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1.
Abstract

Recently, researchers have used multilevel models for estimating intervention effects in single-case experiments that include replications across participants (e.g., multiple baseline designs) or for combining results across multiple single-case studies. Researchers estimating these multilevel models have primarily relied on restricted maximum likelihood (REML) techniques, but Bayesian approaches have also been suggested. The purpose of this Monte Carlo simulation study was to examine the impact of estimation method (REML versus Bayesian with noninformative priors) on the estimation of treatment effects (relative bias, root mean square error) and on the inferences about those effects (interval coverage) for autocorrelated multiple-baseline data. Simulated conditions varied with regard to the number of participants, series length, and distribution of the variance within and across participants. REML and Bayesian estimation led to estimates of the fixed effects that showed little to no bias but that differentially impacted the inferences about the fixed effects and the estimates of the variances. Implications for applied researchers and methodologists are discussed.  相似文献   

2.
Multilevel Structural equation models are most often estimated from a frequentist framework via maximum likelihood. However, as shown in this article, frequentist results are not always accurate. Alternatively, one can apply a Bayesian approach using Markov chain Monte Carlo estimation methods. This simulation study compared estimation quality using Bayesian and frequentist approaches in the context of a multilevel latent covariate model. Continuous and dichotomous variables were examined because it is not yet known how different types of outcomes—most notably categorical—affect parameter recovery in this modeling context. Within the Bayesian estimation framework, the impact of diffuse, weakly informative, and informative prior distributions were compared. Findings indicated that Bayesian estimation may be used to overcome convergence problems and improve parameter estimate bias. Results highlight the differences in estimation quality between dichotomous and continuous variable models and the importance of prior distribution choice for cluster-level random effects.  相似文献   

3.
We compared six common methods in estimating the 2-1-1 (level-2 independent, level-1 mediator, level-1 dependent) multilevel mediation model with a random slope. They were the Bayesian with informative priors, the Bayesian with non-informative priors, the Monte-Carlo, the distribution of the product, the bias-corrected, and the bias-uncorrected parametric percentile residual bootstrap. The Bayesian method with informative priors was superior in relative mean square error (RMSE), power, interval width, and interval imbalance. The prior variance and prior mean were also varied and examined. Decreasing the prior variance increased the power, reduced RMSE and interval width when the prior mean was the true value, but decreasing the prior variance reduced the power when the prior mean was set incorrectly. The influence of misspecification of prior information of the b coefficient on multilevel mediation analysis was greater than that on coefficient a. An illustrate example with the Bayesian multilevel mediation was provided.  相似文献   

4.
对正交频分复用(OFDM)系统中的两种导频辅助信道估计方法，即最大似然估计(MLE)和贝叶斯最小均方误差估计(MMSEE)，在估计误差性能方面的特性进行了详尽的比较研究。理论分析及计算机仿真试验表明，当信道噪比较低时，MMSEE有比MLE更好的估计误差性能，但有比MLE复杂得多的计算复杂度。而当信道信噪比较高或插入导引符号序列数目足够多时，MMSEE与MLE误差估计性能基本一致。  相似文献   

5.
Appropriate model specification is fundamental to unbiased parameter estimates and accurate model interpretations in structural equation modeling. Thus detecting potential model misspecification has drawn the attention of many researchers. This simulation study evaluates the efficacy of the Bayesian approach (the posterior predictive checking, or PPC procedure) under multilevel bifactor model misspecification (i.e., ignoring a specific factor at the within level). The impact of model misspecification on structural coefficients was also examined in terms of bias and power. Results showed that the PPC procedure performed better in detecting multilevel bifactor model misspecification, when the misspecification became more severe and sample size was larger. Structural coefficients were increasingly negatively biased at the within level, as model misspecification became more severe. Model misspecification at the within level affected the between-level structural coefficient estimates more when data dependency was lower and the number of clusters was smaller. Implications for researchers are discussed.  相似文献   

