# General approach for non-parametric two-way ANOVA

Apologies for the almost text-book like question.

I have a 2x2 design with fixed categories and a continuous response variable.

If the variances are equal between groups (Bartlett test) and residuals are normally distributed (Shapiro test), ok I can do standard ANOVA.

Otherwise:

1. Try transforming the data (e.g: arcsin(sqrt), or log(), or even rank()). If transformed data is homoscedastic & normal residues, do normal ANOVA.

2. One option: Kruskal test (tells you whether any means differ between groups) followed by many pairs of wilcox tests (to identify which means differ). If all are significant, all factors (and interactions are significant).

3. Another option: Use the bootstrap approach (permuting residuals) outlined here: Is there an equivalent to Kruskal Wallis one-way test for a two-way model?

Is this correct?

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The proportional odds ordinal logistic model is a generalization of the Wilcoxon and Kruskal-Wallis tests that extend to multiple covariates, interactions, etc. It is a semiparametric method that only uses the ranks of Y. It handles continuous Y, creating k-1 intercepts where k is the number of unique Y values.

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+1, A question, though: If Y were initially continuous (just quite badly behaved), and you converted them to ordinal categories via a rank transform, you could end up with a large number of categories--easily N. My intuition is troubled by this. Would ordinal log reg be reliable in such a situation? Typically, we think of ord reg with situations with a small number of categories (e.g., in surveys, 4) with many observations per category. –  gung Dec 9 '11 at 3:41
Having almost as many intercepts as N is only a computational problem. JMP software even uses a shortcut so that computational time is trivial for this case - someday I'll implement that in the rms package's lrm function in R. These intercepts don't have any other costs because they are forced to be in order and don't spend any degrees of freedom. Think of this like Cox proportional hazards regression for continuous survival time. If you run into computer time problems you can group Y into 101 percentile groups. Note that the prop. odds model elegantly handles excessive ties in Y. –  Frank Harrell Dec 9 '11 at 12:13