Sample size for unequal groups in logistic regression

Question

I am conducting a study, in which I will be using Logistic Regression for analysis. I will be collecting data in an education setting (about cancer prevention), then I will follow subjects to collect data on who took action after their education (cancer prevention screening) among those who were educated. The two groups (DV) will be "screened" and "not screened." Among the surveys that are administered, I am sure the number of subjects in a group that were screened will be much higher than those who did not. Without unequal groups considerations, my sample size, using Green's formula, i.e. $N\geq 104+m$, is 114.

Do the number of subjects in each group have to be equal? How much of a difference is considered not equal? In the case when groups will be unequal, how should it be dealt with for sample size calculation or analysis?

Answer 1

The two groups do not have to be equal. Problems can arise if one group becomes very small. King and Zeng have an interesting paper on "Logistic Regression in Rare Events Data". When talking about rare events they have situations in mind where one group makes up 1% or less in a sample.

On the one hand King and Zeng propose estimation techniques to overcome this problem. There is software implementing these techniques.

On the other hand King and Zeng also discuss data collection strategies to avoid rare events. This might be interesting in your case. Note that these methods are not always innocuous. King and Zeng show how to deal with them and discuss the potential pitfalls.

Before bringing out the heavy artillery, I would try to find out how unequal the distribution of 0's and 1's could be. You probably have some knowledge about that, such as e.g. from previous studies on the same topic. Some expert knowledge about the population and programs you want to assess might be useful too. Then you will see if rare events become an issue and if you need to correct whatsoever.

Answer 2

It does not matter much that your groups are balanced. What is important is the smallest of (the number of zeroes, the number of ones), let's call this number $\nu$, is substantially greater than the number of explanatory variables. If $\nu=$ the number of explanatory variables, then the model is likely to be exactly identified, and the coefficient estimates will diverge to infinity. You would want to stay away from this situation.

Note that your study design, at least the way you described it, does not allow you to estimate the effect of your education treatment. The coefficients that you will estimate will have a strange interpretation of the effect of age, gender, color of the eyes, or whatever explanatory variables you will put into your regression, conditional on having received screening -- i.e., who responds to treatment among the treated, not in the population. Is that what you want to estimates? To estimate the effect of screening at the population level, you should have randomized your individuals into receiving or not receiving treatment, and then using the treatment indicator as an explanatory variable in your regression. In that situation, having balanced sample sizes between treatment and control groups will affect efficiency of the estimate, with 50-50 split providing the greatest precision. But this is a different situation.