The Assumption of Equality

I promise I’m going to review Pinker’s The Blank Slate sometime soon – if I read enough background material and feel brave enough to skewer the guy on a blog he reads, I’ll do that on the 29th on 3QD – but now I’ll just focus on one point of his, the assumption of equality.

Feminists, he says, don’t need to assume that men and women are equal in any way. All they need is to do things like send identical resumés to firms and see if there’s systematic bias in hiring. But mere imbalances in pay or the number of women should not be taken as evidence of discrimination in themselves, because they could be due to things other than discrimination.

In fact, the assumption of equality is crucial. The studies that show systematic bias in hiring can never implicate a single employer. In the seminal study on racial discrimination in hiring in the US, each employer was sent four resumés, two for each race. Sending any more was impossible due to strict controls on the content of each resumé. Overall, the basic result was significant at a p-value of 0 to four decimal places, but for an individual employer, it could never be lower than 0.25.

Update: in the comments, Bruce explains the definition of a p-value to the lay reader. The p-value is the probability of getting a result at least as extreme as the one in the experiment. If you toss four coins and all four are heads, then the p-value is 1/16, since no result is more extreme than all-heads. But if you get three heads, the p-value isn’t the probability of getting three heads, 1/4, but the probability of getting at least three heads, 5/16.

More detailed studies within the same employer could in principle discern discrimination, but even then it’s impossible to finger specific culprits. Fingering specific culprits isn’t necessary if all we want to learn is how much discrimination there is, but is critical if we want to enforce anti-discrimination laws.

For a concrete example, take a big law firm that hires eight lawyers every year. Let’s say that the talent pool consists of all graduates of top 10 law schools, who are 50% female. Let’s also say that there’s systematic discrimination in hiring, so that only 20% of all people hired are women. Looking at the firm’s hiring pattern over the last ten years will quickly confirm that, since there will be 16 women and 64 men hired, which is significant with a p-value of 0.00000003.

Now, thanks to a large pile of research in sociology, psychology, and economics, we can be reasonably certain that it’s not because women are just bad lawyers. We could even look at class performance, and conclude that indeed women are as qualified as men, which allows us to conclude said firm is violating equal rights laws.

Then we could impose a quota, say 6 women over the next two years (which has a p-value of 0.23; 5 would have 0.11); ordinarily quotas should give more leeway, say a p-value of 0.05, but when the discrimination is obvious and blatant, a more stringent quota is in order.

Without research telling us that the assumption of equality is correct, we could never correct such cases. In a single year, hiring 2 women out of 8 is insignificant, with p = 0.14. Even sending matched resumés over several years wouldn’t help. To avoid making the firm suspicious, we’d have to limit ourselves to, say, two of each gender in each year.

If 20% of people hired are female, we need 8 or 9 successful applications, or callbacks (assuming the bottleneck is in callbacks rather than interview results), to get a p-value under 0.1, and 11 or 12 to get a value under 0.05. A hiring or callback rate of one in four means it will take 11-12 years of tracking to discover the discrimination; a rate of one in ten means it will take almost 30. In other words, it makes equal rights laws toothless.

In contrast, once we establish that the assumption of equality makes sense, we could get a p-value under 0.05 in two years, and under 0.005 in three. The lower the p-value, the easier it is to build a case against the firm, and the more it makes sense to impose more stringent quotas, which rectify the problem sooner.

There are also entirely different avenues of discrimination, which become entirely invisible without the assumption:

First, there are ostensibly neutral standards, like fireman exams that emphasize physical toughness more than is needed on the job. Minneapolis’s fire department got better after its first female head took the fireman exam apart and removed the parts that weren’t really necessary, but kept women out. Although the actual changes to the exam did not require any assumption, it took the heuristic that differences in results probably underlay discrimination to know that the exam might be biased.

