Updated: Sep 10
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As another 'angel of death' case gets underway in Manchester, the spotlight is turned once again on use of statistics by prosecutors and investigation objectivity
There have been a number of cases over the years in which doubts emerged about the guilt of caregivers accused - and in some cases convicted - of murdering those in their care. Some doubts led to accusations being dropped, some to convictions being overturned, and some still linger, with those convicted remaining in prison. In many if not most cases, the doubts related to whether the prosecution had used statistics correctly at trial and/or whether the original investigation had been biased against the accused.
This week the trial of nurse Lucy Letby got underway in Manchester. Letby is accused of murdering seven babies in her care and of attempting to murder ten others. I am not suggesting she is innocent. It is possible that there is incontrovertible evidence against her as was the case with Harold Shipman. However, what I do know is if there isn't, this is yet another case in which there is scope for bad statistics and a biased investigation to be presented to the jury. And therefore for another miscarriage of justice.
There is a recognisable pattern in situations that lead to a caregiver being accused, rightly or wrongly, of malevolence. First, someone recognises what they think is a statistical anomaly in data - for example, a high number of deaths on a particular ward over a certain period compared with what would be considered normal. Then, one of two things happens: either direct evidence of malevolence is found - as was the case with Shipman and his forged wills etc - or it isn't. If no direct evidence of malevolence is found, the 'cluster' can either be attributed to chance, to a non-malevolent cause, or to malevolence.
The last of these generally involves:
an erroneous belief that the cluster in question could not possibly have been due to chance,
there being no 'non-malevolent' causes identified, investigated even,
other flawed reasoning/conclusions.
Looking at each of these in turn:
1. Disbelief in possibility of chance
Let's say on a hypothetical ward the expected number of deaths per year, based on the national average for the type of ward in question, is 20, and that this is consistent with what the ward in question has experienced - most years the number of deaths has been between 15 and 25 and very occasionally a little outside this range, never less than 10 and never more than 30 . Then, one year, the ward experiences 40 deaths. If the number of deaths per year is assumed to follow a Poisson Distribution - one in which events occur at a constant mean rate i.e. 20 per year, and are independent of each other, another example of which being the number of calls received in a particular hour of the day at a call centre - as is reasonable, the chance of such an occurrence is 0.0028%, one in 36,000. Many would think this is so unlikely that the cluster must be attributable to something other than chance.
The mistake that is made here is to think only of the ward in question. If there are 1,000 such wards across the country, then in a 20 year period - 20,000 ward years - there is a very high probability, well over 50/50, that one of them will experience 40 deaths in one year. Nothing suspicious whatsoever. Just luck. Or, rather, bad luck.
Think about the case of a rollover lottery that gets won by Ms XYZ. The chance of Ms XYZ specifically winning is minute, one in tens of millions perhaps. However, the chance of somebody winning is 100%. In caregiver cases, it is natural for many of those involved - the hospital administrator, the police, the prosecution, juries, the public etc - to focus on the 0.0028% number not the 50/50. In which case, unsurprisingly, the possibility of chance is dismissed. There must be a cause, they say.
2. No 'non-malevolent' causes having been identified
Let's say that, instead of 40 deaths in one year on the above mentioned hypothetical ward, there were 100. Assuming a Poisson Distribution, the probability of this occurring on a particular ward in a particular year is one in 3,571,854,227,384,530,000,000,000,000,000,000,000. Not even a trillion wards and a trillion years would give the cluster even the slightest chance of occurring! Thus, in this instance, it is reasonable to assume that it must have had an active cause.
