# A Revealing Experiment

Consider this scenario.

ME: “Hey, thanks for coming over! If you’re thirsty, I’ve got your choice of Pepsi, A&W Root Beer, and Mountain Dew.”
YOU: “I’d prefer Mountain Dew, but I’ll take anything.”
ME: “Gotcha, I’ll be right back!”
ME: [leaves, then returns with a Pepsi]
YOU: “Thanks. Too bad you ran out of Mountain Dew.”
ME: “Oh no, I’ve got tonnes.”
YOU: “… but it was too tough to reach, right?”
ME: “No, the Dew was right next to the Pepsi.”

Have I done anything wrong here? No, at least technically. You said you were fine with any soft drink, and I gave you a soft drink. At the same time, though, you expressed a preference for Mountain Dew over the other choices. I could be forgiven for ignoring your preference if I wasn’t able to fulfill it, or doing so would have been inconvenient for me, but in this fictional scenario both of those were off the table. My choice to hand you a Pepsi instead of a Mountain Dew reveals something about me, most likely that I think other people should prefer Pepsi over other soft drinks. You could point to the ordering of my list as further evidence: I’d be more likely to list my preferred option first, as it would be more prominent in my mind than the other choices, and then rattle off others as they came to me. This isn’t strong evidence, but you’d be justified in suspecting my motives as you enjoyed your Pepsi.

robbe de Boer: Hey EOT, I had a question not related to this video, what pronouns do you prefer? I couldn’t find anything real quick.
EssenceOfThought: They. But I’m fine with both she and he as well. It’s all on the channel description/Facebook page ‘About’ section. 😛

By merely existing, EssenceOfThought has set up a very similar situation. They have a pronoun preference, and when listing their pronoun choices put “he” last, but also say they aren’t offended if you pick another reasonable one. The difficulty in typing two extra letters is practically zero, and “they” as a singular pronoun has been in the English language for six centuries, so there isn’t any obstacle to its use beyond your hang-ups. Hell, even the guy who became famous for refusing to use transgender people’s pronouns is perfectly capable of using singular “they.”

Jordan Peterson: I don’t recognize another person’s right to decide what words I’m going to use, especially when the words they want me to use, first of all, are non-standard elements of the English language and they are constructs of a small coterie of ideologically motivated people. They might have a point but I’m not going to say their words for them.

So if we encounter someone calling EssenceOfThought “he,” we’re justified in raising an eyebrow. While they’re not technically in the wrong, the use of “he” is suggestive that they’d overrule someone’s pronoun preference if they thought they could get away with it.

Steve McRae: Essence of thought bullies Rachel Oates and demonstrates himself to probably one of the worse humans ever to be on Twitter or YouTube.

[12:00] Noel Plum: The truth of the matter is, is that effectively in saying what he’s saying, Essence of Thought has said, to the majority of people who have sided with him, you are transphobes. Your position is, transphobic unless you adopt this position that anyone who identifies as a woman gets to compete in this category, then you are holding a transphobic position. And his masterstroke is that it seems to have done the trick, and nobody’s arguing – none of his supporters are arguing with him.

Rachel Oates: In regards to Essence of Thought calling the police and claiming to be the ‘only one’ actually helping me. It didn’t. He called the police. Who apparently turned up at my old flat, broke the door down and then they wasted hours and many resources trying to find me. Meanwhile, I was at home with my friends around me, having all my Youtube friends send me love and support and check in on me. My family phoned me. People were there for me, helping.

The analogy isn’t a perfect fit. In there, I asked for your choice and received it. EssenceOfThought will mention their preferred pronoun if asked, but does not put it on blast nor do they bother to correct people who don’t pick their preference. Ignorance is more of an option than the analogy presents.

Provided, of course, these people were ignorant. Some of them claim to care about transgender people, though. They should know not to screw up someone’s pronouns, and thus be willing to do a little extra legwork to get things right. Even if they only use “he” because their friends and peers do so, that means their social circle is overwhelmingly dominated by transphobic people or people with a high tolerance to transphobia.

Rachel Oates: Also, I’m really sorry if I got EoT’s pronouns wrong in this thread – I’ve heard different things about which pronouns they prefer & may have slipped up here.

A good way to rule out ignorance is to correct them on EoT’s preferred pronouns. If that person responds with something like this…

Noel Plum: If he drops the “he” then I will drop it too. As it stands he accepts he, she or they. I couldn’t give a toss which he “prefers” as i don’t like him.

… then you’re pretty justified in believing the “cloaked transphobia” hypothesis.

“He” is also a strange choice given how EssenceOfThought presents. You see someone with no facial hair and long flowing locks in front of a transgender flag, and you immediately jump to “he?” C’mon, even Rationality Rules splits the baby and uses “she.” These people didn’t settle on “he” by accident, their choice reveals something about their internal opinion of transgender people, their peer group, or their ignorance.

And it isn’t very refreshing.

Apparently I know the solar system very well?

I attended a lecture on Carl Sagan, hosted by the Atheist Society of Calgary, and part of the event was a trivia challenge. While I wasn’t the only person at my table offering answers, my answers seemed to be the ones most consistently endorsed by the group. Assisted by some technical issues, our team wound up with a massive lead over the second-place finisher. The organizer from ASC surprised us all by saying everyone at our table could pick up a free T-shirt. I wasn’t terribly keen on wearing their logo, but I wandered over to the merch table anyway.

Sitting among the other designs was one that stopped me cold.