And second, there are cultural biases. Pinker tries to argue that women are hardwired to like different things from men based on the fact that in the US at least, math departments have fewer female professors than physics departments, but it’s incredible to believe that mathematicians are more bigoted than physicists. The likeliest explanation is that the American educational system steers girls away from science and especially math, which is nigh impossible to detect with the studies Pinker promotes.

Now, you might ask, how do I know that this assumption of equality in abilities, interests, and desires holds?

The answer is, there are multiple pieces of evidence, or lack thereof. First, research into cognitive differences has failed to find any innate racial differences. Any solid ingrained difference has been traced to culture; for example, the use of Chinese characters sharpens spatial perception, which improves mathematical abilities. Eric Turkheimer disposed of the idea that the black/white IQ gap is genetic once and for all in a 2003 paper.

Innate cognitive differences between women and men do exist, but are far smaller than people like Pinker implies. The only social effect that has been reliably traced to them is the fact that young women drive language change, on account of women’s better linguistic perception. Men’s domination of the hard sciences has never been traced to any cognitive difference.

Second, international data holds biology constant while varying culture. If girls are innately less interested in math than boys, then we’d see a similar effect of female underrepresentation in math throughout the world. But in fact, this effect varies hugely by country. In the US and Japan, women are indeed grossly underrepresented in math and science. In Sweden, India, and Thailand, they’re still somewhat underrepresented, but by a margin that doesn’t even come close to the American one.

It might be that the natural level of female representation in science isn’t 50% but 40%, but given that the US is at 13%, dismissing attempts to encourage girls to explore math more as doomed social engineering is unwarranted.

With race, the proper international comparison is of dominant to oppressed groups. As Pinker notes, the IQ gap is found all over the world to correlate with ethnic inequality, even when the ethnicity isn’t defined by race. White Americans have higher IQs than black Americans, and Protestant North Irelanders have higher IQs than Catholic North Irelanders.

Third, large-scale surveys of discrimination of the kind Pinker approves of can function as pilot studies. These studies can’t implicate single employers, but can implicate industries, or trends. When every industry where there is a gender or race gap is found to engage in discrimination once an appropriate study is done, it’s safe to conclude that a firm with a large gender or race gap is guilty of sexism or racism until proven innocent.

And fourth, even when gaps are found not to result from discrimination but from a smaller talent pool, it’s almost always possible to trace the effect to sexism or racism, and seldom to innate factors. People who believe in large, socially significant cognitive differences based on gender have never been able to agree on what these social effects precisely are; in most cases, each person’s views are very close to what we’d expect to find if he were motivated by sexism rather than science.

For instance, take elections. In Canada, female candidates for Parliament are slightly less likely to win than male candidates, but the effect is statistically insignificant, with p = 0.14. There are numerous plausible sexism-based reasons why Canada’s Parliament is only 20% female: unsupportive party leaders, lack of role models, cultural expectations of male leaders, and so on. In contrast, there’s no plausible innate reason, since solid gender differences in cognition don’t include a higher male capacity for leadership.

Pinker berates Bella Abzug for insisting that equality means that women must have fifty percent representation everywhere. But that assumption of equality is exactly true. Nobody’s saying that women should comprise seventy or eighty percent of linguistics professors because of their superiority in handling language. It’s assumed that the slight difference still means the proper gender distribution is roughly fifty-fifty. By the same standard, equality means exactly proportional representation for women and minorities.

8 Responses to The Assumption of Equality

  1. Bruce says:

    For those who have an interest in the topic, but a less developed understanding of statistics, a p-value is the likelihood of obtaining an event at least as extreme as that actually obtained.

  2. […] in the comments on Abstract Nonsense, Bruce explains the definition of a p-value to the lay reader. The p-value is the probability of […]

  3. Axel says:

    The linked article “Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination” isn’t available from for non-subscribers except for people coming from developing countries or transition economies. You can download another version from Mullainathan’s hompepage:

    Sorry for being so pedantic, just a little correction: The p-value is the probability of obtaining a result at least as extreme as that obtained, assuming the truth of the null hypothesis that the finding was the result of chance alone.. This is crucial to the understanding of hypothesis testing in the Fisherian tradition because the p-value doesn’t say anything about the “probability” of the null hypothesis.

