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I am trying to do statistical analysis on data where my x axis is categorical. I have five different treatment groups on the x axis. My y axis is continuous (mRNA levels of a certain gene responding to the different treatment groups). I was planning on using one-way ANOVA and post-hoc testing.

However, I have a lot of variation between patients (I'm looking at mRNA level responses in cells from 7 different patients). Because of this, my data does not follow a normal distribution. Once I log transform the y axis data it does follow a normal distribution. However, my statistical knowledge is pretty basic- can I log transform the y axis and keep the x axis as treatment categories (bar graph)? Or is it a rule that to be able to log transform the data both x and y axes must be log transformed? Obviously I can't log transform categorical data so I'm not sure how to approach this..

Any help would be very much appreciated!!

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    $\begingroup$ Hint: Where does ANOVA use the numerical codes you employ to distinguish the categories on the "x axis"? (Answer: it doesn't.) $\endgroup$ – whuber Jan 10 '14 at 22:16
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You only need to log-transform the mRNA expression, not $x$. If you didn't log-transform, you could use the rank-based Kruskal-Wallis (KW) non-parametric variant of ANOVA. (Kruskal-Wallis is much less sensitive to skewness and outliers, and doesn't require homoscedasticity).

If you go the ANOVA route with log(mRNA), you have to ensure the variance of log-mRNA is equal across the groups (homoscedasticity-->equal variance of log-$y$ between groups). You also can't have outliers when using ANOVA, since they will bias the results. For outlier effect on averages, consider the average of 1, 2, 3, and 1000000, which is near 250000. However, if you use ranks, the the 1,2,3,1000000 turns into 1,2,3,4 for which the average is 2.5. KW is based on ranks.

You can also assign a rank to the mRNA levels (without a log-transform), and use the ranks in ANOVA.

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