6.
The purpose of this simulation study was to assess the performance of latent variable models that take into account the complex sampling mechanism that often underlies data used in educational, psychological, and other social science research. Analyses were conducted using the multiple indicator multiple cause (MIMIC) model, which is a flexible and effective tool for relating observed and latent variables. The data were simulated in a hierarchical framework (e.g., individuals nested in schools) so that a multilevel modeling approach would be appropriate. Analyses were conducted accounting for and not accounting for the nested data to determine the impact of ignoring such multilevel data structures in full structural equation models. Results highlight the differences in modeling results when the analytic strategy is congruent with the data structure and what occurs when this congruency is absent. Type I error rates and power for the standard and multilevel methods were similar for within-cluster variables and for the multilevel model with between-cluster variables. However, Type I error rates were inflated for the standard approach when modeling between-cluster variables.  相似文献   

7.
This article examined the role of centering in estimating interaction effects in multilevel structural equation models. Interactions are typically represented by product term of 2 variables that are hypothesized to interact. In multilevel structural equation modeling (MSEM), the product term involving Level 1 variables is decomposed into within-cluster and between-cluster random components. The choice of centering affects the decomposition of the product term, and therefore affects the sample variance and covariance associated with the product term used in the maximum likelihood fitting function. The simulation study showed that for an interaction between a Level 1 variable and a Level 2 variable, the product term of uncentered variables or the product term of grand mean centered variables produced unbiased estimates in both Level 1 and Level 2 models. The product term of cluster mean centered variables produced biased estimates in the Level 1 model. For an interaction between 2 Level 1 variables, the product term of cluster mean centered variables produced unbiased estimates in the Level 1 model, whereas the product term of grand mean centered variables produced unbiased estimates for the Level 1 model. Recommendations for researchers who wish to estimate interactions in MSEM are provided.  相似文献   

8.
Multilevel modeling is a statistical approach to analyze hierarchical data that consist of individual observations nested within clusters. Bayesian method is a well-known, sometimes better, alternative of Maximum likelihood method for fitting multilevel models. Lack of user friendly and computationally efficient software packages or programs was a main obstacle in applying Bayesian multilevel modeling. In recent years, the development of software packages for multilevel modeling with improved Bayesian algorithms and faster speed has been growing. This article aims to update the knowledge of software packages for Bayesian multilevel modeling and therefore to promote the use of these packages. Three categories of software packages capable of Bayesian multilevel modeling including brms, MCMCglmm, glmmBUGS, Bambi, R2BayesX, BayesReg, R2MLwiN and others are introduced and compared in terms of computational efficiency, modeling capability and flexibility, as well as user-friendliness. Recommendations to practical users and suggestions for future development are also discussed.  相似文献   

9.
With the increasing use of international survey data especially in cross-cultural and multinational studies, establishing measurement invariance (MI) across a large number of groups in a study is essential. Testing MI over many groups is methodologically challenging, however. We identified 5 methods for MI testing across many groups (multiple group confirmatory factor analysis, multilevel confirmatory factor analysis, multilevel factor mixture modeling, Bayesian approximate MI testing, and alignment optimization) and explicated the similarities and differences of these approaches in terms of their conceptual models and statistical procedures. A Monte Carlo study was conducted to investigate the efficacy of the 5 methods in detecting measurement noninvariance across many groups using various fit criteria. Generally, the 5 methods showed reasonable performance in identifying the level of invariance if an appropriate fit criterion was used (e.g., Bayesian information criteron with multilevel factor mixture modeling). Finally, general guidelines in selecting an appropriate method are provided.  相似文献   

10.
This simulation study assesses the statistical performance of two mathematically equivalent parameterizations for multitrait–multimethod data with interchangeable raters—a multilevel confirmatory factor analysis (CFA) and a classical CFA parameterization. The sample sizes of targets and raters, the factorial structure of the trait factors, and rater missingness are varied. The classical CFA approach yields a high proportion of improper solutions under conditions with small sample sizes and indicator-specific trait factors. In general, trait factor related parameters are more sensitive to bias than other types of parameters. For multilevel CFAs, there is a drastic bias in fit statistics under conditions with unidimensional trait factors on the between level, where root mean square error of approximation (RMSEA) and χ2 distributions reveal a downward bias, whereas the between standardized root mean square residual is biased upwards. In contrast, RMSEA and χ2 for classical CFA models are severely upwardly biased in conditions with a high number of raters and a small number of targets.  相似文献   

