利用變數轉換後的數據估算均數對比信賴區間之研究

A Study of Interval Estimation for Contrasts of Treatment Means Based on the Data under Variancestabilizing Transformation

王裕文(Yue-Wen Wang) ； 歐尚靈(Shang-Ling Ou) ； 汪家琪(Chia-Chi Wang) ； 彭雲明(Yun-Ming Pong)

《作物、環境與生物資訊》 8卷1期 (2011/03) Pp. 15-27

https://doi.org/10.30061/CEB.201103.0002

摘要

線型統計分析例如變方分析與回歸分析在進行分析程序前需要檢測分析的前提，前提一共有三項，第一是變數需具有常態性，第二是變異數需具有穩定性，第三是模式效應需具有加成性。其中，若是變方隨均數而改變的情形，則需要做變數轉換用以穩定其變方，常用的變數轉換有：平方根轉換，對數轉換以及Box-Cox的一般冪轉換。變數轉換後的數據通常僅用於顯著性檢定，例如檢測t個處理的處理效應之間是否有顯著差異，或者是數個處理效應間之對比是否顯著，或是進行處理效應間之成對比較。以上這些檢定都是在轉換後的尺度上做檢測，例如作對數轉換後進行處理效應間之成對比較，這個比較是在對數的尺度下進行的，因此其結論在對數尺度上也能成立，本研究說明其結論在原尺度下也是成立的。這個事實證實了以變數轉換後的數據做顯著性檢定的合理性。一般而言在變數轉換下做完變方分析與對比檢定後，統計分析的工作即結束，對於進一步的效應差值的區間估計，尚未有任何文獻提出有關的做法。本文提出利用變數轉換後所估計的共同變方，並藉由效應估值的近似常態性以及蒙地卡羅的模擬方式求出處理效應對比之信賴區間。其做法是對處理效應估值的常態分布以蒙地卡羅的方式重複抽樣，而後經由逆轉換求出我們關心的對比估值，進而由此對比估值的分布求出信賴區間。所求得的信賴區間，其信賴水準接近宣稱的95%信賴水準，而且其區間之寬度窄於簡易法所求得者。

關鍵字

變數轉換 ； 線型統計分析 ； 變方穩定 ； 區間估計 ； 蒙地卡羅法

並列摘要

Three assumptions required by ordinary linear statistical analysis. They are: 1) normality of the data collected, 2) variance homogeneity, and 3) additivity of model effects. In the condition that variance homogeneity is violated we need variable transformation to stabilize the variance in case that the variances (or standard deviation) are proportional to the means. Common transformations used by scientists are square root, logarithmic and general power transformations. Once the data is transformed, the significance test on: 1) homogeneity of treatment effects, 2) contrasts of treatment effects, and 3) pairwise comparisons of treatment effects are performed on the transformed scale. Fortunately, the conclusion obtained on the transformed scale implies its equivalence on the original scale approximately. This justifies the significance test performed on the transformed scale. However, for the confidence interval estimation after the preliminary significance tests, no practical procedures are available so far. In this article, we propose a simple but effective method to the interval estimation for contrasts of treatment effects on the original scale. This method makes use of 1) the common mean square, 2) the approximate normality of treatment effect estimate, and 3) Monte Carlo simulation to generate random sample means and then by inverse transformation to estimate the contrasts that we are interested on the original scale. Through repeated sampling, a distribution of the contrast estimate can be established and the confidence interval for this contrast can be found based on this empirical sampling distribution. Simulation results show that this method is effective in achieving the nominal confidence level and having narrower width than the naive method.

並列關鍵字

Variable transformation ； Linear statistical analysis ； Variance stabilizing ； Interval estimation ； Monte Carlo method