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Trying to make sense of nonsense

March 1, 2013

Posted by on We read about statistics every day: be it the predicted winner of a football league, the association between the weather and mortality, or a newly discovered link between an inanimate object and cancer. Statistics are everywhere. And perhaps even more so this year, as 2013 has been hailed as the International Year of Statistics. Despite all this attention for numbers, we generally don’t know a lot about the people hiding behind their computers churning them out. With media attention for people like Nate Silver and Hans Rosling, some are now able to name at least one statistician, but, stepping it up a level, could you name a female statistician?

Statistics is definitely not the only branch of STEM subjects suffering from a lack of distinguished women. Just take a look at the list of Nobel Prize winners (44 out of 839 Laureates), fellows of the Royal Society (currently 5%), or scientists on television. This is not due to a lack of women in statistics, there are many. So with this being the year of statistics, I thought it might be the perfect timing to highlight some of the women who work(ed) in statistics.

**Dr Janet Lane-Claypon: epidemiologic pioneer**

Dr Janet made quite a few important contributions to epidemiology by using and improving its use of statistics. Born in 1877 in Lincolnshire, she moved to London to study physiology at the London School of Medicine for Women (today part of UCL). She spent a few years there collecting an impressive list of titles: a BSc, DSc and MD, making her one of the first, irrespective of gender, Doctor-doctor’s.

All very exciting of course, but what has she got to do with statistics? Her run-in with statistics started in 1912, when she published a *Report to the Local Government Board upon the Available Data in Regard to the Value of Boiled Milk as a Food for Infants and Young Animals*. It’s an impressive report (available at the British Library, in case you’d like to leaf through it on a rainy Saturday afternoon), and the first of its kind. In it, Lane-Claypon compares the weights of infants fed on breast and cows’ milk, to find whether the type of milk had an effect on how fast babies grew. To answer this question, she used, for the very first time, a retrospective cohort study, description of confounding, and the *t*-test.

Before she started she study, Dr Janet realised she would need a large number of healthy babies who had been fed cows’ milk and a similar number of babies on breast milk. More importantly, she realised that in order to compare the two groups, she would need the babies to be “as far as possible” from the same social environments. She ended up travelling to Berlin, where babies from the working classes regularly attended Infant Consultations where their diet and weight was registered, resulting in the perfect dataset to answer her question.

This visit resulted in data on just over 500 infants making up the first retrospective cohort study (many others would follow, but not till some 30 years later), which was ready to be analysed. However, although all babies came from working class parents, Dr Janet realised that their social environments could still be slightly different, leading to different rates of weight gain between the groups. She explains:

“It does not, however, necessarily follow that the difference of food has been the causative factor, and it becomes necessary to ask whether there can be any other factor at work which is producing the difference found. The social class of the children seemed a possible factor, and it was considered advisable to investigate the possible significance of any difference which existed between the social conditions of the homes.”

Dr Janet compared the wages from the fathers of the infants, for the first time taking confounding into account, and found that they looked the same for the two groups. Still not satisfied whether the difference she had found between breast and cows’ milk fed children was real, she decided to use a new complicated technique that had been published 4 years earlier, but hadn’t been used in epidemiology up till then: Student’s *t*-test. Chances are that you’ve heard about this test, as it is now one of the most commonly used tests in any branch of science. Although it was developed to monitor the quality of stout by W.S. Gosset, Janet Lane-Claypon was the first to use it in medical statistics.

Dr Janet’s pioneering didn’t stop there. She went on to conduct the first ever case-control study in 1926, which possibly used the first ever questionnaire to gather health data (so think about her next time you see a pop-up window/email asking if you’ve got a few minutes to spare) on the causes of breast cancer. Her results were used by two other famous statisticians: Nathan Mantel and William Haenszel. They developed the Mantel-Haenszel test to adjust results for confounding. Her findings included most of the currently recognised risk factors for breast cancer. Dr Janet continued to work till 1929, when she had to retire at 52 due to the silly reason that married women weren’t allowed to work in the civil service.

Some further reading on Dr Janet:

Lane-Claypon JE. Report to the Local Government Board upon the Available Data in Regard to the Value of Boiled Milk as a Food for Infants and Young Animals. 1912

Lane-Claypon JE. A Further Report on Cancer of the Breast with Special Reference to its Associated Antecedent Conditions. Reports on Public Health and Medical Subjects. 1926

Winkelstein W. Vignettes of the history of epidemiology: Three firsts by Janet Elizabeth Lane-Claypon. American Journal of Epidemiology 2004;160(2)97

Winkelstein W. Janet Elizabeth Lane-Claypon: a forgotten epidemiologic pioneer. Epidemiology 2006;17(6)705

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November 9, 2012

Posted by on By predicting the outcome of the US elections correctly in 50 out of 50 states (after an already impressive 49/50 in the 2008 elections), Nate Silver of the NY Times’ FiveThirtyEight blog has managed to convince even the most sceptical data deniers of his prediction models. So much so that his perfect prediction started a twitter trend (#natesilverfacts) and led to him being labelled a witch. So how impressive was this feat really? Is Nate Silver really a wizard from the future aiming for world domination through the power of numbers? Let’s use some stats to assess his stats!