# The Crisis of the Mediocre Man

I was browsing YouTube videos on PyMC3, as one naturally does, when I happened to stumble on this gem.

Tech has spent millions of dollars in efforts to diversify workplaces. Despite this, it seems after each spell of progress, a series of retrograde events ensue. Anti-diversity manifestos, backlash to assertive hiring, and sexual misconduct scandals crop up every few months, sucking the air from every board room. This will be a digest of research, recent events, and pointers on women in STEM.

Lorena A. Barba really knows her stuff; the entire talk is a rapid-fire accounting of claims and counterclaims, aimed to directly appeal to the male techbros who need to hear it. There was a lot of new material in there, for me at least. I thought the only well-described matriarchies came from the African continent, but it turns out the Algonquin also fit that bill. Some digging turns up a rich mix of gender roles within First Nations peoples, most notably the Iroquois and Hopi. I was also depressed to hear that the R data analysis community is better at dealing with sexual harassment than the skeptic/atheist community.

But what really grabbed my ears was the section on gender quotas. I’ve long been a fan of them on logical grounds: if we truly believe the sexes are equal, then if we see unequal representation we know discrimination is happening. By forcing equality, we greatly reduce network effects where one gender can team up against the other. Worried about an increase in mediocrity? At worst that’s a temporary thing that disappears once the disadvantaged sex gets more experience, and at best the overall quality will actually go up. The research on quotas has advanced quite a bit since that old Skepchick post. Emphasis mine.

In 1993, Sweden’s Social Democratic Party centrally adopted a gender quota and imposed it on all the local branches of that party (…). Although their primary aim was to improve the representation of women, proponents of the quota observed that the reform had an impact on the competence of men. Inger Segelström (the chair of Social Democratic Women in Sweden (S-Kvinnor), 1995–2003) made this point succinctly in a personal communication:

At the time, our party’s quota policy of mandatory alternation of male and female names on all party lists became informally known as the crisis of the mediocre man

We study the selection of municipal politicians in Sweden with regard to their competence, both theoretically and empirically. Moreover, we exploit the Social Democratic quota as a shock to municipal politics and ask how it altered the competence of that party’s elected politicians, men as well as women, and leaders as well as followers.

Besley, Timothy. “Gender Quotas and the Crisis of the Mediocre Man: Theory and Evidence from Sweden.” THE AMERICAN ECONOMIC REVIEW 107, no. 8 (2017): 39.

We can explain this with the benefit of hindsight: if men can rely on the “old boy’s network” to keep them in power, they can afford to slack off. If other sexes cannot, they have to fight to earn their place. These are all social effects, though; if no sex holds a monopoly on operational competence in reality, the net result is a handful of brilliant women among a sea of iffy men. Gender quotas severely limit the social effects, effectively kicking out the mediocre men to make way for average women, and thus increase the average competence.

As tidy as that picture is, it’s wrong in one crucial detail. Emphasis again mine.

These estimates show that the overall effect mainly reflects an improvement in the selection of men. The coefficient in column 4 means that a 10-percentage-point larger quota bite (just below the cross-sectional average for all municipalities) raised the proportion of competent men by 4.4 percentage points. Given an average of 50 percent competent politicians in the average municipality (by definition, from the normalization), this corresponds to a 9 percent increase in the share of competent men.

For women, we obtain a negative coefficient in the regression specification without municipality trends, but a positive coefficient with trends. In neither case, however, is the estimate significantly different from zero, suggesting that the quota neither raised nor cut the share of competent women. This is interesting in view of the meritocratic critique of gender quotas, namely that raising the share of women through a quota must necessarily come at the price of lower competence among women.

Increasing the number of women does not also increase the number of incompetent women. When you introduce a quota, apparently, everyone works harder to justify being there. The only people truly hurt by gender quotas are mediocre men who rely on the Peter Principle.

Alas, if that YouTube like ratio is any indication, there’s a lot of them out there.

# Rationality Rules DESTROYS Women’s Sport!!1!

I still can’t believe this post exists, given its humble beginnings.

The “women’s category” is, in my opinion, poorly named given our current climate, and so I’d elect a name more along the lines of the “Under 5 nmol/l category” (as in, under 5 nanomoles of testosterone per litre), but make no mistake about it, the “woman’s category” is not based on gender or identity, or even genitalia or chromosomes… it’s based on hormone levels and the absence of male puberty.

The above comment wasn’t in Rationality Rules’ latest transphobic video, it was just a casual aside by RR himself in the YouTube comment section. He’s obiquely doubled-down via Twitter (hat tip to Essence of Thought):

Of course, just as I support trans men competing in all “men’s categories” (poorly named), women who have not experienced male puberty competing in all women’s sport (also poorly named) and trans women who have experienced male puberty competing in long-distance running.

To further clarify, I think that we must rename our categories according to what they’re actually based on. It’s not right to have a “women’s category” and yet say to some trans women (who are women!) that they can’t compete within it; it should be renamed.

The proposal itched away at me, though, because I knew it was testable.

There is a need to clarify hormone profiles that may be expected to occur after competition when antidoping tests are usually made. In this study, we report on the hormonal profile of 693 elite athletes, sampled within 2 h of a national or international competitive event. These elite athletes are a subset of the cross-sectional study that was a component of the GH-2000 research project aimed at developing a test to detect abuse with growth hormone.

Healy, Marie-Louise, et al. “Endocrine profiles in 693 elite athletes in the postcompetition setting.” Clinical endocrinology 81.2 (2014): 294-305.

