Probability of two cot deaths in one family

Question

In the UK, the probability of a cot death is about 1:8000.

In the case of Sally Clark, two of her children died, apparently of cot death. But she was prosecuted for murder. An expert at her trial multiplied 1:8000 by 1:8000 to give 1:64000000, which was said to be too small for the deaths to be due to cot death. (The figures are slightly more than 1:8000 for her socio-economic group.) Two cot deaths are hardly "independent" events.

This is intuitively wrong; I understand that the 1:8000 figure is chance, a random event. But even if the causes of cot death are unknown, it's still reasonable to say that there are causes, genetic or environmental; and that it can be argued that one cot death in a family can be associated with a higher risk in subsequent children.

What I can't find out is the mathematics of calculating the chance of two cot deaths in one family.

Help please!

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Thank all for your comments. This is an expanded version of my original question, as well as a response to what you have said.

I'm a retired consultant surgeon, so I understand the medical problems; further, our second daughter died from 'cot death', so my interest in the case of Sally Clark is professional and personal. I did do a course in statistics as part of an intercalated BSc during my medical training; but as this was so long ago, before even pocked calculators and personal computers, I have forgotten far more than I ever learned.

I was aware from press reports on the original trial, and was disturbed by what I read, but wasn't entirely sure why I should be unsettled. Of course, a press report is only a partial view of a trial; but still, Sir Roy Meadow's calculations seemed intuitively wrong. I followed the subsequent appeals, and her aquittal, and saw her death from acute alcoholic poisoning a couple of years later. I was confused by the probability of cot death, thinking it was a way of describing how often an event might occur due to "chance"; and if I was familiar with the idea that chance has no memory, I was also aware that, for example, lightening can and does strike twice. (So if the chance is 1:8000, why isn't it the same a second time, after all these are not two wholly independent events, though they aren't quite the same event twice?)

I'm familiar with the Wikipedia article.

I've recently read Math on Trial by Leila Schneps and Coralie Colmez. Its about the use and abuse of maths in court; messages include that advocates and judges have a very deficient understanding of statistics, even to the extent of ignoring them when they seem to be too hard. And also that experts in one field very clearly shouldn't give an opinion on a subject outside of it.

Their first chapter is about Sally Clark. They point out the error in multiplying two variables when these are not independent; but I found their reasoning very hard to follow, and they did not provide any estimate of what probability there might be for two cot deaths in one family.

Sally's first appeal failed, as the judges reckoned that even if Sir Roy's reasoning was false, it wouldn't impress the jury. The second appeal succeeded not because of falsity of Sir Roy's statistics, but because the second baby had evidence of significant bacterial infection. This finding had apparently not been noticed by any expert before; but whether this was because the didn't see it or the records given to them were incomplete isn't clear to me.

I hadn't seen Professor Dawid's report previously, thanks for the link. He notes that a smoker in the family, the mother's age, and unemployment all "result" in increased rates of cot death; but these are neither necessary nor sufficient factors, and they are associations rather than causes. (These details were known at the time of the first trial.)

Meanwhile, the "Back to Sleep" campaign started, following an almost chance observation in Hong Kong; babies there sleep on their backs, and cot death is almost unknown. Getting parents to put their infant on its back (rather than face down) is reported to halve the risk of cot death.

I apologise for the non-statistical reasoning above. But I'm drawn to the conclusion that cot death is closer to the same event happening twice, rather than two entirely independent events. And as the causes of cot death are still unclear (there is complex evidence about breathing patterns, and patterns of brain wave activity), it's impossible to know how strong the association between two of them is.

That is, the question of the probability of two cot deaths in one family cannot be calculated mathematically (because of the lack of knowledge of the strength of the factors involved); and that all that can be said at present is that it lies between 1:73,000,000 (Sir Roy's figure) and less than 1:8000.

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I have also discovered a paper on multiple cot deaths; it suggests a dependency figure of between 5 and 10 for a second event. It also confirms that the 1:8000 or so ratio is incorrect; a more accurate figure is 1:1300. I take this dependency figure to mean that the probability of a second cot death is between 5:1300 and 10:1300. The paper also argues that a second murder is very much less likely than a second cot death.

Again, thanks for your help.

The paper is here: http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf

Answer 1

A large amount has been written about the Sally Clark case and the Wikipedia article has several links.

For one approach to looking at which probabilities matter in such a calculation and why relative probabilities are more important than absolute probabilities, you could start with Professor Philip Dawid's expert evidence.

A better approach would be to try to look for empirical evidence of multiple child deaths in a family and the causes of this, rather than trying to calculate something from false assumptions about independence or trying to guess whether separate events of infanticide are more or less correlated within a family than separate events of cot deaths.

Answer 2

I've investigated further. There were multiple errors of logic and statistics.

The 1:8543 refers only to a subset of the whole population. There were roughly 400,000 births in the UK at this time; using simplified figures, 1:8000 corresponds to 50 cot deaths. But there were in excess of 200. This wasn't made clear at the trial; the error is comparing a subset with the whole. So, the actual chance is nearer to 1:2000 overall.

There were between 20 and 30 infants known to have been murdered in a year. Sir Roy certainly thought that there were more than this, but I'm wary of his maths.

Anyhow, 20 is only a tenth of 200, so you might say that SIDS is much more likely than murder. 20 in 400,000 is 1:20,000. If you then square this and 1:8000, it is apparent that SIDS is about 64,000,000 to 400,000,000, or about 1:6.3, so that SIDS is still more likely. Of course, this is a gross misuse of statistics.

None of these data about murders seem to have been put to the jury; I suspect that neither the barristers nor the judge really understood what was going on.

Again, thanks for your help.

Answer 3

If I understand your question correctly, the infant's $\textit{individual}$ characteristics may be seen as a condition, i.e. one of the infants has some medical condition $A$ such that the risk of death $D$ $\textit{given}$ this condition is much higher: $P(D|A) = p >>\frac{1}{8000}$.