appendix
APPENDIX. NOTES TO CHAPTER 20. STATISTICS.
These quotes are from the following report (available on The Internet)
The Royal Statistical Society. Healthcare serial killer or coincidence? Statistical issues in investigation of suspected medical misconduct by the RSS Statistics and the Law Section September 2022
Seemingly improbable patterns of events can often arise without criminal behaviour
ITEM 1. Seemingly unlikely coincidences occur regularly – in other words rare events do happen. A California couple won two separate lotteries in a single day. The probability of winning both lotteries by chance was approximately one in 26 trillion.
ITEM 2. An unusual number of deaths will be observed somewhere among the patients of one of the millions of medical professionals in the world; It is unlikely that a 1-in-10-million coincidence will happen to any particular medical professional, but given the very large number of medical professionals in the world, it is likely, perhaps even inevitable, that such a coincidence will affect the patients of some medical professional somewhere in the world. (My highlighting.)
(My comment:- Similarly, if you toss a coin twenty times, and it come up heads every time, this would be vastly improbable. However, if millions of people tossed a coin twenty times, sooner or later ONE or more of these people would get twenty heads in a row. The “lucky” person would not necessarily have a biased coin.)
The number of deaths observed while a particular nurse is on duty is influenced by chance, in the form of natural sampling variation: comparing one period to another, the numbers of deaths will differ purely due to coincidence.
ITEM 3. Confounding Variables.
The increase in death rate cannot, in itself, prove that the nurse in question was engaging in misconduct, because other factors, known as confounding variables, might offer alternative explanations.
The Lucia de Berk case (in which she was initially convicted):- The rapid increase in deaths on her ward coincided with (the confounding variable of) a decision to transfer babies with genetic birth defects to the ward.
A cluster of deaths in a neo-natal ward in Toronto was initially associated with a nurse. Later it was discovered (as a confounding variable) that new artificial latex products in feeding tubes might have been responsible. Similarly, an apparent increase in deaths on a neonatal ward in England raised suspicions until a statistician identified the date at which the death rate rose, and this date was recognized as the date when the supplier of milk formula was changed. (Again a confounding variable!)
In the United States, Rule 702 of the Federal Rules of Evidence:- If the underlying investigation failed to consider relevant causal factors that might provide an alternative explanation for a cluster of deaths, it might well be appropriate for a judge to find that the resulting evidence does not meet the requirements of Rule 702 and should be excluded.
ITEM 4. PROFESSOR O’QUIGLEY’S QUOTES.
The following quotes are from (The) Amazing Academics (U-Tube video series). Serial Killers and Statistical Blunders - Why Lucy Letby might be wrongly imprisoned: John O'Quigley
https://www.youtube.com/watch?v=AbN6j-IPQAU
The following are quotes from Professor John O’Quigley, Professor of Statistics at University College, London, as seen in the above U-Tube video:-
QUOTE:- (In the Lucy Letby Case) “The statistical evidence is completely wrong.”
QUOTE:- “There is no evidence of a crime”.
QUOTE:- “None of it stands up - - - Her guilt was assumed by some very strange statistical arguments”.
QUOTE:- “She is one hundred percent innocent!”
QUOTE:- “The statistical analysis of the Letby case does not hold water”.
QUOTE:- The Letby trial involved “a massive problem of selection bias”.
Doctor Brearey had a “drawer of doom” in which he compiled events specifically involving Lucy Letby.
ITEM 5. QUOTE:- When Doctor Dewi Evans came to the division of the 61 cases into 25 suspicious and 36 non-suspicious, he claimed for some time that he did not know which cases involved Lucy Letby, although recently he has admitted that he did know, because the names of the nurses involved in each case were noted in the medical case-notes. Doctor Evans “knew exactly which cases involved Lucy Letby” (before he decided which incidents were suspicious, and which were not).
The question is – what constitutes a suspicious event? QUOTE:- “What constitutes a suspicious event is that nurse (ie:- Lucy Letby) being present - - - a completely circular argument”.
Professor O’Quigley demonstrates this by the following example:- One baby died of asphyxia, when Lucy Letby was NOT on the unit, BUT THIS WAS CATEGORISED AS “NOT SUSPICIOUS”. (My highlighting.) (My comment:- Asphyxia suggests smothering, and SHOULD have been classified as “suspicious”. The fact that it was classified as NOT suspicious clearly demonstrates that the classification of the incidents was heavily biased so as to incriminate Lucy Letby.)
QUOTE:- “There is no real evidence of a statistically significant spike (in deaths on the unit) when you take into account the poor prognostic factors”.
Let me include this following quote as a parenthesis to Professor O’Quigley’s comments. This quote confirms the above comment.
