# Dependent variable is a bounded between 0 and 1

I need to test some hypotheses for a social sciences dissertation. In my description below, I refer to the independent as the Xs and the dependent variables as the Ys.

I have jotted down what models/tests I think I can use for this hypothesis. However, due to the nature of variables, I am uncertain if these violate some of the assumptions of the underlying model. I have done some research on the models, but I am still confused. Any guidance and recommendations will be highly appreciated.

I am expecting an inverted U-shape relationship between the Xs and the Ys. The Xs are Count variables that can take the values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The Ys can be any number between 0 and 1, including 0 and 1.

I am thinking of using a Quadratic regression to test this. However, I am uncertain if I can use this as my X variables are count variables and my Y variable is bounded. Also, the Y variable can only fall in the range [0, 1] - it is not a case of it only being observable within this range.

I have been recommended a zero-one-inflated Beta model as a possible alternative as my Ys can take the value 0 or 1. Ideally I am looking for the simplest solution as my word count is restricts me from doing anything too complicated.

Thank you

• is it by any chance the case that Y represents a percentage of some kind? Jan 23 at 23:54

While I recognize that you may have word count limitations, it makes very little sense to choose your model based on word count. If you can't fit a description of your model in the word count, it would be better to change the hypothesis (or otherwise reduce word count).

Zero-one inflated beta regression does seem appropriate for your hypothesis. You may be able to describe this in a couple sentences, with citations. Or, depending on where you are submitting, you might be able to put it in an online appendix (many journals now do this).

A quadratic term also seems right.

The problem with using a regular, OLS regression is that a) You will violate the assumptions of the model and b) The model may (in fact, probably will) give predictions outside the range of 0 and 1.

If you are using R, there is the zoib pacakage which does this in a Bayesian framework. If you are using SAS then see a SUGI paper by Swearingen, Castro and Bursac from 2012.

Regardless of word count limitations, these are complex models. You may want to hire a consultant.

• Take a look at semiparametric ordinal models. Jan 23 at 12:33
• Thanks for the response. So zero and one can occur, however, I am not expecting there to be many occurrences of these. Possibly one or two of each across 30 to observations. Should I still lean towards applying a zero-one inflated beta regression or could a normal beta regression suffice? Jan 23 at 13:12
• To reduce the complexity of the model, could I group my dependent variables? Say from [0,0.1) [0.1,0.2) ... [0.8,0.9) [0.9, 1.0] and then use a logistic regression? Jan 23 at 13:36
• The usual Beta regression excludes 0 and 1; I believe these models are to address that. But others may know more. Binning your DV isn't a great idea and ordinal logistic isn't all that simple, although it is more familiar. Jan 23 at 18:06
• Thank you Peter. I have read the original paper that I based my theory on and they use a Tobit model even though this data is naturally bounded between 0 and 1. I have read (but not fully observed) the relevant section in Wooldridge's book. However, I cannot understand when to use the Two-Limit Tobit Model (given that this requires a corner solution as noted by Wooldridge) or the Zero-One Inflated Beta model. Jan 24 at 20:06