臺灣博碩士論文加值系統

在生物學上Richards函數已被廣泛地應用在描述各種生長動態。由於Richards函數本身的非線性特質，傳統上在對其參數進行統計推論時，都假設其對數轉換後的模式機差項呈變方均質的常態分布。本研究之目的在藉由模擬試驗評估上述前提成立或不成立時，以近似信賴區間（approximate confidence interval ,ACI）估計Richards函數之四個參數的表現，同時比較偏差校正百分位靴環信賴區間（bias-corrected percentile confidence interval ,BPCI）與靴環-t信賴區間（bootstrap-t confidence interval ,BTCI）的表現。
我們先由文獻中找到參數估值差異甚大的四個由實測數據配適而得的Richards函數，做為假設模式。並假設經對數轉換後的Richards模式之機差項可能為：(1)呈變方均質的常態分布; (2)呈常態分布，但變方與生長量成比例; (3)呈高度偏歪的Gamma分布，但變方均質; (4)呈高度偏歪的Gamma分布，且變方與生長量成比例等四種情況。針對每個假設模式在上述四種假設條件下各模擬生產1000組生長數據，再以Causton法（1969）所找到的參數起始估值循Marquardt的折衷演算法進行配適，建立各參數之ACI、BPCI及BTCI三種信賴區間。
當對數轉換後的模式機差項之變方均質（第一及第三種假設條件）時，ACI的表現令人非常滿意。而在變方不均質的情況（第二及第四種假設條件），代表最大生長量的參數之95﹪ACI的經驗覆蓋率會落到85﹪左右；但是對其他三個參數而言，ACI依然達到名義上的信賴水準。BTCI在本模擬研究中的表現大致與ACI相似。而BPCI在四種假設條件下的表現幾乎都是最差的。其次，ACI及BTCI都有不對稱的偏移現象：其「下限太大」的經驗頻率與「上限太小」的經驗頻率呈現明顯的差異。就此而言，BPCI的表現較為理想，但其經驗覆蓋率通常較低。據此，我們認為ACI法對常態性的前提具有極高的穩健性，而對變方均質的前提則較為敏感；而且，當變方均質的前提不成立時，本研究所涉及的其他兩種靴環信賴區間並無法取代ACI。

Richards function has been widely used to describe growth dynamics in biological science. Being nonlinear in nature, inference on the parameters of Richards function was traditionally made by the approximate methods based on the assumptions that the errors of the log-transformed model were normally distributed and had a constant variance. This simulation study attempted to evaluate the performance of the usual approximate confidence interval (ACI), and the bias-corrected percentile confidence interval (BPCI) and bootstrap-t confidence interval (BTCI) obtained via bootstrap procedures, for the four parameters of Richards function with and without the above assumptions.
Four fitted Richards functions, differed widely in their parameter estimates, were collected from literature to work as the basic hypothetical models. The errors in the log-transformed model were assumed (1)to be normally distributed and have a constant variance, (2)to be normally distributed and have variances proportional to the growth size, (3)to follow a highly skewed gamma distribution and have a constant variance, or (4)to follow a highly skewed gamma distribution and have variances proportional to the growth size. One thousand sets of data generated from each hypothetical model under each of the above four situations were then fitted to Richards function by Marquardt’s compromise algorithm, using initial estimates given by Causton’s method (1969), and the ACI, BPCI, and BTCI for the parameters were thus calculated.
ACI produced very satisfactory performance in the 1st and 3rd situations, i.e. when the errors of the log-transformed model had a constant variance. In case of heteroscedaticity (the 2nd and 4th situations), the empirical coverage rate of the 95% ACI for the parameter of the maximum growth size fell to as low as 85%, but the coverage rates remained above the nominal confidence level for the other three parameters. BTCI performed similarly as ACI in all the four situations. BPCI gave the poorest performance for each of the four parameters in almost all the four situations. Both ACI and BTCI showed asymmetrical distribution, the frequency of “lower limit too high” differed substantially from that of “upper limit too low”. BPCI was better in this regard, though its coverage rate was the lowest in general. We conclude that the method of ACI is very robust to non-normality, but rather sensitive to heteroscedasticity. And the two bootstrap methods considered here could not take the place of ACI when the assumption of homoscedasticity was violated.

