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    政大機構典藏 > 商學院 > 統計學系 > 學位論文 >  Item 140.119/51556
    Please use this identifier to cite or link to this item: http://nccur.lib.nccu.edu.tw/handle/140.119/51556


    Title: 二獨立卜瓦松均數之比較
    Superiority or non-inferiority testing procedures for two independent poisson samples
    Authors: 劉明得
    Liu, Mingte
    Contributors: 薛慧敏
    Hsueh, Huey Miin
    劉明得
    Liu, Mingte
    Keywords: 近似的假設檢定法
    Barnard凸面
    正確的假設檢定法
    不偏性
    Asymptotic test
    Barnard convexity condition
    exact test
    Date: 2009
    Issue Date: 2011-10-11 16:50:04 (UTC+8)
    Abstract: 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.
    Reference: Barnard, G. A.(1947) Significance Test for 2 $\times$ 2 Tables. {\it Biometrika}, {\bf 34}, 123-138.
    Berger, R. L. and Boos, D. D.(1994) $P$ Values Maximized Over a Confidence Set for the Nuisance Parameter, {\it Journal of the American Statistical Association}, {\bf 89}, 1012-1016.
    Casella, G. and Berger, R. L.(1990) Statistical Inference. Pacific Grove, CA: Wadsworth.
    Corinna, M. and Jochen, M. C.(2005) Power Calcualtion for
    Non-inferiority Trials Comparing Two Poisson Distributions. {\it SAS Phuse Papers }, http://www.lexjansen.com/Phuse/2005/pk/pk01.pdf.
    Gail, M.(1974) Power Computations for Designing Comparative Poisson Trails. {\it Biometrics}, {\bf 30}, 231-237.
    Gu, K., Ng, H. K., Tang, M. L. and Schucany, W. R.(2008) Testing the Ratio of Two Poisson Rates. {\it Biometrical Journal}, {\bf 50}, 283-298.
    Krishnamoorthy, K. and Thomson, J.(2004) More Power Test for Two Poisson Means. {\it Journal of Statistical Planning and Inference}, {\bf 119}, 23-35.
    Lehmann, E. L.(1986) {\it Testing Statistical Hypotheses}, {\bf 2nd edition}, Wiley, New York.
    Lui, K. J.(2005) Sample Size Calculation for Testing Non-inferiority and Equivalence Under Poisson Distribution. {\it Statistical Methodology}, {\bf 2}, 37-48.
    Ng, H. K. and Tang, M. L.(2005) Testing The Equality of Two Poisson Means Using The Rate Ratio. {\it Satistics in Medicine}, {\bf 24}, 955-965.
    Przyborowski, H. and Wilenski, H.(1940) Homogeneity of Results in Testing Samples from Poisson Series. {\it Biometrika}, {\bf 31}, 313-323.
    Pirie, W. R. and Hamdan, M. A.(1972) Some Revised Continuity
    Corrections for Discrete Distributions. {\it Biometrics}, {\bf 28}, 3, 693-701.
    R\"{o}hmel, J and Mansmann, U.(1990) Unconditional Non-Asymptotic One-Sided Tests for Independent Binomial Proportions When the Interest Lies in Showing Non-Inferiority and/or Superiority. {\it
    Biometrical Journal}, {\bf 41}, 149-170.
    Shiue, W. and Bain, L. J.(1982) Experiment Size and Power
    Comparisons for Two-Sample Poisson Tests. {\it Applied Statistics}, {\bf 31}, 130-134.
    Storer, B. E. and Kim, C.(1990) Exact Properties of Some Exact Test Statistics for Comparing Two Binomial Proportions. {\it Journal of the American Statistical Association}, {\bf 85}, 409, 146-155.
    Song, J. X.(2009) Sample Size for Simultaneous Testing of Rate Differences in Non-inferiority Trials With Multiple Endpoints. {\it Computational Statistics and Data Analysis}, {\bf 53}, 1201-1207.
    Thode, H. C.(1997) Power and Sample Size Requirements for Tests of Differences Between Two Poisson Rates. {\it The Statistican}, {\bf 46}, 227-230.
    Description: 博士
    國立政治大學
    統計研究所
    92354504
    98
    Source URI: http://thesis.lib.nccu.edu.tw/record/#G0923545041
    Data Type: thesis
    Appears in Collections:[統計學系] 學位論文

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