Model fitting for categorical data and handling over-dispersion

在这项研究中，对有分类回答的数据的方法进行了研究。特别是，为了评估这类数据集的协变量效应，发展和研究了处理超离散性的分析方法。为了纳入基于多项分布的模型无法解释的超离散性，我们考虑了Dirichlet-多项分布。考虑了有/无有序信息的关联指数关系的建模方法，如多项对数、累积对数、连续率对数、相邻类别对数、互补对数和立体型模型，以及由协变量构成的线性预测因子。研究的基本方法是基于分布延拓的极大似然方法，如狄利克雷-多项分布及其延拓，分布均值-方差结构的广义估计方程，以及计算机密集的方法，如刀刃方法，我们开发了这类数据的分析系统，并对实际发表的数据进行了分析。通过这些分析，明确了订单信息的过度分散和建模的影响，以及基于该方法的差异。同时也发现了Dirichlet多项式分布的局限性，特别是在处理欠分散的问题上。通过这些模拟研究，我们得出的结论是，在数据的基线分布为Dirichlet多项式的情况下，这些方法在平均结构参数的偏差和方差估计方面是一致的，并且基于刀刃的方法具有相当的能力来考虑过度离散性的影响。

In this study, methods for data that have categorical responses are investigated. Especially, analytical methods to deal with over-dispersion are developed and investigated in order to evaluate covariate effects for such data sets.To incorporate the over-dispersion that cannot be explained by models based on multinomial distribution, we considered Dirichlet-multinomial distribution. Methods which model the relations of indices of association with/without ordered information, such as multinomial logits, cumulative logits, continuation ratio logits, adjacent category logits, complimentary log-log and stereo type model, and linear predictors constructed from the covariates are considered. Fundamental approaches for the work are the maximum likelihood methods based on the distributional extension of the multinomial distribution, such as Dirichlet-multinomial distribution and its extension, generalized estimation equations for the mean-variance structure of the distribution, and computer intensive methods such as the Jackknife method.We developed analysis systems for such data and analyzed several actual published data. With these analyses, effects of the over-dispersion and modeling of the order information, as wen as differences based on the approach were made clear. The limitations for the Dirichlet-multinomial distribution, especially to handle under-dispersion, were also detected.In order to study performance of the developed methods, some simulation studies were conducted as well. With these simulation studies, we concluded that the methods were in good agreement in terms of biases and variance estimates of the mean structure parameters in the case where the baseline distributions of the data were Dirichlet-multinomial, and that the method based on Jackknife had comparable abilities to incorporate the effects of over-dispersion.