The use of transformation can usually simplify the analysis of data, especially when the original observations deviate from the underlying assumption of linear model. Box-Cox transformation receives much more attention than others. In this dissertation,. we will review the theory about the estimation, hypotheses test on transformation parameter and about the sensitivity of the linear model parameters in Box-Cox transformation. Monte Carlo simulation is used to study the performance of the transformations. We also display whether Box-Cox transformation make the transformed observations satisfy the assumption of linear model actually.

The purpose of this dissertation was to empirically investigate the Type I error and power rates of four data transformations that produce a variety of non-normal distributions. Specifically, the transformations investigated were (a) the g-and-h, (b) the generalized lambda distribution (GLD), (c) the power method, and (d) the Burr families of distributions in the context of between-subjects and within-subjects analysis of variance (ANOVA). The traditional parametric F tests and their nonparametric counterparts, the Kruskal-Wallis (KW) and Friedman (FR) tests, were selected to be used in this investigation. The four data transformations produce non-normal distributions that have either valid or invalid probability density functions (PDFs). Specifically, the data generating procedures will produce distributions with valid PDFs if and only if the transformations are strictly increasing - otherwise the distributions are considered to be associated with invalid PDFs. As such, the primary objective of this study was to isolate and investigate the behaviors of the four data transformation procedures themselves while holding all other conditions constant (i.e., sample sizes, effect sizes, correlation levels, skew, kurtosis, random seed numbers, etc. all remain the same). The overall results of the Monte Carlo study generally suggest that when the distributions have valid probability density functions (PDFs) that the Type I error and power rates for the parametric (or nonparametric) tests were similar across all four data transformations. It is noted that there were some dissimilar results when the distributions were very skewed and near their associated boundary conditions for a valid PDF. These dissimilarities were most pronounced in the context of the KW and FR tests. In contrast, when the four transformations produced distributions with invalid PDFs, the Type I error and power rates were more frequently dissimilar for both the parametric F and nonparametric (KW, FR) tests. The dissimilarities were most pronounced when the distributions were skewed and heavy-tailed. For example, in the context of a parametric between subjects design, four groups of data were generated with (a) sample sizes of 10, (b) standardized effect size of 0.50 between groups, (c) skew of 2.5 and kurtosis of 60, (d) power method transformations generating distributions with invalid PDFs, and (e) g-and-h and GLD transformations both generating distributions with valid PDFs. The power results associated with the power method transformation showed that the F-test (KW test) was rejecting at a rate of .32 (.86). On the other hand, the power results associated with both the g-and-h and GLD transformations showed that the F-test (KW test) was rejecting at a rate of approximately .19 (.26). The primary recommendation of this study is that researchers conducting Monte Carlo studies in the context described herein should use data transformation procedures that produce valid PDFs. This recommendation is important to the extent that researchers using transformations that produce invalid PDFs increase the likelihood of limiting their study to the data generating procedure being used i.e. Type I error and power results may be substantially disparate between different procedures. Further, it also recommended that g-and-h, GLD, Burr, and fifth-order power method transformations be used if it is desired to generate distributions with extreme skew and/or heavy-tails whereas third-order polynomials should be avoided in this context.

The use of transformation can usually simplify the analysis of data, especiallywhen the original observations deviate from the underlying assumption of linearmodel. Box-Cox transformation receives much more attention than others. Inthis dissertation, we will review the theory about the estimation, hypotheses test on transformation parameter and about the sensitivity of the linear model parameters. Monte Carlo simulation is used to study the performance of the transformation. We also display whether Box-Cox transformation makes the transformed observations satisfy the assumption of linear model actually.

在傳統的線性迴歸分析當中，當基本假設不滿足時，有時可考慮變數轉換使得資料能夠比較符合基本假設。在眾多的轉換方法當中，以Box和Cox(1964)所提出的乘冪轉換(Box-Cox power transformation)最為常用，乘冪轉換可將某些複雜的系統轉換成線性常態模式。然而當資料存在離群值(outlier)時，Box-Cox Transformation會受到影響，因此不是一種穩健方法。 在本篇論文當中，我們利用前進演算法(forward search algorithm)求得最小消去平方估計量(Least trimmed squares estimator)，在過程當中估計出穩健的轉換參數。