Modeling for Categorical Data and Handling Over-dispersion via Computer Intensive Methods

当我们分析协变量对多(包括三个以上)类别反应数据的影响时，研究了使用计算机密集方法处理过离散性的可能性和性能。利用Dirichlet-多项分布的Beta-二项分解，建立了一个基于物理随机数发生器给出的均匀随机数生成Dirichlet-多项随机数的系统。利用该系统，我们在分类数据分析中对协变量效应评估中处理过度离散性的影响进行了模拟研究。特别是与多项式极大似然法(ML)、Dirichlet-多项式极大似然法或基于二阶矩的广义估计方程(GEE)相比，我们着重说明了Bootstrap方法和刀切方法等计算机密集型方法的有效性。模拟结果表明，计算机密集型方法可与Dirichlet多项式最大似然法相媲美，即使后者在研究环境下是最优的。基于这些模拟结果，我们将上述方法应用于一些实际数据，发现存在Dirichlet多项式最大似然方法不能拟合数据而GEE和计算机密集型方法能够成功拟合数据的情况。由于计算密集型方法本身的计算量很大，因此评估其性能的仿真研究需要大量的计算负担。因此，我们研究了通过分布式并行处理方法同时使用多台计算机来有效使用计算资源来缓解这一问题。这个问题需要进一步探讨。

Possibilities and performances of the use of computer intensive methods for handling over-dispersion are studied when we analyze effects of covariates on multi (including more than three) category response data. We set up a system to generate Dirichlet-multinominal random numbers based on the uniform random numbers given by a physical random number generator, using beta-binomial decomposition of the Dirichlet-multinominal distribution. Using this system, we carried out simulation studies for examining effects of handling over-dispersion regarding covariate effect evaluation in the categorical data analysis. Especially we focus on elucidating effectiveness of computer intensive methods such as bootstrap and jackknife methods, compared to the alternatives like multinomial maximum likelihood method (ML), Dirichlet-multinominal ML or generalized estimating equations (GEE) based on their moments up to 2nd order. The simulation results showed the computer intensive methods were comparable to the Dirichlet-multinominal ML, even in the case where the latter is the optimum for the study setting. Furthermore, it is indicated that the computer intensive methods are robust to the departure of the postulated models from the true structure.Based on these simulation results, we applied above methods to some real data and we found that there was a case where the Dirichlet-multinominal ML failed to fit the data while the GEE and the computer intensive methods were able to fit the data successfully.Since the calculations of the computational intensive methods themselves are costly, the simulation studies to evaluate their performances require substantial amount of computational burden. We therefore investigated efficient use of computational resources to ease this problem with simultaneous use of several computers via distributed parallel processing methods. This issue needs to be explored further.