This review targeted different techniques used in the analysis of experiments, when no assumptions about the underlying distribution of the observations is made and interactions are present. Table 1 presents a review of nonparametric methods for factorial designs with interactions as well as hypothesis testing. The search engines used were Web of ScienceTM and Google ScholarTM. The search was focused on interactions and univariate factorial analysis topic-related papers that used the fixed effect model from the classical statistics approach. This implied the exclusion of multivariate response models, case studies, Bayesian methods, nested models, and regressions approaches. Additionally, related references from each selected paper were also considered to increase the scope of the research. The corresponding search is by no means exhaustive, it is limited to results obtained from the process previously mentioned, but it is believed to be comprehensive. Some readers might think that the amount of documents reviewed is small, but that seem to be the reality in the field of interaction analysis when this paper was written.
Key words used through the search were: nonparametric, interaction, factorial, test, and hypothesis. Only papers that have these key words were considered. In addition, an initial screening was made to avoid papers with the following key words: multivariate, Bayesian, regression, case study, and nested. The search in the Web of ScienceTM was extended to results that included previous restrictions to avoid the exclusion of relevant works that didn’t pass the initial screening. Due to the large amount of results in Google ScholarTM this second search was not feasible.
In reviewing the literature, it was found that most authors used rank related methods. Shah and Madden , for instance, made a literature review on the problem of ordinal data when performing a factorial analysis. Afterward, they presented a method proposed by Brunner et al. . At the same time, they mentioned that for the test of interactions Wald Type Statistics (WTS) and ANOVA Type Statistics (ATS) could be used. The ATS can be found on Box . Meanwhile, the WTS can be found on works developed by Akritas & Brunner ; Brunner & Domhof & Puri ; and Brunner & Puri . Also, under the scope of ranks, Scheirer, Ray, and Hare  presented a method derived from the Kruskal-Wallis test. In their method, interactions were taken into account by using ANOVA algebra applied to ranks as if they were independent from each other. Ranks are no independent from each other. However, by multiplying the statistic obtained with ANOVA algebra with a correction factor, the null distribution becomes asymptotically a Chi-square variable.
Wobbrock, Findlater, Gergle, and Higgins  used a rank transform method. The Aligned Rank Transform (ART) is presented in their paper for nonparametric factorial data. The method consists of aligning the data before assigning the...