My main supervisory is a relatively prominent figure in the field, but as I've gotten to know him better over time, I've discovered some rather critical issues. It's common knowledge that articles often have co-authors, and my supervisor publishes both quantitative and qualitative research. However, to my surprise, I've found from some of our conversations that his knowledge of statistics is actually less than mine... and at times, he even provides incorrect advice on fundamental statistical matters. I'd like to ask how I should handle this situation, as it involves some political aspects. If I were to address this directly, for example, in an email, I'm concerned that my main supervisor might feel embarrassed in front of my other supervisor (who is relatively new). But not addressing this issue could have serious consequences, impacting our research publications. I'm unsure about how my main supervisor managed to publish in those top journals.
In no particular order:
This is when you should talk to a co-superviser. That's why they're there!
Distinguished academics often have a "bird's-eye", wide-ranging view of the field, while not remembering the specific details of a topic unless they're personally researching it at the moment. Their perspective is invaluable mainly to draw unexpected connections, which is vital for innovative research.
By contrast a PhD student will often develop a deep understanding of their particular topic and methodology, since they focus on it for several years at the very start of their academic career.
A good supervisor will know the above two points well, and will be open to your feedback. Furthermore, you need to establish an open and strong communicative relationship with your supervisor, so you need to know early on whether they can handle this sort of feedback and how you should deliver it.
Especially the first few times, it can be helpful to show a bit of deference. I have found the "A/B" technique useful: do it their way, do it your way, and show the difference. (If there isn't a clear difference, then maybe it's not as big a deal as you're making it!) Showing them that you didn't outright dismiss their advice can help them be receptive to you.
It would seem to me to be that his assets are elsewhere and you should look to him for that other advice. Some people are much better at the big ideas than at detail. Einstein, for example, was said to be a terrible (and dangerous) experimenter. Bright enough overall, of course.
But, you have learned something valuable and can look elsewhere, or within your own knowledge for the statistical detail and guidance you need.
I don't see a need to "address" the issue directly. He can probably speak for himself. If others find collaboration with him valuable, you can too, within the limits you have noticed. Everyone has strengths and weaknesses.
Don't think about it as an issue over the competency of your supervisor. Think about it as a scientific issue. If the methods your supervisor is suggesting are not adequate, you should be able to have a scientific debate about it with your supervisor, preferable in person.
I've had really good and really bad supervisors in the past, but I've never had one that I couldn't debates science and methodology with.
The debate over scientific methodology should never feel like a judgment on someone's competency; it should be over finding the methods that suits the problem the best. If it's done right, everybody would learn from the discussion.
I learn new things from my PhD students all the time. There is nothing wrong with that.