calculating your chances

[Tiglath-Pileser III]

This is a very interesting thread, especially useful was the link. I did find out that most of the successful applicants in my field received 2 offers on average. Last year my probability was 1.40 and this year it is 2.14. So, that is a significant difference, which at least gives me some reason for hope.

[mostlygoo]

This is obviously wonderful. And to me it's very Lit-oriented, too -- I'm thinking Gravity's Rainbow.

Some individual programs do have GPA/GRE scores, too, if you want to weight the probabilities a bit. I know Georgetown's M.A. program does, listed by year for the past 7 or 8 years. A little more work to investigate each program you're applying to... and maybe badger the admissions office for data if it's not online, and explain it's for a statistical analysis... and do that anonymously so they don't link your weird behavior to your application... but I don't think that's unreasonable.

[nostimost]

Hoooly crap. I just had my one-and-only math class of my college career last semester, and we did a lot with Bayes' theorem, so I kiiinda got what was going on, but oh man, I never want to have to work out that formula ever again. Thanks for doing all that work, though! Definitely cool.

I do have one stupid question, though: On the Petersons site, where are you guys finding the acceptance rates? Am I just dumb? I searched for specific schools and programs, and it just told me the application requirements and stuff.

[newms]

On the Petersons site, where are you guys finding the acceptance rates?

It's on the left - click on 'Admissions'. The acceptance rate is there on that page beside 'Student Statistics'.

[nostimost]

It's on the left - click on 'Admissions'. The acceptance rate is there on that page beside 'Student Statistics'.

Ah! Thank you. I was, in fact, being dumb.

[newms]

I was, in fact, being dumb.

Not at all!

[ʕ •ᴥ•ʔ]

Some individual programs do have GPA/GRE scores, too, if you want to weight the probabilities a bit. I know Georgetown's M.A. program does, listed by year for the past 7 or 8 years. A little more work to investigate each program you're applying to... and maybe badger the admissions office for data if it's not online, and explain it's for a statistical analysis... and do that anonymously so they don't link your weird behavior to your application... but I don't think that's unreasonable.

I think the following toy model might be an unbiased estimator of chances given GPA/GRE - or other factors, for that matter, like number of pubs - on the unlikely chance that you have both the averages for the applicant and admittant pool.

Suppose that every applicant has either a low or a high score - say, a 3.5 or a 4.0. (Th average of both pools should be between these scores, and your own score should be equal to or between them.) Then you find what the ratios between applicants must be and the ratios between admittants must be. (If the applicant average is 3.7 and the admittant average is 3.8, then 40% of applicants have high scores, as do 60% of admittants.) Then consider the admissions rate - if it's 20%, say, then that's composed of 12% high scorers and 8% low scorers, meaning that 12/40 (30% of) high scorers and 8/60 (13.3% of) low scorers are admitted. Then give yourself (your score - low score)/(high score - low score) probability that you are in the high-scoring category. So if you have exactly the applicant average your chances remain equal to the admissions rate (as we would expect), and if you have a 3.8, your chances are 0.6*0.4+0.4*.133 = 29%.

I'm certain that this model is an unbiased estimator if you really are only measuring a binary factor - if there's gender-based affirmative action (or simple bigotry), say, or measures that are effectively binary, like if you know the percentage of applicants and admittants who are polylingual and either (a) that you're monolingual, (b ) that you speak a typical number of languages for a polyglot (presumably not in this example, since it's probably one-point-something), or © if you speak some number of foreign language and don't have good reason to suspect it's better or worse than the number typical for polyglots (again unlikely, since it's probably one-point-something, but.) For gradients like GPA - and especially the GRE, which has a known and very wonky distribution - my intuition is that something like the toy model above is a good basis for a true unbiased model, but there's some other set of equations, based on what you know otherwise about their distributions, that will tell you 1) what to set as the high and low scores (possibly even ones which are impossible for real students to have) and 2) how to weight your score relative to each, based on something other than p(high)=(yours - low)/(high - low) (possibly weighing one of them more than one and the other negatively).

Actually, for a quick proof that your choice of high and low scores matters: suppose we selected a 3.0 and 4.0 for the high and low scores above. In that case 70% of applicants and 80% of admittants have high scores, meaning their 20% admissions rate is composed of 4% low scorers and 16% high scorers, such that 16/70 (22.85% of) high scorers and 4/30 (13.3% of) low scorers are admitted. A cookie to whomever can tell me how to determine the proper high and low scores.

(Probably easier cookies: given everything in the OP (or your own, better model) and that you've been accepted into institution Q, what is the conditional probability that you were admitted to institution R? Given that you were rejected from Q?

