I can still do math, even when exhausted
Below is a letter to the WSJ I just popped off, on a class action sex discrimination case against Walmart.
The recent article on Walmart's appeal as the the class action status of the sex discrimination suit puts me in mind of a pervasive problem with using statistics to "prove" discrimination on various bases.
The issue is called Simpson's Paradox (see Wikipedia article here: http://en.wikipedia.org/wiki/Simpson%27s_paradox). It is entirely possible that in every single Walmart store women do better than men with regards to promotions, salaries, etc. but on a companywide basis they do worse. This may seem illogical, but this crops up all the time when there isn't a uniform distribution of various groups amongst the individual stores.
A famous example of this was a study of Berkeley grad school admissions in the 70s. It seemed that men were far more likely to be admitted to grad school than women. Clear sexual discrimination, right? Sorry, no. You see, admissions to grad school are decided on a department-by-department basis. So if there were sex bias, you'd have to see which departments were the culprits. It turned out that in every department, women had a higher admissions rate than men. What had happened is that, in general, women tended to apply to departments that had the highest rejection rates and men tended to apply to departments that had the lowest rejection rates.
According to the article "Sex Bias in Graduate Admissions: Data from Berkeley" (Science 7 February 1975: Vol. 187. no. 4175, pp. 398 - 404 http://www.sciencemag.org/cgi/content/abstract/187/4175/398 ), "The bias in the aggregated data stems not from any pattern of discrimination on the part of admissions committees, which seem quite fair on the whole, but apparently from prior screening at earlier levels of the educational system. Women are shunted by their socialization and education toward fields of graduate study that are generally more crowded, less productive of completed degrees, and less well funded, and that frequently offer poorer professional employment prospects."
I don't quite agree with their own bias here, because they're also not taking into consideration that one of the largest culprits in this study, education, is oversubscribed because most of the people going for graduate studies here are public school teachers and administrators for whom promotions and raises are contingent on higher degrees (no matter how irrelevant to their own job performance), even back in the 70s. And though it's probably still doctrine in the ivory tower that women are vastly "overrepresented" in education, out in the real world people recognize that women are more inclined to working with small children than are men, and that many mothers like the convenience of the teaching profession.
It is more than possible that a similar dynamic is going on at Walmart. I doubt that their wages are the same across the country, as most retail employees are going to be drawn from the local population (is someone going to relocate to be a store manager? Perhaps, but doubtful on the salaries they make), so the wages are going to be very dependent on the local wage market. Wages in rural Missouri are likely to be lower than wages in suburban Chicago, for instance. It could be that stores where retail wages are lower find far more women as employees, and perhaps these areas also have fewer promotion opportunities. It could be that men are more likely to work at Walmart only for the higher wage stores,where due to volume, there are more managers.
I agree with Walmart that as promotion and wage decisions are madelocally, the proper comparison should be on a local basis, so as to prevent the kind of confounding result one gets from Simpson's paradox. I hope whoever Walmart brought in as an expert witness did explain the disparity between the plaintiff's evidence of a national disparity and the store-by-store evidence. This is not to say that sex discrimination is going on in none of the Walmart stores (and it could even be that in some stores, women are preferred over men), but that national statistics mean nothing without looking at a store-by-store account.
Mary Pat Campbell
Sr. Actuarial Associate
Kew Garden Hills, NY
--------
I always made sure to point up Simpson's paradox in every stats and probability course I taught, because I really love that stuff. Also, so many policy decisions are being made on stats without taking base rates into consideration, and I want to inculcate into the "next leaders" (yeah, yeah) to be a little more inquisitive when it comes to stats.
You know, people take lots of classes in high school and college where they don't really remember any of the content, but may come away with a few "take home" messages that stick. Though some would say that the Central Limit Theorem is the most important take home lesson from intro stats, I don't agree at all (especially as they're likely to misremember, and think that everything out there is normally distributed, which is total crap.)
Here are the take home messages from intro stats I think are important:
1. Correlation does not equal causation (look for hidden variables, common causes, and possibility of total chance, think of alternate explanations and were they ruled out)
2. Base rates are very important (Simpson's paradox and the Bayesian probability -- the second especially with regards to medical tests and false positives for rare conditions)
3. What does 95% confidence mean in those telephone polls, and how could they be screwed up with biased questioning
These are all "real world" considerations of stats, and not so much the mathematical detail. The problem with stats for most people is that they either swallow them uncritically, or totally dismiss them, also uncritically. Stats are very powerful when used correctly, and one doesn't need to be an expert to think about some of the issues. Good science/medical reporters will bring up these aspects (I do see this in the exclusive of the WSJ and Science Times, but not much from the AP wire. Remember the whole hormone therapy scare a few years ago? How many highlighted that these therapies were started on women generally almost 20 years after menopause, and not those most likely to use the therapy, women actually going through menopause? )
The recent article on Walmart's appeal as the the class action status of the sex discrimination suit puts me in mind of a pervasive problem with using statistics to "prove" discrimination on various bases.
