It's important to recognize that we've never successfully answered Hume's general skepticism about the existence of causality as something that is real outside of our own minds. So causality can't be positively confirmed to
exist at all, let alone be detected and confirmed through statistics.
Before you dismiss that is philosophical non-sense, causality is closely related to the "Arrow of Time" which is considered an unsolved problem in physics [0]. From what we've observed time appears to be the only asymmetric physical process, largely due to entropy (that is you can immediately tell if a video of a jar breaking is being played backwards because we don't expected a jar to "fall together"). There are Quantum processes, for example, that are time symmetric, that is playing a video of these processes would look the same reversed and forward.
That said, as someone who does a lot of statistics, in practice what we do is model the causal process with a directed acyclic graph and see how well our models behave under causal assumptions. These work okay for answering practical questions about does A cause B. By controlling for correlating variables we can see the impact on what we believe to be a causal variable when we condition on these other correlating factors.
Worth mention that nearly all of the work is done by linear models in practice.
0. https://en.wikipedia.org/wiki/Arrow_of_time