The GH-2000 project had already done the hard work of collecting and analyzing blood samples from athletes, so checking RR’s proposal was no tougher than running some numbers. There’s all sorts of ethical guidelines around sharing medical info, but fortunately there’s an easy shortcut: ask one of the scientists involved to run the numbers for me, and report back the results. Aggregate data is much more resistant to de-anonymization, so the ethical concerns are greatly reduced. The catch, of course, is that I’d have to find a friendly researcher with access to that dataset. About a month ago, I fired off some emails and hoped for the best.

I wound up much, much better than the best. I got full access to the dataset!! You don’t get handed an incredible gift like this and merely use it for a blog post. In my spare time, I’m flexing my Bayesian muscles to do a re-analysis of the above paper, while also looking for observations the original authors may have missed. Alas, that means my slow posting schedule is about to crawl.

But in the meantime, we have a question to answer.

# What Do We Have Here? ¶

Total Assigned-female Athletes = 239
Height, Mean           = 171.61 cm
Height, Std.Dev        = 7.12 cm
Weight, Mean           = 64.27 kg
Weight, Std.Dev        = 9.12 kg
Body Fat, Mean         = 13.19 kg
Body Fat, Std.Dev      = 3.85 kg
Testosterone, Mean     = 2.68 nmol/L
Testosterone, Std.Dev  = 4.33 nmol/L
Testosterone, Max      = 31.90 nmol/L
Testosterone, Min      = 0.00 nmol/L

Total Assigned-male Athletes = 454
Height, Mean           = 182.72 cm
Height, Std.Dev        = 8.48 cm
Weight, Mean           = 80.65 kg
Weight, Std.Dev        = 12.62 kg
Body Fat, Mean         = 8.89 kg
Body Fat, Std.Dev      = 7.20 kg
Testosterone, Mean     = 14.59 nmol/L
Testosterone, Std.Dev  = 6.66 nmol/L
Testosterone, Max      = 41.00 nmol/L
Testosterone, Min      = 0.80 nmol/L

The first step is to get a basic grasp on what’s there, via some crude descriptive statistics. It’s also useful to compare these with the original paper, to make sure I’m interpreting the data correctly. Excusing some minor differences in rounding, the above numbers match the paper.

The only thing that stands out from the above, to me, is the serum levels of testosterone. At least one source says the mean of these assigned-female athletes is higher than the normal range for their non-athletic cohorts. Part of that may simply be because we don’t have a good idea of what the normal range is, so it’s not uncommon for each lab to have their own definition of “normal.” This is even worse for those assigned female, since their testosterone levels are poorly studied; note that my previous link collected the data of over a million “men,” but doesn’t mention “women” once. Factor in inaccurate test results and other complicating factors, and “normal” is quite poorly-defined.

Still, Rationality Rules is either convinced those complications are irrelevant, or ignorant of them. And, to be fair, that 5nmol/L line implicitly sweeps a lot of them under the rug. Let’s carry on, then, and look for invalid data. “Invalid” covers everything from missing data, to impossible data, and maybe even data we think might be made inaccurate due to measurement error. I consider a concentration of zero testosterone as invalid, even though it may technically be possible.

Total Assigned-male Athletes w/ T levels >= 0        = 446
w/ T levels <= 0.5      = 0
w/ T levels == 0        = 0
w/ missing T levels     = 8
that I consider valid   = 446

Total Assigned-female Athletes w/ T levels >= 0      = 234
w/ T levels <= 0.5    = 5
w/ T levels == 0      = 1
w/ missing T levels   = 5
that I consider valid = 229

Fortunately for us, the losses are pretty small. 229 datapoints is a healthy sample size, so we can afford to be liberal about what we toss out. Next up, it would be handy to see the data in chart form.

I've put vertical lines at both the 0.5 and 5 nmol/L cutoffs. There's a big difference between categories, but we can see clouds on the horizon: a substantial number of assigned-female athletes have greater than 5 nmol/L of testosterone in their bloodstream, while a decent number of assigned-male athletes have less. How many?

Segregating Athletes by Testosterone
Concentration  aFab  aMab
> 5nmol/L    19   417
< 5nmol/L   210    26
= 5nmol/L     0     3

8.3% of assigned-female athletes have > 5nmol/L
5.8% of assigned-male athletes have < 5nmol/L
4.4% of athletes with > 5nmol/L are assigned-female
11.0% of athletes with < 5nmol/L are assigned-male

Looks like anywhere from 6-8% of athletes have testosterone levels that cross Rationality Rules' line. For comparison, maybe 1-2% of the general public has some level of gender dysphoria, though estimating exact figures is hard in the face of widespread discrimination and poor sex-ed in schools. Even that number is misleading, as the number of transgender athletes is substantially lower than 1-2% of the athletic population. The share of transgender athletes is irrelevant to this dataset anyway, as it was collected between 1996 and 1999, when no sporting agency had policies that allowed transgender athletes to openly compete.

That 6-8%, in other words, is entirely cisgender. This echoes one of Essence Of Thought's arguments: RR's 5nmol/L policy has far more impact on cis athletes than trans athletes, which could have catastrophic side-effects. Could is the operative word, though, because as of now we don't know anything about these athletes. Do >5nmol/L assigned-female athletes have bodies more like >5nmol/L assigned-male athletes than <5nmol/L assigned-female athletes? If so, then there's no problem. Equivalent body types are competing against each other, and outcomes are as fair as could be reasonably expected.

What, then, counts as an "equivalent" body type when it comes to sport?

