# Regression to Predict a 'Discrete' Ratio

I am to build a model $y_i \sim f(X_i, n_i) + error_i$. The regressor $y_i \in [0, n_i]$. $y_i$ and $n_i$ are positive integers. Each observation $i$ has a different known $n_i$, and $n_i$ varies from 2 to 50 for different observations.

I can transform $y_i$ such that $y'_i=\frac{y_i}{n_i} \in [0, 1]$ and try to model it as a proportion. The problem is that when $n_i$ is small ($n_i=2,3,4,5$), the resulting proportion $y'_i$ is very discrete. For example, when $n_i=3$, the only values possible for $y_i$ are 0,1,2.

Does anyone have some suggestions on how to model?

• If $y_i$ is unknown (as you state), how could you even compute $y_i'$? Can you clarify what you mean by 'unknown' there? – Glen_b -Reinstate Monica Jul 12 '15 at 0:45
• It's common to use logistic regression for count proportions, though other binomial GLMs are used, such as probit models and cloglog models. Try logistic regression as a search term both here and more widely. – Glen_b -Reinstate Monica Jul 12 '15 at 0:48
• Sorry I was not clear. 'Unknown' means the dependent variable is not known at time of prediction. It will be known afterwards. – Tom Bennett Jul 12 '15 at 1:54
• Oh, okay -- it's not necessary to specify that you don't know the values you're predicting. – Glen_b -Reinstate Monica Jul 12 '15 at 7:14
• Is there any comment on the discrete nature of the proportions? For certain data points, $n=2$, and the only valid proportions are $0, /frac{1}{2}, 1$, but for other data points, let's say $n=10$, and the valid proportions are $0, \frac{1}{10}, \frac{2}{10},..,1$. I wonder it makes sense to treat them the same and try to fit with one logistic regression. – Tom Bennett Jul 12 '15 at 23:02