*Originally posted on http://blogs.warwick.ac.uk/simongates/entry/sample_size_and/ on 17 June 2013.*

Performing a sample size calculation has become part of the rigmarole of randomized trials and is now expected as a sign of “quality”. For example, the CONSORT guidelines include reporting of a sample size calculation as one of the items that should be included in a trial report, and many quality scales and checklists include presence of a sample size calculation as one of the quality markers. Whether any of this is right or just folklore is an interesting issue that receives little attention. [I’m intending to come back to this issue in future posts]

For now I want to focus on one aspect of sample size calculations that seems to me not to make much sense.

In the usual idealized sample size calculation, a treatment effect that it is desired to detect is assumed. Ideally this should be the “minimum clinically important difference” (MCID); the smallest difference that it would be worthwhile to know about, or the smallest difference that would lead to one treatment being favoured over the other in clinical practice. Obviously this is not an easy thing to calculate, but leaving practical issues to one side for the moment, in an ideal situation you would have a good idea of the MCID. Having established the MCID, this is used as the treatment effect in a standard sample size calculation, based on a significance test (almost always at the 5% significance level) and a specified level of power (almost invariably 80% or 90%). This gives a number of patients that need to be recruited. This number will give a “statistically significant” difference the specified percentage of the time (power) if the true difference is the MCID.

The problem here is that the sample size calculation is based on finding a statistically significant result, not demonstrating that the difference is larger than a certain size. But if you have identified a minimum clinically important difference, what you want to be able to say with a high degree of confidence is whether the treatment effect exceeds it. However, the standard sample size calculation is based on statistical significance, which is equivalent to finding that the difference that is non-zero. Obviously, the upper confidence limit is likely to be close to zero and will only rarely be far enough from zero to exclude the MCID. Hence the standard sample size may have adequate power to show whether there is a non-zero difference, but has very little power to show that the difference exceeds the MCID. Hence most results will be inconclusive; they will show that there is evidence of benefit, but uncertainty that it large enough to be clinically important.

As an example, imagine the MCID is thought to be a risk ratio of 0.75 (a bad outcome occurs in 40% of the control group and 30% of the intervention group). A standard sample size calculation gives 350 participants per group. So you do the trial and (unusually!) the proportions are exactly as expected: 40% in the control and 30% in the intervention group. The calculated risk ratio is 0.75 but the 95% confidence interval around this is 0.61 to 0.92. So you can conclude that the treatment has a non-zero effect but you don’t know whether it exceeds the minimum clinically important difference. With this result you would only have a 50% chance that the real treatment effect exceeded the MCID.

So sizing a trial based on the MCID might seem like a good idea, but in fact if you use the conventional methods, the result is probably not going to give you much information about whether the treatment effect really is bigger than the MCID or not. I suspect that in most cases the excitement of a “statistically significant” result overrides any considerations of the strength of the evidence that the effect size is clinically useful.