具相加性的無母數迴歸模式在一般性設計中的估計問題

這個研究的動機來自於將微陣列實驗(microarray experiment)所得到之基因表現值正規化(normalization), 我們考慮具有基因與區塊(block)二個因子(factor)的實驗. 由於每個基因複製(replication)的次數遠小於區塊的個數, 所以我們考慮的二因子實驗為不完整的(incomplete). 當對由微陣列實驗所產生的資料配適(fit)具相加性的二方式分類模型(additive two-way classification model), 是否所有由未之參數所形成的對比(contrast)皆為可估(estimable)是一個值得討論的問題. 假設第i個因子為不完整的N-因子實驗中主要感興趣的因子. 當對由實驗所產生的資料配適具相加性的N-方式分類模型, 我們討論消去與其他因子有關的未知參數(unknown parameter)所獲得之縮小的正規方程(reduced normal equation)的係數矩陣(coefficient matrix)之結構. 然後, 以此係數矩陣之分解為起點, 我們提出一個演算法使得具相加性的N-方式分類模型中的有效參數(effective parameter)可以被確認. 針對具相加性的二方式分類模型, 我們提出得到具一致性(consistency)之未知參數估計的充分且必要條件(sufficient and necessary condition). 最後, 我們以每一個格子(cell)至少具有一個觀查值的行列設計(row-column design)與一組由微陣列實驗所產生的資料來闡明研究中所提出之係數矩陣的分解與演算法.

關鍵字

分類模型；可估計；對比；有效的參數；連接性；不完整的N-因子實驗；行列設計；一致估計

並列摘要

Motivated by 'local normalization' to remove bias in the measured gene expressions of microarray experiments, we consider the two-factor experiment with the factors gene and block. Since the number of replications of each gene is much smaller than the number of blocks, the considered two-factor experiment is incomplete. As an additive two-way classification model is fitted to the microarray data, whether all contrasts formed by unknown parameter are estimable has to be discussed. Suppose that the ith factor of the incomplete N-factor experiment is the factor of interest. As an additive N-way classification model is fitted to the data, the structure of the coefficient matrix of reduced normal equation obtained by eliminating the parameters of the other factors is discussed. Then, an algorithm is proposed to identify the effective parameters in an additive N-way classification model based on the proposed decomposition of the coefficient matrix. A necessary and sufficient condition for getting consistent estimates of unknown parameters in an additive two-way classification model is provided. The new decomposition and algorithm are illustrated by a row-column design with at least one observation per cell and 'normalization' for microarray data in which the pin and dye bias are considered to be corrected.

並列關鍵字

Classification model ； estimable ； contrast ； effective parameter ； connectedness ； incomplete N-factor experiment ； row-column design ； consistent estimator

參考文獻

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具相加性的無母數迴歸模式在一般性設計中的估計問題

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