On the analysis of unreplicated factorial designs

摘要

This thesis deals with the problems associated with the analysis of unreplicated fractional factorial designs. These are mainly the estimation of the residual variance and the identification of active contrasts by corresponding tests. In this thesis, only orthogonal designs are studied. This work consists of five chapters and is organized as follows: Chapter 1 introduces the problems in the analysis of unreplicated factorial designs and reviews some existing solutions and their limitations. Chapter 2 reviews twelve existing quantitative methods for analyzing unreplicated fractional factorial designs in detail. These are the directed methods, which use an estimate of the standard deviation as the denominator of a test statistic (Lenth (1989), Juan and Pena (1992), and Dong (1993)); outlier­detection techniques (Seheult and Tukey (1982), Le and Zamar (1992)); Bayesian methods (Box and Meyer (1986)); hybrid procedures (Benski (1989), Lawson, et al. (1998)); non­parametric method (Loughin and Nobel (1997)); generalized likelihood ratio test (Al­Shiha and Yang (1999)); the method based on display ratio (Johnson and Tukey (1987)); and the method based on variance homogeneity test (Bissell (1989)), respectively. Chapter 3 presents a new quantitative method, which we call MaxU r ­method. Some of the statistical properties, e.g. the exact null distribution and the power function of this method are discussed and simulated critical values of MaxU r are presented. The exact quantiles and power of the test MaxU r are also derived in the special case when there are only three contrasts (e.g., in the 2 2 design). Finally, four examples are given to illustrate the use of the new method. In chapter 4, all thirteen methods are compared by means of a simulation study. In this simulation study, not only cases where the active contrasts all have the same magnitude are considered, but also cases where the active contrasts differ in magnitude. Chapter 5, the last chapter, summarizes the results of the simulation study, and provides a comparison to those reported before by other authors. It concludes with discussions and an outlook for possible future investigation.