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）类别响应数据的影响时，研究了使用计算机密集方法处理过度分散的可能性和性能。在物理随机数生成器给出均匀随机数的基础上，利用dirichlet -多项式分布的-二项分解，建立了生成dirichlet -多项式随机数的系统。利用该系统，我们进行了模拟研究，以检验在分类数据分析中协变量效应评估中处理过度分散的影响。与多项式极大似然方法（ML）、dirichlet -多项式ML或基于二阶矩的广义估计方程（GEE）等替代方法相比，我们特别关注于阐明计算机密集型方法（如bootstrap和jackknife方法）的有效性。仿真结果表明，计算机密集型方法与dirichlet -多项式ML相当，即使后者是研究环境的最佳选择。此外，计算机密集型方法对于假设模型与真实结构的偏离具有较强的鲁棒性。基于这些模拟结果，我们将上述方法应用于一些实际数据，我们发现存在dirichlet -多项式ML无法拟合数据的情况，而GEE和计算机密集型方法能够成功拟合数据。由于计算密集型方法本身的计算成本很高，因此评估其性能的仿真研究需要大量的计算负担。因此，我们研究了计算资源的有效使用，通过分布式并行处理方法同时使用多台计算机来缓解这个问题。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.