MIMIC) model. A Monte Carlo simulation study demonstrates that Bayesian estimation with diffuse priors, under various conditions, produces results quite comparable with ML estimation in the single- and multilevel MIRT and MIMIC models. Additionally, an empirical example utilizing the Multistate Bar Examination is provided to compare the practical utility of the MIRT and MIMIC models. Structural relationships among the ability factors, covariates, and a binary outcome variable are investigated through the single- and multilevel measurement models. The article concludes with a summary of the relative advantages of Bayesian estimation over ML estimation in MIRT and MIMIC models and suggests strategies for implementing these methods.     

A Robust Bayesian Approach for Structural Equation Models with Missing Data

In this paper, normal/independent distributions, including but not limited to the multivariate t distribution, the multivariate contaminated distribution, and the multivariate slash distribution, are used to develop a robust Bayesian approach for analyzing structural equation models with complete or missing data. In the context of a nonlinear structural equation model with fixed covariates, robust Bayesian methods are developed for estimation and model comparison. Results from simulation studies are reported to reveal the characteristics of estimation. The methods are illustrated by using a real data set obtained from diabetes patients.     

A Bayesian Multi-Level Factor Analytic Model of Consumer Price Sensitivities Across Categories

Identifying price sensitive consumers is an important problem in marketing. We develop a Bayesian multi-level factor analytic model of the covariation among household-level price sensitivities across product categories that are substitutes. Based on a multivariate probit model of category incidence, this framework also allows the researcher to model overall price sensitivity (i.e., indicated by higher-order factor scores) as a function of household-level covariates. All model parameters are estimated simultaneously to circumvent the downward bias resulting from two-stage estimation. The modeling framework is illustrated using scanner panel data from multiple categories of instant coffee.     

Exploring the Dynamics of Dyadic Interactions via Hierarchical Segmentation

In this article we present an exploratory tool for extracting systematic patterns from multivariate data. The technique, hierarchical segmentation (HS), can be used to group multivariate time series into segments with similar discrete-state recurrence patterns and it is not restricted by the stationarity assumption. We use a simulation study to describe the steps and properties of HS. We then use empirical data on daily affect from one couple to illustrate the use of HS for describing the affective dynamics of the dyad. First, we partition the data into three periods that represent different affective states and show different dynamics between both individuals’ affect. We then examine the synchrony between both individuals’ affective states and identify different patterns of coherence across the periods. Finally, we discuss the possibilities of using results from HS to construct confirmatory dynamic models with multiple change points or regime-specific dynamics.     

A Multivariate Multilevel Approach to the Modeling of Accuracy and Speed of Test Takers

Response times on test items are easily collected in modern computerized testing. When collecting both (binary) responses and (continuous) response times on test items, it is possible to measure the accuracy and speed of test takers. To study the relationships between these two constructs, the model is extended with a multivariate multilevel regression structure which allows the incorporation of covariates to explain the variance in speed and accuracy between individuals and groups of test takers. A Bayesian approach with Markov chain Monte Carlo (MCMC) computation enables straightforward estimation of all model parameters. Model-specific implementations of a Bayes factor (BF) and deviance information criterium (DIC) for model selection are proposed which are easily calculated as byproducts of the MCMC computation. Both results from simulation studies and real-data examples are given to illustrate several novel analyses possible with this modeling framework. The authors thank Steven Wise, James Madison University, and Pere Joan Ferrando, Universitat Rovira i Virgili, for generously making available their data sets for the empirical examples in this paper.     

