# How do you interpret ANOVA with transformed response?

Let's say I have a quantitative variable Y that ranges between 0 and 1. I have a categorical variable X that has 3 levels, A, B, C. Since Y ranges between 0 and 1, I transform all of Y using the logit transformation so that logit(Y) = ln(Y / (1 - Y))

I build the model logit(Y) = A + B + C, with no intercept, and thus the coefficients basically tell me mean logit(Y) given A.

However, I don't want mean logit(Y) given A, I want mean Y given A. Should I do my ANOVA without transforming my Y. Note that my final model will have both categorical and continuous predictors. That is, lets say Z is a continuous variable. Should I do:

• logit(Y) = A + B + C + Z
• Y = A + B + C + Z

Thanks

• Likely I would do neither, but in some situations possibly either one -- there's not really enough information. Where possible, I'd look at something like a generalized linear model so that your model will both respect the bounds on the variable and give you a way to get inference about the mean response fairly directly. Can you say more about $Y$, please. What makes it stay between 0 and 1? e.g. Is it a count proportion, or a continuous proportion, or maybe something else? Can it be exactly 0 and/or exactly 1 but is otherwise continuous? Sep 24 '21 at 2:45
• It's a continuous proportion, i.e. amount out of 100. I'm just trying to get the mean Y given X controlling for a bunch other predictors. Sep 24 '21 at 17:10
• Again, can you say more about $Y$, please. Can you observe 0's and 1's? What sort of thing is it? (e.g. is it a spatial thing like proportion of land area? Is it a mixture of substances or chemicals? something else?) Sep 24 '21 at 21:07
• Its the proportion of a loan that is lost due to defualt. Sep 25 '21 at 0:08
• Ah, useful. It would be worth adding a lot of these details to your question. Presumably you can't have 0's (if you paid the whole loan presumably it's not a default), but you might have 1's. My first thought would be a beta regression (possibly 1-inflated or 0-1 inflated). Sep 25 '21 at 0:32