On the Performance of Bayesian Approaches in Small Samples: A Comment on Smid, McNeish, Miocevic, and van de Schoot (2020)

Zitzmann, Steffen; Luedtke, Oliver; Robitzsch, Alexander; Hecht, Martin

doi:10.1080/10705511.2020.1752216

Cited by 57 publications

(48 citation statements)

References 40 publications

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“…Note that pulling the variance estimates away from zero corresponds to adding a value to these estimates. A formal argument for why such a prior reduces the MSE was only recently presented by Zitzmann et al ( 2020 ). For reasons of completeness and comparability with the two previously presented estimators, we illustrate the strategy here once more, using the example model from above.…”

Section: The Indirect Strategymentioning

confidence: 99%

“…As Zitzmann et al ( 2020 ) showed in their Appendix C, the mean of this distribution can be approximated as:…”

Section: The Indirect Strategymentioning

confidence: 99%

“…Only recently, Smid et al ( 2020 ) discussed how priors can be “thoughtfully” constructed on the basis of previous knowledge about the parameter of interest (e.g., on the basis of a previous study or a meta-analysis) in order to reduce small-sample bias. However, it has been argued that the variability of an estimator should not be ignored when evaluating the quality of a method (e.g., Greenland, 2000 ; Zitzmann et al, 2020 ), particularly when the sample size is small. Therefore, other suggestions for specifying the prior have been aimed at reducing the Mean Squared Error (MSE), which combines bias and variability: MSE = bias 2 + variability.…”

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confidence: 99%

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Prior Specification for More Stable Bayesian Estimation of Multilevel Latent Variable Models in Small Samples: A Comparative Investigation of Two Different Approaches

2021

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Bayesian approaches for estimating multilevel latent variable models can be beneficial in small samples. Prior distributions can be used to overcome small sample problems, for example, when priors that increase the accuracy of estimation are chosen. This article discusses two different but not mutually exclusive approaches for specifying priors. Both approaches aim at stabilizing estimators in such a way that the Mean Squared Error (MSE) of the estimator of the between-group slope will be small. In the first approach, the MSE is decreased by specifying a slightly informative prior for the group-level variance of the predictor variable, whereas in the second approach, the decrease is achieved directly by using a slightly informative prior for the slope. Mathematical and graphical inspections suggest that both approaches can be effective for reducing the MSE in small samples, thus rendering them attractive in these situations. The article also discusses how these approaches can be implemented in Mplus.

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Section: The Indirect Strategymentioning

confidence: 99%

“…As Zitzmann et al ( 2020 ) showed in their Appendix C, the mean of this distribution can be approximated as:…”

Section: The Indirect Strategymentioning

confidence: 99%

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confidence: 99%

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Prior Specification for More Stable Bayesian Estimation of Multilevel Latent Variable Models in Small Samples: A Comparative Investigation of Two Different Approaches

2021

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“…Lüdtke et al (2018) introduced a Bayesian approach to estimating the univariate STARTS model and showed that the Bayesian approach avoids estimation problems with appropriate prior distributions. Furthermore, the Bayesian approach allows for the stabilization of parameter estimates by specifying weakly-informative prior distributions (see Depaoli & Clifton, 2015;Smid et al, 2020;Zitzmann et al, 2020). This study proposes a Bayesian approach to estimating the bivariate STARTS model and implementing it in the software Stan (Stan Development Team, 2019).…”

Section: Model Using Markov Chain Monte Carlomentioning

confidence: 99%

A Bayesian Approach to Estimating Reciprocal Effects with the Bivariate STARTS Model using Markov Chain Monte Carlo

Lüdtke¹,

Robitzsch²,

Ulitzsch³

2021

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The bivariate Stable Trait, AutoRegressive Trait, and State (STARTS) model provides a general approach for estimating reciprocal effects between constructs over time. However, previous research has shown that this model is difficult to estimate using the maximum likelihood (ML) method (e.g., nonconvergence). In this article, we introduce a Bayesian approach for estimating the bivariate STARTS model and implement it in the software Stan. We discuss issues of model parameterization and show how appropriate prior distributions for model parameters can be selected. Specifically, we propose the four-parameter beta distribution as a flexible prior distribution for the autoregressive and cross-lagged effects. Using a simulation study, we show that the proposed Bayesian approach provides more accurate estimates than ML estimation in challenging data constellations. An example is presented to illustrate how the Bayesian approach can be used to stabilize the parameter estimates of the bivariate STARTS model.

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“…For mitigating estimation problems, it can be advantageous to choose a parameterization of the CFA that transforms the optimization problem of estimating unbounded parameters into an optimization involving bounded parameters. Furthermore, a parameterization with bounded or standardized parameters has the advantage that it is often more convenient for applied researchers for specifying thoughtful prior distributions Zitzmann et al, 2020). Thus, we suggest employing a parameterization of the CFA 1 In the joint estimation approach, the joint likelihood…”

Section: Parameterizationmentioning

confidence: 99%

A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models with Small Sample Sizes

Lüdtke¹,

Ulitzsch²,

Robitzsch³

2020

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With small to modest sample sizes and complex models, maximum likelihood (ML) estimation of confirmatory factor analysis (CFA) models can show serious estimation problems such as nonconvergence or parameter estimates that are outside the admissible parameter space. In the present article, we discuss two Bayesian estimation methods for stabilizing parameter estimates of a CFA: Penalized maximum likelihood (PML) estimation and Markov Chain Monte Carlo (MCMC) methods. We clarify that these use different Bayesian point estimates from the joint posterior distribution—the mode (PML) of the joint posterior distribution, and the mean (EAP) or mode (MAP) of the marginal posterior distribution—and discuss under which conditions the two methods produce different results. In a simulation study, we show that the MCMC method clearly outperforms PML and that these performance gains can be explained by the fact that MCMC uses the EAP as a point estimate. We also argue that it is often advantageous to choose a parameterization in which the main parameters of interest are bounded and suggest the four-parameter beta distribution as a prior distribution for loadings and correlations. Using simulated data, we show that selecting weakly informative four-parameter beta priors can further stabilize parameter estimates, even in cases when the priors were mildly misspecified. Finally, we derive recommendations and propose directions for further research.

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On the Performance of Bayesian Approaches in Small Samples: A Comment on Smid, McNeish, Miocevic, and van de Schoot (2020)

Cited by 57 publications

References 40 publications

Prior Specification for More Stable Bayesian Estimation of Multilevel Latent Variable Models in Small Samples: A Comparative Investigation of Two Different Approaches

Prior Specification for More Stable Bayesian Estimation of Multilevel Latent Variable Models in Small Samples: A Comparative Investigation of Two Different Approaches

A Bayesian Approach to Estimating Reciprocal Effects with the Bivariate STARTS Model using Markov Chain Monte Carlo

A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models with Small Sample Sizes

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