I'm interested in knowing about the difference in interpretation between (1) linear regression on a logit transformed variable with values between 0 and 1 and (2) beta regression where the values between 0 and 1 are untransformed.
I'm reading a following paper about the use of beta regression:
Specifically, I'm trying to figure out how my interpretation of my results will be different if I take a percentage outcome variable I have and either (1) use the logit transformation and use a normal model or (2) use beta regression. This is what the authors have to say on the matter:
"How should one perform a regression analysis in which the dependent variable (or response variable), y, assumes values in the standard unit interval (0, 1)? The usual practice used to be to transform the data so that the transformed response, say ˜y, assumes values in the real line and then apply a standard linear regression analysis. A commonly used transformation is the logit, ˜y = log(y/(1 − y)). This approach, nonetheless, has shortcomings. First, the regression parameters are interpretable in terms of the mean of ˜y, and not in terms of the mean of y (given Jensen’s inequality)."
Could somebody give me a less technical explanation of the author's point here? I'm not really sure what Jensen's inequality is or why it applies here.
Here's another paper that makes a similar point:
"The logistic-normal model in , which assumes normal distribution for logit-transformed proportion responses, can provide a computationally convenient framework, but it suffers from an interpretation problem given that the expected value of response is not a simple logit function of the covariates."
I think this quote is probably referring to the issue identified in the first one but I'm still not quite grasping how.
This issue issue is closed. See the comments on the first response for the answer.