Skewness transformation for one but not the other variable?

Question

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?

Answer 1

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.