分散性效應存在下位置效應之 D-最適部分複因子設計之研究

Abstract

Although homogeneity of variance is a basic assumption in most ANOVA analyses, it is not uncommon to encounter the situations that the variance of the response variable changes from one experimental setting to another. In factorial designs, the factors responsible for such change are called dispersion factors. Recently, several articles study on how to identify the dispersion fac- tors from experimental data. Clearly, it is still important to address the design issue concerning the estimation of location effects when there exist dispersion factors. This study focuses on regular 2n−p fractional factorial designs (FFDs). We simply consider the situation that there is exact one dispersion factor in the experiment. The task is to estimate a set of specified location effects in this situation. The BLUE (Best Linear Unbiased Estimator) of using GLSE (Generalized Least Square Estimation) is applied and its information matrix is shown to have a special pattern when using 2n−p FFDs. Namely, we estab- lish a connection between the D-efficiency for with the alias relations of the used 2n−p FFDs. Specifically, we show that there is no orthogonal design for provided that the general mean and all location main effects are included in it. An algorithm modified from that of Franklin and Baily (1977) is given to search for D-optimal designs for any specified within the class of 2n−p FFDs. Moreover, the minimum aberration criterion is used to determine the final de- sign if there are multiple equally D-optimal designs for a . The algorithm is implemented in R language. Some classes of designs generated from the algorithm are also reported. The existence of dispersion factor results in heterogeneity of variance when analyzing experimental data. According to the method proposed by Box and Meyer (1986), we first use normal plotting to identify unusually large location effects and dispersion effects, simultaneously. Then we apply MLE (Maximum likelihood Estimation) for the identified location and dispersion effects. Con- sequently, the Wald’s test is used for the significance test of location effects. Some data set is given to illustrate this analysis approach.