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I am reminded of the famous Berkley admissions Simpon's Paradox example (overall, women had a lower rate of admission than men, but in each individual program the admission rate for women was either higher than or not significantly different from that for men.)

In this case, the question "why is the admission rate lower for women?" was equivalent to "why do women apply to programs for which admission is more competitive?". That's not exactly analogous to your example, lacking the symmetry between questions, but the common thread is that the causal model implied by framing is consistent with one aspect of the observed data but not all aspects.

Frances: thanks. The logic of my point here is similar to the logic of a point you explained to me once, a few years back, about average numbers of lifetime sexual partners for men and women. Things have to add up! I think you had already understood my point here, implicitly.

Yep, pay for graduates matters. But so does the probability of graduating in finite time. And whether or not you enjoy that subject, and feel "at home" there. And those questions need looking at for both men and women. And especially whether women will look at men, or stay with men, who are not "successful men", as defined by women. ("Men are so shallow, in only considering women's looks!")

Phil: interesting example. In economics, it looks like a reverse Simpson's Paradox? Men stick with economics even if they get lower grades so it's more competitive for them.

And it reminds me of that really weird result, from a decade or so back, that high school economics grades have a *negative* "effect" on university economics grades for girls, but not for boys. Did I remember that right, Frances?

Simpson's Paradox occurs when a statistical relationship present in an aggregate dataset reverses when the dataset is disaggregated (stratified.) For example, in the aggregate data, putative cause X may be positively related to effect Y but when conditioned by covariate Z it has a negative relationship in all strata.

In the Berkley's case most people agree that the correct view is the stratified one. This is the familiar procedure "conditioning on a confounding factor." Unfortunately, many explanations of Simpson's Paradox stop here, leaving the impression that conditioning is always correct. Nor is it always the naive who think this; Judea Pearl has quoted Don Rubin (an eminent statistician) as saying "to avoid conditioning on some observed covariates,... is nonscientific ad hockery.”

In fact, this is completely wrong. The choice of conditioning covariates is dependent on one's causal model and cannot be determined by examining the data. It is correct to condition on Z when Z may have a causal influence on both X and Y. On the other hand, if X may instead influence Z, then conditioning on Z gives the wrong answer. I would call this a "reverse Simpson's Paradox" because the problem seems to escape many people. It is possible for a dataset to support relationship reversal each time one conditions on covariates {C1}, {C1,C2}, {C1,C2,C3} ... etc. That is one reason why conclusions from observational studies are often unreliable; one must choose exactly the correct subset of covariates for conditioning and the inherent difficulty of this is usually exacerbated by not explicitly considering what the causal model is supposed to be.

Finally, if both X and Z are influenced by some third covariate W, then neither conditioning on Z nor not conditioning on Z will give the right answer. Larry Wasserman explains all this very well.

Nick, I remember the result that high school econ grades have a negative effect on uni econ grades (I'm thinking perhaps Dwayne Benjamin wrote that up for something? Someone at U of T anyways). But am ashamed to say I don't remember that gender effect.

Found the published version of the paper: http://www.jstor.org/stable/view/1183277 . Scanned it but couldn't find a regression run separately for men and women - perhaps there was a gender*high school grade interaction term I didn't spot. I think the result is more that taking high school in econ had a negative effect on uni econ grades, unless the student did really really well in high school econ.

But they did find gender differences in high school econ courses, and spend a lot of time discussing them.

Phil: I *think* I understand Larry Wasserman's post (just). Wondering if it's related to my old post here.

Frances: my memory is bad too. I remember thinking "maybe the boys weren't paying attention in high school, so it had no effect". But a more likely explanation is some sort of sample selection effect.

OT

http://monetaryrealism.com/money-creation-in-the-modern-economy-bank-of-england/

I think that putting numbers on this matters...
if men are under-represented in arts but 40% of the arts stream is male then that's different to 10% of the science stream is female.

Two weeks into out academic year, an incredibly talented female I know just got head-hunted out of a post-graduate science course. Which means the female participation rate in science goes down, average female science performance goes down but she's happy because it's the job of her dreams. Maybe the reverse happens in the arts with men especially economics.

mpedger: I don't have the numbers in "arts" and "science" offhand, but fewer men than women go to university (and fewer still graduate). Social work, for example, has a very small percentage of men at my university (less than 10%, IIRC). I have seen the numbers, but it would take me some time to dig them up.

Male Female Male Female Male Female Male Female
Program Dimension summary (opens new window) Acadunit Dimension summary (opens new window) Sort ascendingSort descending Sort ascendingSort descending Sort ascendingSort descending Sort ascendingSort descending Sort ascendingSort descending Sort ascendingSort descending Sort ascendingSort descending Sort ascendingSort descending
Total Item summary (opens new window) Total Item summary (opens new window) 14,601 13,262 14,384 12,857 14,168 12,623 13,564 12,353
Arts & Social Sciences 2,578 4,625 2,612 4,519 2,685 4,546 2,663 4,512
Public Affairs 3,504 4,290 3,556 4,176 3,496 4,072 3,259 4,022
Business 1,322 955 1,293 949 1,303 935 1,294 967
Science 2,266 1,493 2,120 1,329 2,051 1,199 1,912 1,069
Engineering & Design 4,090 1,146 3,879 1,078 3,671 1,025 3,447 922

OK, I just did it for Carleton. The table is a total mess. But the numbers are (left to right) male then female, then male then female for the previous year, etc.
So, for the most recent year, 2,578 males and 4,625 females in Arts and Social Science, and 2,266 males and 1,493 females in Science, etc.

Another example of this is 'women are paid 77 cents of each dollar men make' rather than 'men make 1.30 of each dollar women make', and women's salaries need raising rather than men's lowering even though there has been at least as much of the latter as the former. There is a paternalistic male standard, and wanting to frame goals as desirable.

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