"Fifty-six subjects were recruited (32 CD and 24 NCD). One participant from the CD group was excluded due to imaging artifacts, rendering a final sample of 31 CD and 24 NCD."
Feels too small of a sample size to make any strong conclusions
Feels too small of a sample size to make any strong conclusions
Whether N = 50 is too small sample really depends on the strength of the effect you are trying to detect. For strong effects it’s plenty enough.
In R, we can do this with: binom.test(8, 10, p=0.50, alternative="greater") which returns a p-value of 0.05469 and a 95% confidence interval of 0.4930987, 1 [see note below]. This doesn't meet the traditional threshold of 95% confidence, which (in a simplified way) says that we're ok with a 5% chance of a false positive if we repeated this testing procedure many times. So yes, this isn't a p-value of 1%.
If we expand this to 20/25 successes, we have a lot more evidence. The p-value shrinks to 0.002039 and a 95% CI becomes 0.6245949, 1 [see note below]. So we're almost certain that the procedure has a success rate greater than 0.5, but without going deeply into the confusing aspects of how to correctly interpret confidence intervals and instead interpreting them as they are generally used, anywhere from 62% to 100% is reasonable if we're ok with a 5% false positive rate, which is still a wide range (the width is what I'd say is important, ignoring the way people use CIs in practice).
Another way to do this would be to get a p-value as to how feasible a less-than 80% success rate is for 20/25 successes. That has a p-value of 0.5793, which isn't significant at any commonly used level.
In the end, you'd probably use something more sophisticated than a binomial test so that you can account / control for other factors, but hopefully this illustrates what you can do with small sample sizes.
Math details: https://en.wikipedia.org/wiki/Binomial_test#Usage
Note: The upper range is always 1 because of details around calculating a one-sided binomial confidence interval (we're doing "greater than" rather than "different than").