Haseman與Elston (H-E) [Behav Genet 2:3-19, 1972]提出一個方法　檢定待定位數量性狀基因座與標識基因座間是否連鎖來推論待定位數量性狀基因座在染色體上的位置、該方法利用數量性狀基因座與標識基因座在連鎖時的特性　建立同胞對數量性狀平方差與同胞對標識基因同源全等比率估計值兩者之間的直線迴歸關係；顯著的負迴歸係數代表該數量性狀基因座與標識基因連鎖、雖然此方法在執行上相當地簡便　但用於實際的連鎖分析上仍有一些限制、由文獻中的研究發現　其中的一些限制可透過將H-E迴歸模式中的依變數改用以群體數量性狀平均值調整後之同胞對數量性狀乘積而獲得改善‧並且　由模擬的結果得知　此新提出的H-E模式確實可提高連鎖檢定的檢定力、在本論文中　我們証明了調整後之同胞對數量性狀乘積與同胞對標識基因同源全等比率估計值兩者之間的直線關係在理論上仍然成立　因此　利用新的H-E模式中的迴歸係數以進行連鎖檢定及推論　在理論上是正確且可行的、另一方面　為改進在某些情況下對同源全等基因數目判別或估計的困難　我們並提出對H-E模式做一拓展；將模式中的解釋變數改以同胞對標識基因型態全等比率代替　建立同胞對數量性狀平方差與標識基因型態全等比率兩者之間的關係、

For mapping a quantitative trait locus (QTL), several approaches have been developed. Haseman and Elston (H-E) [Behav Genet 2:3-19, 1972] proposed a method to detect genetic linkage between a QTL and a polymorphic marker. They used the linkage relation between the QTL and the marker to construct a regression model, which regresses the squared sib-pair trait difference on the estimated proportion of genes identical by descent (IBD) at a marker locus: a significantly negative regression coefficient suggests linkage. Although this procedure is easily implemented, there are some limitations in some aspects. The H-E model can be modified by changing the dependent variable from the squared sib-pair difference to the mean-corrected product of the sibs trait values. It has been shown, by simulation, that this new procedure definitely makes sense in view of increasing a considerable power of detecting linkage. In this thesis, we will show that the linear relation still holds theoretically between the mean-corrected product and the estimated proportion of genes IBD at a marker locus so that there exists an informative regression coefficient in the new H-E model for detecting linkage as it is in the original model. On the other hand, for improving the difficulty in the IBD status identification or estimation in some scenarios, we make the extension of the H-E model by relaxing the requirement that the IBD relations be completely specified. We propose to use genes IBS (identical by state) for coping with the ambiguous problem in identification of IBD. The relation between the squared sib-pair trait difference and the proportion of marker allele shared IBS is established.

Contents
1 Introduction 1
1.1 Definition of the notation 4
1.2 The genetic model 6
1.2.1 Assumption 6
2 Review of the Haseman-Elston model 8
2.1 Conditional expectation of the squared sib-pair
differences 9
2.2 The joint distribution of pjt and pjm 13
2.3 The joint distribution of pjm and 14
2.4 Deriving the expected value of the regression
coefficient in the H-E model 16
3 Two extended issues on the Haseman-Elston model 22
3.1 Linearity of the new H-E model 23
3.1.1 Conditional expectation of the mean-corrected
product of the sib pair phenotypes 24
3.1.2 Deriving the expected value of the regression coefficient in the
NHE model 28
3.2 The extension of the H-E method by only requiring IBS information
at the marker locus 33
3.2.1 Conditional expectation of the squared sib-pair
differences 35
3.2.2 The joint distribution of sjt and sjm 40
3.2.3 Deriving the relation between SQDj and sjm 42
4 Conclusion 48
Figure 1-2 50
Table 1-9 52
Appendix 1 60
Appendix 2 63
References 71

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