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Nature and Nature Communications recently started publishing post-publication commentary in specially published peer-reviewed comments called as "Matters Arising" https://www.nature.com/ncomms/submit/matters-arising

It's a great initiative.

Does anyone know how often these commentaries lead to changes in the published article? Any stats on how often authors respond with new data/analysis.

I could not even find an example.

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Many journals publish commentary articles on published papers; sometimes these are even invited directly by the journal editor to allow other researchers in the field to respond and are published simultaneously as an editorial, especially if a new finding is challenging established dogma.

Sometimes these commentaries are mostly critical, sometimes they are mostly positive. When negative, the original authors often will reply.

Like the commentary, that reply is typically another separately published article rather than an amendment to the original article. It could include additional analyses, further perspectives, references to additional literature, or agreement with critiques of the original paper.

Nature isn't inventing something new here, they are just adding a title to an established practice.

You can try to find some of these comments and replies by searching Google Scholar for 'comment on' since those words are often added to these articles, though this search term is of course not perfectly sensitive nor specific. I think it is unlikely you will find statistics on how common these sorts of articles or replies to them are, and I suspect it varies a lot with field and journal.

One example paper of authors replying to a comment on their original work comes up on the first page of the search linked above:

Wang, J., Scholl, A., Zheng, H., Ogale, S. B., Viehland, D., Schlom, D. G., ... & Neaton, J. (2005). Response to comment on" Epitaxial BiFeO3 multiferroic thin film heterostructures". Science, 307(5713), 1203-1203.

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