# Newton's First Law of Athletics ¶

One reasonable measure of equivalence is height. It's one of the stronger sex differences, and height is also correlated with longer limbs and greater leverage. Whether that's relevant to sports is debatable, but height and correlated attributes dominate Rationality Rules' list.

[19:07] In some events - such as long-distance running, in which hemoglobin and slow-twitch muscle fibers are vital - I think there's a strong argument to say no, [transgender women who transitioned after puberty] don't have an unfair advantage, as the primary attributes are sufficiently mitigated. But in most events, and especially those in which height, width, hip size, limb length, muscle mass, and muscle fiber type are the primary attributes - such as weightlifting, sprinting, hammer throw, javelin, netball, boxing, karate, basketball, rugby, judo, rowing, hockey, and many more - my answer is yes, most do have an unfair advantage.

Fortunately for both of us, most athletes in the dataset have a "valid" height, which I define as being at least 30cm tall.

Out of 693 athletes, 678 have valid height data.

The faint vertical lines are for the mean adult height of Germans born in 1976, which should be a reasonable cohort to European athletes that were active between 1996 and 1999, while the darker lines are each category's mean. Athletes seem slightly taller than the reference average, but only by 2-5cm. The amount of overlap is also surprising, given that height is supposed to be a major sex difference. We actually saw less overlap with testosterone! Finally, the height distribution isn't quite Gaussian, there's a subtle bias towards the taller end of the spectrum.

Height is a pretty crude metric, though. You could pair any athlete with a non-athlete of the same height, and there's no way the latter would perform as well as the former. A better measure of sporting ability would be muscle mass. We shouldn't use the absolute mass, though: bigger bodies have more mass and need more force to accelerate as smaller bodies do, so height and muscle mass are correlated. We need some sort of dimensionless scaling factor which compensates.

And we have one! It's called the Body Mass Index, or BMI.

$$BMI = \frac w {h^2},$$

where $$w$$ is a person's mass in kilograms, and $$h$$ is a person's height in metres. Unfortunately, BMI is quite problematic. Partly that's because it is a crude measure of obesity. But part of that is because there are two types of tissue which can greatly vary, body fat and muscle, yet both contribute equally towards BMI.

That's all fixable. For one, some of the athletes in this dataset had their body fat measured. We can subtract that mass off, so their weight consists of tissues that are strongly correlated with height plus one that is fudgable: muscle mass. For two, we're not assessing these individual's health, we only want a dimensionless measure of muscle mass relative to height. For three, we're not comparing these individuals to the general public, so we're not restricted to using the general BMI formula. We can use something more accurate.

The oddity is the appearance of that exponent 2, though our world is three-dimensional. You might think that the exponent should simply be 3, but that doesn't match the data at all. It has been known for a long time that people don't scale in a perfectly linear fashion as they grow. I propose that a better approximation to the actual sizes and shapes of healthy bodies might be given by an exponent of 2.5. So here is the formula I think is worth considering as an alternative to the standard BMI:

$$BMI' = 1.3 \frac w {h^{2.5}}$$

I can easily pop body fat into Nick Trefethen's formula, and get a better measure of relative muscle mass,

$$\overline{BMI} = 1.3 \frac{ w - bf }{h^{2.5}},$$

where $$bf$$ is total body fat in kilograms. Individuals with excess muscle mass, relative to what we expect for their height, will have a high $$\overline{BMI}$$, and vice-versa. And as we saw earlier, muscle mass is another of Rationality Rules' determinants of sporting performance.

Time for more number crunching.

Out of 693 athletes, 227 have valid adjusted BMIs.
663 have valid weights.
241 have valid body fat percentages.

Total Assigned-female Athletes = 239
total with valid adjusted BMI = 86

Total Assigned-male Athletes = 454
total with valid adjusted BMI = 141
adjusted BMI, Median   = 20.28

The bad news is that most of this dataset lacks any information on body fat, which really cuts into our sample size. The good news is that we've still got enough to carry on. It also looks like there's a strong sex difference, and the distribution is pretty clustered. Still, a chart would help clarify the latter point.

Whoops! There's more overlap and skew than I thought. Even in logspace, the results don't look Gaussian. We'll have to remember that for the next step.

# A Man Without a Plan is Not a Man ¶

Just looking at charts isn't going to solve this question, we need to do some sort of hypothesis testing. Fortunately, all the pieces I need are here. We've got our hypothesis, for instance:

Athletes with exceptional testosterone levels are more like athletes of the same sex but with typical testosterone levels, than they are of other athletes with a different sex but similar testosterone levels.

If you know me, you know that I'm all about the Bayes, and that gives us our methodology.

1. Fit a model to a specific metric for assigned-female athletes with less than 5nmol/L of serum testosterone.
2. Fit a model to a specific metric for assigned-male athletes with more than 5nmol/L of serum testosterone.
3. Apply the first model to the test group, calculating the overall likelihood.
4. Apply the second model to the test group, calculating the overall likelihood.
5. Sample the probability distribution of the Bayes Factor.

"Metric" is one of height or $$\overline{BMI}$$, while "test group" is one of assigned-female athletes with >5nmol/L of serum testosterone or assigned-male athletes with <5nmol/L of serum testosterone. The Bayes Factor is simply

$$\text{Bayes Factor} = \frac{ p(E \mid H_1) \cdot p(H_1) }{ p(E \mid H_2) \cdot p(H_2) } = \frac{ p(H_1 \mid E) }{ p(H_2 \mid E) },$$

which means we need two hypotheses, not one. Fortunately, I've phrased the hypothesis to make it easy to negate: athletes with exceptional testosterone levels are less like athletes of the same sex but with typical testosterone levels, than they are of other athletes with a different sex but similar testosterone levels. We'll call this new hypothesis $$H_2$$, and the original $$H_1$$. Bayes factors greater than 1 mean $$H_1$$ is more likely than $$H_2$$, and vice-versa.

