I am a PhD student and the most junior person on our small team (me, a lab manager, a statistical analyst, a wet-bench scientist, a senior statistician, and our PI). The statistical analyst has been on this team for over 15 years--we'll call her Emily. Emily trained me in some areas when I started, but I've learned a lot and I have become very independent in the 5 years I’ve been a trainee.
Emily and I have been working closely on a project lately. Because we work so closely, I often ask her for documents related to relevant analyses that she has previously done. I have come to realize that Emily is very error-prone, which I believe our PI already knows. Lately, many of these errors have been discovered when I say something like "that estimate doesn't seem quite right to me" in a lab meeting. I nearly always turn out to be correct and Emily is livid with me for pointing it out. She is typically very defensive and it's very difficult to persuade her to re-check her work. She is dismissive of pretty much everything I say to her directly.
When we went to publish a paper for which I wrote the entire first draft, contributed half of the figures/table, and designed ~90% of the experimental approach, my PI said that Emily and I should be "co-first" author. I agreed because Emily had done ~80% of the experimental work (she wrote most of the code). When my boss made the suggestion that we equally share first-authorship, Emily was so upset that she wasn't the sole first author that she literally screamed in my PI's office about it and then didn't come into lab for 3 days after. These things have set up a poor working relationship between me and Emily.
That paper was published, and I'm currently working on several papers that build on those published findings. In working on these new projects, I've discovered an important error that Emily made in the published paper that we are co-first author on.
I made the decision to go to Emily about the error first so that she had a chance to address it with our PI. She denies that there is a mistake and tells me that I'm "too confident about myself" when I say that I'm 100% certain that there is a mistake and lay out all of the evidence that definitively proves it. Unfortunately, I lost my cool a little bit and some of my responses to her were harsh and less-than-professional.
So, I have an email thread that documents Emily's mistake, that I approached her about it directly, and that she actually doesn’t understand what she was doing and therefore denies that the mistake is real; however, we both made some statements that were unprofessional.
The mistake needs to be corrected in order for me to continue with publishing my recent projects (but fortunately does not change the main takeaway of the paper). Emily plans on telling our PI that I’m just a trainee and that I don't know what I'm talking about and that she did not make a mistake (again, I can definitively prove that she did with the email chain). She also has some clout with our PI because she's been there much longer than I have and is more experienced.
How should I proceed?