当结构假设和分布假设不满足时的验证性因子分析：稳健极大似然法估计和贝叶斯估计的比较研究

结构方程模型已被广泛应用于心理学、教育学、以及社会科学领域的统计分析中。结构方程模型分析中最常用的估计方法是基于正 态分布的估计量,比如极大似然估计法。这些方法需要满足两个假设。第一, 理论模型必须正确地反映变量与变量之间的关系,称为结构假 设。第二,数据必须符合多元正态分布,称为分布假设。如果这些假设不满足,基于正态分布的估计量就有可能导致不正确的卡方指数、不 正确的拟合度、以及有偏差的参数估计和参数估计的标准误。在实际应用中,几乎所有的理论模型都不能准确地解释变量与变量之间的关系, 数据也常常呈非多元正态分布。为此,一些新的估计方法得以发展。这些方法要么在理论上不要求数据呈多元正态分布,要么对因数据呈非 正态分布而导致的不正确结果进行纠正。当前较为流行的两种方法是稳健极大似然估计和贝叶斯估计。稳健极大似然估计是应用 Satorra and Bentler (1994) 的方法对不正确的卡方指数和参数估计的标准误进行调整,而参数估计和用极大似然方法得出的完全等同。贝叶斯估计方法则是 基于贝叶斯定理,其要点是：参数的后验分布是由参数的先验分布和数据似然值相乘而得来。后验分布常用马尔科夫蒙特卡洛算法来进行模拟。 对于稳健极大似然估计和贝叶斯估计这两种方法之间的优劣比较,先前的研究只局限于理论模型是正确的情境。而本研究则着重于理论模型 是错误的情境,同时也考虑到数据呈非正态分布的情境。本研究所采用的模型是验证性因子模型,数据全部由计算机模拟而来。数据的生成 取决于三个因素：8 类因子结构,3 种变量分布,和3 组样本量。这三个因素产生72 个模拟条件（72=8x3x3）。每个模拟条件下生成2000 个 数据组,每个数据组都拟合两个模型,一个是正确模型、一个是错误模型。每个模型都用两种估计方法来拟合：稳健极大似然估计法和贝叶 斯估计方法。贝叶斯估计方法中所使用的先验分布是无信息先验分布。结果分析主要着重于模型拒绝率、拟合度、参数估计、和参数估计的 标准误。研究的结果表明：在样本量充足的情况下,两种方法得出的参数估计非常相似。当数据呈非正态分布时,贝叶斯估计法比稳健极大 似然估计法更好地拒绝错误模型。但是,当样本量不足且数据呈正态分布时,贝叶斯估计在拒绝错误模型和参数估计上几乎没有优势,甚至 在一些条件下,比稳健极大似然法要差。     

多维题组效应Rasch模型

首先, 本文诠释了“题组”的本质即一个存在共同刺激的项目集合。并基于此, 将题组效应划分为项目内单维题组效应和项目内多维题组效应。其次, 本文基于Rasch模型开发了二级评分和多级评分的多维题组效应Rasch模型, 以期较好地处理项目内多维题组效应。最后, 模拟研究结果显示新模型有效合理, 与Rasch题组模型、分部评分模型对比研究后表明：(1)测验存在项目内多维题组效应时, 仅把明显的捆绑式题组效应进行分离而忽略其他潜在的题组效应, 仍会导致参数的偏差估计甚或高估测验信度; (2)新模型更具普适性, 即便当被试作答数据不存在题组效应或只存在项目内单维题组效应, 采用新模型进行测验分析也能得到较好的参数估计结果。     

Residual Normality Assumption and the Estimation of Multiple Membership Random Effects Models

While conventional hierarchical linear modeling is applicable to purely hierarchical data, a multiple membership random effects model (MMrem) is appropriate for nonpurely nested data wherein some lower-level units manifest mobility across higher-level units. Although a few recent studies have investigated the influence of cluster-level residual nonnormality on hierarchical linear modeling estimation for purely hierarchical data, no research has examined the statistical performance of an MMrem given residual non-normality. The purpose of the present study was to extend prior research on the influence of residual non-normality from purely nested data structures to multiple membership data structures. Employing a Monte Carlo simulation study, this research inquiry examined two-level MMrem parameter estimate biases and inferential errors. Simulation factors included the level-two residual distribution, sample sizes, intracluster correlation coefficient, and mobility rate. Results showed that estimates of fixed effect parameters and the level-one variance component were robust to level-two residual non-normality. The level-two variance component, however, was sensitive to level-two residual non-normality and sample size. Coverage rates of the 95% credible intervals deviated from the nominal value assumed when level-two residuals were non-normal. These findings can be useful in the application of an MMrem to account for the contextual effects of multiple higher-level units.     