Calculating all that would be easy if I was using Stan or PyMC3, but I ran into problems translating the former's probability distributions into charts, and I don't have any experience with the latter. My next choice, emcee, forces me to manually convolve two posterior distributions. Annoying, but not difficult.

# I'm a Model, If You Know What I Mean ¶

That just leaves one thing left: what models are we going to use? The obvious choice for height is the Gaussian distribution, as from previous research we know it's a great model.

Fitting the height of lT aFab athletes to a Gaussian distribution ...
0: (-980.322471) mu=150.000819, sigma=15.000177
64: (-710.417497) mu=169.639051, sigma=8.579088
128: (-700.539260) mu=171.107358, sigma=7.138832
192: (-700.535241) mu=171.154151, sigma=7.133279
256: (-700.540692) mu=171.152701, sigma=7.145515
320: (-700.552831) mu=171.139668, sigma=7.166857
384: (-700.530969) mu=171.086422, sigma=7.094077
ML: (-700.525284) mu=171.155240, sigma=7.085777
median: (-700.525487) mu=171.134614, sigma=7.070993

Alas, emcee also lacks a good way to assess model fitness. One crude metric is look at the progression of the mean fitness; if it grows and then stabilizes around a specific value, as it does here, we've converged on something. Another is to compare the mean, median, and maximal likelihood of the posterior; if they're about equally likely, we've got a fuzzy caterpillar. Again, that's also true here.

As we just saw, though, charts are a better judge of fitness than a handful of numbers.

If you were wondering why I didn't make much of a fuss out of the asymmetry in the height distribution, it's because I've already seen this graph. A good fit isn't necessarily the best though, and I might be able to get a closer match by incorporating the sport each athlete played.

            Assigned-female Athletes
sport              below/above 171cm
Power lifting:  1 / 0
Football:  0 / 0
Swimming: 41 /49
Marathon:  0 / 1
Canoeing:  1 / 0
Rowing:  9 /13
Cross-country skiing:  8 / 1
Alpine skiing: 11 / 1
Weight lifting:  7 / 0
Judo:  0 / 0
Bandy:  0 / 0
Ice Hockey:  0 / 0
Handball: 12 /17
Track and field: 22 /27

Basketball attracts tall people, unsurprisingly, while skiing seems to attract shorter people. This could be the cause of that asymmetry. It's no guarantee that I'll actually get a better fit, though, as I'm also dramatically cutting the number of datapoints to fit to. The model's uncertainty must increase as a result, and that may be enough to dilute out any increase in fitness. I'll run those numbers for the paper, but for now the Gaussian model I have is plenty good.

Fitting the height of hT aMab athletes to a Gaussian distribution ...
0: (-2503.079578) mu=150.000061, sigma=15.001179
64: (-1482.315571) mu=179.740851, sigma=10.506003
128: (-1451.789027) mu=182.615810, sigma=8.620333
192: (-1451.748336) mu=182.587979, sigma=8.550535
256: (-1451.759883) mu=182.676004, sigma=8.546410
320: (-1451.746697) mu=182.626918, sigma=8.538055
384: (-1451.747266) mu=182.580692, sigma=8.534070
ML: (-1451.746074) mu=182.591047, sigma=8.534584
median: (-1451.759295) mu=182.603231, sigma=8.481894

We get the same results when fitting the model to >5 nmol/L assigned-male athletes. The log likelihood, that number in brackets, is a lot lower for these athletes, but that number is roughly proportional to the number of samples. If we had the same degree of model fitness but doubled the number of samples, we'd expect the log likelihood to double. And, sure enough, this dataset has roughly twice as many assigned-male athletes as it does assigned-female athletes.

The updated charts are more of the same.

Unfortunately, adjusted BMI isn't nearly as tidy. I don't have any prior knowledge that would favour a particular model, so I wound up testing five candidates: the Gaussian, Log-Gaussian, Gamma, Weibull, and Rayleigh distributions. All but the first needed an offset parameter to get the best results, which has the same interpretation as last time.

Fitting the adjusted BMI of hT aMab athletes to a Gaussian distribution ...
0: (-410.901047) mu=14.999563, sigma=5.000388
384: (-256.474147) mu=20.443497, sigma=1.783979
ML: (-256.461460) mu=20.452817, sigma=1.771653
median: (-256.477475) mu=20.427138, sigma=1.781139
Fitting the adjusted BMI of hT aMab athletes to a Log-Gaussian distribution ...
0: (-629.141577) mu=6.999492, sigma=2.001107, off=10.000768
384: (-290.910651) mu=3.812746, sigma=1.789607, off=16.633741
ML: (-277.119315) mu=3.848383, sigma=1.818429, off=16.637382
median: (-288.278918) mu=3.795675, sigma=1.778238, off=16.637076
Fitting the adjusted BMI of hT aMab athletes to a Gamma distribution ...
0: (-564.227696) alpha=19.998389, beta=3.001330, off=9.999839
384: (-256.999252) alpha=15.951361, beta=2.194827, off=13.795466
ML    : (-248.056301) alpha=8.610936, beta=1.673886, off=15.343436
median: (-249.115483) alpha=12.411010, beta=2.005287, off=14.410945
Fitting the adjusted BMI of hT aMab athletes to a Weibull distribution ...
0: (-48865.772268) k=7.999859, beta=0.099877, off=0.999138
384: (-271.350390) k=9.937527, beta=0.046958, off=0.019000
ML: (-270.340284) k=9.914647, beta=0.046903, off=0.000871
median: (-270.974131) k=9.833793, beta=0.046947, off=0.011727
Fitting the adjusted BMI of hT aMab athletes to a Rayleigh distribution ...
0: (-3378.099000) tau=0.499136, off=9.999193
384: (-254.717778) tau=0.107962, off=16.378780
ML: (-253.012418) tau=0.110751, off=16.574934
median: (-253.092584) tau=0.108740, off=16.532576


