I first mentioned the problems associated with small samples and noisy data in late August. That post demonstrated that you’d get the sign wrong almost half of the time with a small sample, even though a t-test would tell you that the result is statistically significant. The next two posts on the topic (September 9th and 19th) pointed out that being Bayesian won’t save you, even if you use fairly informative priors.
It turns out that I’m not alone in pointing out these problems. Caroline Tucker discusses a new paper in Ecology by Nathan Lemoine and colleagues that points out the same difficulties. She sums the problem up nicely.
It’s a catch-22 for small effect sizes: if your result is correct, it very well may not be significant; if you have a significant result, you may be overestimating the effect size.
There is no easy solution. Lemoine and his colleagues focus on errors of magnitude, where I’ve been focusing on errors in sign, but the bottom line is the same:
Be wary of results from studies with small sample sizes, even if the effects are statistically significant.
Lemoine, N.P., A. Hoffman, A.J. Felton, L. Baur, F. Chaves, J. Gray, Q. Yu, and M.D. Smith. 2016. Underappreciated problems of low replication in ecological field studies. Ecology doi: 10.1002/ecy.1506