This is true when we are dealing with an uninformative prior, but published research is known to be biased toward positive results and uncorrected multiple comparisons. This situation leads to small sample studies with high random variance being paradoxically correlated with significant results. High random variance appears as a false large effect size in the published result, so if the power is low when calculated with a smaller (adjusted) effect, there is reason to believe that the p-value is inflated. See e.g. Andrew Gelman's work on small sample studies, garden of forking paths or [0].
> Not always. Lots of studies publish as "significant" as soon as they get a p-value just under .05. Inflated effect sizes are certainly a sign that something could be wrong, but it's just one indicator.
Exactly! The implication being the above.
[0] https://en.wikipedia.org/wiki/Why_Most_Published_Research_Fi...
All a power analysis does is reduce the chance that the result is a false negative. It doesn't reduce the chance of a false positive.
> False positives produce inflated effect sizes
Not always. Lots of studies publish as "significant" as soon as they get a p-value just under .05. Inflated effect sizes are certainly a sign that something could be wrong, but it's just one indicator.
Regardless, even if you have a power analysis at the conventional threshold of 80%, and a p-value of .05, you're still going to get spurious positive results 5% of the time, and spurious negative results 20% of the time, by definition.