Looks like the Gamma distribution is the best of the bunch, though only if you use the median or maximal likelihood of the posterior. There must be some outliers in there that are tugging the mean around. Visually, there isn't too much difference between the Gaussian and Gamma fits, but the Rayleigh seems artificially sharp on the low end. It's a bit of a shame, the Gamma distribution is usually related to rates and variance so we don't have a good reason for applying it here, other than "it fits the best." We might be able to do better with a per-sport Gaussian distribution fit, but for now I'm happy with the Gamma.

Time to fit the other pool of athletes, and chart it all.

Fitting the adjusted BMI of lT aFab athletes to a Gamma distribution ...
0: (-127.467934) alpha=20.000007, beta=3.000116, off=9.999921
384: (-128.564564) alpha=15.481265, beta=3.161022, off=12.654149
ML    : (-117.582454) alpha=2.927721, beta=1.294851, off=14.713479
median: (-120.689425) alpha=11.961847, beta=2.836153, off=13.008723

Those models look pretty reasonable, though the upper end of the assigned-female distribution could be improved on. It's a good enough fit to get some answers, at least.

# The Nitty Gritty ¶

It's easier to combine step 3, applying the model, with step 5, calculating the Bayes Factor, when writing the code. The resulting Bayes Factor has a probability distribution, as the uncertainty contained in the posterior contaminates it.

Summary of the BF distribution, for the height of >5nmol/L aFab athletes
n       mean   geo.mean         5%        16%        50%        84%        95%
19      10.64       5.44       0.75       1.76       5.66      17.33      35.42

Percentage of BF's that favoured the primary hypothesis: 92.42%
Percentage of BF's that were 'decisive': 14.17%

That looks a lot like a log-Gaussian distribution. The arthithmetic mean fails us here, thanks to the huge range of values, so the geometric mean and median are better measures of central tendency.

The best way I can interpret this result is via an eight-sided die: our credence in the hypothesis that >5nmol/L aFab athletes are more like their >5nmol/L aMab peers than their <5nmol/L aFab ones is similar to the credence we'd place on rolling a one via that die, while our credence on the primary hypothesis is similar to rolling any other number except one. About 92% of the calculated Bayes Factors were favourable to the primary hypothesis, and about 16% of them crossed the 19:1 threshold, a close match for the asserted evidential bar in science.

That's strong evidence for a mere 19 athletes, though not quite conclusive. How about the Bayes Factor for the height of <5nmol/L aMab athletes?

Summary of the BF distribution, for the height of <5nmol/L aMab athletes
n       mean   geo.mean         5%        16%        50%        84%        95%
26   4.67e+21   3.49e+18   5.67e+14   2.41e+16   5.35e+18   4.16e+20   4.61e+21

Percentage of BF's that favoured the primary hypothesis: 100.00%
Percentage of BF's that were 'decisive': 100.00%

Wow! Even with 26 data points, our primary hypothesis was extremely well supported. Betting against that hypothesis is like betting a particular person in the US will be hit by lightning three times in a single year!

That seems a little too favourable to my view, though. Did something go wrong with the mathematics? The simplest check is to graph the models against the data they're evaluating.

Nope, the underlying data genuinely is a better fit for the high-testosterone aMab model. But that good of a fit? In linear space, we multiply each of the individual probabilities to arrive at the Bayes factor. That's equivalent to raising the geometric mean to the nth power, where n is the number of athletes. Since n = 26 here, even a geometric mean barely above one can generate a big Bayes factor.

26th root of the median Bayes factor of the high-T aMab model applied to low-T aMab athletes: 5.2519
26th root of the Bayes factor for the median marginal: 3.6010

Note that the Bayes factor we generate by using the median of the marginal for each parameter isn't as strong as the median Bayes factor in the above convolution. That's simply because I'm using a small sample from the posterior distribution. Keeping more samples would have brought those two values closer together, but also greatly increased the amount of computation I needed to do to generate all those Bayes factors.

With that check out of the way, we can move on to $$\overline{BMI}$$.

Summary of the BF distribution, for the adjusted BMI of >5nmol/L aFab athletes
n       mean   geo.mean         5%        16%        50%        84%        95%
4   1.70e+12   1.06e+05   2.31e+02   1.60e+03   4.40e+04   3.66e+06   3.99e+09

Percentage of BF's that favoured the primary hypothesis: 100.00%
Percentage of BF's that were 'decisive': 99.53%
Percentage of non-finite probabilities, when applying the low-T aFab model to high-T aFab athletes: 0.00%
Percentage of non-finite probabilities, when applying the high-T aMab model to high-T aFab athletes: 10.94%

This distribution is much stranger, with a number of extremely high BF's that badly skew the mean. The offset contributes to this, with 7-12% of the model posteriors for high-T aMab athletes assigning a zero percent likelihood to an adjusted BMI. Those are excluded from the analysis, but they suggest the high-T aMab model poorly describes high-T aFab athletes.