The issue is called Simpson's Paradox (see Wikipedia article here: http://en.wikipedia.org/wiki/Simpson%27s_paradox). It is entirely possible that in every single Walmart store women do better than men with regards to promotions, salaries, etc. but on a companywide basis they do worse. This may seem illogical, but this crops up all the time when there isn't a uniform distribution of various groups amongst the individual stores.
A famous example of this was a study of Berkeley grad school admissions in the 70s. It seemed that men were far more likely to be admitted to grad school than women. Clear sexual discrimination, right? Sorry, no. You see, admissions to grad school are decided on a department-by-department basis. So if there were sex bias, you'd have to see which departments were the culprits. It turned out that in every department, women had a higher admissions rate than men. What had happened is that, in general, women tended to apply to departments that had the highest rejection rates and men tended to apply to departments that had the lowest rejection rates.
According to the article "Sex Bias in Graduate Admissions: Data from Berkeley" (Science 7 February 1975: Vol. 187. no. 4175, pp. 398 - 404 http://www.sciencemag.org/cgi/content/abstract/187/4175/398 ), "The bias in the aggregated data stems not from any pattern of discrimination on the part of admissions committees, which seem quite fair on the whole, but apparently from prior screening at earlier levels of the educational system. Women are shunted by their socialization and education toward fields of graduate study that are generally more crowded, less productive of completed degrees, and less well funded, and that frequently offer poorer professional employment prospects."
I don't quite agree with their own bias here, because they're also not taking into consideration that one of the largest culprits in this study, education, is oversubscribed because most of the people going for graduate studies here are public school teachers and administrators for whom promotions and raises are contingent on higher degrees (no matter how irrelevant to their own job performance), even back in the 70s. And though it's probably still doctrine in the ivory tower that women are vastly "overrepresented" in education, out in the real world people recognize that women are more inclined to working with small children than are men, and that many mothers like the convenience of the teaching profession.
It is more than possible that a similar dynamic is going on at Walmart. I doubt that their wages are the same across the country, as most retail employees are going to be drawn from the local population (is someone going to relocate to be a store manager? Perhaps, but doubtful on the salaries they make), so the wages are going to be very dependent on the local wage market. Wages in rural Missouri are likely to be lower than wages in suburban Chicago, for instance. It could be that stores where retail wages are lower find far more women as employees, and perhaps these areas also have fewer promotion opportunities. It could be that men are more likely to work at Walmart only for the higher wage stores,where due to volume, there are more managers.
I agree with Walmart that as promotion and wage decisions are madelocally, the proper comparison should be on a local basis, so as to prevent the kind of confounding result one gets from Simpson's paradox. I hope whoever Walmart brought in as an expert witness did explain the disparity between the plaintiff's evidence of a national disparity and the store-by-store evidence. This is not to say that sex discrimination is going on in none of the Walmart stores (and it could even be that in some stores, women are preferred over men), but that national statistics mean nothing without looking at a store-by-store account.
Mary Pat Campbell
Sr. Actuarial Associate
Kew Garden Hills, NY
--------
I always made sure to point up Simpson's paradox in every stats and probability course I taught, because I really love that stuff. Also, so many policy decisions are being made on stats without taking base rates into consideration, and I want to inculcate into the "next leaders" (yeah, yeah) to be a little more inquisitive when it comes to stats.
You know, people take lots of classes in high school and college where they don't really remember any of the content, but may come away with a few "take home" messages that stick. Though some would say that the Central Limit Theorem is the most important take home lesson from intro stats, I don't agree at all (especially as they're likely to misremember, and think that everything out there is normally distributed, which is total crap.)
Here are the take home messages from intro stats I think are important:
1. Correlation does not equal causation (look for hidden variables, common causes, and possibility of total chance, think of alternate explanations and were they ruled out)
2. Base rates are very important (Simpson's paradox and the Bayesian probability -- the second especially with regards to medical tests and false positives for rare conditions)
3. What does 95% confidence mean in those telephone polls, and how could they be screwed up with biased questioning
These are all "real world" considerations of stats, and not so much the mathematical detail. The problem with stats for most people is that they either swallow them uncritically, or totally dismiss them, also uncritically. Stats are very powerful when used correctly, and one doesn't need to be an expert to think about some of the issues. Good science/medical reporters will bring up these aspects (I do see this in the exclusive of the WSJ and Science Times, but not much from the AP wire. Remember the whole hormone therapy scare a few years ago? How many highlighted that these therapies were started on women generally almost 20 years after menopause, and not those most likely to use the therapy, women actually going through menopause? )