Permutation testing for goodness of fit and stochastic ordering

Problems of testing for ordered categorical variables are of great interest in many application disciplines, where a finite number of Q 1 of such variables are observed on each individual unit (Pesarin and Salmaso (2006) (Pesarin and Salmaso, 2010a) and (Pesarin and Salmaso, 2010b)). In particular, Goodness of Fit tests are used to measuring how well do the observed data correspond to the assumption model. Several parametric solutions to univariate case have been proposed in literature. In particular, when dealing with categorical variables, the most used methods are Pearson’s Chi-squared and Deviance statistic. However, these methods, usually based on the restricted maximum likelihood ratio test, are generally criticized because their asymptotic null and alternative distributions are mixtures of chi-squared variables whose weights essentially depend on underlying population distribution F and so the related degree of accuracy is difficult to assess and to characterize; thus their use when F is unknown is somewhat questionable in practice. Moreover, is well known the difficulty or impossibility to use them in multivariate cases. In many situations it can be of interest testing for a set of restricted alternatives to H0 (Kim and Foutz (1997) and Chapman (1958)). In these cases we can refer to Stochastic Ordering. Parametric solutions don’t allow this kind of tests. By working within the Non-parametric combination of dependent permutation tests, it is possible to find exact solutions to these problems. The NPC approach works as a general methodology for most multivariate situations, as for instance in cases where sample sizes are smaller than the number of observed variables, or where there are non-ignorable missing values, or when some of the variables are categorical (ordered and nominal) and others are quantitative and in many other complex situations. In this work, NPC tests for stochastic dominance are presented, both for two sample directional testing and for testing for a stochastic ordering in a multivariate setting. A simulation study is reported to show the NPC approach efficacy.