# Fitting a regression model on bounded data

Typically, in a regression equation, the dependent variable is unbounded i.e. follows normal distribution. But sometimes it may happen that dependent variable is bounded i.e. dependent variable is ratio of something bounded by $$[0, 1]$$ with both values included.

There are some regression model available to handle such case e.g. Beta distribution, regression with quasi binomial distribution etc.

My question is about application of quasi binomial distribution in such case. I have come across a model where the Recovery rate from a typical class of investment (e.g. bond) is assumed to be dependent on some exogenous variables, and a regression equation is fitted to quantify the dependence of Recovery rate w.r.t. those variables using quasi binomial distribution.

My question is if we can fit regression using quasi binomial for any kind of ratio. I understand that such fit is mostly valid for k/n case, where there are n experiments and k success are observed but the success ratio k/n is actually recorded and regression equation is fitted with quasi binomial on k/n.

However for other cases of ratio where k/n structure may not be straightforward, like the present case of modelling of Recovery rate from investments, is fitting regression with quasi binomial is conceptually valid?

Any insight and book/resources will be very helpful where such non k/n structure for fitting regression with quasi binomial is discussed with valid examples

• An alternative approach would be to transform the dependent variable, e.g. $y'=y/(1-y)$ ("odds") or the logarithm therefrom. For power transforms, there is even an automated method to obtain an "optimal" power ("Box-Cox transform") by means of a maximum-likelihood approach. Commented Jul 6, 2022 at 10:52
• As the first approach, what you are saying is basically using logit link function. This is default and automatically taken care of when you fit such regression using glm function in R. But my question was really different Commented Jul 6, 2022 at 12:20
• Commented Jul 6, 2022 at 23:57
• @kjetilbhalvorsen Thanks for your links. However my question is not really what is difference between logistic and fractional logistic and estimation methods etc. My question is how can I interpret such model with binomial if my observations do not come from a k/n setup as my example of recovery rate Commented Jul 7, 2022 at 5:58