In such cases where chance can be dismissed, all possible causes should be carefully considered. In practice however this does not always happen - sometimes, people jump to conclusions and assume there is a murderer out there. Two such cases are noted in Green et al. 2022:
A cluster of deaths in a neo-natal ward in Toronto was initially associated with a nurse, who was suspected of malevolent activity. Only later was it discovered that new artificial latex products in feeding tubes and bottles could have been responsible. An apparent increase in death on a neonatal ward in England raised similar suspicions until a medical statistician identified the date at which the death rate rose, and a neonatologist recognized it as the date when the supplier of milk formula was changed. As these examples show, an increase in deaths may be caused by factors that are not immediately apparent, even to those involved. Such factors may require considerable expertise to discover and could be missed entirely in some instances.
In the cases above, fortunately, so-called confounding causes of the clusters were identified, and miscarriages of justice were avoided. But this does not not always happen. For example, a hospital administrator tasked with investigating confounding variables such as a change in hospital practices, product or treatment might naturally prefer the cluster to be attributed to misconduct rather than to an administrative mistake. His perhaps. Indeed, Green et al. 2022 recommends that such investigations be carried out by independent parties.
3. Other flawed reasoning/conclusions
In no particular order:
Suspicion may be directed onto a nurse who is not liked or is deemed to be a bit odd;
Better nurses will tend to notice and signal a death earlier than a worse nurse, so deaths are more likely to be registered in their shifts not later.
Better nurses will tend to clock in earlier and leave later, so deaths are more likely to occur on their watch given the longer time they spend on the ward.
A disproportionate number of deaths occur or are noticed/registered in the morning. Thus suspicion is more likely to fall on nurses who do more morning shifts than others do.
Better nurses will tend to be entrusted with harder tasks, ones perhaps where the risk and thus incidence of death is higher.
A fall in deaths following removal of suspected nurse from the ward may be due to bad publicity and people avoiding that hospital rather than a murderer being no longer present.
During investigations, causes of death get reexamined by pathologists and there may be a tendency or pressure to recategorise deaths as unnatural, driven perhaps by a desire to atone for perceived past error. Incidence of potassium or of elevated insulin levels may be deemed unnatural - i.e. evidence of poisoning - when there are in fact completely natural explanations.
If a particular nurse is already under suspicion, there may be a tendency to recategorise as unnatural only deaths that occurred when the suspected was on duty. As noted in Green et al. 2022, "Regardless of how it occurs, this kind of bias would undermine the fairness of the investigation by causing an increase in the count of “suspicious” deaths associated with the nurse. The higher count would arise from the very suspicions that the investigation is supposed to evaluate – an example of circular reasoning".
It may later be determined that a nurse under suspicion was not in fact on duty when one of the deaths previously deemed unnatural and attributed to them occurred. Rather than this casting doubt on the case against the suspected nurse as it should, and it perhaps introducing the possibility of another perpetrator, there may be a tendency simply to re-re-categorise the death as natural and to press ahead.
All the above have occurred in real cases. Investigations/judgements get conducted/handed down by humans, and humans are fundamentally flawed. These flaws can relate to a poor grasp of probability and statistics, for example conflating the probability of an animal having four legs if it is a dog with the probability of it being a dog if it has four legs, the equivalent of the issues set out in 1. above. Or they can relate to innate bias, for example confirmation bias or the fundamental attribution error. Humans are also influenced by the tabloid media, so prefer lurid explanations to mundane ones.
Caregivers who have either been wrongly accused or convicted, or where there is for good scientific/statistical reason for suspicion of such, include Lucia de Berk, Daniela Poggiali, Jane Bolding, Sally Clark, Susan Nelles, Ben Geen, and Collin Norris*. If it turns out there is no direct evidence against Lucy Letby, let's hope her name doesn't join the list.
* Details of all these cases can be found online. And there I'm sure are others, perhaps many others, that I have not come across and thus did not mention.
The views expressed in this communication are those of Peter Elston at the time of writing and are subject to change without notice. They do not constitute investment advice and whilst all reasonable efforts have been used to ensure the accuracy of the information contained in this communication, the reliability, completeness or accuracy of the content cannot be guaranteed. This communication provides information for professional use only and should not be relied upon by retail investors as the sole basis for investment.
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