I am swamped and don't want to take on another review. But one of my papers is also currently under review in that same journal. Will I upset anyone by refusing?
Declining to review
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Back when I was a journal editor's assistant, I would often suggest picking reviewers who had submitted manuscripts to us. It could go both ways - they might feel obligated to review, but sometimes we'd reject their piece before they turned in a review and they wouldn't have much motivation to help us out after that.
Someone turning down a review request never influenced how I treated their manuscript. I doubt it influenced my boss's final decisions, but of course I can't be certain.
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I think the Roderic's comments match the norm. It shouldn't influence the outcome.
Most importantly, the people who are reviewing your paper don't know that you've been asked to review a paper for the journal.
However, saying that it shouldn't and probably doesn't influence the outcome doesn't mean that it might have a marginal influence. As Roderic indicates, the editor has access to that information and even if someone says that that additional information isn't influential it's hard to ignore other evidence that what people say and what they actually do is often quite different.
It's probably not going to sink, but on the margins, it might make a difference. It's like a good GOTV operation. You hope you don't really need it, but it might make a difference on the margins.
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You should probably ask Dan Nexon since he is the editor who appears here. My guess is this: if you ordinarily do reviews, but decline when you are overwhelmed, then the editor will not read much into it. Plus editors don't desk reject based on whether you review or not. They may unconsciously read more into the reviews if they think you are free-riding uber-shirker, so that borderline calls go against you. But they are not going to consciously reject you because you begged off one time.
To be sure, it is better to say yes most of the time than no most of the time, as karma does probably matter a bit. But the best way to put it is: I cannot do this review in a timely manner given my current set of commitments. Ask me again in x months.