Tracking of nociceptive thresholds using adaptive psychophysical methods

Psychophysical thresholds reflect the state of the underlying nociceptive mechanisms. For example, noxious events can activate endogenous analgesic mechanisms that increase the nociceptive threshold. Therefore, tracking thresholds over time facilitates the investigation of the dynamics of these underlying mechanisms. Threshold tracking techniques should use efficient methods for stimulus selection and threshold estimation. This study compares, in simulation and in human psychophysical experiments, the performance of different combinations of adaptive stimulus selection procedures and threshold estimation methods. Monte Carlo simulations were first performed to compare the bias and precision of threshold estimates produced by three different stimulus selection procedures (simple staircase, random staircase, and minimum entropy procedure) and two estimation methods (logistic regression and Bayesian estimation). Logistic regression and Bayesian estimations resulted in similar precision only when the prior probability distributions (PDs) were chosen appropriately. The minimum entropy and simple staircase procedures achieved the highest precision, while the random staircase procedure was the least sensitive to different procedure-specific settings. Next, the simple staircase and random staircase procedures, in combination with logistic regression, were compared in a human subject study (n = 30). Electrocutaneous stimulation was used to track the nociceptive perception threshold before, during, and after a cold pressor task, which served as the conditioning stimulus. With both procedures, habituation was detected, as well as changes induced by the conditioning stimulus. However, the random staircase procedure achieved a higher precision. We recommend using the random staircase over the simple staircase procedure, in combination with logistic regression, for nonstationary threshold tracking experiments.     

The Performance of Multivariate Methods for Two-Group Comparisons with Small Samples and Incomplete Data

Abstract

In intervention studies having multiple outcomes, researchers often use a series of univariate tests (e.g., ANOVAs) to assess group mean differences. Previous research found that this approach properly controls Type I error and generally provides greater power compared to MANOVA, especially under realistic effect size and correlation combinations. However, when group differences are assessed for a specific outcome, these procedures are strictly univariate and do not consider the outcome correlations, which may be problematic with missing outcome data. Linear mixed or multivariate multilevel models (MVMMs), implemented with maximum likelihood estimation, present an alternative analysis option where outcome correlations are taken into account when specific group mean differences are estimated. In this study, we use simulation methods to compare the performance of separate independent samples t tests estimated with ordinary least squares and analogous t tests from MVMMs to assess two-group mean differences with multiple outcomes under small sample and missingness conditions. Study results indicated that a MVMM implemented with restricted maximum likelihood estimation combined with the Kenward–Roger correction had the best performance. Therefore, for intervention studies with small N and normally distributed multivariate outcomes, the Kenward–Roger procedure is recommended over traditional methods and conventional MVMM analyses, particularly with incomplete data.     

Comparing Risky Decision Making Under Conditions of Real and Hypothetical Consequences

In a series of three experiments, subjects made risky decisions under conditions of hypothetical or real consequences. Task variations across experiments included: (1) type of risk (monetary gambles or investments of time and effort), (2) within-subject and between-subjects manipulations of consequence condition, and (3) single or multiple decisions. The hypothesis of no difference between choices in real and hypothetical consequence conditions was retained in each experiment. Supplemental analyses ruled out various “artifactual” interpretations of the null results. Discussion focused on conditions in which researchers can and cannot infer decision makers’ actual risk preferences from their responses in laboratory tasks.     

