An investigation of hierarchical Bayes procedures in item response theory

Hierarchical Bayes procedures for the two-parameter logistic item response model were compared for estimating item and ability parameters. Simulated data sets were analyzed via two joint and two marginal Bayesian estimation procedures. The marginal Bayesian estimation procedures yielded consistently smaller root mean square differences than the joint Bayesian estimation procedures for item and ability estimates. As the sample size and test length increased, the four Bayes procedures yielded essentially the same result.The authors wish to thank the Editor and anonymous reviewers for their insightful comments and suggestions.     

A Comparison of ML,WLSMV, and Bayesian Methods for Multilevel Structural Equation Models in Small Samples: A Simulation Study

Multilevel structural equation models are increasingly applied in psychological research. With increasing model complexity, estimation becomes computationally demanding, and small sample sizes pose further challenges on estimation methods relying on asymptotic theory. Recent developments of Bayesian estimation techniques may help to overcome the shortcomings of classical estimation techniques. The use of potentially inaccurate prior information may, however, have detrimental effects, especially in small samples. The present Monte Carlo simulation study compares the statistical performance of classical estimation techniques with Bayesian estimation using different prior specifications for a two-level SEM with either continuous or ordinal indicators. Using two software programs (Mplus and Stan), differential effects of between- and within-level sample sizes on estimation accuracy were investigated. Moreover, it was tested to which extent inaccurate priors may have detrimental effects on parameter estimates in categorical indicator models. For continuous indicators, Bayesian estimation did not show performance advantages over ML. For categorical indicators, Bayesian estimation outperformed WLSMV solely in case of strongly informative accurate priors. Weakly informative inaccurate priors did not deteriorate performance of the Bayesian approach, while strong informative inaccurate priors led to severely biased estimates even with large sample sizes. With diffuse priors, Stan yielded better results than Mplus in terms of parameter estimates.     

Bayesian estimation in the three-parameter logistic model

A joint Bayesian estimation procedure for the estimation of parameters in the three-parameter logistic model is developed in this paper. Procedures for specifying prior beliefs for the parameters are given. It is shown through simulation studies that the Bayesian procedure (i) ensures that the estimates stay in the parameter space, and (ii) produces better estimates than the joint maximum likelihood procedure as judged by such criteria as mean squared differences between estimates and true values. The research reported here was performed pursuant to Grant No. N0014-79-C-0039 with the Office of Naval Research. A related article by Robert J. Mislevy (1986) appeared when the present paper was in the printing stage.     

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。     

Bayesian estimation in the two-parameter logistic model

A Bayesian procedure is developed for the estimation of parameters in the two-parameter logistic item response model. Joint modal estimates of the parameters are obtained and procedures for the specification of prior information are described. Through simulation studies it is shown that Bayesian estimates of the parameters are superior to maximum likelihood estimates in the sense that they are (a) more meaningful since they do not drift out of range, and (b) more accurate in that they result in smaller mean squared differences between estimates and true values.The research reported here was performed pursuant to Grant No. N0014-79-C-0039 with the Office of Naval Research.     

潜变量交互效应标准化估计：方法比较与选用策略

标准化估计对模型的解释和效应大小的比较有重要作用。虽然潜变量交互效应的恰当标准化估计公式已经面世超过10年, 国内外都在使用和引用, 但至今未见到关于不同估计方法得到的恰当标准化估计的系统比较。通过模拟实验, 比较了乘积指标法、潜调节结构方程(LMS)、无先验信息和有先验信息的贝叶斯法的潜变量交互效应标准化估计在不同条件下的表现。结果发现, 在正态条件下, LMS和有信息贝叶斯法表现较好; 而在非正态条件下, 乘积指标法比较稳健, 但需要较大的样本(不小于500), 小样本且外生潜变量之间相关很低时可使用无信息贝叶斯法。     

Bayesian Estimation for Item Factor Analysis Models with Sparse Categorical Indicators