What is your expected number of admissions, given that you will be notified of admissions in an unknown order and then rejections, and that the first notice you receive is admission to Q? Given that you expect to hear admissions in a known order, and then rejections, and that the first notice you receive is admission to Q?)

Edited February 8, 2011 by ʕ •ᴥ•ʔ

[ʕ •ᴥ•ʔ]

(Actually, there's one case in which GREs give us a very easy calculation: if there's an unofficial math cutoff which, if you make the cutoff, they don't care about.. Since the quant section has an incredibly uniform distribution - one point is very close a quarter-percentile throughout - means equal medians and, if a school's applicant pool acquires its average from being drawn only from the top whatever% of scorers (who are drawn from that group uniformly, at least with respect to GRE scores), an admittant average of x implies a cutoff of 2x-800, such that your chance of being admitted to the program is zero if you're below the cutoff and (admission rate)(800- applicant average)/(800 - admittant average) if you're above. But that's assumption-heavy.)

[Reviewing]

I have to say that your prior is poor. The chance of admission vary significantly between different applicants. Plus people don't get the same offers. Some have fundings but some not. In a word, the prior at the very beginning makes the calculation far from the facts.

[greengrass2]

Um... is there an app for this?

(You can thank me with stock privileges and a Maserati or two)

Warning: nerdy and completely useless except as game; if I didn't want to spend a totally unreasonable number of hours engaged in pointless intellectual pursuits, I wouldn't be applying to grad school My math or reasoning in general might be off; if it is; call me on it. Note also that I'm being a Bayesian about things, so read "there is x% of y" as "you should estimate the probability of y at x%."

So, you've applied to some set of schools. For most of them, Peterson's lists their number of applicants, admissions rates, and actual number of attenders from each class. How would one produce an unbiased estimate of your chances of universal rejection (and, possibly, a few other things) from just this information? (Of course if you have more information than this, you would want to make the model more complex to incorporate that extra information, and I'd love to see models that incorporate GPA/GREs, &c.)

a : the number of admissions you will actually receive

n : the number of schools to which you have applied

p_i : the admissions rate of the i^th school to which you have applied (the order isn't important)

Prior assumption: for the schools for which you have applied, you have no particular reason to believe that you are especially more or less competitive than the typical applicant. This doesn't mean that you expect to be exactly in the middle - if that case you know you would be universally rejected, assuming admissions rates are all below 50% - but that you expect a 1% chance of being in the first percentile of competitive applicants, a 2% chance of being in 2nd percentile or better, a 3% chance of being in the third percentile or better, and so on. If you can accept this prior, your chances of being accepted into school i is, conveniently enough, p_x, and the average expected number of schools you will get into is

μa = Σⁿ_i=1p_i = p₁ + p₂ +p₃ + ... + p_n

or the additive sum of their admission rates. However, you don't know how well these are correlated with each other. If they're maximally correlated - they all admit students on precisely the same criteria - then your chances of a wipeout are equal to the complement of the most favorable admissions rate among your schools; if they are totally uncorrelated, your chances of a wipeout are equal to the multiplicative sum of their complements; if if maximally negatively correlated, then your chances of a wipeout are min(0,1 - μa). Common sense says that they should be positively but not maximally correlated, but how much? Fortunately you know

b : the number of admittances schools in your field send out divided by their number of graduate students per year, where "field" is selected such that its competitiveness roughly reflects the competitiveness of the set of schools to which you have applied

(Sneaky assumption: the number of those in your field you are admitted to grad school and choose not to go at all is zero, or at least small enough to be ignored.)

Thus we know that a randomly chosen applicant in your field - someone, by the first prior, who is as competitive as you - should expect, given that she is accepted into any schools, to get into b schools on average. If, as would be convenient, her expected total number of admittances including the possibility of wipeout is the same as yours, μa, then your/her chances of a wipeout, p(a=0) are

μa = 0*p(a=0) + b*(1-p(a=0))

μa/b = 1 - p(a=0)

p(a=0) = 1 - μa/b

If you haven't applied to the typical number of schools

However, this randomly chosen applicant, who is as competitive as you, isn't necessarily applying to as many schools as you - she's applying to n̄ of them, which might be more or less - although by assumption we suppose the schools she applies to are as competitive as your own. So in fact her expected number of admissions, μā = μan̄/n and

μan̄/n = 0*p(ā=0) + b*[1-p(ā=0)] = b*[1-p(ā=0)]

μan̄/bn = 1 - p(ā=0)

p(ā=0) = 1 - μan̄/bn

n̄ = bn * [1 - p(ā=0)] / p(ā=0)

