有限差分近似法在數量性狀基因座定位上最大概似估值變異矩陣估算上之應用

Application of Finite Difference Approximation on the Estimation of Dispersion Matrix of Maximum Likelihood Estimates for Mapping and Analysis of Quantitative Trait Loci (QTL)

江欣容(Hsin-Jung Chiang) ； 劉清(Ching Liu)

《作物、環境與生物資訊》 4卷3期 (2007/09) Pp. 201-214

https://doi.org/10.30061/CEB.200709.0003

摘要

在數量性狀基因座的定位與分析上，混合模式的最大概似估值可以EM法(expectation maximization, Dempster et al. 1977)、ECM法(expectation-conditional maximization, Meng and Rubin 1993)、IRLS法(iteratively reweighted least squares)等方法求出，但最大概似估值的漸近變異矩陣因為需要利用概似函數的二次微分式求算，又由於一般混合模式之概似函數的二次微分式相當複雜不易導出，因此如何計算混合模式之最大概似估值的漸近變異矩陣為一相當重要的課題。因為僅有估值卻不知估值的變異，將無法評估根據此估值所做統計推論的可靠性。本研究提出以數值方法上的有限差分近似法(finite difference approximation method)來計算概似函數的近似二次微分式及最大概似估值的漸近變異矩陣。為證實利用有限差分近似法所算之漸近變異矩陣結果的正確性，特將模擬之常態分布、二項分布與卜瓦松分布的F2子代的數量性狀資料，分別以概似函數的二次微分式(analytic derivative)與有限差分近似法之近似二次微分式(finite difference approximation of second order derivative)作計算，並比較兩者之計算結果。模擬結果證實無論數量性狀資料分布為何，利用概似函數的二次微分式與有限差分近似法的近似二次微分式，兩者所算出的漸近變異矩陣幾乎完全相同。因此在數量性狀基因座的定位與分析上，當各類數學模式之概似函數無已知的二次微分式或不易以解析方法導出時，建議可先以EM法、ECM法、IRLS法等方法來計算模式之最大概似估值。一旦求得最大概似估值的解析解，則可利用數值方法上的有限差分近似法來計算概似函數的近似二次微分式及最大概似估值的漸近變異矩陣。

關鍵字

數量性狀基因座 ； 遺傳標識 ； 簡單區間定位法 ； 綜合區間定位法 ； 有限差分近似法 ； 最大概似估值 ； 漸近變異矩陣

並列摘要

For mapping and analysis of quantitative trait loci (QTL), the maximum likelihood (ML) estimates of parameters of mixture model can be calculated via EM, ECM, IRLS or other methods, whereas the asymptotic dispersion matrix of ML estimates requires the second order derivative of likelihood function which is generally complicated and not easily derivable. The calculation of asymptotic matrix of ML estimates is important in that it enables us to evaluate the plausibility of our statistical inference based on ML estimates. This study proposed the calculation of the asymptotic dispersion matrix of ML estimates by the finite difference approximation method if the second order derivative of likelihood function was too complicated to derive. To verify the correctness of the asymptotic dispersion matrix calculated by the finite difference approximation method, the asymptotic dispersion matrix of the simulated normal, binary and Poisson distributed F2 intercross data were calculated by analytical formula and the finite difference approximation of second order derivative. Results from the simulated F2 intercross data indicated that the asymptotic dispersion matrices calculated by the finite difference approximation method were very close to those of the analytical formula. Therefore, if the second order derivatives of likelihood functions under various kinds of mathematical model settings are too complicated to derive, it is suggested to calculate the ML estimates via EM, ECM, IRLS or other methods which do not require the second order derivative. Once the ML estimates is available, the asymptotic dispersion matrix of ML estimates can be calculated by finite difference approximation method.

並列關鍵字

Quantitative trait loci QTL ； Genetic marker ； Simple interval mapping SIM ； Composite interval mapping CIM ； Finite difference approximation method ； Maximum likelihood estimates ； The asymptotic dispersion matr