Out of curiosity, I am wondering if a PhD application decision letter gives the reasons for the decision?
If yes, why? If not, why?
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In my experience in the US, I have never seen any sort of decision letter for an application (whether for Ph.D. or any position) that gives reasons for the decision. It may be different in other countries, but in the US at least, there are many good reasons to never give such information formally.
The actual decision-making process is never as clean and clear-cut as anybody wants it to be, and with the limited information available in applications and even interviews, there are going to be cases where people make mistakes or make good decisions for unclear or hard to state reasons. Lots of good people may get turned down for reasons having nothing to do with them, such as just having too many good candidates in the same area.
If an institution gives real reasons in an official statement (as opposed to something meaningless and bland like "not an appropriate fit at this time"), then it is opening up the door for all sorts of potential problems and disputes, not to mention potential legal liability if something could be construed as relating to a protected category such as gender, age, religion, or ethnicity. They also have to worry about seriously unbalanced people who may take it as a personal insult and begin some sort of stalking or harassment campaign (it happens!). Not giving reasons may feel unsatisfying and cowardly, but it is an easy and safe path for an institution, and as such is generally official policy.
Now, if you have an informal trusted relationship with somebody who was involved in the decision process, then it is sometimes possible to get a strictly unofficial take on what may have happened. You might learn things you didn't want to know about how sausage is made, however.
No, there is usually no specific reason given.
What would even constitute such a reason for rejecting someone from grad school? People sometimes imagine that there might be a clear-cut explanation, perhaps numerical. "The applicant pool was strong enough this year that we were able to set a GRE verbal cutoff of 165" or "We prefer applicants who have taken eleven courses in their major, but you've taken only eight" or "You lost out to someone who published one more paper than you did" or "We accept applicants from Yale only when we run out of applicants from Harvard." That's generally not how admissions decisions work. Instead, the judgment is holistic, based on the entire application, including intangible aspects like how compelling the letters of recommendation are. There's not much to say beyond "We got many strong applications and managed to fill the incoming class with applicants who impressed us even more than you did", and there's nothing to be gained by spelling that out.
Of course, there's occasionally a simple reason, when you really can refuse to admit someone because of something specific. However, that reason is sometimes insulting, and it would be dangerous to give the impression that the applicant could be admitted in the future if they just fixed this one thing. (It's rare to see an otherwise wonderful application with one utterly unacceptable aspect. Instead, if you're unacceptable in one respect you're likely to be at least questionable in others.) It would just be adding insult to injury to say "Well, your application probably wasn't good enough overall in any case, but here's one aspect we found particularly appalling."
I think what applicants often want isn't an explanation of the decision so much as constructive advice. Should they give up on the idea of attending graduate school? Do they need additional preparation? Are they not presenting themselves in the best light? It's completely reasonable to seek this sort of advice, but preferably from a mentor. Brief comments from admissions committees are unlikely to be useful.
At my UK institution it is now required that all applications have a reason for the decision, which is communicated to the applicant. This is meant to ensure and document a clear decision making process for quality assurance purposes.
However, these officially documented reasons are often bland, boilerplate and uninformative. The useful information usually requires direct comparison with other candidates (for the relative ranking), which is exactly the information the institution can't release by law.