I recently discovered that a paper I reviewed a while back and rejected as a result of major methodological flaws had been resubmitted to another journal and published without addressing any of the significant errors that required the paper to be rejected in the first place. Is there anything I can do that doesn't violate the confidentiality of the peer review process to address the problem?
Two things are unclear about your question Is there anything I can do that doesn't violate the confidentiality of the peer review process to address the problem?:
- What is the goal of any intervention?
- Why do you think that anything should be done?
If, for example, you aim for retraction, I guess this could be difficult. You may go for this on the ground that you can prove that the authors were aware of the flaws you pointed out and submitted the paper knowing that there was something wrong. The journal may have some rule that the authors have to certify that the results are true to the best of their knowledge. If this really is so, you may have some angle of attack. If you aim for correction by the authors, it could also be difficult. However, you could consider writing a letter to the editor, pointing out what is wrong and then ask if the journal would publish such a letter. Some journals I know have such formats. You could also do follow-up research on the same topic and write a paper yourself where you refer to the paper with flaws and describe what wrong there.
For the second question, it sounds like you feel that "What's published and peer-reviewed should be true." While this resonates with me, it's not something that is close to true now and probably never will be. Mistakes happen. You may then trust in the scientific community that the flawed paper will be perceived as such in the long run and find peace with this particular paper.
Finally, the premise that confidentiality of the peer review process may permit some actions is not totally clear. As I wrote in a comment, I asked a question about this on MO and got very different responses.
I think the first question you need to ask yourself is what have the authors done wrong. Disagreements between authors and reviewers about methodology are fairly common. I see two situations. The first is you believe the paper is wrong and needs to be retracted. The second is you believe the authors have intentionally mislead the reader (since you told them about the error in your review).
You should be able to make the claim that the methodology is flawed from the published manuscript. As that claim would only be based on the published manuscript, it would not violate the confidentiality of the peer review process. You should follow whatever procedure the publisher has for reporting issues.
If you believe the authors engaged in ethical misconduct by ignoring your review, you should be very careful. You should talk to colleagues to make sure that the methodology is in fact flawed. Once you are sure of the flaw, you then need to approach colleagues about the "hypothetical situation" of whether it is potentially unethical if the authors knew about the flaw prior to publishing the work. Once you are confident that there is a flaw and that it was potentially unethical to publish the work, you should go through the editor at the journal you reviewed for. Let the editor know about the misconduct and hopefully they will follow up on it. If the editor chooses not to follow up on a claim of misconduct, then you should follow up on that. In this case, where the journal refuses to follow up on a claim of misconduct, I think it is reasonable to violate the confidentiality of the peer review process to the extent required to show the misconduct to an outsider (e.g., COPE).
There are plenty of similar cases where a rejected paper gets published in some other journal. But if a reviewer had pointed out some mistakes, then it is a bad attempt to submit to another journal without making a correction or giving proper evidence. In my point of view, the author must have disobeyed the statement during journal submission, if they have anything like,
"whether this manuscript was rejected by another journal. If yes, then why?"
So inform the methodological flaws you noticed to the editor of the published journal with a request to being confidential. And let the editor to reply in that matter.
Another best way would be you can write a commentary note to the published paper to make a discussion. In this case you will have to justify the drawbacks of the used methodology.
Another possible way would be to remain silent.