If we knew n̄, we could know p(ā=0) as well - or visa versa - and thus

p(a=0) = p(ā=0)^(n/n̄)

p(a=0) = (1 - μan̄/bn)^(n/n̄) or p(a=0) = p(ā=0) ^ { p(ā=0) / b[1 - p(ā=0)] }

Can we produce n̄ or p(ā=0) independently? Unfortunately I don't see a way to do so, limiting yourself to the Peterson's data. Choose a number that seems reasonable for one or the other based on anecdotal evidence, or find some publicly available data (and post it here, ideally.) But either way an estimate of one should get you to p(a=0). This should also give you B=μa|_a>0, the expected number of schools you get into in the event that you get into any schools at all:

μa = 0*p(a=0) + B*[1-p(a=0)]

B = μa / [1-p(a=0)]

p(a=0) = 1 - μa/B

Revising in light of results

All of the above assumes that you haven't heard back from any schools yet. If you get an acceptance or rejection, how should that affect your expectations of getting into other schools? Unfortunately the ratio of acceptances to grad students doesn't tell us what the distribution of acceptances among the admitted is.

Suppose you hear back from your first institution, University Q - an acceptance. Will you get into another? According to Bayes' theorem,

p(a>1)|_{into Q} = p_Q|_a>1 * p(a>1) / p_Q

Only p_Q is a known constant, so we need to guess p_Q|_a>1 * p(a>1).

p_Q|_a>1 is the chance that, given that you got into more than one school, one of those schools was Q. This is equal to

p_Q|_a>1 = (p_Q - p_Q|_a=1) / p(a>1)

so

p(a>1)|_{into Q} = [ (p_Q - p_Q|_a=1) / p(a>1) ] * p(a>1) / p_Q

p(a>1)|_{into Q} = (p_Q - p_Q|_a=1) / p_Q

(One intuitive, but clearly wrong, estimate of the chance of admittance to Q given only one admission is

p_Q|_a=1 = (p_Q / μa) leading to

p(a>1)|_{into Q} = [p_Q - (p_Q / μa)] / p_Q

p(a>1)|_{into Q} = 1 - (1 / μa)

This implies that an acceptance from one school is nearly as good a signal as a decision from another, and in fact that getting into an easier school should revise your expectations up more than getting into a harder school - prior to learning anything, you have a higher expectation of getting into at least one school other than your reach than getting into at least one school other than your safety, but in fact getting into your reach and into your safety brings their chances to the same level. In fact if there's any overlap between expected admissions at all then the chance of being admitted to an easier program but not a harder program is not only more likely than the reverse, but in a way that exaggerates their independent probabilities.)

One obvious method is to use recursion: imagine someone, as competitive as yourself, who applied to every program but the one you've just heard back from, i.e. μa₂ = μa - p_Q, n₂=n-1,n̄ remains constant, and her field is your field, such that

B₂ = (μa - p_Q) / [(1- { p(ā=0) ^ [ (n-1) / n̄ ] }]

p(a₂=0) = 1 - [ (μa - pQ)/B₂ ]

In that case, p(a=0) - p(a₂=0) = p_Q|_a=1, and - if we want to write out a big ridiculous equation -

p(a>1)|_Q = {1+ p_Q + {(1- [ p(ā=0)^(n-1)/n̄ ](μa + p_Q)}/(μa - p_Q) - p(ā=0)^n/n̄ } / p_Q

p(I made some sort of obvious arithmetic mistake or worse) > 0.5, so the above is most likely nonsense. If it's right then calculating how to update your chances in case of a rejection should be trivial.

[ʕ •ᴥ•ʔ]

I have to say that your prior is poor. The chance of admission vary significantly between different applicants.

How would you estimate the chances of a given candidate being admitted to a given school, given (whatever information about the applicant and school that you would like to be considered known?)

Plus people don't get the same offers. Some have fundings but some not. In a word, the prior at the very beginning makes the calculation far from the facts.

Well yes, obviously this doesn't attempt to address funding (though you could incorporate it if numbers were available.)

[Reviewing]

How would you estimate the chances of a given candidate being admitted to a given school, given (whatever information about the applicant and school that you would like to be considered known?)

You are right at this point. However, if we consider only chances of a given candidate being admitted to a given school, the calculations are useful for not the particular applicant but the admission committee. I mean, if I want to know my chance, the calculation does not give me special considerations, which loses its point...

Well yes, obviously this doesn't attempt to address funding (though you could incorporate it if numbers were available.)

That is what makes it unlikely to give you a perfect result--the unavailability and variation of those numbers (while # of admitted students may remain relatively stable in the near future, fundings open for students vary greatly ...)

[Reviewing]

One more question, is the student statistics appear on the peterson's complete statistics about student recruitment of the recent year?

[thesnout]

That equation scares me. I'm going to look away now before I self implode. Ahhh math. hehe