On Inference of Partially Correlated Data

the size of each subset. However, in case the missing ness not completely at random (MCAR), Looney and Jones [9] argued that ignoring some of the correlated observations would bias the estimation of the variance of the difference in treatment means and would dramatically affect the performance of the statistical test in terms of controlling type I error rates and statistical power [10]. They propose a corrected z-test method to overcome the challenges created by ignoring some of the correlated observations. However, our preliminary investigation shows that the method of Looney and Jones [9] pertains to large samples and is not the most powerful test procedure. Furthermore, Rempala and Looney [11] studied asymptotic properties of a two-sample randomized test for partially dependent data. They indicated that a linear combination of randomized t-tests is asymptotically valid and can be used for non-normal data. However, the large sample permutation tests are difficult to perform and only have some optimal asymptotic properties in the Gaussian family of distributions when the correlation between the paired observations is positive. Other researchers, such as Xu and Harra [12] and Konietschke et al. [13] also discuss the problem for continuous variables including the normal distribution by using weighted statistics. However, the procedure suggested by Xu and Harra [12] is a functional smoothing to the Looney and Jones [9] procedure. As such, the Xu and Hara procedure is not a practical alternative for the non-statistician researcher. The procedure suggested by Konietschke et al. [13], is a nonparametric procedure based on ranking.