Is there a non-parametric form of a 3-way ANOVA? I am currently in the process of writing a publication about the home range of cat shark species in South Africa. However, I am currently struggling with how to create an interaction model of shark species, maturity level and sex, and their effect on the average travel distance within the study region. I was hoping to gain a test statistic (non-parametric) and p value for each category (e.g. mature male leopard shark, mature female leopard shark, immature male leopard shark, etc.). To do this I have tried a gaussian GLM though this does not give me the output I'm looking for since I'm using 3 fixed factors and no covariates. I've also tried a Scheirer-Ray-Hare test, though this only allows for 2 factors. Does anyone have any ideas of what test I could potentially use?
 A: ANOVA, even a 3-way ANOVA, is a special case of linear regression.
For one-way ANOVA, the typical "nonparametric" flavor is the Kruskal-Wallis test, so it seems like you would want some kind of 3-way Kruskal-Wallis test.
Much as ANOVA is a special case of linear regression, the Kruskal-Wallis test is a special case of proportional odds ordinal logistic regression.
Consequently, there is a sense in which the nonparametric flavor of 3-way ANOVA is a proportional odds ordinal logistic regression model on the variables, their two-way interactions, and their three-way interactions, much as the parametric flavor of 3-way ANOVA would be linear regression on the variables, their two-way interactions, and their 3-way interactions.
This is the first I've heard of the Scheirer-Ray-Hare test, but its Wikipedia article makes it sound like it can handle any number of factors, not just two, so perhaps your inability to include three factors is a software issue. Additionally, the Wikipedia article makes it sound like the Scheirer-Ray-Hare test is another special case of the proportional odds ordinal logistic regression.
A: The exchange in comments now makes this clearer. The OP has three species of shark, two level of maturity, and two sexes of shark. This forms a $3\times2\times2$ design. There will be 2 degrees of freedom for species, 1 for sex, 1 for maturity. There will be 2 for species by sex, 2 for species by maturity, and 1 for sex by maturity. By calculation or by subtraction from the overall total we can see that this leaves 2 for the three way-interaction. So R is correct in printing out just two terms for the three-way interaction. It chooses male mature leopard sharks and male mature pyjama sharks as the comment suggests.
