In 1997, Christopher, the eleven-week-old child of a young lawyer named Sally Clark, died in his sleep: an apparent case of Sudden Infant Death Syndrome (SIDS, or crib death). Such events are terribly sad, but, even with the best of care, they happen. Only in this case, one year later, Sally’s second child, Harry, also died, aged just eight weeks. Sally was arrested, and accused of killing the children. She was convicted of murdering them, and in 1999 was given a life sentence. Now, this is not the place to go into the weakness of the case, the paucity of forensic evidence, or the disagreements about the causes of death. Rather, I want to show how a simple mistaken assumption led to incorrect probabilities. In this case the mistaken evidence came from Sir Roy Meadow, a pediatrician. Despite not being an expert statistician or probabilist, he felt able to make a statement about probabilities in his role as expert witness in Ms. Clark’s trial. He asserted that the probability of two SIDS deaths in a family like Sally Clark’s was 1 in 73 million. A probability as small as this suggests we might apply Borel’s law: we shouldn’t expect to see such an improbable event. If we don’t expect to see it, but we do anyway, then there must be some other explanation—such as, in the present case, that the mother had killed the children. Unfortunately, however, Meadow’s 1 in 73 million probability is based on a crucial assumption: that the deaths are independent; that one such death in a family does not make it more or less likely that there will be another. On average, the chance of a given child dying of SIDS is about 1 in 1,300. Meadow instead (correctly) used the much smaller figure of 1 in 8,543, arrived at by taking account of the fact that Sally Clark was nonsmoking, affluent, and young, all factors which reduce the probability of this kind of infant death. He failed to take account of the fact that both of the Clark children were male, a factor which increases the probability of a SIDS death. Then he made the critical assumption. He assumed that the probability of having a second such death in a family was independent of whether there had already been one. You’ll recall from chapter 3 that if two events are independent, you can find the probability that both of them will occur by multiplying their separate probabilities together. And that’s just what Meadow did. If you assume independence, then the probability of getting two such deaths in a family is 1/8,543 × 1/8,543; about 1 in 73 million, and this is the figure he presented to the court, describing it as the sort of event you would expect to see once every hundred years. Now, you’ll recall how slight changes to what we assume about the shape of a distribution can change probabilities by large amounts. In the present case, perhaps we shouldn’t assume that SIDS deaths within the same family are independent. And, in fact, that assumption does seem unjustified: data show that if one SIDS death has occurred, then a subsequent child is about ten times more likely to die of SIDS. Meadow’s estimated probability of two deaths was wrong. To arrive at a valid conclusion, we would have to compare the probability that the two children had been murdered with the probability that they had both died from SIDS. This would require us carrying out similar calculations for child homicide statistics. I won’t go through the details here, but Professor Ray Hill of the University of Salford in the UK calculated that “single [SIDS] deaths outweigh homicides by about 17 to 1, double [SIDS] deaths outweigh double homicides by about 9 to 1, and triple [SIDS] deaths outweigh triple homicides by about 2 to 1.”12 There is a factor-of-ten difference between Meadow’s estimate and the estimate based on recognizing that SIDS events in the same family are not independent, and that difference shifts the probability from favoring homicide to favoring SIDS deaths. Professor Hill added, “[O]ne wonders whether the Clark jury would have convicted if, instead of being given the ‘once in a hundred years figure,’ they had been told that second [SIDS] deaths occur around four or five times a year and indeed happen rather more frequently than second infant murders in the same family.” Later evidence also showed that at the time of his death, the second child, Harry, had a blood infection known to cause sudden infant death. Following widespread criticism of the misuse and indeed misunderstanding of statistical evidence, Sally Clark’s conviction was overturned, and she was released in 2003.
This is one of the more popular case studies that I have seen in multiple books. This is usually used to illustrate how an incorrect understanding of statistics can adversely impact the lives of others. In this case, Sally Clark was wrongfully imprisoned for a few years and never recovered from this ordeal, eventually developing psychiatric problems and dying of alcohol poisoning even though she was acquitted.
Just one of the interesting anecdotes, and something that I thought should be shared in every conversation about statistics!