11.
This article examines whether Bayesian estimation with minimally informed prior distributions can alleviate the estimation problems often encountered with fitting the true score multitrait–multimethod structural equation model with split-ballot data. In particular, the true score multitrait–multimethod structural equation model encounters an empirical underidentification when (a) latent variable correlations are homogenous, and (b) fitted to data from a 2-group split-ballot design; an understudied case of empirical underidentification due to a planned missingness (i.e., split-ballot) design. A Monte Carlo simulation and 3 empirical examples showed that Bayesian estimation performs better than maximum likelihood (ML) estimation. Therefore, we suggest using Bayesian estimation with minimally informative prior distributions when estimating the true score multitrait–multimethod structural equation model with split-ballot data. Furthermore, given the increase in planned missingness designs in psychological research, we also suggest using Bayesian estimation as a potential alternative to ML estimation for analyses using data from planned missingness designs.  相似文献   

12.
This article proposes a novel exploratory approach for assessing how the effects of Level-2 predictors differ across Level-1 units. Multilevel regression mixture models are used to identify latent classes at Level 1 that differ in the effect of 1 or more Level-2 predictors. Monte Carlo simulations are used to demonstrate the approach with different sample sizes and to demonstrate the consequences of constraining 1 of the random effects to 0. An application of the method to evaluate heterogeneity in the effects of classroom practices on students is used to show the types of research questions that can be answered with this method and the issues faced when estimating multilevel regression mixtures.  相似文献   

13.
Structural equation modeling (SEM) is now a generic modeling framework for many multivariate techniques applied in the social and behavioral sciences. Many statistical models can be considered either as special cases of SEM or as part of the latent variable modeling framework. One popular extension is the use of SEM to conduct linear mixed-effects modeling (LMM) such as cross-sectional multilevel modeling and latent growth modeling. It is well known that LMM can be formulated as structural equation models. However, one main difference between the implementations in SEM and LMM is that maximum likelihood (ML) estimation is usually used in SEM, whereas restricted (or residual) maximum likelihood (REML) estimation is the default method in most LMM packages. This article shows how REML estimation can be implemented in SEM. Two empirical examples on latent growth model and meta-analysis are used to illustrate the procedures implemented in OpenMx. Issues related to implementing REML in SEM are discussed.  相似文献   

14.
Latent class analysis is an analytic technique often used in educational and psychological research to identify meaningful groups of individuals within a larger heterogeneous population based on a set of variables. This technique is flexible, encompassing not only a static set of variables but also longitudinal data in the form of growth mixture modeling, as well as the application to complex multilevel sampling designs. The goal of this study was to investigate—through a Monte Carlo simulation study—the performance of several methods for parameterizing multilevel latent class analysis. Of particular interest was the comparison of several such models to adequately fit Level 1 (individual) data, given a correct specification of the number of latent classes at both levels (Level 1 and Level 2). Results include the parameter estimation accuracy as well as the quality of classification at Level 1.  相似文献   

15.
The present study examined students' attitudes toward science and associated constructs, based on the theories of reasoned action and planned behavior, and explored relationships between individual and school-related variables common to the research literature. Responses from 1,291 students in Grades 5 through 10 were collected using the 30-item Behaviors, Related Attitudes, and Intentions toward Science (BRAINS) Survey along with background information questions. Additional self-report data were collected from teachers (n = 56; 82.4%) in participating schools (n = 68) to obtain information about their education and experience, characteristics and practices, as well as other classroom variables, which could influence students' outlook. Student information, teacher data collected, and other data compiled about participating schools, were used to explore patterns in students' attitudes, beliefs, and intentions. These variables were used to generate multivariate multilevel models through a forward construction process. The final model presented favors individual variables to explain differences in students' responses on all five of the BRAINS subscales, more than group-level variables captured. Of the predictor variables explored, students' perceived science ability and frequency of talk with family were influential on all subscales, and increasing these variables had a positive effect on the estimated mean scores according to the final model presented. Findings from this study also include commonly observed relationships, such as the decline in attitudes over time, but these were found to be less pervasive in this sample. The paper concludes with a discussion about the comparative ineffectiveness of teacher and school-related variables in explaining students' attitudes toward science in this study, in light of design decisions and limitations, to guide future investigations.  相似文献   

