Studies with small sample sizes I'm asking myself the question of why studies with small sample sizes are not as convincing as those with larger sample sizes, and when this becomes a statistical issue. A complaint I've heard a lot with studies is while they show a certain "desired" result (e.g. the treatment group had lower cholesterol than the placebo group), the sample size was small.
Statistically, if you have the sample size to detect an effect, then the sample size was large enough in that sense. So at that point, on what grounds could you claim a higher sample size would be better? To get more exact confidence intervals if they are asymptotic?
Personally I would feel more convinced with a sample of 1000 rather than 500, but I can't identify technically where this makes a difference aside from precision.

In frequentist statistics this is a matter of $\alpha$ (significance level) and $1-\beta$ (power) of a hypothesis test. Usually you would fix $\beta$ (for ex. $80 \%$) and then determine the number of samples required to obtain such power for a desired $\alpha$ (for ex. $0.05$).
A specific example: power analysis for a simple t-test given $d = 1$, $1-\beta=0.8$, and $\alpha=0.05$ returns a required sample size of $n \approx 17$. Increasing $1-\beta=0.9$ yields $n \approx 22$. The significance level has stayed the same in both cases, i.e. the CI obtained from such tests will cover the true values with the same proportions, what has changed is the power, the second test has more power than the first (and narrower CI's).
