Model fitting for categorical data and handling over-dispersion

在本研究中，研究了具有分类反应的数据的方法。特别是，为了评估这些数据集的协变量效应，开发和研究了处理过度分散的分析方法。为了考虑基于多项分布的模型无法解释的过分散，我们考虑了dirichlet -多项分布。考虑了多项logits、累积logits、连续比logits、邻类logits、互补对数-对数模型和立体模型以及由协变量构建的线性预测模型等对有/无有序信息关联指标关系的建模方法。这项工作的基本方法是基于多项分布扩展的极大似然方法，如dirichlet -多项分布及其扩展，分布的均值-方差结构的广义估计方程，以及计算机密集型方法，如Jackknife方法。我们为这些数据开发了分析系统，并分析了几个实际发表的数据。通过这些分析，明确了该方法对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。dirichlet -多项分布的局限性，特别是在处理欠分散时，也被发现。为了研究所开发方法的性能，还进行了一些仿真研究。通过这些模拟研究，我们得出结论，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很一致的，并且基于Jackknife的方法具有相当的能力来考虑过度分散的影响。

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.