Psychometric models for item-level data are broadly useful in psychology. A recurring issue for estimating item factor analysis (IFA) models is low-item endorsement (item sparseness), due to limited sample sizes or extreme items such as rare symptoms or behaviors. In this paper, I demonstrate that under conditions characterized by sparseness, currently available estimation methods, including maximum likelihood (ML), are likely to fail to converge or lead to extreme estimates and low empirical power. Bayesian estimation incorporating prior information is a promising alternative to ML estimation for IFA models with item sparseness. In this article, I use a simulation study to demonstrate that Bayesian estimation incorporating general prior information improves parameter estimate stability, overall variability in estimates, and power for IFA models with sparse, categorical indicators. Importantly, the priors proposed here can be generally applied to many research contexts in psychology, and they do not impact results compared to ML when indicators are not sparse. I then apply this method to examine the relationship between suicide ideation and insomnia in a sample of first-year college students. This provides an important alternative for researchers who may need to model items with sparse endorsement.     

Bayesian data analysis as a tool for behavior analysts

Bayesian approaches to data analysis are considered within the context of behavior analysis. The paper distinguishes between Bayesian inference, the use of Bayes Factors, and Bayesian data analysis using specialized tools. Given the importance of prior beliefs to these approaches, the review addresses those situations in which priors have a big effect on the outcome (Bayes Factors) versus a smaller effect (parameter estimation). Although there are many advantages to Bayesian data analysis from a philosophical perspective, in many cases a behavior analyst can be reasonably well‐served by the adoption of traditional statistical tools as long as the focus is on parameter estimation and model comparison, not null hypothesis significance testing. A strong case for Bayesian analysis exists under specific conditions: When prior beliefs can help narrow parameter estimates (an especially important issue given the small sample sizes common in behavior analysis) and when an analysis cannot easily be conducted using traditional approaches (e.g., repeated measures censored regression).     

Spurious Latent Class Problem in the Mixed Rasch Model: A Comparison of Three Maximum Likelihood Estimation Methods under Different Ability Distributions

Recent research has shown that over-extraction of latent classes can be observed in the Bayesian estimation of the mixed Rasch model when the distribution of ability is non-normal. This study examined the effect of non-normal ability distributions on the number of latent classes in the mixed Rasch model when estimated with maximum likelihood estimation methods (conditional, marginal, and joint). Three information criteria fit indices (Akaike information criterion, Bayesian information criterion, and sample size adjusted BIC) were used in a simulation study and an empirical study. Findings of this study showed that the spurious latent class problem was observed with marginal maximum likelihood and joint maximum likelihood estimations. However, conditional maximum likelihood estimation showed no overextraction problem with non-normal ability distributions.     

2PL模型的两种马尔可夫蒙特卡洛缺失数据处理方法比较

马尔科夫蒙特卡洛（MCMC）是项目反应理论中处理缺失数据的一种典型方法。文章通过模拟研究比较了在不同被试人数,项目数,缺失比例下两种MCMC方法（M-H within Gibbs和DA-T Gibbs）参数估计的精确性,并结合了实证研究。研究结果表明,两种方法是有差异的,项目参数估计均受被试人数影响很大,受缺失比例影响相对更小。在样本较大缺失比例较小时,M-H within Gibbs参数估计的均方误差（RMSE）相对略小,随着样本数的减少或缺失比例的增加,DA-T Gibbs方法逐渐优于M-H within Gibbs方法     

Mixture-modelling-based Bayesian MH-RM algorithm for the multidimensional 4PLM

Several recent works have tackled the estimation issue for the unidimensional four-parameter logistic model (4PLM). Despite these efforts, the issue remains a challenge for the multidimensional 4PLM (M4PLM). Fu et al. (2021) proposed a Gibbs sampler for the M4PLM, but it is time-consuming. In this paper, a mixture-modelling-based Bayesian MH-RM (MM-MH-RM) algorithm is proposed for the M4PLM to obtain the maximum a posteriori (MAP) estimates. In a comparison of the MM-MH-RM algorithm to the original MH-RM algorithm, two simulation studies and an empirical example demonstrated that the MM-MH-RM algorithm possessed the benefits of the mixture-modelling approach and could produce more robust estimates with guaranteed convergence rates and fast computation. The MATLAB codes for the MM-MH-RM algorithm are available in the online appendix.     

