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標題: 無重複因子試驗的廣義最大熵分析
A Generalized Maximum Entropy Approach to Analyzing Unreplicated Factorials
作者: 張英超
Chang, Ying-Chao
關鍵字: Unreplicated Factorials;無重複因子試驗;Generalized Maximum Entropy Estimator;Lenth法;廣義的最大熵估式
出版社: 農藝學系所
引用: Al-Shiha, A. A. and S. S. Yang. 1999. A multistage procedure for analyzing unreplicated factorial experiments. Biom. J. 41: 659-670. Al-Shiha, A. A. and S. S. Yang. 2000. Critical values and some properties of a new test statistic for analyzing unreplicated factorial experiments. Biom. J. 42: 605-616. Benski, C. and E. Cabau. 1995. Unreplicated experimental designs in reliability growth programs. IEEE Trans. Reliability 44: 199-205. Birnbaum, A. 1959. On the analysis of factorial experiments without replication. Technometrics 1: 343-357. Birnbaum, A. 1961. A multi-decision procedure related to the analysis of single degrees of freedom. Ann. Stat. Math. 12: 227-236. Box, G. E. P. and R. D. Meyer. 1986. An analysis for unreplicated fractional factorials. Technometrics 28: 11-18. Chen, Y. 2003. On the analysis of unreplicated factorial designs. Doctoral thesis, Fachbereich Statistik der Universität Dortmund. ( Daniel, C. 1959. Use of half-normal plots in interpreting factorial two-level experiments. Technometrics 1: 311-341. Dong, F. 1993. On the identification of active contrasts in unreplicated fractional factorials. Stat. Sin. 3: 209-217. Fraser, I. 2000. An application of maximum entropy estimation: the demand for meat in the United Kingdom. Appl. Econ. 32: 45-59. Golan, A., G. Judge and D. Miller. 1996. Maximum Entropy Econometrics: Robust Estimation with Limited Data. John Wiley & Sons, New York. Hamada, H. and N. Balakrishnan. 1998. Analyzing unreplicated factorial experiments: a review with some new proposals. Stat. Sin. 8: 1-41. Jaynes, E. T. 1957a. Information theory and statistical mechanics. Phys. Rev. 106: 620-630. Jaynes, E. T. 1957b. Information theory and statistical mechanics. II. Phys. Rev. 108: 171-190. Judge, G and A. Golan. 1992. Recovering information in the case of ill-posed inverse problems with noise. Mimeo, Dept. of Agriculture and Resource, University of California. Berkeley, CA. Kesavan, H. K. and J. N. Kapur. 1989. The generalized maximum entropy principle. IEEE Trans. Syst. Man. Cybern. 19: 1042-1052. Kullback, S. 1959. Information Theory and Statistics. John Wiley, New York. Kullback, S. and R. A. Leibler. 1951. On information and sufficiency. Ann. Math. Stat. 22: 79-86. Kunert, J. 1997. On the use of the factor-sparsity assumption to get an estimate of the variance in saturated designs. Technometrics 39: 81-90. Lenth, R. V. 1989. Quick and easy analysis of unreplicated factorials. Technometrics 31: 469-473. Montgomery, D. C. 2005. Design and Analysis of Experiments. 6th ed. Wiley, New York. Pukelsheim, F. 2004. The three sigma rule. Am. Stat. 48: 88-91. SAS Institute. 2004. The ENTROPY Procedure (Experimental). In: SAS/ETSR User''s Guide, Version 9.1, pp. 731-782. SAS Institute Inc, Cary, NC. SAS Institute. 2008a. The ENTROPY Procedure. In: SAS/ETSR User''s Guide, Version 9.2, pp. 601-664. SAS Institute Inc, Cary, NC. SAS Institute. 2008b. The GLM Procedure. In: SAS/STATR User''s Guide, Version 9.2, pp. 2429-2617. SAS Institute Inc, Cary, NC. Shannon, C. E. 1948. A mathematical theory of communication. Bell Sys. Tech. J. 27: 379-423, 623,659. Ye, K. Q., M. Hamada and C. F. J. Wu. 2001. A step-down Lenth method for analyzing unreplicated factorial designs. J. Qual. Technol. 33: 140-152. Zahn, D. A. 1975. Modifications of and revised critical values for the half-normal plot. Technometrics 17: 189-200.
當一項試驗涉及的因子非常多時,通常都必須先進行「篩選試驗(screening experiment)」,選出對反應變數有具體效應的重要因子,而後再以這些因子進行「最佳化試驗(optimization experiment)」,以推定這些重要因子的最佳投入水準(optimum input levels)。由於試驗資源的限制,篩選試驗往往會採取無重複的設計(unreplicated design)。但在分析這類試驗數據時,由於沒有剩餘的自由度可以用來估計試驗誤差變方,因此也就無法藉由傳統的t測驗或F測驗來辨認顯著的效應。就此,最早是由Danial(1959)提出以對比(contrast)或回歸係數(regression coefficient)的常態或半常態機率圖(normal or half-normal plot)來辨識顯著的效應,稱之為活動效應(active effect)。此後的半個世紀以來,就不斷地有學者在效應稀疏性(effect sparsity)的假設下提出不同的分析方法。其中Lenth(1989)的方法是目前最為流行的方法。