Introduction and application of the multiscale coefficient of variation analysis

Quantifying how patterns of behavior relate across multiple levels of measurement typically requires long time series for reliable parameter estimation. We describe a novel analysis that estimates patterns of variability across multiple scales of analysis suitable for time series of short duration. The multiscale coefficient of variation (MSCV) measures the distance between local coefficient of variation estimates within particular time windows and the overall coefficient of variation across all time samples. We first describe the MSCV analysis and provide an example analytical protocol with corresponding MATLAB implementation and code. Next, we present a simulation study testing the new analysis using time series generated by ARFIMA models that span white noise, short-term and long-term correlations. The MSCV analysis was observed to be sensitive to specific parameters of ARFIMA models varying in the type of temporal structure and time series length. We then apply the MSCV analysis to short time series of speech phrases and musical themes to show commonalities in multiscale structure. The simulation and application studies provide evidence that the MSCV analysis can discriminate between time series varying in multiscale structure and length.     

An overview of Bayesian reasoning in the analysis of delay‐discounting data

Statistical inference (including interval estimation and model selection) is increasingly used in the analysis of behavioral data. As with many other fields, statistical approaches for these analyses traditionally use classical (i.e., frequentist) methods. Interpreting classical intervals and p‐values correctly can be burdensome and counterintuitive. By contrast, Bayesian methods treat data, parameters, and hypotheses as random quantities and use rules of conditional probability to produce direct probabilistic statements about models and parameters given observed study data. In this work, we reanalyze two data sets using Bayesian procedures. We precede the analyses with an overview of the Bayesian paradigm. The first study reanalyzes data from a recent study of controls, heavy smokers, and individuals with alcohol and/or cocaine substance use disorder, and focuses on Bayesian hypothesis testing for covariates and interval estimation for discounting rates among various substance use disorder profiles. The second example analyzes hypothetical environmental delay‐discounting data. This example focuses on using historical data to establish prior distributions for parameters while allowing subjective expert opinion to govern the prior distribution on model preference. We review the subjective nature of specifying Bayesian prior distributions but also review established methods to standardize the generation of priors and remove subjective influence while still taking advantage of the interpretive advantages of Bayesian analyses. We present the Bayesian approach as an alternative paradigm for statistical inference and discuss its strengths and weaknesses.     

Multilevel SEM with random slopes in discrete data using the pairwise maximum likelihood

Pairwise maximum likelihood (PML) estimation is a promising method for multilevel models with discrete responses. Multilevel models take into account that units within a cluster tend to be more alike than units from different clusters. The pairwise likelihood is then obtained as the product of bivariate likelihoods for all within-cluster pairs of units and items. In this study, we investigate the PML estimation method with computationally intensive multilevel random intercept and random slope structural equation models (SEM) in discrete data. In pursuing this, we first reconsidered the general ‘wide format’ (WF) approach for SEM models and then extend the WF approach with random slopes. In a small simulation study we the determine accuracy and efficiency of the PML estimation method by varying the sample size (250, 500, 1000, 2000), response scales (two-point, four-point), and data-generating model (mediation model with three random slopes, factor model with one and two random slopes). Overall, results show that the PML estimation method is capable of estimating computationally intensive random intercept and random slopes multilevel models in the SEM framework with discrete data and many (six or more) latent variables with satisfactory accuracy and efficiency. However, the condition with 250 clusters combined with a two-point response scale shows more bias.     

贝叶斯题组随机效应模型的必要性及影响因素

题组模型可以解决传统IRT模型由于题目间局部独立性假设违背时所导致的参数估计偏差。为探讨题组随机效应模型的适用范围, 采用Monte Carlo模拟研究, 分别使用2-PL贝叶斯题组随机效应模型(BTRM)和2-PL贝叶斯模型(BM)对数据进行拟合, 考虑了题组效应、题组长度、题目数量和局部独立题目比例的影响。结果显示：(1) BTRM不受题组效应和题组长度影响, BM对参数估计的误差随题组效应和题组长度增加而增加。(2) BTRM具有一定的普遍性, 且当题组效应大, 题组长, 题目数量大时使用该模型能减少估计误差, 但是当题目数量较小时, 两个模型得到的能力估计误差都较大。(3)当局部独立题目的比例较大时, 两种模型得到的参数估计差异不大。     

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.