Generalizability of evidence-based assessment recommendations for pediatric bipolar disorder

Bipolar disorder is frequently clinically diagnosed in youths who do not actually satisfy Diagnostic and Statistical Manual of Mental Disorders (4th ed., text revision; DSM-IV-TR; American Psychiatric Association, 1994) criteria, yet cases that would satisfy full DSM-IV-TR criteria are often undetected clinically. Evidence-based assessment methods that incorporate Bayesian reasoning have demonstrated improved diagnostic accuracy and consistency; however, their clinical utility is largely unexplored. The present study examines the effectiveness of promising evidence-based decision-making strategies compared with the clinical gold standard. Participants were 562 youths, ages 5 to 17 and predominantly African American, drawn from a community mental health clinic. Research diagnoses combined a semistructured interview with youths' psychiatric, developmental, and family mental health histories. Independent Bayesian estimates that relied on published risk estimates from other samples discriminated bipolar diagnoses (area under curve = .75, p < .00005). The Bayes and confidence ratings correlated at rs = .30. Agreement about an evidence-based assessment intervention threshold model (wait/assess/treat) was κ = .24, p < .05. No potential moderators of agreement between the Bayesian estimates and confidence ratings, including type of bipolar illness, were significant. Bayesian risk estimates were highly correlated with logistic regression estimates using optimal sample weights (r = .81, p < .0005). Clinical and Bayesian approaches agree in terms of overall concordance and deciding next clinical action, even when Bayesian predictions are based on published estimates from clinically and demographically different samples. Evidence-based assessment methods may be useful in settings in which gold standard assessments cannot be routinely used, and they may help decrease rates of overdiagnosis while promoting earlier identification of true cases.     

A Bayesian Power Analysis Procedure Considering Uncertainty in Effect Size Estimates from a Meta-analysis

In conventional frequentist power analysis, one often uses an effect size estimate, treats it as if it were the true value, and ignores uncertainty in the effect size estimate for the analysis. The resulting sample sizes can vary dramatically depending on the chosen effect size value. To resolve the problem, we propose a hybrid Bayesian power analysis procedure that models uncertainty in the effect size estimates from a meta-analysis. We use observed effect sizes and prior distributions to obtain the posterior distribution of the effect size and model parameters. Then, we simulate effect sizes from the obtained posterior distribution. For each simulated effect size, we obtain a power value. With an estimated power distribution for a given sample size, we can estimate the probability of reaching a power level or higher and the expected power. With a range of planned sample sizes, we can generate a power assurance curve. Both the conventional frequentist and our Bayesian procedures were applied to conduct prospective power analyses for two meta-analysis examples (testing standardized mean differences in example 1 and Pearson's correlations in example 2). The advantages of our proposed procedure are demonstrated and discussed.     

Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.     

基于增长模型的非随机缺失数据处理:选择模型和极大似然方法

对含有非随机缺失数据的潜变量增长模型,为了考察基于不同假设的缺失数据处理方法:极大似然(ML)方法与DiggleKenward选择模型的优劣,通过Monte Carlo模拟研究,比较两种方法对模型中增长参数估计精度及其标准误估计的差异,并考虑样本量、非随机缺失比例和随机缺失比例的影响。结果表明,符合前提假设的Diggle-Kenward选择模型的参数估计精度普遍高于ML方法;对于标准误估计值,ML方法存在一定程度的低估,得到的置信区间覆盖比率也明显低于Diggle-Kenward选择模型。     

Conditional estimation of exponential random graph models from snowball sampling designs

A complete survey of a network in a large population may be prohibitively difficult and costly. So it is important to estimate models for networks using data from various network sampling designs, such as link-tracing designs. We focus here on snowball sampling designs, designs in which the members of an initial sample of network members are asked to nominate their network partners, their network partners are then traced and asked to nominate their network partners, and so on. We assume an exponential random graph model (ERGM) of a particular parametric form and outline a conditional maximum likelihood estimation procedure for obtaining estimates of ERGM parameters. This procedure is intended to complement the likelihood approach developed by  Handcock and Gile (2010) by providing a practical means of estimation when the size of the complete network is unknown and/or the complete network is very large. We report the outcome of a simulation study with a known model designed to assess the impact of initial sample size, population size, and number of sampling waves on properties of the estimates. We conclude with a discussion of the potential applications and further developments of the approach.     

