Identifying Active Effects With Nonparametric Method And The Test Of Outlier

When the manpower,material and financial resources,and time are limited in experi-ments,unreplicated factorial designs are widely used in our actual life.However,unreplicated factorial experiments have a defect—we have no degree of freedom to estimate the variance of the error.So,we cannot derive the analysis of variance to identify the active effects unless leaving some factorial effects of high order interaction unestimated.Therefore,it is significant to identify the active effects in unreplicated factorial experiments.Several methods are introduced to identify the active effects in literatures,such as normal plot(Daniel,1976),half-normal plot(Daniel,1959),Lenth's method(Lenth,1989),Dong's method(Dong,1993a,1993b)and so on.The application of normal plot is simple but this method is subjective and may arise visual deviation.The effects in this method depend on the setting of the levels of the factors.So,there may be several different normal plots for a group of data;The effects do not depend on the setting of the levels of the factors in half-normal plot.So,there is only one half-normal plot,but this method is still subjective.Only senior statisticians can identify an effect which strays from the line to be active or not;Lenth(1989)defined a robust estimate of the standard deviation of the effects through the trimming median.This method is rather conservative since it usually overestimates the standard deviation of the effects.Further,the degree(k-1)/3 of the t-statistic may not be integer,where k-1 is the number of the effects;Dong(1993a,1993b)proposed a method which was simple.However,when the active effects are not sparse,it's frequency of correctly selecting the active effects is low and it's mean square error is large.The defect is ubiquitous in the existing methods on account of their common precondition of the effect sparsity which means the active effects not exceed 20%.However,it cannot be denied that the situation unmatched effect sparsity exists in reality.This thesis proposes a new method to identify the active effects for the situation that the active effects are not sparse,i.e.,the Robust Z test combined with the test of outlier.First,initially determine the proportion,of the active effects preliminarily by Robust Z test.Then,tentatively identify the active effects through the test of outlier according to the proportion.Finally,estimate the standard deviation of the effects using the nonactive effects and construct a statistic to identify the active effects.The frequency of correctly selecting the active effects and the mean square error of the new method compared with Lenth's method and Dong's method can be obtained by 10000 Monte Carlo experiments.The simulated results show that,when the proportion of the active effects is larger than 20%,the frequency of correctly selecting the active effects of the new method is the highest,and the mean square error of the new method is the lowest.When the proportion of the active effects is smaller than 20%,Dong's method has the highest frequency of correctly selecting the active effects,while it has little difference with the new method.And the mean square error of the new method is very close to that of Dong's method which has the lowest one.The frequencies of correctly selecting the active effects and containing all active effects of the new method are compared with those of Lenth's method and Dong's method through 10000 Monte Carlo experiments with the data in t,uniform,and F distributions.The simulated results indicate that the new method is more applicable.The application of the new method,half-normal plot,Lenth's method and Dong's method in 5 examples demonstrates that the new method can identify obviously and moderately active effects and performs pretty well in the situation of effect sparsity.