# Skewness transformation for one but not the other variable?

I have two dependent variables, one is positively skewed (significantly), the other is negatively skewed (not significant). I can apply a log10 transformation to improve the skewness on the first one.

Questions:

1. Can I then statistically compare the two variables in a mixed ANOVA test (the two variables are steps of a factor "emotion" and I compare between factor "group"), given that one is transformed and the other isn't?
2. If not, what options do I have?
• First, is this an independent samples comparison or a paired comparison? I Guess the first, but that information should be in your post! – kjetil b halvorsen Nov 3 '16 at 11:28
• Sorry about that! I was trying to keep it simple, but I should have realised that that may be essential information. Have updated the text now. – Spica Nov 3 '16 at 11:45

I assume this is an independent samples comparison, and that you want to compare the expectation of the two groups the samples are drawn from. If it is a paired sample, the distribution of the two groups separately is irrelevant, what is relevant is the distribution of the difference.

Let us write a model.
$$\DeclareMathOperator{\E}{\mathbb{E}} Y_{11},\dots,Y_{1n} \sim \text{iid with mean \mu_1, variance \sigma^2} \\ Y_{21},\dots,Y_{2m} \sim \text{iid with mean \mu_2, variance \sigma^2}$$ Now, if you use some transform (logarithm, whatever, say $$g$$) of, say , $$Y_2$$, then you can bet that $$\E g(Y_2) \not = \mu_2$$, so that, even if the null hypothesis is true, that is, $$\mu_1=\mu_2$$, the corresponding null hypothesis after the transformation will not be true, so the results of the t-test applied after transforming only one group will be meaningless (it will test another null than what you want). That answer your point 1.

For the second point, multiple options:

1. Find a compromise transformation for both groups
2. Use the t-test with bootstrapping
3. Use some nonparametric test
4. Use a permutation test
5. Probably other ideas.

For choosing between these ideas, we would need to know about your context.

• Tahnks @kjetil! If I understand your first point correctly, you are saying that transforming one and not the other is meaningless and should not be done. From the 2nd point I gather a compromise transformation would be a good idea (boostrapping for ANOVA in SPSS is not easy/an actual option, I know of no nonparametric tests equivalent for a mixed repeated measures ANOVA and I don't know what a permutation test ist :) – Spica Nov 3 '16 at 11:51
• If you found this answer helpful youcan upvote and accept it. If it was not useful, please explain why you still have a problem. – kjetil b halvorsen Feb 14 '17 at 20:58