Doing the Math on Prison Sex

I spend a lot of time on this blog debunking bad statistics reporting, but this time I actually want to commend some good statistics reporting. A recent study on reported sexual improprieties in the prison system included some interesting stats about the involvement of prison staff in such cases:

Roughly half of all sexual impropriety reported in U.S. prisons and jails last year was perpetrated by correctional staff, not inmates.

Female staff were the offenders in two-thirds of the prison cases, and two-thirds of the victims of prison staff were male inmates, according to the U.S. Bureau of Justice Statistics.

Based on these results, the fun-because-counterintuitive conclusion to draw would be that female guards are more dangerous or unethical than the male ones. (As Fark reports, presumably in jest, “Females [are] 66% hornier than males.”) Fortunately, reporter Frank Green does not reach that conclusion. Instead, he quotes an academic who explains why the results aren’t that strange after all:

[Brenda V.] Smith [a professor at American University’s Washington School of Law and a member of the National Prison Rape Elimination Commission] said it is not surprising that a larger number of female staff in prisons are involved in sex offenses. Male inmates outnumber female inmates more than 10-to-1. The federal report did not break down the data for homosexual versus heterosexual misconduct, but assuming most staff and most inmates are heterosexual, you would expect to find more female staff reported as perpetrators and more male inmates as victims, she said.

So congratulations to Mr. Green for including a sensible interpretation of an odd-sounding statistic.

Still, the article left me wondering. Given the large number of male inmates, we should expect that most sexual relations between staff and inmates would involve female staff. But how many, exactly? What fraction would be large enough for us justifiably to conclude that female staff commit a disproportionate share of offenses? What’s the benchmark statistic?

The question is more difficult than it sounds, because it turns on both the male-to-female ratio of inmates and the male-to-female ratio of staff. It also depends on the frequency of homosexuality among both males and females. And it depends on how many of the interactions are actually consensual, since a non-consensual homosexual interaction need involve only one homosexual. Or even less than one, if the act arose out of sheer desperation or a desire to terrorize the victim. (Even consensual sex acts between staff and inmates are considered violations of prison policy.) The article didn’t include all that information, but I took a shot at deriving a rough benchmark anyway. [Warning: Remainder of post probably interesting only to math dorks.]

According to the article, inmates are 91% men and 9% women, while staff are 67% men and 33% women. I assumed (a) that 5% of both male and female guards are homosexual, (b) that only the sexual orientation of staff is relevant in determining whether a sexual encounter takes place, (c) that only sexual encounters consistent with the staff member’s orientation occur, and (d) that staff-prisoner interactions are independent events; that is, there is no sorting designed to reduce the frequency of interactions between male prisoners and female staff or vice versa.

Here are the expected frequencies for different types of interactions:

Male prisoner, male homosexual staff: 0.030485
Male prisoner, male heterosexual staff: 0.579215
Male prisoner, female homosexual staff: 0.015015
Male prisoner, female heterosexual staff: 0.285285
Female prisoner, male homosexual staff: 0.003015
Female prisoner, male heterosexual staff: 0.057285
Female prisoner, female homosexual staff: 0.001485
Female prisoner, female heterosexual staff: 0.0282125

Assumptions (b) and (c), above, imply that only the italicized interactions above could result in sexual encounters; I refer to these as “matched” interactions. So the question is, what fraction of all “matched” interactions involve female staff? The answer, it turns out, is 0.7656, or about 77%. Since the observed percentage is only about 67%, this seems like a “dog bites man” story: it appears that female staff are disproportionately less likely to be involved.

I hasten to add, given the many assumptions made, that my calculations should be taken with a hefty grain of salt.