# Two-way robust ANOVA

My struggle with non-parametric methods continues... I'd like to apply a median polish instead of two-way ANOVA (normality and homoscedascity assumptions are violated, and $n_{ij}$ are small, so I can't use CLT as an excuse). I've never used median polish so far, and our course in statistics taught us to worship ANOVA and forget about robust methods if basic assumptions are not met. I saw this post and it seems that median polish can be applied for two-way factorial design. Which technique do you find appropriate in case of violation of ANOVA assumptions?

Now, what are the basic data considerations for median polish (or any other technique you find appropriate in this case)? Same shape, homoscedascity? Any resource (link/reference) is appreciated.

P.S.

Note that I'm aware of medpolish function in R.

• John Tukey describes median polish, with many examples and exercises, in his EDA book. It's easily carried out by hand (although a spreadsheet helps for doing the subtractions correctly :-). It's important that there not be too many missing data (empty cells) in the array, for then median polish can fail to converge and can be pretty biased. Otherwise, there are no other requirements. The proof of the pudding is in the analysis of the residuals. Tukey describes several very clever ways to extract additional information (e.g., interactions, transformations) from them.
– whuber
Mar 28, 2011 at 22:57
• You should provide this as an answer instead of a comment. Mar 28, 2011 at 23:02

## 1 Answer

How is normality violated? Medians are more sensitive to skew than means as n gets low. Be careful of that. It would be very problematic if small n's varied in a systematic way.

How much is homoscedascity violated? If the n's are about equal it won't matter much for quite large differences.

• Well... AFAICT homoscedascity is not so problematic, but normality is. I used shapiro.test, lillie.test and ad.test, and $p < 0.01$. Density plot shows clear bi/multimodality and positive skewness. Mar 28, 2011 at 16:36
• @aL3 Normality of the data is irrelevant: normality of the residuals is what matters in ANOVA.
– whuber
Mar 28, 2011 at 22:58
• @whuber, to follow this up (from a stats-struggler), does this mean that if I follow up my 2-way ANOVA with the 'results graphs' my software provides, including a histogram, a Normal probability plot (y-axis: Cumulative frequency, x-axis: Residual Value) and THEY look more or less normal, then despite my raw data being quite non-normal, a 2-way ANOVA is ok and actually fits my data?
– Mog
May 31, 2011 at 2:44
• @Mog Yes, that's a standard check. Actually, normality is needed primarily to trust the p-values from the F test. The goodness of fit is established by the tendency of the residuals to be near zero and not depart too far from zero.
– whuber
May 31, 2011 at 4:09
• This string of comments is somewhat orthogonal to my answer. Medians are biased in the direction of skew to a degree that's dependent upon N. As N becomes smaller they become substantially more biased. IF the conditions in the experiment have correlated small N's then they will have biased median estimates (i.e., the smaller N condition will reflect skew more). I was cautioning that this might affect the median polish. It WILL affect estimates based on medians.
– John
May 31, 2011 at 16:14