  4. Alon Levy says:

    I’m not sure if the correct reaction is “I hate being on a university network” or “I love being on a university network.”

  5. Lanoire says:

    I found your review of Pinker accurate and, if anything, too generous. Very well written.

  6. Chi says:

    Some interesting propaganda here. Turkheimer’s study doesn’t show much because the children were aged 7 or under. This is well before the shared environment factor vanishes.

    Also, note that East Asians average above whites even when raised in white homes. This is the case even where there has been severe malnourishment.

    “Contrary to “culture” theory, the ethnic academic gaps are almost identical for transracially adopted children, and to the extent they are different they go in the opposite direction predicted by culture theory. The gap between whites and Asians fluctuated from 19 to .09 in the NAEP data while the gap in the adoption data is from 1/3 to 3 times larger. This is consistent with the Sue and Okazaki paper above which showed that contrary to popular anecdotes, the values that lead to higher academic grades are actually found more often in white homes. In other words Asian-Americans perform highly despite their Asian home cultural environment not because of it. And though the sample is meager, I find it interesting that the gap between the black and white adopted children was virtually identical (within just 4-6 points) to the gap between whites and blacks in the general population, just like in the Scarr adoption study.”

    Some problems with the Turkheimer study:

    1 – The study included only young children and does not make any attempt to extrapolate that all other findings of significant increases in h^2 by age 17 are in any way invalid. The effects of the shared environment vanish at around age 12.

    2 – Turkheimer began his paper by recognizing that the heritability of cognitive ability in childhood is well established.

    3 – Turkheimer made no attempt whatsoever to determine what components of SES he was measuring. There are three obvious items to consider: macro environmental, micro environmental, and genetic. All work to date indicates that the first of these can be found in children, but that it is absent in late adolescents; by late adolescence, all of the environmental component is of the second type; and that genetic intelligence is the largest determinant of SES.

    4 – Turkheimer says that the effect he observed was related to the homes in which the children were raised. This is interesting, since it relates to the adoption studies which show that after childhood there is no adult IQ correlation between biologically unrelated children who were reared together in the same home.

    5 – Turkheimer discusses in some detail that SES is not strictly an environmental variable, since it is known to be (statistically) caused by the intelligence of the parents. He points out that the models he used “cannot determine which aspect of SES is responsible for the interactions” observed.

  7. johny says:

    “First, research into cognitive differences has failed to find any innate racial differences. Any solid ingrained difference has been traced to culture; for example, the use of Chinese characters sharpens spatial perception, which improves mathematical abilities. Eric Turkheimer disposed of the idea that the black/white IQ gap is genetic once and for all in a 2003 paper.”

    The first sentence alone shows that you have not seriously investigated this issue; there are dozens of studies showing significant racial IQ differences. The black/white gap in particular is so massive that it is practically impossible to account for by other than genetic means, and is greater for Africans/whites than for African-Americans/whites. There is even recent research that has begun to unearth some of the specific genetic causes. Furthermore, most of the standard counter-arguments, e.g. test bias, were refuted years or even decades ago; yet some people still refuse to face reality.

    Second, on the “chinese characters” hypothesis. I actually laughed out loud when I saw this. I am a teacher and have been working in several different Asian countries, including China and Vietnam, for over a decade, with literally thousands of students, in all subjects including math. If characters are the explanation for Asian math performance, how do you explain the equally high performance of Vietnamese, who use a latin-based script (quoc ngu)? One might argue that literate Vietnamese used characters until the 19th century, so that the characters affected the population in a similar way to Chinese. Except, whoops, that would be a genetic explanation also. Believe me, if this “chinese characters” idea is your most compelling piece of evidence in support of the environment-only hypothesis, you are in serious trouble.

  8. Patsy says:

    Thanks for finally talking about >The Assumption of Equality
    | Abstract Nonsense <Loved it!

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