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My analysis involved some behavioral data on swine. One measure we had was standing time (min) for pigs using accelerometers. Using SAS, I checked for normality, and results showed data to be non-normal (Shapiro–Wilk < 0.05). I then performed residual analysis, which again showed non-normal data. Skewness was -0.42. The reason for the negative skewness was probably because there was a set upper limit (60 min) for the variable measured. So I reflected the data and did a reflected SQRT. I then fit the transformed data through the model and re-checked the residuals. Results were still non-normal. Any suggestion on what I can do?

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    $\begingroup$ What model are you fitting? What questions are you asking of the data? What sample size do you have? What do the residuals look like (e.g. on a QQ plot)? $\endgroup$ – Glen_b May 12 '14 at 23:46
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    $\begingroup$ Hi Glen, I am fitting my datas with mixed linear model using SAS's Proc Mixed (with assumptions that data are normally distributed). My experiment design.We would like to see if the stand time for each pig differs in the different treatments (3 treatment groups) with 20 pigs in each treatment. Q-q plot has a curved pattern $\endgroup$ – FTan May 13 '14 at 0:17
  • $\begingroup$ Is your question necessarily about the mean or just a general location, or is something like $P(X>Y)>\frac{_1}{^2}$ sufficient for your purpose? What does the distribution within each treatment look like? $\endgroup$ – Glen_b May 13 '14 at 0:25
  • $\begingroup$ Please note that "has a curved pattern" isn't much help; that it would be curved is obvious from your earlier discussion. How much it curves and where matters. It would help if you'd show rather than tell. $\endgroup$ – Glen_b May 13 '14 at 4:28
  • $\begingroup$ Hi Glen. I am basically just comparing the means of the different treatment groups. May I know how I can share the curve graph here? I have it in words form ( copied from SAS). thx! $\endgroup$ – FTan May 14 '14 at 16:19
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Trying to get to normality is usually a means to an end. Understanding the end will really help in understanding what is the best path forward, and oftentimes whether or not transforming to get to normality is even needed. There are tons of natural distributions that are not normal and do not need to be transformed to normal.

The set upper limit of 60 is problematic because it creates a false upper end of the distribution. You may be getting a bad Shapiro-Wilk simply because there are too many 60's in your dataset. I would recommend bootstrapping a right-tail or concatenating the upper and lower ends of your dataset. These are not ideal methods and must all be put in context of what you're trying to do, so as to see if they make sense.

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    $\begingroup$ Hi Gary. A fellow post doc in this behavior field has suggested that if re-fitting the transformed data does not give normality in residuals, we can then to a simple non-parametric such as Kruskal Wallis and compare results to a Mixed/Glimmix ( SAS mixed models). WPlease see my above comments on the purpose of my data just in case you are wondering. Thanks $\endgroup$ – FTan May 13 '14 at 0:25
  • $\begingroup$ FTan Questions in both my comments relate to whether or not a Kruskal Wallis might be suitable, but your responses didn't relate to those particular questions. $\endgroup$ – Glen_b May 13 '14 at 4:29

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