Let’s start by toning down Silver’s amazing feat of predicting the election outcomes in 50 separate states. In most US states, the outcome of the election didn’t need complex prediction models to come to a reliable estimate of the election outcome: some results, such as in the District of Columbia where over 90% of the population voted Obama, were uncontested. The same goes for other red Obama-voting states as California (59%), Hawaii (71%), Maryland (62%) or New York (63%) or blue Romney states as Oklahoma (67% voted GOP), Utah (73%), Alabama (61%) or Kansas (60%).

Only in swing states, that could go either way, Nate Silver would have needed his number crunching to decide on a future winner. If we go by the NY times’ numbers, only 7 states were a toss-up between the Democrats and Republicans: Colorado, Florida, Iowa, New Hampshire, Ohio, Virginia and Wisconsin. Treating those 7 states as coin tosses – each outcome has an equal 50% probability – we can test the hypothesis that Nate Silver is a witch, *H*_{witch}, against the competing hypothesis that he is a completely non-magical human being, *H _{muggle}*. If Nate is a witch, we assume he predicts each state’s election results correct, witches having a perfect knowledge of all future events. The probability for this happening is expressed in a fancy maths equation like this:

Whatever the truth is about Nate Silver, it appears he’s pulled off something pretty extraordinary. Unfortunately for him, he’s still one step removed from being the world’s best predictor as Paul the psychic octopus managed to correctly predict the outcomes of 8 football matches at the 2010 World Cup. World’s best *human* predictor will have to do for now then.

However, as with Paul, Nate wasn’t the only person making predictions. Paul only gained the street cred necessary to be taken seriously as a clairvoyant cephalopod after a bout of predicting Eurocup results (and getting one wrong), and the same could be said for Nate Silver. If he hadn’t pulled off a similar feat in the previous elections, no one would have paid much attention to his blog this time round. His 2008 prediction was perhaps even more impressive than his latest one: he might have missed Indiana, but got the results for the remaining 10 swing states right.

As polls get about the same amount of coverage (if not more) as the actual elections, there are a lot of people who try to pitch in. Let’s take a guess and say there were 50 people trying to predict the state-by-state 2008 election outcomes. Chances that at least someone would get at least 8 of the 11 swing states correct (assuming this would be the threshold to attract the attention of witch hunters) are 1-(255/256)^{50}= 0.18 (for the reasoning behind this calculation, read David Spiegelhalter’s blog on the numbers behind Paul being a completely normal, if not slightly lucky, octopus). So there was an about 1 in 5 chance of at least someone coming up with some remarkably correct predictions.

So we now know that frequentist statisticians would label Silver as a witch, but what about much cooler Bayesians? (no bias at all here…) Bayesian statistics differ from frequentist statistics in that it takes prior knowledge into account when putting a probability on an event. Or : Bayesian statistics is probably a cool branch of stats, but if you know XKCD thinks so too, it’s suddenly a lot more probable to be true (the coolness of a specific branch of statistics is conditional on XKCD endorsement).

To calculate the posterior probability of Nate Silver being a witch, we need to know a few things:

*p(W)*, or the prior probability that Nate Silver is a witch, regardless of any other information. This will depend on the prevalence of witches in Silver’s hometown, New York. According to this NY Meetup page, there are 3023 witches in NY. Considering the population of the whole city (8,244,910 according to the US census), the prior probability of a random person in NY being a witch is 0.0004.*p(W’)*, or the probability that Nate is a muggle regardless of any other information, and that’s 1 – 0.0004 or 0.9996 in this case.*p(P|W)*, the probability of Nate making a perfect prediction, given that he’s a witch: 100%, or 1.*p(P|W’)*, the probability of Nate making a perfect prediction as a muggle, which we put at or 0.008 earlier.*p(P)*, the probability of making a perfect prediction, regardless of any other information. Using the law of total probability – all probabilities have to add up to 1 or 100% – this is 1×0.0004 + 0.008×0.9996 = 0.0084

Now that we know all this we can fill out the formula for calculating posterior probability:

*That’s pretty slim, though at 5%, we can’t be sure he isn’t a witch. However, going back to the 2008 elections, there were already some suspicions of Nate Silver’s potential Wiccan background. If we start with the 0.18 probability we arrived at earlier, the posterior probability of Nate Silver being a witch rises to 0.96 or 96%. So yes, Nate Silver is probably a witch. Alternatively, you could of course exchange ‘witch’ with ‘statistician’ and conclude with 96% confidence that he’s just very good at his job.*