# Logistic/Probit Regression if the response variable is not a probability

I am working on a model which involves predicting a ratio between 0 and 1 using a number of variables. The ratio in question cannot be thought of as a probability. I am wondering if a logistic regression or profit regression is appropriate.

For example: $Y \sim a_0 + a_1 X_1 + a_2 X_2 +... + error$, where $Y \in [0, 1]$ and Y is not a probability.

If the zeros and ones come from the same process as the others proportions, then you can model how the expected proportion relates to explanatory variables with a fractional logit, which is essentially a GLM with a logit link, binomial family, and het-robust errors.

The standard reference is Papke, L.E. and J.W. Wooldridge (1996) "Econometric Methods for Fractional Response Variables With an Application to 401 (K) Plan Participation Rates," Journal of Applied Econometrics. Vol 11, No. 6, pp. 619-632.

Tobit regression and beta regression could be used. Beta regression is often used to model proportions.

http://www.ime.usp.br/~sferrari/beta.pdf

• Wouldn't the beta regression approach exclude the zeros and the ones (unless you rescale them in some way)? Also, I am not sure that tobit is appropriate: the data are not censored, since values outside the [0, 1] interval are not feasible for proportions. – Dimitriy V. Masterov Jun 30 '15 at 5:48
• Yes, I think that range is (0,1) and not [0,1] but it is often used to model proportions. Tobit model is often used when there are significant amount of zero proportions. As is in when looking data about the investment activity of households. – Analyst Jun 30 '15 at 10:42