**[StATS]: Sample size for a binary endpoint (August
12, 2005)**

Someone sent me an email asking for the sample size needed to detect a 10% shift in the probability of recurrence of an event after one of two different surgical procedures is done.

There were several things that I had to get clarification on.

The first thing, though I already suspected the answer, is whether the 10% is a relative change or an absolute change. If the control group has a recurrence rate of 30%, then a 10% increase could mean a relative change (up to 33%) or an absolute change (up to 40%).

I had already guessed that this was an absolute change, because very few people are interested in detecting a 10% relative change. It’s just too small to be of clinical interest most of the time. But it is a point that is always worth clarifying. A 25% increase, for example, is far more ambiguous, because a relative increase (to 37.5%) is plausible, though a bit on the low end, and an absolute increase (to 55%) is also plausible, though a bit on the high end.

But also of equal importance is is that actual rate of recurrence in the control group. A 10% shift could mean that A has a 20% recurrence rate and B has 30%. Or it could mean that A has an 45% rate and B has a 55% rate. Or it could mean quite a few other things. It makes a huge difference in the sample size calculation.

In general, an absolute change of a given size requires a larger sample size when the rates are close to 50% compared to when the probabilities are small. In contrast, though, a relative change is easier to detect when the rates are close to 50% than when the rates are small. you normally don’t talk about relative changes for large probabilities, because doubling a probability like 80% leads to a nonsensical result.

In response, he told me that the rate in the control group was likely to be less than 40%, maybe as low as 20%.

I also had to ask whether to use a one sided test or a two sided test. He told me he wanted a two sided test.

I should have asked what alpha level he wanted to use, but everyone always says 0.05, so I didn’t bother asking.

Finally, I have always gotten in the habit of asking how many patients the researcher expects to accrue per year. I’m glad I asked, because in this case, the researcher only sees 6-12 patients per year who would be candidates for one of these two procedures.

This told me that the researcher. a serious problem. As a general rule of thumb, you need to accumulate at least 25 to 50 events (recurrence) in order to have enough power to detect a large difference. With rates around 20 to 30%, that means that you need several hundred patients. And in this situation, it is even worse, because a change from 20% to 30% is not a large change in a statistical sense. It is less than a doubling of the rate, for example. So unless you rethink how you want to do the experiment, you are looking at a couple of decades to finish the research.

The problem is that events like recurrence either happen or they don’t. There’s no gray area. You’ve probably notices that studies of mortality (another event where there is no gray area) usually require hundreds or even thousands of patients.

Often if you can use a continuous outcome variable, one that allows for gradual rather than abrupt changes, you can get by with a smaller sample size. So that’s why most studies of prevention of heart attacks focus on a surrogate outcome like blood cholesterol levels instead of a more direct and relevant measure like number of hospitalizations due to a heart attack.

You can also browse for pages similar to this one at Category: Sample size justification.

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