廣泛膨脹卜瓦松回歸模型之分散性分數檢定統計量

Abstract

計次資料配適卜瓦松回歸模型中，一個常見的問題為過度分散現象，而零膨脹為造成此現象的原因之一。Cox(1983)是第一個提出方法來檢定過度分散存在與否；Dean(1992)稍後提出了另一種方法。而Dean(1992)提出的方法比Cox(1983) 略勝一籌的原因是:Cox(1983)的檢測過程中沒有共變數，而Dean則有。此外，除了過度分散的問題之外，另一個統計問題則是不足分散現象。當在資料中發現變異數大於期望值，稱之為過度分散；反之，則為不足分散。由此兩種現象被統稱為非均等分散。雖然已經有許多檢定方法提出檢定過度分散現象，但僅以有限分數檢定檢定量方法檢定不足分散現象，且少考有考慮零膨脹現象之外所造成之過度分散與不足分散情況。因此本論文，使用COM-卜瓦松分布來求得非均等分散檢定統計量，並延伸其方法到其它膨脹狀況，最後以模擬來驗證此方法。

A common problem in fitting count data with Poisson regression model is overdispersion, zero-inflation is one of the causes. Cox(1983) is the first one who proposed tests for the existence of overdispersion. Dean(1992) proposed another method. The reason that what Dean(1992) proposed is better than Cox(1983) is that Cox(1983) method proceeds tests without covariates, but Dean’s method did. In addition to the overdispersion problem, another statistical issue is underdispersion. When data show that variance is greater than mean, it is called overdispersion. In contrast, call underdispersion. Both types of data are called unequi-dispersed. Although many methods had been proposed to test overdispersion, limited methods have dealt with test for underdispersion using score test statistic and take into the account situations beyond zero-inflated over- or underdispersion. In this thesis, we use COM-Poisson distribution to derive unequi-dispersion test statistic, and extend the methods to other inflated cases, and conduct simulations to justify our methods.