# regression when a dependent variable is a proportion (with known numerator/denominator)

I have a question concerning regression analysis with a dependent proportional variable.

In our study we have an outcome variable called Study-quality.

Study-quality is determined by looking at 11 criteria which are present or absent. When a criterium is met, a study can gain +1 point. So a study can gain a total score between 0 and 11. higher amount of points = better studyquality.

Some criteria are not applicable for certain studies, when this is the case, the total amount of points that can be gained is 9 or 10.

To make a fair comparision, we counted the total gained points per study and devided them by the total points one study could gain. (so for example 7/10 or 4/11).

this proportion is our outcome variable Studyquality.

leading to my question: we have collected several variables (year, sample size, funding source, study setting, etc.) we think are associated with Studyquality.

e.g. the more recent the study is published, the more likely studyquality would be higher

What type of regression analysis could be best used to test possible associations with a proportional outcome var? (wihtin STATA software?)

Thank you!

## 1 Answer

Your data can be simply modeled using logistic regression. Logistic regression can be used also for data in other format then 0's and 1's, for example in R you can use it with three different (but equivalent) formats of data. In fact you can view your data as successes and failures in some number of trials. Even if you have only the aggregated data (total number of successes) accompanied with the number of criteria that were used, you can still use logistic regression for such data. The model that you assume is modeling $k$ successes in $n$ Bernoulli trials, where the probability of success $p$ is unknown and to be estimated, i.e. you are dealing with binomial distribution and logistic regression is designed for such problems.

The overall effect may be viewed as the per-study effect estimated using logistic regression. Moreover, logistic regression, or alternatively generalized mixed effects model with logistic link, may enable you to study the influence (control for) of other variables.