QANOVA: quantile-based permutation methods for general factorial designs

Abstract

Population means and standard deviations are the most common estimands to quantify effects in factorial layouts. In fact, most statistical procedures in such designs are built toward inferring means or contrasts thereof. For more robust analyses, we consider the population median, the interquartile range (IQR) and more general quantile combinations as estimands in which we formulate null hypotheses and calculate compatible confidence regions. Based upon simultaneous multivariate central limit theorems and corresponding resampling results, we derive asymptotically correct procedures in general, potentially heteroscedastic, factorial designs with univariate endpoints. Special cases cover robust tests for the population median or the IQR in arbitrary crossed one-, two- and higher-way layouts with potentially heteroscedastic error distributions. In extensive simulations, we analyze their small sample properties and also conduct an illustrating data analysis comparing children’s height and weight from different countries.

Inferring median survival differences in general factorial designs via permutation tests

“…Extensions to higher-way crossed or hierarchically nested layouts can be obtained similarly. 22,25 Estimates of the survival medians will be based on the Kaplan-Meier estimators given bŷ S i ðtÞ ¼…”

“…. ; n k are not large enough, the type-1 error rates of the tests based on Wald-type statistics are often inflated; 22,25 see also the simulation section below. To solve this problem, we suggest to conduct the new Wald-type test as a robust permutation test.…”

“…However, for non-censored observations, extensions to general factorial designs have recently been proposed. 22 These are based on the idea of using heterogeneity robust studentized permutation tests [23][24][25] in statistics of Wald-type. Thus, it appeared natural to transfer these ideas to time-to-event situations.…”

Permutation test for the multivariate coefficient of variation in factorial designs

“…For the statistical analysis, the projection matrix T = H ⊤ (HH ⊤ ) + H is usually preferred over H itself (Brunner et al, 1997;Smaga, 2017;Ditzhaus et al, 2019;Dobler and Pauly, 2019), where (HH ⊤ ) + denotes the Moore-Penrose inverse of HH ⊤ . It is easy to check that both matrices describe the same null hypothesis, but T has some favorable properties as being unique, symmetric and idempotent.…”

“…The finite exactness cannot be preserved but permuted studentized statistics were shown to be still asymptotically exact for various non-exchangeable two-sample scenarios (Neuhaus, 1993;Janssen, 1997;Janssen and Pauls, 2003;Pauly, 2011). Recently, the success story of this idea has been continued in the framework of oneway layouts Romano, 2013, 2016) and even general factorial designs (Pauly et al, 2015;Friedrich et al, 2017;Smaga, 2017;Harrar et al, 2019;Ditzhaus et al, 2019;Dobler and Pauly, 2019).…”