Distribution from -1 to positive infinity? I'm performing a regression analysis with a proportions that ranges from 0 to positive infinity, but it's currently centered on 1 (values < 1 indicate a negative relationship, values > 1indicate a positive relationship). To center the response on 0, we're simply subtracting 1 from all values, giving us a distribution of -1 to positive infinity. Is there a statistical distribution that restricts our predicted values to that range? Or any other way to achieve that?
My advisor is recommending just using general linear regression, meaning we'll get predicted values less than -1, which would be uninterpretable. I don't really like that approach, but I'm at a loss as to what my other options are.
Any citations or references would be much appreciated.
EDIT for details about the study:
The response is a ratio of two frequencies of animal captures. We have a bunch of sites split into two categories, and we count the number of animal captures per day in each category in each site. So the response is just (Capture Frequency 1 ÷ Capture Frequency 2) for each site. If CF1<CF2, we get a value less than 1. If CF1>CF2, we get a value greater than 1.
If we do a regression with our response centered on 1, it's going to predict some change in our response per unit predictor. So let's say that for some unit of a given predictor, our results show that the ratio increases by 0.3. So when the predictor equals 3, our ratio should equal 0.9 (3 x 0.3 = 0.9), meaning we expect fewer captures in our area of interest relative to the comparison area. Now when the predictor equals 4, our ratio would equal 1.2. All of sudden it's saying we have more captures in our area of interest, the opposite as before, but the value of the response is still positive. The relationship flips without the signs changing, which is just hard to interpret. It'd be much easier to interpret a negative response value as meaning fewer captures and a positive response value meaning more captures. If we center our response on 0 instead of 1, we can achieve that interpretability.
We'd like to forecast the ratio of A to B given a suite of predictors. A and B represent geographic zones. We have a big dataset with bunch of data for values of B, but no data for values of A. I went out an collected data on animal captures in both A and B. We'd like to correct our big dataset with my data (i.e., we know we have X many captures in B from this big dataset, but how many captures in A would there have been?).


we count the number of animal captures per day in each category in each site.

If you are analyzing count data, then use a model appropriate for count data like a Poisson generalized linear model. You model the actual capture numbers, including the site as a categorical predictor variable in the model along with the other predictors. Then the coefficient for site in the model will be related to the difference in capture numbers between siteA and siteB (with a log link to capture numbers under a standard Poisson model). You could include interactions of site with the predictors, and include an offset term if there is something like site area that would necessarily be associated with capture numbers.
It sounds like you are using your study to impute missing values for A-type captures in a large data set with B values. If so, pay attention to the issues in Stef van Buuren's Flexible Imputation of Missing Data. You will probably be best off using your current study to help impute several versions of the larger data set for future analysis.
