# Regression Model for a Ratio Outcome with Count Variables as Numerator and Denominator

I have a dataset where each row corresponds to a country and contains independent variables such as "per capita income" and "mean education status." Additionally, there are two variables of interest: "number of restaurants" and "number of restaurants with Michelin star."

Our goal is to predict the ratio of "number of restaurants with Michelin star" to the total "number of restaurants" using the independent variables "per capita income" and "mean education status." However, it's important to note that the number of restaurants varies significantly among countries, ranging from as low as 3 to over 100. This raises the question of whether we should consider weighted regression based on the number of restaurants.

Given that both the numerator and denominator are count variables, What is the true regression model that i should use?

• +1. The natural approach would be a Binomial GLM, but chances are the rates of stars are so low that a Poisson GLM with the total number of restaurants as an "offset" would do just fine.
– whuber
May 16, 2023 at 23:21
• @whuber thank you for the quick response. But let's assume the chance of getting michelin star is not low. And also isn't binomial GLM same to the logistic regression? But my outcome is not binary, it's a ratio between 0 and 1. I do not have independent variable values for each individual restaurant, but rather for each country as a whole. May 16, 2023 at 23:43
• Your outcome clearly is binary: each restaurant either receives a star or not. Your ratio is a derived value that, in reducing two pieces of information to one, loses essential information. It's okay to have values for the country as a whole: that's a minimal sufficient statistic, provided you are doing country-level regressions.
– whuber
May 17, 2023 at 13:43
• @whuber Thank you very much! I got it. But with that method i will have many observations, should i fix the weights for maximum likelihood estimation? What should i do? My all calculations will be significant because of the many follicles coming from one patient were counted as many observation. Jun 5, 2023 at 20:03
• Where did "follicles" and "patient" come from in a question about countries? Please pose the question you actually have. Generally, the particulars of your situation will determine appropriate models and the analogy between countries and patients might be imperfect.
– whuber
Jun 5, 2023 at 20:19