想要探討化療對和粒線體DNA 上的突變個數是否有相關，我們所觀察的是母親和其後代身上粒線體DNA 的突變個數，因此屬於成對樣本，為了建立成對的計數型模型，我們利用二變量卜瓦松模型來解釋成對樣本之間的相關性。但是，因為所觀察到的粒線體DNA 長度不完整，以至於可能有部分的突變個數無法被觀察到，如果忽略此部分，會導致分析上會有偏誤。用來調整無法被觀察的部分的調整項，稱作平移調整項(offset)。然而，在傳統的二變量卜瓦松模型上對於這類的調整方式，是在每對成對樣本加上相同的平移調整項。在本文中，我們將推廣二變量卜瓦松分配，考慮的是每對樣本間所觀察的長度不同，因此是針對每對樣本間給予不同的平移調整項(varying offset)，藉由此調整所估計出的參數估計偏誤會更，我們將以模擬的方式來證明此調整方式會比原本的更好。而在本文所分析的資料中，我們有興趣的是母親是否接受化療對於粒線體DNA 上的突變個數是否有相關。

The goal of this research is to investigate the relationship between DNA and whether the chemo. To built the model for a paired count data, a bivariate Poisson regression is used to examine the relationship between samples taking account the paired correlation. However,
if the length of the accessible genome is not complete, some mutation counts will not be observed. In such a case, the length of accessible genome, or we call offset term, vary within each pairs (e.g. mother and their offspring). The classical bivariate Poisson regression model treats the offset term as equal within pairs; thus, it cannot be applied directly. In this thesis, we propose an extended bivariate Poisson regression model with more general offset terms to adjust the length of the accessible genome for each observation. In application, we are interested in whether the mother received chemotherapy and the number of mutations in mitochondrial DNA is relevant.

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