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.     

Evaluation of the Bayesian and Maximum Likelihood Approaches in Analyzing Structural Equation Models with Small Sample Sizes

The main objective of this article is to investigate the empirical performances of the Bayesian approach in analyzing structural equation models with small sample sizes. The traditional maximum likelihood (ML) is also included for comparison. In the context of a confirmatory factor analysis model and a structural equation model, simulation studies are conducted with the different magnitudes of parameters and sample sizes n = da, where d = 2, 3, 4 and 5, and a is the number of unknown parameters. The performances are evaluated in terms of the goodness-of-fit statistics, and various measures on the accuracy of the estimates. The conclusion is: for data that are normally distributed, the Bayesian approach can be used with small sample sizes, whilst ML cannot.     

Time-specific Errors in Growth Curve Modeling: Type-1 Error Inflation and a Possible Solution with Mixed-Effects Models

Growth curve modeling (GCM) has been one of the most popular statistical methods to examine participants’ growth trajectories using longitudinal data. In spite of the popularity of GCM, little attention has been paid to the possible influence of time-specific errors, which influence all participants at each timepoint. In this article, we demonstrate that the failure to take into account such time-specific errors in GCM produces considerable inflation of type-1 error rates in statistical tests of fixed effects (e.g., coefficients for the linear and quadratic terms). We propose a GCM that appropriately incorporates time-specific errors using mixed-effects models to address the problem. We also provide an applied example to illustrate that GCM with and without time-specific errors would lead to different substantive conclusions about the true growth trajectories. Comparisons with other models in longitudinal data analysis and potential issues of model misspecification are discussed.     

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.     

高校教师教学水平评价概化理论预算限制下最佳样本量估计

使用“高校教师教学水平评价问卷”,要求566名学生对19名教师进行评价,对收集到的数据作不同的概化设计,包括t×i、（s：t）×i、（s：t）×（i：v）和（s：t）×（i：v）×o四种设计。基于概化理论,结合预算限制,统一LaGrange乘法公式,自行推导不同设计的最佳样本量公式,联合估计的方差分量,计算出不同设计的最佳样本量。结果表明：（1）LaGrange乘法统一公式表现出较强的通用性,能够适用于预算限制下各种概化设计;（2）评价场合是影响高校教师教学水平评价一个相当重要的因素;（3）（s：t）×（i：v）×o是高校教师教学水平评价概化理论预算限制下最优概化设计;（4）高校教师教学水平评价概化理论预算限制下,每位教师最佳评价学生人数为20人,每个维度最佳评价题目数为3题。     

When the Third is Dead: Memory,Mourning, and Witnessing in the Aftermath of the Holocaust1

The origins of psychoanalysis, as well as the concerns of our daily endeavors, center on engagement with the fate of the unbearable – be it wish, affect, or experience. In this paper, I explore psychological states and dynamics faced by survivors of genocide and their children in their struggle to sustain life in the midst of unremitting deadliness. Toward this continuous effort, I re‐examine Freud’s theoretical formulations concerning memory and mourning, elaborate André Green’s concept of the ‘Dead Mother’, and introduce more recent work on the concepts of the ‘third’ and ‘thirdness’. Throughout, my thoughts are informed by our clinical experience with the essential role of witnessing in sustaining life after massive trauma. I bring aspects of all these forms of knowing to reflections about a poem by Primo Levi entitled Unfinished business and to our own never finished business of avoiding denial while living in an age of genocide and under the aura of uncontained destructiveness.