Nonparametric analysis of covariance in block designs
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Analysis of Covariance (ANOCOVA) has been considered as a more effective approach in data analysis than an ordinary Analysis of Variance (ANOVA) for two reasons: (1) increasing the power and precision of the tests through the reduction in error variance, and (2) providing a means of statistically adjusting for pre-existing differences between treatment groups. The parametric analysis of covariance theory was developed with the assumption of normality. If this assumption underlying the parametric model is uncertain, then the applicability of the parametric test is doubtful. Nonparametric ANOCOVA is a robust competitor of the parametric method with less restrictive distributional assumptions. In the past two decades, several nonparametric ANOCOVA tests for one-way layouts have been suggested and shown to be robust and powerful through simulation studies. A number of nonparametric ANOCOVA procedures for higher way layouts have also been studied and the limiting results are usually based on increasing cell sizes. The model that is considered in this research is the ANOCOVA model for Randomized Block Designs (RBD) with one observation per cell, as Yij=ƒÊ+ƒÀi+„„j+0(Xij-X..)+cij, i=1,c,n;j=1,c,c, where cij's are either iid or exchangeable and have continuous cdf's. Some nonparametric aligned rank test procedures are proposed in this paper for detecting the treatment effects for the above model. The first test proposed in this paper for ANOCOVA in RBD is based on the ranks of the block-mean-aligned observations. The overall ranking is used and the proposed test statistic has asymptotically a x^ distribution. Based on the same principle in developing the above test, a test for ANOCOVA in Incomplete Block Designs is also proposed. The second test proposed in this paper for ANOCOVA in RBD also uses overall ranking on the aligned observations. A test procedure using within-block rankings on the covariate-aligned observations is also proposed. The Rank Transformation test of Conover and Iman (1981) is briefly discussed in this paper. Through a Monte Carlo simulation study, the aligned rank test based on within block rankings and the Rank Transformation test are shown to be very robust and powerful and are strongly recommended for the model we discussed. It is found that the parametric test does not present good results in most of the non-normal cases and its use is discouraged when the underlying distribution is unknown.