Comparações múltiplas bayesianas em modelos normais homocedásticos e heterocedásticos

Abstract

RESUMOProcedimentos de comparações múltiplas são utilizados para comparar médias de níveis de um fator, porém, os testes mais populares apresentam problemas de ambiguidade dos resultados e de controle do erro tipo I, além de terem seus desempenhos influenciados negativamente no caso de heterogeneidade de variâncias e não balanceamento. Objetivou-se, neste trabalho propor alternativas bayesianas para comparações múltiplas considerando os casos de homogeneidade e heterogeneidade de variâncias. A metodologia utilizada nesse trabalho foi baseada na distribuição a posteriori t multivariada. Foram geradas k cadeias de médias, utilizando o método de Monte Carlo. Foi obtida a amplitude padronizada sob H 0 , obtida na distribuição a posteriori das médias, contemplando a possibilidade de se analisar tanto o caso de variâncias heterogêneas como o caso de variâncias homogêneas. Os procedimentos de comparações múltiplas bayesianos foram propostos com sucesso. Termos para indexação:Simulação, erro tipo I, poder, Monte Carlo, não balanceamento. ABSTRACTMultiple comparison procedures are used to compare factor's levels means, since the most popular tests show problems related to ambiguous results and to the control of the type I error rates. Moreover, their performance is worst in heterocedastics and unbalanced cases. The objective of this work is to propose a Bayesian alternative for multiple comparisons considering the homocedastic and heterocedastic normal models. The methodology adopted in this paper was based on a posteriori multivariate t distribution. It was used k Monte Carlo chains of the mean factor to make inferences. The standardized range was obtained, under H 0 , from the posteriori distribution of the means, for the analysis of homocedastic and heterocedastic cases. The bayesian procedures of multiple comparisons were successfully proposed.

Implementation of Bayesian tests pbayes and dbayes for randomized block design in R code

“…The dbayes test calculates the difference between the pairs of the means and compares the absolute value with the least significant difference (lsd). The null hypothesis, H0: μi = μi', is rejected when the modulus of the difference between the pairs is greater than the lsd [12].…”

“…In addition, the qpostbayes function generates a sample of size n of the multivariate t distribution, this step is essential for performing the pbayes and dbayes tests. By means of the Monte Carlos method, k chains of means based on the multivariate a posteriori distribution are obtained [12]. After, densities of the standardized amplitude distribution q, expression (1), are obtained by means of the same function.…”

“…The pbayes test calculates the probability of the intervals to contain the value zero, if the zero is contained in the interval, the treatments are considered equal. These intervals are determined by the lower (LI ii ') and upper (LS ii ') limits according to expression (6) [12]:…”

“…Andrade et al [11] implemented in R code the Bayes function, which allows Bayesian tests (pbayes and dbayes) to be performed in the context of completely randomized design (CRD). These were proposed by Andrade and Ferreira [12] and can be used for both balanced and unbalanced data, with analysis of variance being valid or not, for both types of data.…”