# Two-way ANOVA on not completely normal residuals

I have 2 factors, one (Page type) of 2 levels and the other (Intensity) of 4 levels. I did a 2x4 Full Factorial Design with 10 replications, so I did 80 experiments. Then I wanted to do a two-way ANOVA on these data. Correct me if I'm wrong, but what I have to check is if the residuals are indipendent, homoschedastic and normally distributed.

The indipendence and the homoschedasticity is verified by the "Residual by Predicted Plot" because there are no visibile trends or increasing spreads.

The normality assumption is not verified (second plot), but it seems not a big deal because:

Furthermore, normality is the least important assumption of a linear model (e.g., an ANOVA); the residuals may not need to be perfectly normal. (https://stats.stackexchange.com/a/28635/36058)

It seems like they are almost normal.

So, are my thoughts correct? Can I correctly do a two-way ANOVA on these data?

(Full ANOVA: http://i.imgur.com/7tR3Kv2.png)

As Peter said you probably have little to worry about as the normality assumption would seem reasonable given your plots. Remember, not only is the normality assumption not too big a deal (as studies have indicated that the F-tests are fairly robust to non-normality), but also that the residuals will never be exactly normal anyway. In fact, technically, they never will not least because the standard normal is bounded from negative infinity to positive infinity.

Remember, the question is whether the normal distribution would seem like a reasonable model for the residuals, and therefore whether your data conditional on your model (so the cells of your ANOVA) are reasonably normally distributed themselves. The assumption of the ANOVA is that all your groups represent samples from normal populations with the same variance, and therefore the only differences to be assessed are the location of their means. You just need to be happy that you think your data reflects this theoretical scenario.

Looks pretty normal to me.

If you are concerned, you can also run a robust regression (ROBUSTREG in SAS) and see if the results vary.

I noticed though, that you said it is a 4x2 ANOVA, yet it looks like you have only 7 predicted values (first graph). Or are some predictions just really close to each other?

• Thanks for the answer, I'll try the robust regression soon. I enlarged the plot and there are 8 values: i.imgur.com/dus792n.png
– HBv6
Dec 22 '13 at 13:16