The question was recently posed: how to choose between Ph.D. programs? The answer can be reduced to a trivial mathematical assessment, whose solution lies in the determination of both (a) the function form and (b) the value of an unknown function, V.
Whatever the form of V, you want to maximize its specific form V(h,m), where h is a test statistic that evaluates the probability of being professionally and personally happy and m is a test statistic that evaluates the probability of your program being well funded and monetarily well supported. Please note that in principle, the function is an approximation – it can have a correlation on m’, the amount of money you will be paid to be part of the program (note! m’ can go negative in certain Ph.D. programs, so don’t choose based on unsigned quantity|m’| but on the actual value m’). However, in principle h and m’ are correlated and that correlation can be factored into h during the maximization process.
I should say that the form of the function V(h,m) is not analytically known, but it has certain boundary conditions. I repeat them here for completeness:
- V(h,m) -> 🙁 as h->0 and/or m->0
- V(h,m) -> 🙂 as h->infinity and m->infinity
The form of the function in the intermediate condition that h is finite and m->infinity, or m is finite and h->infinity, is not well described.
This author considers the static approximation of V, which is insufficient. Function V depends on time, since both m and h depend on time. Presumable the author is considering the time-averaged values of m(t) and h(t).
In addition to the boundary conditions of V(m,h), one must consider the negative poles of V(m(t),h(t)), which is typically due to h(t)-> -infty. While exceedingly narrow negative poles are frequent and have minimal impact, a figure of merit can be constructed by integrating the values of h(t) that are below a given threshold. This value should be considered, independently of other (likely earlier) values of h(t) and certainly independently of m(t).
This commenter has pointed out an important point, but not one that necessarily changes the argument. Indeed, m and h can be functions of time. The act of maximization of the function V(m(t),h(t)) is usually considered within a range of values of t, t = [t0, t1]. This can be considered a local maximization, as global maximization (where to = 0 and t1 = dead) has been repeatedly shown to be nearly unsolvable.
Indeed, it is important to consider the possible maximization of V under three time windows: T1 = [t_{ug}, t_{g1}], T2 = [t_{g1}, t_{g2}], and T3 = [t_{g2},t_{gX}], where t_{ug} is the last year of your undergraduate career, t_{g1} is the beginning of your first year of graduate school, t_{g2} is the beginning of the second year of graduate school, and t_{gX} is the year you graduate from graduate school. For a discussion of the time-dependence of the boundary condition t_{gX}, c.f. “Graph – Motivation Level.” Published in Ph.D. Comics. 1999..