To answer some of your questions, here's how we do things in my department. I would expect that things are not too different elsewhere:
- The admissions committee is composed of about 5 faculty.
- All members read each PhD application; there is no "pre-filtering", though it's unlikely your application will be read in great detail if your "top-line" numbers (GPA, GRE, TOEFL for international students) are way out of line with department norms. We receive translations of international transcripts, and can usually get a decent handle on how good these students are.
- Some number of applicants are "obvious admits", their profiles being simply outstanding in all or most respects. Usually these "slam dunks" occupy about 20-40% of the offered spots. The "good" and "very good" applicants compete for the remaining spots.
- After scoring the applicants, the committee meets to focus on the applicants who are "in the discussion" for PhD admission. Factors working in favor of (e.g., really positive letters) and against (e.g., lower grade in an advanced math class) each applicant are discussed and weighed. Sometimes, a faculty member will "go to bat" for a student they think highly of, even if that student ranked a bit lower in the initial scoring.
- As I've noted before, applicants worry way too much about research experience and the personal statement. This is not to say that having research experience isn't helpful, or that a strong personal statement isn't an asset, but rather that it is generally much easier (and, in my opinion, more reliable) to rank applicants based on other factors like grades and letters of recommendation. This is particularly true in stat/biostat, where meaningful research experience is relatively rare and it's considered completely acceptable for incoming PhD students to not have much of an idea what they'd like to do research on.
Overall, the process really isn't that mysterious: we are trying to identify the most talented students, with an eye towards balancing research "upside" with likelihood of success in the program. The strongest predictors of success in graduate school remain grades, letters of recommendation, and to a lesser extent standardized test scores. Nobody wants to hear that because applicants would like to think that they can dramatically alter their prospects by crafting the perfect personal statement, but the reality is that your fate is mostly sealed by the time you come to prepare your application.
A lot of people contend that "admissions is a crapshoot," but that attitude is simply inconsistent with the remarkably high intra-individual correlation observed in individual admissions results across programs. With a few exceptions, an applicant who applies to a set of schools of similar strength is likely to get into either all/most of them or none of them.
CommonerCoffee reacted to Anakin in 2015 Admissions General