How should I address a possible mistake to co-authors in a submitted paper

If you believe there is an issue with a paper you are a co-author of, then you absolutely should discuss it with your co-authors. In most (all?) fields, having read and approved the final manuscript is a necessary condition for authorship of a publication. I assume that you cannot really approve of a manuscript that you suspect has methodological flaws, so you should speak up now. Whether that is best done by emailing everyone or just contacting the lead author depends on a lot of external and contextual factors, so I cannot offer advice there.

Waiting for reviewers to spot the mistake seems like a very bad strategy. After all, what will happen if they don't spot the mistake? Either you bring up the issue after receiving the reviews (which is surely not better than bringing it up now) or you remain silent and become co-author to a paper that may have serious methodological problems.

In the end, if you bring up the issue now, there are three possible outcomes (presumably after some arguments and back-and-forth):

• You convince your co-authors that there is an error. Since the paper is already submitted, all authors will have to discuss how they want to handle the issue in that case. This may range from updating the paper with the necessary corrections after receiving the reviews (regardless of whether or not the reviewers found the same issue) to retracting the paper.
• Your co-authors convince you that the issue is in fact not important or not an issue at all, which may also help to improve the paper (clarify why x is not an issue in this particular context).
• Neither you nor your co-authors are convinced by the others. In that case, you may have to withdraw as an author, painful as that may be.

I would argue that each of these results (even the last one) is preferable to just hoping for peer review to catch the issue you already know is there.

Edit: I missed that the manuscript was already submitted. However, I don't thnk this fundamentally changes things.

I think you have only one course of action that is both proper and safe.

You should immediately contact your co-authors pointing out your reservations in detail. It is but a momentary embarrassment if you are wrong. Then, if your colleagues agree that there might be a serious methodological error, especially one that might change the conclusions, then you need to immediately contact the editor, asking for advice and suggesting withdrawal while you fix the issue.

The down sides of other actions (or inaction) seem to me to be too severe to contemplate. If the paper is sent to review and the reviewers find the errors, complex as you suggest, they will most likely suggest rejection, rather than trying to work how it should be fixed. Some journals won't accept a future version of a rejected paper, so you would be back to the start in finding a publisher, and you would need to spend the time to fix the paper in any case.

But the consequences of the reviewers not finding the error are equally bad or worse. If, they don't find it and you, then report it, then you still need to fix it and the editor may not have the patience you require. The paper will need, then, to be sent out once more for review, delaying the publication, at best. Of course, you have no guarantee what that review will suggest.

Finally, if the reviewers don't find find the error and you don't report it, then the paper will likely be published. If it doesn't have an error then all is well, but you are probably one of the best people to know if there is really an error, and if you suspect it, then there is an issue whether you have an error or not. But if the paper is published with an error, then it will probably be found eventually, damaging everyone's reputation, yours as well as the journal's. Editors won't be very happy to work with you in the future, feeling that they have been misled.

Swallow your pride. Start the wheels in motion. Your reputation is at stake.