D-optimal Regular 2n−p Fractional Factorial Designs for Location Effects with A Single Dispersion Factor
Date Issued
2005
Date
2005
Author(s)
Lin, Ming-Yu
DOI
zh-TW
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.
Subjects
分散性效應
位置效應
最適設計
Dispersion effect
Location effect
Optimal design
Type
thesis
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