BackSeeing patterns where none appear
The police arrested nurse Lucia de Berk on charges of murdering five children
PremiumSomething often said about mathematics has always appealed to me—that fundamentally, it involves a search for patterns. Patterns in numbers, of course. But patterns in everything else that mathematics touches as well. Like the weather, or home runs in baseball, or possible sightings of the Tasmanian Tiger.
Or like evidence in criminal trials.
But there can be problems with searching for patterns. A professor of statistics at the University of Bristol, Peter Green, was recently quoted thus: “We humans are terribly good at seeing patterns when they’re not there." And sometimes when they’re not there and we still see them, that leads to trouble.
Let me explain what Prof. Green was referring to.
Lucia de Berk was a nurse at a children’s hospital in The Hague, Netherlands. One day in 2001, when she was on duty, a baby died there. That death, it would appear, was the last straw for one of her colleagues. This person told hospital authorities that during de Berk’s periods on duty, there had been several deaths in the hospital. The authorities, in turn, told the police about this. When the police looked into the complaint, they found 10 suspicious “incidents" at the hospital while de Berk was on duty. Checking her previous jobs at different hospitals, they found 10 more such.
The police promptly arrested de Berk on charges of murdering five children. Their contention was that this pattern of “incidents" while she was on duty was not possible to ascribe to pure chance. Why? Because, they said, the probability that all these “incidents" had happened by chance—that it was all a coincidence—was one in seven billion.
That’s a tiny probability indeed. It’s not clear how the police came up with the number. But at de Berk’s trial, the prosecution called in an expert witness, a psychologist called Henk Elffers. He had done his own analysis of de Berk’s “crimes" and offered a different probability: one in 342 million. That’s about 20 times larger than the police number, but still tiny indeed. How did Elffers calculate this figure?
He counted the overall number of nurse shifts at each hospital de Berk had worked at during her employment there. Of those, he identified how many were hers. Then he totted up the “incidents" at each hospital while she was employed, and the ones that happened during her shift. Combining these in what any serious statistician would call, at best, a naïve manner, he concluded that the chance of a nurse being present at all these incidents, across three different hospitals, was one in 342 million. In fact, in court he actually rounded even that minuscule probability off: The probability of all these deaths happening purely by chance, he said, “is nil".
With this “expert" pronouncement, de Berk was doomed. The court decided that she had poisoned four of her patients and had attempted to murder three more. In March 2003, she was sentenced to life imprisonment with no possibility of parole. An appeal the next year produced a worse result: she was convicted for seven murders and three attempted ones.
But other statisticians were disturbed by this naïve calculation. Elffers had simply multiplied probabilities across different wards in different hospitals. As one report put it: “This would make any nurse look guiltier with each job change. For example, even a mundane one in 20 chance at one hospital, and the same chance at the next, would transform into a more suspicious one in 400 chance." (https://bit.ly/3Bj0NnY). Do this for the several shifts involved and the chance spirals downwards quickly, to one in 342 million.
The same report explains cautions that were likely overlooked by Elffers in de Berk’s case. For example, should we consider only those unexplained deaths that happened on her shifts? What of the similar deaths when she was not in the hospital? Related to that, shouldn’t investigations happen without investigators being aware of which nurse is on what shift? And what about factors like her duty timings? For various reasons, deaths at these hospitals were statistically more likely to happen in the mornings—so if de Berk happened to be doing mostly morning shifts during the time period in question, that would only add to the suspicion against her.
Take into account all this and more, and the chance of a coincidence rises from one in 342 million to...wait for it...one in 26. That’s the number that statisticians Richard Gill and Piet Groeneboom calculated.
Over 1,734 total shifts, there were 26 unexplained “incidents". de Berk did 203 of those shifts. With those numbers, you’d normally expect three of those incidents to happen on her watch. Some simple probability calculations, then, will show that there’s a 25% chance that five deaths happened on her watch, or a 14% chance that seven did. There’s nothing remarkable there. We’ve gone from an almost impossible occurrence (one in 342 million) to a mildly unusual one (one in 7). Considering the data more closely, Gill and Groeneboom made “some less favourable choices" for her. But that only decreased the chance to one in 26.
With all this analysis, Gill began a petition to reopen de Berk’s case. That happened in 2008. In April 2010, she was formally pronounced not guilty.
All in all, her experience remains a signal case of the misinterpretation and misuse of statistics. Mathematicians, but humans in general, have a remarkable ability to see patterns. But that facility can sometimes go horribly wrong: there was simply no pattern in de Berk’s work as a nurse that should have suggested she was murdering patients. But once a pattern was suspected, half-baked statistical analysis “proved" her guilt.
So yes, a cautionary tale. I was drawn to it after running across some totally different numbers. A few days ago, Assam’s chief minister, Himanta Biswas Sarma, told a Karnataka election rally that a Muslim man will marry four women, and that such a woman is told to have 10-12 children and becomes a baby-making machine (https://bit.ly/41tIerR).
What should we make of the pattern Sarma claims to see? Well, take these numbers: In India, 1.9% of Muslim men have more than one wife; so do 2.1% of Christians, and 1.3% of Hindus. In India, there are about 90 million Muslim men, 15 million Christian men, and 500 million Hindu men.
Now what do you make of the pattern Sarma claims to see?
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.
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