How to conduct the Two-way ANOVA study when the data was non-normal distributions?

[Dennisky]

Hi all, 

We want to do a study to evaluate the effect of different drugs on bone mineral density in mouse.

（each mouse is tested for bone mineral density only once in the study）.

It was design to analyze the data for measuring three time point for mouse in five treatment groups.

The purpose was to explore whether the BMD value is different between 5 drug types at different age groups (3 months, 5 months，7 months). There are 5 different drug types in the study (control group, drug A, drug B, drug C and drug D).

Although we should examine the BMD in the three time points, each mouse is tested for bone density only once. Because we must dissect the mice for getting the precise data of BMD.

Thus, we need 60 mice and divided into 15 groups (4/group, see table 1 for Simulated data).

We will conduct the data by using theTwo-way ANOVA analysis.(not Repeated-Measures Analysis ) 

However, we found our data not follow a normal distribution (although we log transform this data or using the Box-Cox Transformation, it was non-normal distributions).

Could we still estimate the data by using theTwo-way ANOVA analysis?

If the data was non-normal distributions, how to conduct the study?

Thanks a lot !

[gcjfernandez]

Please check the residual of the Two-way ANOVA (not the data) for normal distribution. Before and after the transformation reevaluate the residual for normality.

[SteveDenham]

I wouldn't rely exclusively on the Shapiro-Wilk test for normality.  Also look at the QQ plot of the residuals, and the histogram or PDF plot of both the data and the residuals.  These may provide clues to the distribution that best fits your data.

SteveDenham

[Norman21]

You might want to consider a non-parametric equivalent of the 2-way ANOVA, such as the Friedman test, as described on this page:

https://stats.idre.ucla.edu/sas/whatstat/what-statistical-analysis-should-i-usestatistical-analyses-...

[SteveDenham · Accepted solution]

The Friedman test depends on the time factor being measured within subject, so I don't think that it is quite what would fit this design.  There is a test called the "aligned ranks transformation".  This link provides a SAS approach:

https://libres.uncg.edu/ir/uncg/f/S_Richter_C463_2002.pdf 

This will allow you to correctly test for a time by treatment interaction, although probably not as powerfully as a transformed or generalized linear model.

SteveDenham

PS Here is another paper that isn't so SAS-centric.

https://www.researchgate.net/publication/221514774_The_Aligned_Rank_Transform_for_Nonparametric_Fact... 

[PaigeMiller]

This was stated above by @gcjfernandez but it is worth repeating.

The data does not have to be normally distributed for ANOVA. The errors have to be normally distributed, you need to check the distribution of the residuals.