16.
Abstract

Applied researchers often find themselves making statistical inferences in settings that would seem to require multiple comparisons adjustments. We challenge the Type I error paradigm that underlies these corrections. Moreover we posit that the problem of multiple comparisons can disappear entirely when viewed from a hierarchical Bayesian perspective. We propose building multilevel models in the settings where multiple comparisons arise. Multilevel models perform partial pooling (shifting estimates toward each other), whereas classical procedures typically keep the centers of intervals stationary, adjusting for multiple comparisons by making the intervals wider (or, equivalently, adjusting the p values corresponding to intervals of fixed width). Thus, multilevel models address the multiple comparisons problem and also yield more efficient estimates, especially in settings with low group-level variation, which is where multiple comparisons are a particular concern.  相似文献   

17.
This simulation study examines the efficacy of multilevel factor mixture modeling (ML FMM) for measurement invariance testing across unobserved groups when the groups are at the between level of multilevel data. To this end, latent classes are generated with class-specific item parameters (i.e., factor loading and intercept) across the between-level classes. The efficacy of ML FMM is evaluated in terms of class enumeration, class assignment, and the detection of noninvariance. Various classification criteria such as Akaike’s information criterion, Bayesian information criterion, and bootstrap likelihood ratio tests are examined for the correct enumeration of between-level latent classes. For the detection of measurement noninvariance, free and constrained baseline approaches are compared with respect to true positive and false positive rates. This study evidences the adequacy of ML FMM. However, its performance heavily depends on the simulation factors such as the classification criteria, sample size, and the magnitude of noninvariance. Practical guidelines for applied researchers are provided.  相似文献   

18.
Small samples are common in growth models due to financial and logistical difficulties of following people longitudinally. For similar reasons, longitudinal studies often contain missing data. Though full information maximum likelihood (FIML) is popular to accommodate missing data, the limited number of studies in this area have found that FIML tends to perform poorly with small-sample growth models. This report demonstrates that the fault lies not with how FIML accommodates missingness but rather with maximum likelihood estimation itself. We discuss how the less popular restricted likelihood form of FIML, along with small-sample-appropriate methods, yields trustworthy estimates for growth models with small samples and missing data. That is, previously reported small sample issues with FIML are attributable to finite sample bias of maximum likelihood estimation not direct likelihood. Estimation issues pertinent to joint multiple imputation and predictive mean matching are also included and discussed.  相似文献   

19.
In applied research, such as with motivation theories, typically many variables are theoretically implied predictors of an outcome and several interactions are assumed (e.g., Watt, 2004). However, estimation problems that might arise when several interaction and/or quadratic effects are analyzed simultaneously have not been investigated because simulation studies on interaction effects in the structural equation modeling framework have mainly focused on small models that contain single interaction effects. In this article, we show that traditional approaches can provide estimates with low accuracy when complex models are estimated. We introduce an adaptive Bayesian lasso approach with spike-and-slab priors that overcomes this problem. Using a complex model in a simulation study, we show that the parameter estimates of the proposed approach are more accurate in situations with high multicollinearity or low reliability compared with a standard Bayesian lasso approach and typical frequentist approaches (i.e., unconstrained product indicator approach and latent moderated structures approach).  相似文献   

20.
Measurement bias can be detected using structural equation modeling (SEM), by testing measurement invariance with multigroup factor analysis (Jöreskog, 1971;Meredith, 1993;Sörbom, 1974) MIMIC modeling (Muthén, 1989) or restricted factor analysis (Oort, 1992,1998). In educational research, data often have a nested, multilevel structure, for example when data are collected from children in classrooms. Multilevel structures might complicate measurement bias research. In 2-level data, the potentially “biasing trait” or “violator” can be a Level 1 variable (e.g., pupil sex), or a Level 2 variable (e.g., teacher sex). One can also test measurement invariance with respect to the clustering variable (e.g., classroom). This article provides a stepwise approach for the detection of measurement bias with respect to these 3 types of violators. This approach works from Level 1 upward, so the final model accounts for all bias and substantive findings at both levels. The 5 proposed steps are illustrated with data of teacher–child relationships.  相似文献   

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