Many a male scientist would have you believe that, when it comes to pursuing an academic career in a scientific field, female candidates at various levels enjoy a preferential treatment, often being chosen over equally or better qualified male applicants. This is allegedly due to a concerted effort taking place in many countries in the western world, aimed at increasing academic female representation in fields of science and engineering where women have traditionally been vastly outnumbered by men.
On the other hand, many if not most female scientists take issue with the above contention. They maintain that such actions, if they occur at all, have little or no effect in an environment that is dominated by and strongly biased in favor of men, and where women attempting to establish a career routinely face more or less overt discrimination.
Aside from anecdotal evidence  (more or less accurately recollected and recounted), is there any way of assessing such a statement quantitatively ? In other words, is it possible to determine whether, say, a female postdoc in a scientific discipline is more likely to land a tenure-track assistant professorship than a male counterpart ? Or whether she is more or less likely to make it all the way to full professorship, once on the tenure track ?
At least as far as the North American academic science scene is concerned, data are rather easily available. The US National Science Foundation publishes annually the Science and Engineering indicators. Appendix table 5-19 of the 2008 edition offers a fairly detailed breakdown of the employment figures for male and female doctorate holders across the various disciplines.
With the aid of Bayesian statistics, one may seek to provide an answer to the above questions .
The basic aspects of Bayesian statistics are quite intuitive. Elementary rules of probability assert that, given two independent events A and B, each one characterized by a well-defined probability of occurring (let us call them p(A) and p(B)), the probability p(A,B) that both of them will occur is given by
i.e., by the product of the individual probabilities.
For two events that are not independent, i.e., the occurrence of one is affected by that of the other, one introduces the conditional probability p(A|B) for event A to occur, given that event B is known to have taken place. An analogous definition is given for p(B|A).
A fundamental result known as Bayes’ Theorem asserts the following:
p(A,B) = p(A|B) p(B) = p(B|A) p(A)
Stated in common language, the above means that the probability of occurrence of two events, can be conceptually broken down into the product of the probability for either one of the two to occur (typically referred to as the prior probability), with the conditional probability that the second one take place if the first one has happened. It does not matter which one of the two events is considered “first”.
For example, the probability of being in a car accident can be regarded as the product of the prior probability for one to be in a car, multiplied by the conditional probability of being in an accident if inside a moving car.
First case: postdocs landing tenure-track faculty jobs
It is essentially the norm, nowadays, for newly hired tenure-track (TT) faculty in most scientific disciplines to have undergone a period of a few years of postdoctoral training. The probability for a postdoctoral researcher to be female and to land a TT faculty position can be expressed as
p(TT,F) = p(F) p(TT|F)
where p(F) is the prior probability that a postdoctoral researcher be female in the first place, and p(TT|F) is the conditional probability that the postdoctoral researcher make the transition to tenure-track faculty given that her gender is female. Analogously, for male researchers one can write
p(TT,M) = p(M) p(TT|M)
On taking the ratio of the above two expressions one obtains
g = [p(TT|F)/p(TT|M)] = [p(M)/p(F)] [p(TT,F)/p(TT,M)]
The quantity g can be regarded of a measure of hiring bias either in favor or against female researchers, as it expresses the relative probability that a female postdoc will land a a faculty job, compared to that of a male postdoc. A value of g equal to, or sufficiently close to one, indicates a substantially gender-blind hiring, whereas a value of g significantly greater (lower) than one indicates hiring bias in favor of (against) female postdocs.
Because data for the prior probabilities p(M) and p(F), as well as for the joint probabilities p(TT,M) and p(TT,F) can be inferred from the above mentioned NSF data, which keep track of the female fractions of the postdoctoral and tenure-track populations, one can estimate the “gender bias ratio” g .
