Suppose I have two datasets, $\mathbf{a}$ and $\mathbf{b}$. I want to test whether the two datasets are different in a statistically significant way.
To compute the F-test, I take the ratio of the variances of each dataset and compare this to F values based on some significance level (e.g. $\alpha = 0.05$) and the number of degrees of freedom. If the F value I computed lies outside the bounds of $1\pm\alpha$, then the null hypothesis is rejected (i.e. the two datasets are different in a statistically significant way).
To compute the KS test, I find the ECDF of each dataset and the find the maximum vertical distance between the ECDFs to compute the D-statistic. Similar, to the F-test, if the D-statistic is greater than some critical value, the null hypothesis is rejected (i.e. the two datasets are different in a statistically significant way).
My intuition is that the tests should generally give similar results. If something is statistically significant, it should be statistically significant for both tests, no? Perhaps this intuition is wrong. But, at the very least, I thought that the KS test was more sensitive than the F-test. As such, if the F-test rejects the null hypothesis, then I thought for sure, the KS test would also reject the null.
But I have found many cases where this is not true. I have some examples where the F-test results in rejection of the null hypothesis while the KS test does not!
Any explanation of why this could be is appreciated.