Our credence in the primary hypothesis here is similar to our credence that an elite golfer will not land a hole-in-one on their next shot. That's surprisingly strong, given we're only dealing with four datapoints. More data may water that down, but it's unlikely to overcome that extreme level of credence.

Summary of the BF distribution, for the adjusted BMI of <5nmol/L aMab athletes
n       mean   geo.mean         5%        16%        50%        84%        95%
9   6.64e+35   2.07e+22   4.05e+12   4.55e+16   6.31e+21   7.72e+27   9.81e+32

Percentage of BF's that favoured the primary hypothesis: 100.00%
Percentage of BF's that were 'decisive': 100.00%
Percentage of non-finite probabilities, when applying the high-T aMab model to low-T aMab athletes: 0.00%
Percentage of non-finite probabilities, when applying the low-T aFab model to low-T aMab athletes: 0.00%

The hypotheses' Bayes factor for the adjusted BMI of low-testosterone aMab athletes is much better behaved. Even here, the credence is above three-lightning-strikes territory, pretty decisively favouring the hypothesis.

Our final step would normally be to combine all these individual Bayes factors into a single one. That involves multiplying them all together, however, and a small number multiplied by a very large one is an even larger one. It isn't worth the effort, the conclusion is pretty obvious.

# Truth and Consequences ¶

Our primary hypothesis is on quite solid ground: Athletes with exceptional testosterone levels are more like athletes of the same sex but with typical testosterone levels, than they are of other athletes with a different sex but similar testosterone levels. If we divide up sports by testosterone level, then, roughly 6-8% of assigned-male athletes will wind up in the <5 nmol/L group, and about the same share of assigned-female athletes will be in the >5 nmol/L group. Note, however, that it doesn't follow that 6-8% of those in the <5 nmol/L group will be assigned-male. About 41% of the athletes at the 2018 Olymics were assigned-female, for instance. If we fix the rate of exceptional testosterone levels at 7%, and assume PyeongChang's rate is typical, a quick application of Bayes' theorem reveals

\begin{align} p( \text{aMab} \mid \text{<5nmol/L} ) &= \frac{ p( \text{<5nmol/L} \mid \text{aMab} ) p( \text{aMab} ) }{ p( \text{<5nmol/L} \mid \text{aMab} ) p( \text{aMab} ) + p( \text{<5nmol/L} \mid \text{aFab} ) p( \text{aFab} ) } \\ {} &= \frac{ 0.07 \cdot 0.59 }{ 0.07 \cdot 0.59 + 0.93 \cdot 0.41 } \\ {} &\approx 9.8\% \end{align}

If all those assumptions are accurate, about 10% of <5 nmol/L athletes will be assigned-male, more-or-less matching the number I calculated way back at the start. In sports where performance is heavily correlated with height or $$\overline{BMI}$$, then, the 10% of assigned-male athletes in the <5 nmol group will heavily dominate the rankings. The odds of a woman earning recognition in this sport are negligible, leading many of them to drop out. This increases the proportion of men in that sport, leading to more domination of the rankings, more women dropping out, and a nasty feedback loop.

Conversely, about 5% of >5nmol/L athletes will be assigned-female. In a heavily-correlated sport, those women will be outclassed by the men and have little chance of earning recognition for their achievements. They have no incentive to compete, so they'll likely drop out or avoid these sports as well.

In events where physicality has less or no correlation with sporting performance, these effects will be less pronounced or non-existent, of course. But this still translates into fewer assigned-female athletes competing than in the current system.

But it gets worse! We'd also expect an uptick in the number of assigned-female athletes doping, primarily with testosterone inhibitors to bring themselves just below the 5nmol/L line. Alternatively, high-testosterone aFab athletes may inject large doses of testosterone to bulk up and remain competitive with their assigned-male competitors.

By dividing up testosterone levels into only two categories, sporting authorities are implicitly stating that everyone within those categories is identical. A number of athletes would likely go to court to argue that boosting or inhibiting testosterone should be legal, provided they do not cross the 5nmol/L line. If they're successful, then either the rules around testosterone usage would be relaxed, or sporting authorities would be forced to subdivide these groups further. This would lead to an uptick in testosterone doping among all athletes, not just those assigned female.

Notice that assigned-male athletes don't have the same incentives to drop out, and in fact the low-testosterone subgroup may even be encouraged to compete as they have an easier path to sporting fame and glory. Sports where performance is heavily correlated with height or $$\overline{BMI}$$ will come to be dominated by men.

# Let's Put a Bow On This One ¶

[1:15] In a nutshell, I find the arguments and logic that currently permit transgender women to compete against biological women to be remarkably flawed, and I’m convinced that unless quickly rectified, this will KILL women’s sports.

[14:00] I don’t want to see the day when women’s athletics is dominated by Y chromosomes, but without a change in policy, that is precisely what’s going to happen.

It's rather astounding. Transgender athletes are a not a problem, on several levels; as I've pointed out before, they've been allowed to compete in the category they identify for over a decade in some places, and yet no transgender athlete has come to dominate any sport. The Olympics has held the door open since 2004, and not a single transgender athlete has ever openly competed as a transgender athlete. Rationality Rules, like other transphobes, is forced to cherry-pick and commit lies of omission among a handful of examples, inflating them to seem more significant than they actually are.

