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**Implementation and performance issues in the Bayesian and likelihood fitting of multilevel models.**
*(English)*
Zbl 1037.62013

The paper presents results of a large simulation study directed to comparison of (i) Bayesian and likelihood fitting methods, and (ii) computational efficiency of several estimation procedures based on Markov Chain Monte Carlo (MCMC) for several types of multilevel models, namely random slopes regression (RSR) and random-effects logistic (RELF) models. Being the co-developers of the Bayesian MCMC capabilities in the multilevel modelling package MlwiN, the authors examine the relative performance, in the sense of point and interval estimation accuracy, of likelihood and Bayesian fitting procedures in two-level RSR. In the case of RELF models they compare (in terms of required CPU time to achieve given accuracy) a number of MCMC fitting methods, including adaptive rejection sampling and a specially for MLwiN developed procedure based on adaptive hybrid Metropolis-Gibbs sampling.

It is demonstrated that the Bayesian approach with a particular choice of diffuse inverse Wishart prior distribution for (co)variance parameters performs at least as well as maximum likelihood in RSR models with medium sample sizes, but neither approach performs sufficiently well for small samples. An adaptive hybrid Metropolis-Gibbs sampling method outperforms the Gibbs sampling in RELF models, which have been considered.

It is demonstrated that the Bayesian approach with a particular choice of diffuse inverse Wishart prior distribution for (co)variance parameters performs at least as well as maximum likelihood in RSR models with medium sample sizes, but neither approach performs sufficiently well for small samples. An adaptive hybrid Metropolis-Gibbs sampling method outperforms the Gibbs sampling in RELF models, which have been considered.

Reviewer: N. M. Zinchenko (Kyïv)

### MSC:

62F15 | Bayesian inference |

65C40 | Numerical analysis or methods applied to Markov chains |

62J99 | Linear inference, regression |

65C60 | Computational problems in statistics (MSC2010) |

### Keywords:

multilevel models; Markov Chain Monte Carlo methods; Bayesian approach; hybrid Metropolis-Gibbs sampling; Gibbs sampling; maximum likelihood methods; random-effects; logistic regression; random slopes regression; variance components; hierarchical models### Software:

MLwiN
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\textit{W. J. Browne} and \textit{D. Draper}, Comput. Stat. 15, No. 3, 391--420 (2000; Zbl 1037.62013)

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