StatsG0d

I agree with @bayessays. Most students will basically have taken up to Real Analysis and not much extra. Some students even get admitted without RA, although this is becoming more rare. I do think that the more you have, the stronger your application will be. So if you're fighting for the top programs, it definitely won't hurt to take more math (provided you earn good grades in those classes).

I think it matters more that you have letters that attest to your mathematics ability than your statistics ability.

Your math background is extremely broad, and you've received a bunch of high grades in graduate level math courses. That said, it's hard to know how adcoms will view you because of your undergrad GPA and getting low grades in calculus / linear algebra, which are the two most important prerequisite courses in statistics / biostatistics. You will have to address these discrepancies in your SOP. Given your math background, I recommend that you take the math subject GRE. I think you could do quite well on it, and if you do, no one is going to care about your undergrad grades as a lot of material on the math subject GRE is calculus / linear algebra.

I don't think biostats is more competitive than stats. Probably more the contrary. Very few students have any relevant research experience coming in. To me, it sounds like your research experience does relate to biostats, as high dimensional data is common (e.g., genomics) and so is shrinkage (e.g., Bayesian analysis). To formulate your interests, I recommend you read some papers or just google some fields that are popular in biostatistics and relate to the branch of statistics you are interested in, e.g. Computation - genomics Machine learning - precision medicine Spatial/temporal - Image analysis (e.g., diagnosing cancers based off of an MRI) Bayesian - design/analysis of clinical trials Virtually every subfield of statistics has applications in biostatistics.

If you are interested in applied statistics, I don't think UPenn would be a very good fit. The other schools on your list seem reasonable (maybe not Iowa state though). Have you considered biostatistics? I think you would have a very strong profile.

Although your GPA is quite low, you have several published manuscripts and your math background is very deep. I think you have a shot at the schools in the 5-10 range. I would apply to all of those (replacing your least 5 favorite schools in the above), and see what happens. Most of the programs you listed are very unknown. It's not clear they would want someone who appears equipped to be a methodologist, as many of these programs are applied. If applications are your interest, I would clearly state this on your SOP, because based on your math background I would think you would want to be a methodologist.

I agree if the OP meant stats departments then they should apply to those. I assumed they meant biostats

I apologize if I confused you. I didn't mean the "applied real analysis" course specific--just a general course in what most institutions call real analysis. Advanced Calculus is the way to go. Perhaps that's the confusion surrounding your advisor's saying no one takes it.

The problem with NYU and Northwestern is that both programs are highly selective relative their ranking. This is typically because some international students care more about university prestige rather than departmental prestige. The other schools don't seem bad. If being in a somewhat large metro area is important to you, you can apply to some of the larger programs in those areas such as NCSU (downtown Raleigh), OSU (Columbus, OH), Minnesota (Minneapolis, but cold). Some of the larger state schools in smaller areas tend to have really sophisticated bus systems that can get you around relatively efficiently without a car (e.g., Florida State, Iowa State, South Carolina, etc.) You can browse through each school's / community's transit page and see where the buses go and how convenient it would be to e.g. live somewhere within walking distance to a grocery store and being able to take a bus to campus.

I think you're selling yourself really short. First, I wouldn't bother applying to UCSD, TAMU, Rutgers, or Rice. NCSU doesn't have a PhD in biostatistics, although they do have a Biostatistics concentration in their Statistics department. Also, not sure if UW refers to Wisconsin or Washington. You're competitive for any of the biostats programs in the top-5. In fact, I would be shocked if you didn't at least get into one of UNC or Michigan. I would say apply to all the top-7 biostats programs and maybe those Canadian schools if you're interested. I'd maybe add McGill if you're interested in precision medicine. I'd also add Berkeley if you're interested in causal inference. It sort of depends. If you've taken all the qualifying exam courses, the department *might* let you take the qualifying exam the summer you arrive. Otherwise, they may force you to retake their versions of the courses and then take the qualifying exam at the end of the first year. It usually depends on both the department and the specific case. If the qualifying exam is taken in the 2nd year (e.g., how it is at UNC), they'll let you skip the first year curriculum, but you'll have to take the 2nd year curriculum and take the qualifying exam the summer after your first year.

It's really hard to say. Typically, a lot of non-US universities have sort of a pipeline with various US-based PhD programs. If yours doesn't have that, it's honestly a crapshoot. Your best bet is to apply to schools that tend to admit a lot of international students (e.g., University of Florida comes to mind). There are probably many others, but that's the one I know for sure tends to admit many international students. Browse this forum--I'm positive others have mentioned other programs.

You should check out mathematicsgre.com, as this forum is really biased more towards statistics than math.

It's possible (and highly likely) that the other students had already taken the course (e.g., in undergrad). You definitely need to take real analysis for stats. You *might* be able to get away without taking it if you're doing biostats (although, this is becoming less common at the top 5-7 programs, as the field is becoming more competitive). Even if you managed to get into a program without taking analysis, you would have wished that you'd taken it. Even in Casella-Berger level Math Stats, real analysis is very useful for making mathematically rigorous arguments / proofs. I feel like you (or anyone without real analysis) would struggle in a pure stats program without it. Any/all of those courses will be useful when you reach the dissertation stage, but the reality is adcoms don't really care much about how many statistics courses are taken (unless they're mathematically rigorous courses e.g., linear models, probability theory, (martingale-based) survival analysis, etc.). If I'm on an adcom and I see that you've taken these stats courses, I'll think "OK, it's nice that they clearly have shown an interest in statistics, but how prepared are they to be successful in the program?" I'd look at the GRE and see a lower score relative to other applicants, and then think "well, perhaps this student had a lower GRE score, but has demonstrated mathematical maturity through courses." Then, when I see the lack of a single proof-based course on the profile, I would almost certainly reject the applicant. I think it's important for you to reflect deeply and see if you know what you're getting yourself into. If you are trying to avoid taking real analysis because you dislike theory, then I can assure you that you will not like doing a stats PhD, and you will burn out really quickly. The courses / qualifying exam is difficult even for those that have taken real analysis, and I truthfully can't imagine an individual doing well without it, especially relative to peers. If you are more interested in the application of statistics, there are other fields you can consider that utilize advanced statistical methods (e.g., epidemiology, psychology, quantitative methods in the social sciences) without the need to dive into the theory. The purpose of a stats PhD is to make you equipped to develop your own methods.

This is exactly what I was trying to say regarding Kosorok vs. Hudgens and causal inference / precision medicine above, but is a more elegant and general answer. Couldn't agree more. I totally agree with you, but I figured I'd let the OP decide / figure out which programs are suitable.

It's a fair question. To me, most of causal inference is concerned with identifying a population average treatment effect (typically not adjusted for covariates), while precision medicine is mostly concerned with which treatment for which individual at what time. Most of traditional causal inference utilizes classical statistical techniques (e.g., regression, GLM, etc.), albeit with some adjustments to account for confounding. In causal inference, it's really important to prove things such as consistency and asymptotic normality. In precision medicine, a lot of the methods are more machine learning focused. They might prove consistency, but asymptotic normality is a bit rarer. I guess I just feel precision medicine, while a specific case of causal inference, is a lot different than other fields with specific cases.