I was working with a client who had a limited population of medical residents to choose from, and it would be a struggle to get even 60 of them. The primary outcome was binary: whether a certain medical procedure was run properly in a test setting. The intervention was special training on a model; the control was normal training without the model. I got a phone call back that said, what would the power be if I used three groups rather than two? I thought to myself “Good grief!” You can’t say that to a client, of course, so here’s what I said.

For reference, here is the original power calculation. With a sample of 30 subjects per group, you would have 80% power for detecting a three fold decline in error rates: from 50% in the control group to 16.7% in the treatment group. I should note here that this is a pretty big decline, and declines that might be almost as important, such as a two fold decline, would not be readily detected using this sample size. We had gone over this at the earlier consulting session.

When I was presented with the proposed modification, here’s what I explained:

Adding a third group to the study adds to the complexity of the study and would require more than 30 observations per group. Since it was a struggle for you to find 60 total subjects, I presume that any study that would require quite a bit more than 90 total subjects would be impossible to conduct.

I can work out the exact numbers if you like, but as a general rule, any change to the design that increases the complexity will require a substantial increase in sample size.

The problem here is that your primary outcome measure is binary, and binary outcomes typically require larger sample sizes than other types of outcome measures.

There is a temptation to try the experiment by splitting the 60 available subjects into three groups, making the sample size 20 per group rather than 30. This is unadvisable. We are already looking at a research design which can only detect a three fold change in the proportions. If you were to further decrease the sample size, you would not be able to detect even a three fold change in proportions with reasonable power.

The only time that a sample size as small as 20 per group makes sense in a study like this, looking at binary outcomes, is if you expect something close to an “all or nothing” response. This means, for example, that almost no subject in your control group inserts the screw correctly and almost every subject in your treated group inserts the screw correctly. It’s not quite that extreme, but it’s close. Do you really want to run a research study that is only capable of detecting differences that even the best possible intervention is unable to achieve?

If the focus of the study were solely on a continuous variable, such as the time of the procedure, then perhaps a sample size of 20 per group MIGHT work. But I strongly suspect that your binary outcome (failure to conduct the procedure properly) is far more important than time.

Now, if you decide that you want to run the experiment with three groups and 20 subjects per group, I would be glad to continue to work with you on the project. Just because I advise against a certain choice during the design phase, I won’t abandon you if you make a different choice.

Final thoughts: The person running the research didn’t really want three groups, but was being pushed in that direction by other members of the research team. So he was happy with my explanation. The moral of the story is that when you have a borderline sample size to begin with, there is no way that you should think about adding additional complexities to the research design.

You can find an earlier version of this page on my blog.