Bayesian multi-way balanced nested MANOVA models with random effects and a large number of the main factor levels
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Bayesian multi-way balanced nested MANOVA models with random effects and a large number of the main factor levels Chun-Lung Su1 Received: 7 October 2019 / Accepted: 15 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This article considers the balanced nested multi-way multivariate analysis of variance (MANOVA) models with random effects and a large number of main factor levels under certain prior assumptions. Two different parametrizations for the MANOVA models with random effects and the corresponding explicit asymptotics are established. The asymptotic approximations are then compared with those obtained from the classical large-sample approximation and Markov chain Monte Carlo method via a balanced nested two-way MANOVA model with random effects. Simulation results demonstrate that our approach is superior to the classical approximation method on estimating the posterior standard deviations of variance component parameters. Keywords Asymptotic posterior · Balanced nested model · MANOVA · Sufficient reduction statistics
1 Introduction In certain experimental designs, nested models are commonly used when constraints prevent us from crossing every level of one factor with every level of the other factor. In particular, the balanced nested case has the advantage of minimizing the sensitivity of statistical assumptions and maximizing the power of the test (Montgomery 2012). Classical large-sample inferences on testing treatment or sub-treatment effects are usually made under the assumption that the number of levels of main and nested factors is small and replication is large at each level of the nested factor (Arnold 1980). For instance, in the two-way balanced nested ANOVA or MANOVA models comparisons are made between the products from two suppliers with three shifts nested within each supplier and large replications made in each shift; in mammography or colonoscopy examination, the number of levels of clinics and laboratory scientists
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Chun-Lung Su [email protected] Department of Statistics, Tunghai University, Taichung, Taiwan
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nested within clinics is small while the number of physical examination for human subjects is large. However, in some experimental design, data may consist of a larger number of levels of the main factor and a small number of levels of nested factor and replications. For example, in some 24-h automatic factories, data consist of hundreds of machines (main factor) and a small number of shift operators (nested factor) and in-process destructive tests (replications). For this type of data, some authors, from the non-Bayesian perspective, have introduced asymptotic results for test statistics under univariate ANOVA models without normality assumption (see Boos and Brownie 1995; Akritas and Arnold 2000; Bathke 2002; Wang and Akritas 2006); similar asymptotics of three common test statistics for MANOVA models under non-normality have been shown in Gupta et al. (2006). For a Bayesian counterpart, Su (2017) has developed large sample results
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