Abstract
In this paper, the asymptotic and permutation testing procedures are developed in general factorial designs without assuming homoscedasticity or a particular error distribution. The one-way layout, crossed and hierarchically nested designs are contained in our general framework. New test statistics are modifications of Wald-type statistic, where a weight matrix is a certain diagonal matrix. Asymptotic properties of the new solutions are also investigated. In particular, the consistency of the tests under fixed alternatives or asymptotic validity of the permutation procedures are proved in many cases. Simulation studies show that, in the case of small sample sizes, some of the proposed methods perform comparably to or even better in certain situations than the Wald-type permutation test of Pauly et al. (2015). Illustrative real data examples of the use of the tests in practice are also given.
Highlights
Let us consider the following general factorial design introduced by Pauly et al [20]
The Wald-type permutation statistic (WTPS) test tends to result in accurate test decision for small sample sizes in many cases, but it is more or less liberal for extremely skewed distributions in the case of unequal variances
Compared to other test statistics, the Wald-type one has the advantage that it is applicable in general factorial designs without assuming homoscedasticity or a particular error distribution
Summary
Let us consider the following general factorial design introduced by Pauly et al [20]. The Wald-type permutation statistic (WTPS) test tends to result in accurate test decision for small sample sizes in many cases, but it is more or less liberal for extremely skewed distributions (like log-normal one) in the case of unequal variances. Smaga [23] considered the asymptotic and permutation tests based on modified WTS, where the Moore-Penrose inverse is replaced by a. Heteroscedastic designs and small sample sizes, the methods based on {2}-inverses seem to be a more conservative replacement for the WTPS. They may perform worse under symmetric distributions, i.e., they may be more conservative or more liberal than the WTPS test.
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