Since Pearl asserts the first two, do I have to get rid of the idea that more qualifications lead to more pay? I can’t see any other way out of the bind.
I believe so, with the caveat that this could be a reversal effect. That is, qualifications and pay may be positively correlated for the whole group because men have more of both than women, while for each subgroup the correlations are negative.
Consider the following situation:
Men have 60 points to spend at character creation. Each point can either be used on a year of schooling, or a dollar of salary, with a minimum of 10 in each.
Women have 30 points to spend at character creation. Each point can either be used on a year of schooling, or a dollar of salary, with a minimum of 10 in each.
Now Bob says, “Look! If we look at groups determined by salary, each man is more qualified than women in his cohort, by thirty years of schooling.” Barbara says, “Look! If we look at groups determined by schooling, each woman earns less than men in her cohort, by thirty dollars.”
If most people choose to spend their points equally, then the population will be dominated by the points (15,15) and (30,30), and so the Association of Higher Education will say “Look! Schooling and salary are positively correlated.”
The causal diagram in this situation is clear, though: it’s being male that leads to more points while the direct effect of schooling on salary is negative because those two come from the same pool of points.
Thanks for the insightful comment. I hadn’t considered that particular application of Simpson’s paradox. But really, I don’t think this is that likely, is it? I mean, you’re letting me get one statement I like “qualifications correlate with earnings in general” but give up two statements that I find likely: “qualification correlate with earnings for males (resp. females)”.
This paper looks like it says that qualifications are correlated with earnings for each subgroup. See the tables on pages 21 and 22. I say “looks like” since I haven’t actually read it and just skipped to the tables. I hope to get a chance to look at it more in depth soon.
But really, I don’t think this is that likely, is it?
I think that particular reversal is probably unlikely in general, but I can think of several plausible cases when it would exist.
Suppose that IQ positively impacts both education and income. But education has a negative effect on income, because the more educated someone is, the more they will choose to work on abstract tasks which don’t pay as highly. (A salesman earns more than mathematician, say, and the primary function of education is to convince some people that mathematicians are higher status than salesmen.) It looks like the impact of education on income is positive, because of the effect of IQ. (This is basically the same as the reversal effect we discussed, except swapping out sex for IQ.)
Suppose among workers in general, qualification has a positive impact on earnings. For one particular sex at one particular firm, the selection process might be such that qualification has a negative impact on earnings. For small firms in particular, this situation might be likely to arise by chance.
I believe so, with the caveat that this could be a reversal effect. That is, qualifications and pay may be positively correlated for the whole group because men have more of both than women, while for each subgroup the correlations are negative.
Consider the following situation:
Men have 60 points to spend at character creation. Each point can either be used on a year of schooling, or a dollar of salary, with a minimum of 10 in each.
Women have 30 points to spend at character creation. Each point can either be used on a year of schooling, or a dollar of salary, with a minimum of 10 in each.
Now Bob says, “Look! If we look at groups determined by salary, each man is more qualified than women in his cohort, by thirty years of schooling.” Barbara says, “Look! If we look at groups determined by schooling, each woman earns less than men in her cohort, by thirty dollars.”
If most people choose to spend their points equally, then the population will be dominated by the points (15,15) and (30,30), and so the Association of Higher Education will say “Look! Schooling and salary are positively correlated.”
The causal diagram in this situation is clear, though: it’s being male that leads to more points while the direct effect of schooling on salary is negative because those two come from the same pool of points.
That’s a great explanation, thanks for writing it! From now on, I will use your explanation instead of mine.
Thanks! (I am amused that the linked explanation includes evidence of Vaniver_2010 being confused by Simpson’s Paradox.)
Thanks for the insightful comment. I hadn’t considered that particular application of Simpson’s paradox. But really, I don’t think this is that likely, is it? I mean, you’re letting me get one statement I like “qualifications correlate with earnings in general” but give up two statements that I find likely: “qualification correlate with earnings for males (resp. females)”.
This paper looks like it says that qualifications are correlated with earnings for each subgroup. See the tables on pages 21 and 22. I say “looks like” since I haven’t actually read it and just skipped to the tables. I hope to get a chance to look at it more in depth soon.
I think that particular reversal is probably unlikely in general, but I can think of several plausible cases when it would exist.
Suppose that IQ positively impacts both education and income. But education has a negative effect on income, because the more educated someone is, the more they will choose to work on abstract tasks which don’t pay as highly. (A salesman earns more than mathematician, say, and the primary function of education is to convince some people that mathematicians are higher status than salesmen.) It looks like the impact of education on income is positive, because of the effect of IQ. (This is basically the same as the reversal effect we discussed, except swapping out sex for IQ.)
Suppose among workers in general, qualification has a positive impact on earnings. For one particular sex at one particular firm, the selection process might be such that qualification has a negative impact on earnings. For small firms in particular, this situation might be likely to arise by chance.