Is there such a thing called native speaker bias when publishing? This has been a subject that has occupied my thoughts for the past years but haven't really managed to formulate it completely for myself. I have this feeling (and I can't say it's more than a feeling at this point) that native speakers of English have an easier time with getting their article published.

I should make it clear, right away, that I don't mean just because you are a native speaker you can get whatever published, but given a particular project, if the main author is a native speaker the article is more likely to get accepted, or at least draw more positive reviews. I motivate the potential existence of such a bias on the fact that it's likely that a native speaker does a better job in writing than someone who has English as a foreign language. 

Since I do not have any data to back this theory up, I would like to ask if you have come across any numbers/facts regarding any such bias in the publication/acceptance rates in general, as well as high-IF journals specifically. 

EDIT: I should perhaps rephrase and add a bit more details to the question here. I do not refer to small grammatical mistakes, misspellings or anything of that nature. What I am referring to here, is the wider vocabulary a native-speaker has in his/her disposal, the phrases and expressions that they use that might not be readily and easily available to non-native speaker. I could perhaps summarise it the differences as the metaphorical chocolate chips that a native speaker can and most likely will bake into the cookie that is the manuscript.

I am clear on the point that the clearness criteria is still the most important and that's why I did not talk about the rejections (it's natural that non-native speakers get their papers rejected based on language more often compared to native speakers). As a non-native speaker myself, it is beyond any doubt that I need to write my manuscripts with a clearly understandable language in order for them to get published, but again that's not really what I am asking here. I am more interested in whether or not articles written by native speakers come across as "better quality" based on the fact that they are more likely to utilise their edge with the language.

There are of course other factors at play, but it would be interesting to see if there are any stats involved.
Let me share with you my personal experience and feeling about this issue. I am not a native English speaker and I am aware that my word choices and the patterns of my sentences makes it clear to my audiences and especially my readers. However, I have received several positive compliments in the referee reports of my papers about my writings. Therefore I strongly believe that people who are in charge of deciding about accepting a paper mostly consider the logical and grammatical correctness of the writing of the paper ( and of course its scientific value). So I would be very surprised if I see such a phenomenon as you described.