Let us consider the most recent (2006) data in the above-mentioned table, pertaining to all fields of science. Men account for 56.3% of all postdocs and 55.2% of all junior faculty. This means that the value of g across all sciences is
gAS = [56.3/43.7][44.8/55.2] = 1.05
It is easy to obtain estimates for individual fields as well:
gPS = [2.9/1.0][1.9/5.6] = 0.98 (physical sciences)
gLS = [6.7/6.1][9.8/11.2] = 0.96 (life sciences)
gMTH = [0.8/0.2][1.1/2.4] = 1.83 (mathematics)
gENG = [2.4/0.6][1.3/4.8] = 1.08 (engineering)
gPSY = [0.6/1.1][4.8/2.9] = 0.90 (psychology)
gSS = [0.4/0.5][5.2/4.8] = 0.87 (social sciences)
The above numbers would appear to indicate remarkably gender-blind hiring at the tenure-track level. For, the values of g for the various scientific disciplines and engineering, are less than 10% away from unity (which corresponds to lack of gender bias). The only case where some noticeable bias seems to exist (in favor of women) is Mathematics. Perhaps surprisingly, the sciences and engineering seem to be in fact overall friendlier to women at TT hiring time, than psychology and the social sciences, albeit not by a large amount.
Second case: from the TT to full professorship
The same analysis can be carried out to investigate the relative probability that a female tenure-track faculty will eventually become a tenured full professor (FP), with respect to the same probability for a male colleague. Using the same notation as above, we define
p(FP,F) = p(FP,F|TT,F) p(TT,F)
p(FP,M) = p(FP,M|TT,M) p(TT,M)
and consider the “gender bias ratio” h = [p(FP,F|TT,F)/p(FP,M|TT,M)].
The values, computed as above using data from the same source, are the following:
hAF = [33.7/89.8][28.6/23.2] = 0.46 (all sciences)
hPS = [2.7/17.6][5.6/1.9] = 0.45 (physical sciences)
hLS = [13.3/30.5][11.2/9.8]= 0.50 (life sciences)
hMTH = [1.4/9.2][2.4/1.1] = 0.33 (mathematics)
hENG = [1.2/15.1][4.8/1.3] = 0.29 (engineering)
hPSY = [7.0/9.1][2.9/4.8] = 0.46 (psychology)
hSS = [8.7/20.7][4.8/5.2] = 0.39 (social sciences)
The problem here is clear: women on the tenure track are significantly (between two and three times) less likely than men to make it all the way to full professorship. It is at this stage, not at TT hiring time, that problems begin for female academics, who ostensibly face an environment much more hostile to them than to their male counterparts.
It would be interesting to know at what level much of the “weeding out” occurs, i.e., whether it is along the TT, or at the time of applying for tenure, or later. The data provided by the NSF Science and Engineering Indicators are not broken down into further stages (e.g., tenured associate professors), and therefore it is not possible to make further statements.
It is interesting, however, to note how the social sciences and psychology are no better than science and engineering, from the standpoint of female academic advancement. In fact, social sciences are only better than mathematics and engineering (where women appear to have the hardest time), and worse than the physical and life sciences.
This suggests that this may not be so much an issue of women in academic science, as much as one of women in academia. If the above analysis is accurate (as usual I welcome comments and I shall publicly thank anyone who points out any flaw in my reasoning), then it would suggest that perhaps the bulk of the effort aimed at increasing female representation in academic science (or in academia) ought be directed at providing effective mentoring and support on the tenure track and afterwards, rather than ensuring fairness at hiring time.
 I have personally witnessed some half-hearted attempts, on the part of various university administrations, to increase the fraction of female faculty in fields where their under-representation is most severe (e.g., my own, physics). In my experience, such efforts never really go beyond some recommendations, e.g., that search committees pay particular attention to qualified female candidates.
 It must be emphasized that the issue addressed here is only possible bias against women who are already in the science career track. There exists, of course, a much broader (and arguably more important) issue, namely why so few women choose science as a career path in the first place, but that issue is not discussed here.
 The assumption is made that the numbers do not change significantly over the average duration of a postdoctoral stage (which varies across the fields but is roughly between three and four years).