目錄
中文摘要………………………………………………………………...i
英文摘要………………………………………………………………..iii
第一章 前言………………………………………………………… 1
第二章 Richards 函數及其求配的方法…………………………… 4
2.1 Richards 函數………………………………………………… 4
2.2 Richards 函數的求配方法…………………………………… 6
2.3近似信賴區間的估計………………………….……………...10
2.4靴環信賴區間…………………………………………………11
第三章 模擬方法……………………………………………………..15
3.1 假設模式的選定…………………………………….………..15
3.2 模擬方法……………………………………………….……..17
3.3 模擬步驟……………………………………………….……..20
第四章 結果……………………………………………………………22
第五章 討論……………………………………………………………40
參考文獻……………………………………………………………….43

參考文獻
林俊隆。1986。稻熱病菌飛散孢子之增殖動態的數學模式。農藝彙報9：69-80。
廖大經、林俊隆。1997。 Richards函數之求配方法的比較研究（1）--均質的數據。中華農學會報186：68-88。
鄔宏潘、羅其正、錢美華。1977。稻之生長模型及其生長介量變異性之研究。國科會研究彙報.10（2）：15-36。
Causton, D.R. 1969. A computer program for fitting the Richards function. Biometrics.25:401-409.
Causton, D.R. and J.C. Venus. 1981. The Biometry of Plant Growth. Edward Arnold. London.
Davies, O.L., and J.Y. Ku. 1977. Re-examination of the Richards growth function. Biometrics. 33:546-547.
Draper, N.R., and H. Smith. 1998. Applied Regression Analysis. 3th ed. John Wiley & Sons, Inc., New York.
Efron, B. 1979. Bootstrap methods: another look at the jackknife. Ann. Statist. 7:1-26.
Efron, B., and R. J. Tibshirani. 1986. Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Stat. Sci. 1:54-77.
Fisher, N. I., and P. Hall. 1991. Bootstrap algorithms for small samples. J. Statist. Plann. Inf. 27,157-169.
Gallant, A. R. 1987. Nonlinear Statistical Models. John Wiley & Sons, Inc., New York.
Landsberg, J. J. 1977. Some useful equations for biological studies. Expl Agric.13:273-286.
Ratkowsky, D.A. 1983. Nonlinear Regression Modeling-A Unified Practical Approach. Marcel Dekker, Inc.
Richards, F.J. 1959. A flexible growth function for empirical use. J.Exp.Bot. 10:290-300.
Richards, F.J. 1969. The Quantitative analysis of growth. In F. C. Steward (ed.). Plant physiology, A Treatise Vol VA. Academic Press.
Thompson, J.R. 1989. Empirical Model Building. John Wiley & Sons, Inc., New York.
Manly, B.F.J. 1997. Randomization, Bootstrap and Monte Carlo Methods in Biology. 2ed. Chapman & Hill., New York.
Marquardt, D.W. 1963. An algorithm for least-squares estimation of nonlinear parameters. J.Soc.Indust.Appl.Math. 11:431-441.
Mooney, C., and R. Duval. 1993. Bootstrapping A Nonparametric Approach to Statistical Inference. Sage Publications.
Neter, J., H.K. Kutner,. C.J.Nachtsheim,. and W. Wasserman. 1999. Applied Linear Statistical Models., New York: McGraw-Hill.Inc.
SAS Institute Inc. 2002. SAS/STAT user’s Guide, Release 8.02 Edition. Cary, NC.
SAS Institute Inc. 1990. SAS Language: Reference, Version 6, First Edition. Cary, NC.