by On probabilism and statistics ‘Mr Brown has exactly two children. At least one of them is a boy. What is the probability that the other is a girl?’ What could be simpler than that? After all, the other child either is or is not a girl. I regularly use this example on the statistics courses I give to life scientists working in the pharmaceutical industry. They all agree that the probability is one-half. So they are all wrong. I haven’t said that the older child is a boy. The child I mentioned, the boy, could be the older or the younger child. This means that Mr Brown can have one of three possible combinations of two children: both boys, elder boy and younger girl, elder girl and younger boy, the fourth combination of two girls being excluded by what I
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On probabilism and statistics
‘Mr Brown has exactly two children. At least one of them is a boy. What is the probability that the other is a girl?’ What could be simpler than that? After all, the other child either is or is not a girl. I regularly use this example on the statistics courses I give to life scientists working in the pharmaceutical industry. They all agree that the probability is one-half.
So they are all wrong. I haven’t said that the older child is a boy. The child I mentioned, the boy, could be the older or the younger child. This means that Mr Brown can have one of three possible combinations of two children: both boys, elder boy and younger girl, elder girl and younger boy, the fourth combination of two girls being excluded by what I have stated. But of the three combinations, in two cases the other child is a girl so that the requisite probability is 2/3 …
This example is typical of many simple paradoxes in probability: the answer is easy to explain but nobody believes the explanation. However, the solution I have given is correct.
Or is it? That was spoken like a probabilist. A probabilist is a sort of mathematician. He or she deals with artificial examples and logical connections but feel no obligation to say anything about the real world. My demonstration, however, relied on the assumption that the three combinations boy–boy, boy–girl and girl–boy are equally likely and this may not be true. The difference between a statistician and a probabilist is that the latter will define the problem so that this is true, whereas the former will consider whether it is true and obtain data to test its truth.
Statistical reasoning certainly seems paradoxical to most people.
Take for example the well-known Simpson’s paradox.
From a theoretical perspective, Simpson’s paradox importantly shows that causality can never be reduced to a question of statistics or probabilities, unless you are — miraculously — able to keep constant all other factors that influence the probability of the outcome studied.
To understand causality we always have to relate it to a specific causal structure. Statistical correlations are never enough. No structure, no causality.
Simpson’s paradox is an interesting paradox in itself, but it can also highlight a deficiency in the traditional econometric approach towards causality. Say you have 1000 observations on men and an equal amount of observations on women applying for admission to university studies, and that 70% of men are admitted, but only 30% of women. Running a logistic regression to find out the odds ratios (and probabilities) for men and women on admission, females seem to be in a less favourable position (‘discriminated’ against) compared to males (male odds are 2.33, female odds are 0.43, giving an odds ratio of 5.44). But once we find out that males and females apply to different departments we may well get a Simpson’s paradox result where males turn out to be ‘discriminated’ against (say 800 male apply for economics studies (680 admitted) and 200 for physics studies (20 admitted), and 100 female apply for economics studies (90 admitted) and 900 for physics studies (210 admitted) — giving odds ratios of 0.62 and 0.37).
Econometric patterns should never be seen as anything else than possible clues to follow. From a critical realist perspective it is obvious that behind observable data there are real structures and mechanisms operating, things that are — if we really want to understand, explain and (possibly) predict things in the real world — more important to get hold of than to simply correlate and regress observable variables.
Math cannot establish the truth value of a fact. Never has. Never will.