Jodge and Golan(1992)以廣義的最大熵原理(Generalized Maximum Entropy Principle, GMEP)(Kesavan and Kapur, 1989)為基礎,提出廣義的最大熵估式(Generalized Maximum Entropy Estimator, GMEE),以解決估計一般線性模式時所遭遇的ill-posed problem或ill-conditioned problem。而分析無重複因子試驗的困難本質上就是一個ill-posed problem;在SAS/ETS(9.1)的User’s Guide中就出現以GMEE來分析這類數據的個例。

本研究在2^4與2^5的完全設計、2^(7-3)的部分設計、以及27個與35個因子的Plackett-Burman設計下進行了一系列的模擬試驗,藉以比較Lenth法和GMEE在IER(Individual Error Rate)、EER(Experiment-wise Error Rate)及檢驗力(power)上的表現,以了解這兩種方法應用於分析無重複試驗數據的相對效率。


When an experiment involving many factors, a screening experiment is needed to identify those factors that have statistical effects on the response variables, and then proceed optimization experiment to find optimum input levels of the factors. Owing to the limited experimental resources, we usually take an unreplicated design for the screening experiment. When analyzing the data of an unreplicated factorial experiment, there are no degrees of freedom left to estimate the error variance, so that the t-test and the F-test are not applicable. At this point, Daniel (1959) proposed a normal or half-normal probability plot of contrasts (or regression coefficients) to identify significant effects, called active effects. Thereafter, many methods have been proposed under the assumption of effect sparsity. Among them, Lenth's method (1989) is the most popular presently.

Faced with ill-posed problems and ill-conditioned problem in the estimation of a general linear model, Judge and Golan (1992) developed the generalized maximum entropy estimator (GMEE) based on the generalized maximum entropy principle (Kesavan and Kapur, 1989). The difficulty in analyzing unreplicated factorials is virtually an ill-posed problem. SAS Institute (2004) gave an illustration of analyzing unreplicated factorials using GMEE in the User's Guide of SAS/ETS (9.1).

We conducted a series of simulated experiments in 2^4 and 2^5 full designs, 2^(7-3)fractional design, and Plackett-Burman designs with 27 and 35 factors, so as to compare the individual error rate (IER), experiment-wise error rate (EER) and test power between Lenth's method and GMEE for evaluating their relative efficiency.

Lenth's method gave good test power only when effect sparsity was assumed. Although the IER (or EER) of GMEE was not as good as Lenth's method, GMEE performed well for the power regardless of the number of active effects. Even for a weak active effect, it could be detected by GMEE. We conclude that GMEE is an valid method for identification of active effects in unreplicated factorial experiments.
其他識別: U0005-0108200901122100
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