The analysis here and the explanation here is all courtesy of my spouse, who has a statistical background in the social sciences. It’s a little….. dense. I’ll provide some commentary at the bottom. We are both assuming that marking yes for minority status represents an applicant who is nonwhite (Ian, or anyone else who could shed light on this, let me know if that’s not the case).
We wanted to see if philosophy PhD program acceptance is related to the gender and race of applicants. We obtained a sample of 49 individuals who had applied to at least one program each and heard back, and were willing to disclose their race, gender, and the responses they obtained. Our sample included 6 minority students, two of whom were women, and ten white women. We lumped all nonwhite people into a single group because there were not enough cases to reliably estimate the acceptance rate for each race individually. Ideally, we would want to test for an interaction between race and gender to see if the effect of race was different for men and women, or, equivalently, the effect of gender was different for white and nonwhite students. However, with only two nonwhite women applying to programs, we could not reliably estimate this effect. Thus, we modeled only the main effects of gender and race, not their interaction. Our model assumes that race has the same effect regardless of gender and vice versa. The number of programs applied to by each individual varied from 1 to 20. We used binary logistic regression to measure the relationships among these variables, as we were modeling a probability (acceptance rate). We were testing the null hypothesis that program acceptance was unrelated to both race and gender, with the alternative hypothesis that acceptance rates were different for different races and/or genders.
We started by building a logistic regression model using all 49 cases. The Hosmer-Lemeshow test for goodness of fit yielded a significant result, chisq(7) = 14.504, p = 0.043, indicating that the assumptions of the logistic regression model were not met. The logistic regression model assumes a linear relationship between the predictors (in this case race and gender) and the natural log of the odds ratio for the outcome (program acceptance) with random errors following the binomial distribution. Thus, we will not even look at the results of this logistic regression, as they are not considered valid. One possible reason for this assumption being violated is the presence of unusual observations for which the model cannot make a good prediction. We detected four outliers:
ID | Minority? | Gender | Total Applications | Total Admissions | Standardized Residual |
24 | No | Male | 18 | 14 | 5.65 |
14 | No | Male | 11 | 7 | 3.23 |
25 | No | Female | 11 | 8 | 2.57 |
100 | No | Male | 20 | 0 | -2.55 |
These contained the case with the lowest number of unsuccessful applications by far (#100) and the three cases with the highest number of admissions. To show you how unusual these observations are, we have included histograms of successful and unsuccessful applications.
As the graphs show, no other applicants have anywhere near 14 successful or 20 unsuccessful applications. Subjects 14 and 25 have unusually high numbers of successful applications, too, but not obviously separated from the rest. It’s worth noting that the next highest number of successful applications (6) was obtained by #13, a minority student.
ID | Minority? | Gender | Total Applications | Total Admissions | Standardized Residual |
13 | Yes | Male | 9 | 6 | 1.07 |
As we will see, white students tend to have somewhat lower acceptance rates than minority students. The fact that #14 and #25 are white makes their success more remarkable, and #13’s less so. This explains why his residual is so much smaller. We removed these cases and built a new model.
The Hosmer-Lemeshow test was now far from significance, chisq(7) = 6.7333, p = 0.457, indicating that the logistic regression model fits well. The test that all slopes are zero was highly significant, G(2) = 16.867, p < 0.001, indicating that the results observed were highly unlikely given the null hypothesis that program acceptance was unrelated to both race and gender. The Z test for the gender variable was significant at the .05 level, Z = 2.08, p = .038. The Z test for the race variable was highly significant, Z = 3.61, p < .001. This indicates that the associations of race and gender observed with admission are highly unlikely given the lack of such a relationship in the entire population of individuals applying to philosophy PhD programs. We obtained the following regression equation :
Y = -1.44 + 0.66 * female + 1.56 * minority
This equation predicts the log of the odds ratio for a given application being successful given the gender and race of the applicant. The variable “female” takes a value of 1 for a woman and 0 for a man. The variable “minority” takes a value of 1 for a nonwhite person and 0 for a white person. For example, for a white woman applying to a program, the equation would yield a fitted value of Y = -1.44 + 0.66 * 1 + 1.56 * 0 = -.78. To interpret this, we need to antilog the fitted value to get an odds ratio of 0.4584. This is equivalent to a probability of 0.4584 / (1 + 0.4584) = .3143, or a 31% chance of success. The positive coefficients for the two variables representing nonwhite race and female gender indicate that being female and belonging to a racial minority group are both associated with higher odds of acceptance to a given program. The intercept represents the natural log of the odds of success when both predictors equal zero, i.e. for a white male. In this case, the estimated odds of success are e^-1.44 = 0.2369, and the estimated probability is 0.2369 / (1 + .2369) = 0.1915 or 19%. The two coefficients represent the natural log of the increase in the odds of success associated with being female and nonwhite, respectively. Being female is associated with multiplying the odds of success by e^.66 = 1.935, whereas being nonwhite is associated with multiplying the odds of success by e^1.56 = 4.759. Unfortunately, this has no straightforward interpretation in terms of probability. The change in probability depends on what the probability was in the first place.
We could use the regression equation to generate estimates of the probability of a successful application in each of our groups, but it makes more sense to use the observed probability in each group as our estimate. We really just built the logistic regression model so we could test the hypothesis that the group differences appeared by chance, and were able to reject that hypothesis. Because there are only six minority students total, and only two of them are female, the observed acceptance rate in each of these categories should probably not be considered representative of anything. By collapsing them into one group, we get a somewhat more reliable figure which should still be taken with a grain of salt. Here are the observed frequencies in each of these three groups:
students | Actual applications | Actual acceptances | Actual Rate | Predicted Rate | |
white men | 33 | 286 | 66 | 0.23 | 0.19 |
white women | 10 | 67 | 25 | 0.39 | 0.31 |
nonwhite | 6 | 25 | 14 | 0.56 | |
male | 4 | 0.53 | |||
female | 2 | 0.68 | |||
total | 49 | 378 | 105 | 0.28 |
The difference in acceptance rates among these three groups is large enough to have practical significance. All other things being equal, it seems like the nonwhite students who apply have a better chance of being accepted to a given program than white women, who in turn have better odds than white men. It would be tempting to compare the observed frequencies to the predicted ones as a test of how well our model works, but “predicting” the same cases we used to build our model doesn’t really mean anything.
The end result here is that being female and being an underrepresented minority is associated with an increase in the odds of being admitted. However, this is a relatively small sample: only 49 people, only 12 women, only 6 minorities. As I discussed here, we have some good reasons to believe that this data might not be representative of the average student applying to philosophy programs. Further evidence of that: white males have a 19% chance of being accepted per application. We already know that any white male is competing with more than 4 other applicants for each spot, so we’re clearly overstating the success rate here.
The reason for the success of women and minorities also might be explained by other factors not covered here: their GREs, GPAs, publication record, honors, recommendations, statement of purpose, writing samples. Some of these factors are quantifiable (GPA, GRE) and we’d like to look a little deeper at those over the next few days. Some are not (statement of purpose, writing samples), although they are perhaps where the most meaningful differences are.
I hope that no one will see these results and think that the higher rates of acceptance for minority and female students means they’re getting something they didn’t earn. It is well documented that philosophy has a women problem. It has an even bigger minority problem (although it seems like that conversation is rarely heard, perhaps because there are so few voices having it). The discipline is well served if admissions committees are viewing every application from female and minority applicants and making sure the best candidates are offered admissions. After looking at some of the most successful candidates, it seems like the best people are the ones getting multiple offers, regardless of gender or race.