Significant difference between proportions, found at sample size below that suggested by power analysis I'm looking at data from experiments that have been run to look for a difference between proportions, 1-tailed.
In many of these it has been found that there is a significant difference between the two proportions (e.g., effect size of 3%, p<0.05)
However, I have run a post-hoc power analysis, inputting the effect size (as it was observed), desired power (0.8), significance (0.05), and it suggests that a sample size notably larger (for example 3x larger) than that which was used is in fact needed to run the test at this power / significance level.
The problem I have is interpreting this. If the power analysis had been done before and we had just happened to input the actual resulting effect size - the required sample size would have shown as this larger N, so by finding significance at a lower N, how should I interpret the result? How much confidence (if any) is lost?
As an aside: I know it is advised against to do post-hoc power analysis with the observed effect size, but I'm just trying to validate the quality of the test and how confident I can be in the significance that was found... as described above, hypothetically the exact same power analysis could've been done identically beforehand and would have suggested this higher N that was ultimately not reached. 
I just don't know what implications this has for the effect size and p<0.05 that's then found in the data when the sample size < required N.
 A: I am among the voices that advises against post hoc power, as often as I get the chance. It's a silly thing to do, and doesn't add information. Power is the probability of rejecting the null hypothesis. So an interpretation of post hoc power is to answer the question "what is the probability that I rejected the null hypothesis?" Well, duh, you did reject, so it's equal to 1.
That said, the most common practice of computing post hoc power is to ignore the outcome of the test and do a calculation based on the observed effect size. In most cases, if you have borderline significance (commonly, $p$ just less than .05), then the post hoc power computed in this way will be about .50, though in really skewed situations it can be somewhat different. You'll get higher or lower post hoc powers than that depending on how much $p$ is less, or more, respectively, than the stated significance level. 
Put another way, it stands to reason that if you were to repeat exactly the same study, there's about half a chance that your results will be less significant than they were this time. To be 80% confident that the next study will also show significance, you need to up the sample size. But none of this makes any sense unless you truly intend to repeat the study, because power analysis is inherently prospective. 
Returning to the present, you already have data to back you up in confidently stating that the two proportions differ. That's enough - don't look a gift horse in the mouth.
