The Analysis Of Unreplicated Orthogonal Experiments

This article concerns the analysis of unreplicated factorial and fractional factorial designs such as those used in screening experiments. The main advantage of fractional designs is that the necessary information1 can be obtained with fewer runs. However, due to confounding , not all of the effects can be distinguished when this type of design is used.The process of identifying active and inactive factor effects relies heavily on the estimate of the experimental error variance. For fractional factorial designs several different approaches to estimate this variance have been suggested.This article presents an effective alternative method for formal analysis of unreplicated factorials. It is based on a simple formula for the standard error of the contrast estimates. The usual t procedures can be used to interpret the results.The article mainly consists of several parts as follows:Part one: right censored sample moment estimate method. This part presents a new technique to estimate the experimental error variance. To get an idea of the appropriate degrees of freedom, the empirical distribution of the estimator were fitted by scaled chi-squared distributions by matching the first two moments(the fits are quite good). The results suggest the rule that d=m/1 .55 is about right for the number of degrees of freedom.Part two:Comparing of several methods. The effect-sparsity assumption is that most means of effects are equal 0. Now suppose for a moment that they are all equal to 0, then the effects are independent realizations of an N(0,2) random variable . Then suppose that there are just a few active contrasts among the effects, marginally, the effects are independent realizations of a random variable whose distribution is a mixture of the form (1-β)?F + β, where F is N(02), G is some distribution more higher variance or greater mean than F. and βa contamination parameter. Three different methods here are considered to estimate experimental error variance under these different circumstances mentionedabove. The results show the new methods is much more reliable than the other methods.Part three: Examples. Using a data set in example 1, the process of employing this new method(Sh_Mthd) is shown. In example 2, four data sets are used to compare different methods including L Mthd, Sc_Mthd, Sh Mthd and probability plots. The new method results in conclusions similar to those obtained using other methods mentioned above.