泊松分佈(Poisson distribution)是一經常被配適於稀有事件建模的機率分配，其應用領域相當的廣泛，如生物，商業，品質控制等。其中許多的應用均為兩群體均數的比較，如欲檢測一新的處理是否較原本的處理俱優越性(superiority)，或者欲驗證一新的方法相較於舊的方法是否俱有不劣性(non-inferiority)。因此，此研究的目標為發展假設檢定的方法，用於比較兩獨立的泊松樣本是否有優越性及不劣性。一般探討假設檢定方法時，均因干擾參數的出現而導致理論探討及計算上的困難。為因應此困境，本研究由簡入繁，亦即先探討相等式的虛無假設(the null hypothesis of equality)，繼而，再推展至非優越性的虛無假設(the null hypothesis of non-superiority)，最後將這些探究的假設檢定方法應用至檢定不劣性並驗證這些方法的適用性。
兩種Wald 檢定統計量是本研究主要的研究興趣。對應於這兩種檢定統計量的近似的假設檢定法，是利用其極限分配為常態分配的特性而衍生的。此研究裡，可推導得到近似的檢定法的檢定力函數及欲達成某一檢定力水平時所需的樣本數公式。並依據此檢定力函數檢驗此檢定法的有效性(validity)及不偏性(unbiasedness)。並且推廣一連續修正的方法至任何的樣本數組合。另外一方面，此研究亦介紹並推廣兩種p-值的正確(exact)檢定法。其中一種為信賴區間p值檢定法(Berger和Boss, 1994), 另一種為估計的p值檢定法(Krishnamoorthy和Thomson, 2004)。一般正確檢定法較需要繁瑣的計算，故此研究將提出某些步驟以降低計算的負擔。就信賴區間p值檢定法而言，其首要工作為縮減求算p值的範圍，並驗證所使用的檢定統計量是否滿足Barnard凸面(convexity)的條件。若此統計量符合凸面convexity的條件，且在Poisson 的問題上，則此正確的信賴區間p值將出現在屬於虛無假設的參數空間的邊界上。然而，對於估計的p值檢定法而言，因在虛無假設的參數空間上求得Poisson均數的最大概似估計值，並不簡單及無法直接求得，故在此研就，將以一Poisson均數的點估計值代替。對於正確的假設檢定方法，此研究亦提出一個欲達成某一檢定力水平時所需的樣本數的步驟。
此研究將透過一個廣大的數值分析來驗證之前所提出的假設檢定方法。其中，可發現這些近似的假設檢定法之間的差異會受到兩群體之樣本數之比率的影響，而連續性的修正於某些情況下確實能夠使型I誤差較能夠受到控制。另外，當樣本數不夠多時，正確的假設檢定法是較近似的假設檢定法適當，尤其在型I誤差的控制上更是明顯。最後，此研究所提出的假設檢定方法將實際應用於一組乳癌治療的資料。 / The Poisson distribution is a well-known suitable model
for modeling a rare events in variety fields such as biology,
commerce, quality control, and so on. Many applications involve
comparisons of two treatment groups and focus on showing the
superiority of the new treatment to the conventional one, or the
non-inferiority of the experimental implement to the standard
implement upon the cost consideration. We aim to develop statistical
tests for testing the superiority and non-inferiority by two
independent random samples from Poisson distributions. In developing
these tests, both computational and theoretical difficulties arise
from presence of nuisance parameters. In this study, we first
consider the problems with the null hypothesis of equality for
simplicity. The problems are extended to have a regular null
hypothesis of non-superiority next. Subsequently, the proposed
methods are further investigated in establishing the
non-inferiority.
Two types of Wald test statistics are of our main research interest.
The correspondent asymptotic testing procedures are developed by
using the normal limiting distribution. In our study, the asymptotic
distribution of the test statistics are derived. The asymptotic
power functions and the sample size formula are further obtained.
Given the power functions, we justify the validity and unbiasedness
of the tests. The adequate continuity correction term for these
tests is also found to reduce inflation of the type I error rate. On
the other hand, the exact testing procedures based on two exact
$p$-values, the confidence-interval $p$-value (Berger and Boos
(1994)), and the estimated $p$-value (Krishnamoorthy and Thomson
(2004)), are also applied in our study. It is known that an exact
testing procedure tends to involve complex computations. In this
thesis, several strategies are proposed to lessen the computational
burden. For the confidence-interval $p$-value, a truncated
confidence set is used to narrow the area for finding the $p$-value.
Further, the test statistic is verify whether they fulfill the
property of convexity. It is shown that under the convexity the
exact $p$-value occurs somewhere of the boundary of the null
parameter space. On the other hand, for the estimated $p$-value, a
simpler point estimate is applied instead of the use of the
restricted maximum likelihood estimators, which are less
straightforward in this problem. The estimated $p$-value is shown to
provide a conservative conclusion. The calculations of the sample
sizes required by using the two exact tests are discussed.
Intensive numerical studies show that the performances of the
asymptotic tests depend on the fraction of the two sample sizes and
the continuity correction can be useful in some cases to reduce the
inflation of the type I error rate. However, with small samples, the
two exact tests are more adequate in the sense of having a
well-controlled type I error rate. A data set of breast cancer
patients is analyzed by the proposed methods for illustration.
| Identifer | oai:union.ndltd.org:CHENGCHI/G0923545041 |
| Creators | 劉明得, Liu, Mingte |
| Publisher | 國立政治大學 |
| Source Sets | National Chengchi University Libraries |
| Language | 英文 |
| Detected Language | English |
| Type | text |
| Rights | Copyright © nccu library on behalf of the copyright holders |
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