In response to this non-existent problem, Rationality Rules' proposed solution would accomplish the very thing he wants to avoid! You don't get that turned around if you're a rational person with a firm grasp on the science.

No, this level of self-sabotage is only possible if you're a clueless bigot who's ignorant of the relevant science, and so frightened of transgender people that your critical thinking skills abandon you. The vast difference between what Rationality Rules claims the science says, and what his own citations say, must be because he knows that if he puts on a good enough act nobody will check his work. Everyone will walk away assuming he's rational, rather than a scared, dishonest loon.

It's hard to fit any other conclusion to the data.

# Separate and Unequal

Oh HEY, didn’t see you there! Sorry for the absence, but life caught up to me again. I was doggedly plugging away on my next post when I stumbled across a comic that must have been tailor-made for my current headspace, and it simply had to jump the queue.

# Cherry Picking

With the benefit of hindsight, I can see another omission from Rationality Rules’ latest transphobic video. In his citations, he cites two sporting bodies: the International Association of Athletics Federations and the Australian Sports Anti-Doping Authority. He relies heavily on the former, which is strange. The World Medical Association has condemned the IAAF’s policies on intersex and transgender athletes as “contrary to international medical ethics and human rights standards.” The IAAF has defended itself, in part, by arguing this:

The IAAF is not a public authority, exercising state powers, but rather a private body exercising private (contractual) powers. Therefore, it is not subject to human rights instruments such as the Universal Declaration of Human Rights or the European Convention on Human Rights.

Which is A) not a good look, and B) false. If you won’t take my word on that last one, maybe you’ll take the UN’s? [Read more…]

# And the Beat Goes On

Essence of Thought has published a timeline of the Rationality Rules affair. If you’re missed any of the last five months, it’ll bring you up to speed.

Cripes, has it been that long already?! I had a look through my archives, and all but two of my posts over the last two months have been focused on Rationality Rules, and even those two were about transphobia. I know, I know, the constant drumbeat is getting a bit repetitive and boring. But there’s a reason for it.

[11:31] Now, some of the walkouts had formed a support group, which I was later added to, and reading through their accounts is truly horrifying. Many discussed the abuse they suffered thanks to Woodford and his audience. There are numerous discussions on how their sleep was impacted, about how they’re having to see psychiatrists and other specialists. I’ve even seen [a post?] discussing suicide in relation to what had occurred. That’s the level of severity we are talking about with this issue: people discussing suicide. That’s the damage Woodford and his supporters have caused this one group, this one organization.

I don’t have any way to verify this part, but some of it tracks with comments I’ve read elsewhere, the claims have remained consistent over time, and it would explain why ACA members seem willing to talk to Essence of Thought despite the ocean between them.

One thing I do know: the odds of anyone holding Rationality Rules responsible are basically zero. Some big names in the atheo-skeptic sphere, such as Matt Dillahunty and AronRa, either agree with RR or don’t care enough to do their homework. The ACA tried to do the right thing, but it appears RR supporters elected themselves into a majority on the ACA’s board, possibly breaking the rules in the process, and promptly started kissing their abuser’s ass.

In order to remove any ambiguity in the following statement, I wish to make clear that the ACA earnestly and sincerely apologizes to Stephen Woodford (Rationality Rules) for vilifying his character and insinuating that he is opposed to the LGBTQIA+ community. The Board of Directors has officially retracted our original statement.

Rationality Rules was so confident nobody would take him to task, his “improved” video contains the same arguments as his “flawed” one. And honestly, he was right; I’ve seen this scenario play out often enough within this community to know that we try to bury our skeletons, that we treat our minorities like shit, that we “skeptics” are just as prone to being blind followers as the religious/woo crowds we critique. And just like all those other times, I cope by writing words until I get sick of the topic. Sometimes, that takes a while.

This is one of those “a while” times. If it helps, I’m actively trying to avoid covering topics other people already have, and elevating the voices of others to break up the monotony.

# Burden of Proof

I deliberately left this one to stew, in case I could think of a better angle. Alas, you just can’t beat perfection.

# Lies of Omission

In Rationality Rules’ latest transphobic video, one phrase in particular caught my ears.

[3:57] The way that the IAAF put it, was that double-amputee sprinter Oscar Pistorius is ineligible to compete at the Beijing Olympics because his prosthetic racing legs give him a clear competitive advantage; or in other words, the IAAF ruled the Pistorius’ prosthetics disqualify him because they make him faster than what he would have been if he had not lost his legs. Now this reasoning is critical, as it embodies both the principle of “fair play” and the principle of therapeutic use exceptions, otherwise known as TUE’s.

[5:42] … a collection of experts criticized the cited study for only testing Pistorius’s biomechanics at full speed while running in a straight line, unlike a real 400 meter race, and for not accounting for the disadvantages that he suffers, such as having trouble leaving the starting block; and as a result, Pistorius is ineligible status was lifted. He was allowed to compete. … [6:24] as we move on to the transgender athletic debate, please keep in mind the principle of “fair play,” the principle of TUEs, and Pistorius’ case as a whole.

[20:02] I am not opposed to trans women who have experienced male puberty competing in the female category of SOME events because they’re trans. I am opposed because the attributes which are granted from male puberty that play a vital role in some events have not been shown to be sufficiently mitigated by HRT. It’s not about whether or not they’re women, it’s about whether or not “fair play” has been maintained.

Rationality Rules never details what “fair play” is, in fact you’ve just read every mention of the term in that video. At the same time, his argument strongly relies on it. That makes the lack of any definition a curious omission. [Read more…]