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I want to run an ANOVA test. I am therefore testing for normality. I have tested each group and the residuals (group together)for normality. My data sample does not look approximately normal. However I have an outlier (5,95 SD from the mean). This is still a true value, not due to wrong data entering. When I am deleting this number, the data sample looks close to a normal distribution. How should I deal with this value? Is it best to use a non-parametric test? A transformation? Can I just remove the value?

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  • $\begingroup$ One more option: go back to the original experiment to figure out how it could have produced an outlier in the first place. $\endgroup$ – J. M. is not a statistician Jul 23 '16 at 16:08
  • $\begingroup$ Since observations that are many sd's from the mean (I won't say "outliers", since that depends on the model) can occur as part of your process I'd be more inclined to choose some procedure more robust to them than an assumption of normality is -- and for that I might choose any of several options (not all mentioned in your question). For me the 1st step in model choice (which should come before getting the data) would be to consider what kind of thing I am measuring - you don't even mention what it is, but often that immediately suggests a more suitable model.than just jumping to normality... $\endgroup$ – Glen_b -Reinstate Monica Jul 23 '16 at 22:06
  • $\begingroup$ ctd ... one can then (still before collecting data) consider robustness to outliers (and to the other assumptions) and how to get something not too badly affected by the assumptions that appear more likely to fail badly enough to matter / which you see the analysis as most heavily dependent on. Trying to fix your model after you have data is problematic because you run into the problem of making your data fit your hypothesis, and that's going to screw up the behaviour of estimates of location, of spread (etc) and therefore of effect sizes and p-values. Here it's already happened ,,, $\endgroup$ – Glen_b -Reinstate Monica Jul 23 '16 at 22:07
  • $\begingroup$ ctd... you've already collected the data and looked at it and so you're already stuck with the problems of a data-based choice of analysis. About the best you could manage, perhaps is either (a) to attempt to make a choice that would reflect what you would have tried to do from the beginning had you put the consideration of such issues prior to data collection or (b) to assess the impact of such choices, including, but not limited to "normal-ANOVA if you see no outliers, OR whatever else you're going to choose if you see any" on your analysis (do you consider other assumptions than normality?) $\endgroup$ – Glen_b -Reinstate Monica Jul 23 '16 at 22:17
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You could consider Cook's distance as an aid for your decision. It is a measure for the effect that removing this observation would have on your analysis. Values with a large Cook's distance merit further attention, those that have a small Cook's distance despite being far outside the range of your other observations shouldn't do much harm. As you do not say which statistical software you use, I cannot tell you exactly how to do that. I use R myself, I would look at the fourth graph of the plot of an lm-object:

plot(lm.fit, which=4, cook.levels=cutoff)

(where lm.fit is an lm-object)

My apologies, this suggestion would probably have been better suited for a comment, but I lack the reputation to comment.

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The answer depends on a lot of information you omitted, as already mentioned by @Glen_b. For example, which model are you interested? Is it a one-way or two-way ANOVA? Linear models are not as much sensitive to normality assumptions, but are quite sensitive to the assumption of homogeneitty of variance. A simple way to diagnose it is to use Cook's distance, another one is just make a boxplot. Either way, if you find that the value has a high influence on your data, you could run the test with and without the value to show (suppose you find a significant effect without the outlier) that the value is affecting your results.

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