# Non-parametric alternative for 2-way ANOVA [duplicate]

Suppose I have 2 independent variables whereby normal distribution can be assumed. However, there is unequal variance and interaction between the two factors is unknown.

Could someone suggest an alternative to 2-way ANOVA with the above mentioned scenario.

I know one possible non-parametric test is the Friedman's test. Are there any other non-parametric test available?

• Of note, Friedman's test is used when you have a blocking factor. Have you seen Is there an equivalent to Kruskal Wallis one-way test for a two-way model? This suggests that this question is a duplicate. – chl Nov 5 '12 at 17:31
• An important question here is in what way are your distributions non-normal? 1st, note that only the residuals need be normal, not Y & certainly not IVs (see here: what-if-residuals-are-normally-distributed-but-y-is-not, if you need more info). But even if your residuals aren't quite normal, the Central Limit Theorem can cover for you if they aren't too far off & you have enough data. OTOH, if your response variable is non-normal in the sense that it's binary, you need Logistic regression. Etc. – gung - Reinstate Monica Nov 5 '12 at 18:03
• The key is the distribution of the residuals, not independent variables... – Glen Sep 19 '13 at 20:09
• As @Glen suggests (+1), its only the residuals rather than the X's or even the unconditional distribution of the Ys that matter for the distributional assumption. If the residuals were too far from normal for your purposes, one possibility to consider is a parametric but non-gaussian assumption (e.g. a GLM) more suited to the kind of data at hand. – Glen_b Sep 19 '13 at 23:37

The problem you have in analyzing these data is that interactions don't really make sense when you have non-parametric tests. Non-parametric tests consider data to be ranks - that is we know if something is higher or lower, but we don't know the magnitude.

An interaction says does the magnitude of the effect of X1 depend on the level of X2. You're asking to compare the size of the effects, but the size of the effects is not something you can consider in a non-parametric test.

However, unequal variance is a bad reason to do a non-parametric test. Unequal variance is pretty much irrelevant if your group sizes are equal. If they're not, it's really easy to correct for it. You can use survey methods, the Browne-Forsythe correction, the Welch correction, robust estimates, sandwich estimates. Which these is easier depends on the software that you're using, and what you're familiar with.

• You can test for interactions amongst multiple factors w/ ordinal logistic regression. – gung - Reinstate Monica Sep 19 '13 at 22:35
• Yes, but that's not non-parametric. With 7 categories, you're going to need some big samples too. – Jeremy Miles Sep 20 '13 at 17:15
• @JeremyMiles, so does it makes sense to combine two variables into one, e.g. in R interaction(factor1,factor2), in order to do the test? – toto_tico Nov 13 '15 at 4:57
• I don't think that makes a difference. – Jeremy Miles Nov 13 '15 at 15:23

The only method that I know of that has support in the literature for the interaction test is the use of a transformation of the raw data to normal scores, such as the ranking procedure by Van der Waerden and by Blom. Both are available in SAS under proc rank.