Generalized graded unfolding model with structural equation for subject parameters

The generalized graded unfolding model (GGUM) is capable of analyzing polytomous scored, unfolding data such as agree‐disagree responses to attitude statements. In the present study, we proposed a GGUM with structural equation for subject parameters, which enabled us to evaluate the relation between subject parameters and covariates and/or latent variables simultaneously, in order to avoid the influence of attenuation. Additionally, an algorithm for parameter estimation is newly implemented via the Markov Chain Monte Carlo (MCMC) method, based on Bayesian statistics. In the simulation, we compared the accuracy of estimates of regression coefficients between the proposed model and a conventional method using a GGUM (where regression coefficients are estimated using estimates of θ). As a result, the proposed model performed much better than the conventional method in terms of bias and root mean squared errors of estimates of regression coefficients. The study concluded by verifying the efficacy of the proposed model, using an actual data example of attitude measurement.     

A Bayesian Model For The Estimation Of Latent Interaction And Quadratic Effects When Latent Variables Are Non-Normally Distributed

Structural equation models with interaction and quadratic effects have become a standard tool for testing nonlinear hypotheses in the social sciences. Most of the current approaches assume normally distributed latent predictor variables. In this article, we present a Bayesian model for the estimation of latent nonlinear effects when the latent predictor variables are nonnormally distributed. The nonnormal predictor distribution is approximated by a finite mixture distribution. We conduct a simulation study that demonstrates the advantages of the proposed Bayesian model over contemporary approaches (Latent Moderated Structural Equations [LMS], Quasi-Maximum-Likelihood [QML], and the extended unconstrained approach) when the latent predictor variables follow a nonnormal distribution. The conventional approaches show biased estimates of the nonlinear effects; the proposed Bayesian model provides unbiased estimates. We present an empirical example from work and stress research and provide syntax for substantive researchers. Advantages and limitations of the new model are discussed.     

多维测验项目参数的估计：基于SEM与MIRT方法的比较

作者简要回顾了SEM框架下分类数据因素分析(CCFA)模型和MIRT框架下测验题目和潜在能力的关系模型, 对两种框架下的主要参数估计方法进行了总结。通过模拟研究, 比较了SEM框架下WLSc和WLSMV估计方法与MIRT框架下MLR和MCMC估计方法的差异。研究结果表明：(1) WLSc得到参数估计的偏差最大, 且存在参数收敛的问题; (2)随着样本量增大, 各种项目参数估计的精度均提高, WLSMV方法与MLR方法得到的参数估计精度差异很小, 大多数情况下不比MCMC方法差; (3)除WLSc方法外, 随着每个维度测验题目的增多参数估计的精度逐渐增高; (4)测验维度对区分度参数和难度参数的影响较大, 而测验维度对项目因素载荷和阈值的影响相对较小; (5)项目参数的估计精度受项目测量维度数的影响, 只测量一个维度的项目参数估计精度较高。另外文章还对两种方法在实际应用中应该注意的问题提供了一些建议。     

Evaluating Factorial Invariance: An Interval Estimation Approach Using Bayesian Structural Equation Modeling

In this study, we introduce an interval estimation approach based on Bayesian structural equation modeling to evaluate factorial invariance. For each tested parameter, the size of noninvariance with an uncertainty interval (i.e. highest density interval [HDI]) is assessed via Bayesian parameter estimation. By comparing the most credible values (i.e. 95% HDI) with a region of practical equivalence (ROPE), the Bayesian approach allows researchers to (1) support the null hypothesis of practical invariance, and (2) examine the practical importance of the noninvariant parameter. Compared to the traditional likelihood ratio test, simulation results suggested that the proposed Bayesian approach could offer additional insight into evaluating factorial invariance, thus, leading to more informative conclusions. We provide an empirical example to demonstrate the procedures necessary to implement the proposed method in applied research. The importance of and influences on the choice of an appropriate ROPE are discussed.