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Re: PhD admission without interview
First, I know very well that it's very hard to wait for some news, and there isn't much to do about it.
But it's way too early to give up hope. February is a very hot month for interviews, and even after that we see many cases of success. My own interviews when I applied were closer to the end of February, and I remember that very few users had concrete results during that period.
If after a month from now you still don't get any interview, then I think perspectives are grim, but still not enough to give up hope. But in January? Things are just starting to happen.
Maybe Urch does not keep very old threads, but back in 2016/2017 we had more users here, and we had more statistics about that kind of thing. If I'm not mistaken, about 50% of the interviews are in February.
Different universities can follow very different procedures, including how interviews are done (if they are done at all). Even in the same universities, different departments have different ways to work. And within a department, different professors prefer to do things in a different way. There is no way to tell what they are going to do.
From my experience, you also can't really trust what the universities' websites state. And when they say that an interview is not required, it often just means that professors are free to decide if they want to interview applicants or not. So, anything can happen, depending on what the faculty decides.
About professors saying positive stuff, I don't know if it's common. But I guess a lot of applicants would get some positive feedback, because applicants are often very strong. The problem is that you may be strong, but there are also lots of other applicants that are strong too. All we can try to do is to guess if an applicant has a figthing chance, but it's very hard to be sure about who are the ones who will really get offers. This is one of the reasons that, even for the very strong applicants, a common advice here is to apply widely. A lot of factors can influence, many of them unknown.
I see no reason to contact the department or the professors just because you haven't heard from them yet. If you have an offer, but are still waiting for other universities, then it may be worth a shot. I did that after I got my offer, and some universities replied saying I was rejected (but not many).