This quote is from Private Eye Special Report. The Lessons of The Lucy Letby Case, by Doctor Phil Hammond. MD. (available online.), Part 14:- David Rose had access to detailed police notes from meetings with chief prosecution witness Doctor Evans. Multiple “suspicious events” were identified by Evans, but ten that did not involve Letby were disregarded.
Now back to Professor O’Quigley’s comments.
ITEM 6. Professor O’Quigley then puts forward the following explanation of the statistical blunder upon which the case was built, which he describes as being “outcome bias”. Considering only this specific unit, the odds against this spike in deaths occurring by chance are very long odds. However, if you consider ALL the (say 200) (neonatal) units in the country, the spike is not especially improbable. Now if you factor in very vulnerable babies entering a unit that is not equipped to deal with such a level of vulnerability, then such a spike in deaths is not improbable at all.
He then goes on to say:- If you look for the WORST performance of these 200 units, then that WORST performer would be EXPECTED to have an apparently anomalous spike in deaths. In fact, they actually WERE the WORST performer for that year. If you compare the spike in deaths with the average number of deaths of the six WORST performing units, on that basis, this apparent spike in deaths is not statistically significant.
QUOTE:- “A highly selected chart”. (ie:- relating deaths to Lucy Letby’s presence.)
Professor O’Quigley then states that – even assuming “suspicious” events as defined by the prosecution, there are other nurses present equally as often as Lucy Letby.
Here is another Professor O’Quigley quote:- Professor John O’Quigley, Professor of Statistics stated that the statistical chart tabulating the Lucy Letby (alleged) murders is a “crock”, and bogus, and worthless. He says that the spike, which given the conditions in the hospital, is NOT a spike (My capitals). He says that the apparent spike was only to be expected, given the poor conditions in the ward. To confirm this, go to the following web page:-
Episode 5: Bad Science
https://www.youtube.com/watch?v=0c7HPNjIN1I
Here are some more Professor O’Quigley quotes from a U-Tube video.
61 events, only 25 deemed to be “suspicious”.
QUOTE:- “There was one case of asphyxiation that was originally classified as “suspicious”, well of course, sounds like smothering. It would be hard to imagine a case that would be apparently LESS “suspicious” than that - - - - Initially it was deemed as being “suspicious”, but then of course, it was discovered that Nurse Letby was NOT present, and not being present, well then a reassessment was made, and it was deemed actually NOT “suspicious” anymore. - - - - The “suspicious” circumstances were not pointing to Lucy Letby, Lucy Letby was pointing to the “suspicious” circumstances”. END QUOTE
ITEM 7. QUOTE:- “Nurse Letby was working at least one third (of the time) more than average”. END QUOTE (My comment:- Effectively, Lucy Letby was singled out simply because she worked longer hours.)
To verify these above quotes, go to the following U-Tube video
Lucy Letby: Statistical Smoke and Mirrors with John O'Quigley
https://www.youtube.com/watch?v=k8jkl255PWI
That is the end of Professor O’Quigley quotes.
SOME MORE QUOTES FROM STATISTICIANS AND OTHER EXPERTS.
ITEM 8. The following quotes are from the book Baby Killer 2. Is Lucy Letby Innocent, by Stu Armstrong, ISBN 9798301688835, printed by Amazon. (I strongly recommend this book!)
Page 62:- Dr Rachel Pennington, paediatrician and neonatologist is quoted:- “Dr Pennington - - - noted that many medical records lacked - - - long term monitoring data or detailed post-mortem findings - - - - She argued that the analysis of these records appeared to be influenced by hindsight bias - - - interpreting normal medical events as SUSPICIOUS BECAUSE THEY WERE LINKED TO LETBY” (My capitals and highlighting,) (Dr Pennington also suggested alternative explanations for sudden collapses or deaths – eg:- staff shortages, outdated equipment etc.)
ITEM 9. Pages 40 to 45:- Forensic pathologist, Dr Eleanor Hughes states:- “The prosecution presented these deaths as unnatural without addressing the - - - vulnerability of these infants”. She cited several cases with severe infections that were liable to cause collapse. “benign explanations were plausible - - - - for collapses”.
ITEM 10. Another important point is that Lucy Letby was a more experienced nurse, and was therefore allotted the more serious cases, ie:- the babies who were more likely to die. To confirm this fact, here is a quote from the book Baby Killer 2. Is Lucy Letby Innocent, by Stu Armstrong, ISBN 9798301688835, printed by Amazon. (I strongly recommend this book.) Pages 80 to 83:- Statistician Dr Michael Bennett pointed out that, as a senior nurse, Letby was assigned to the most vulnerable infants.
ITEM 11. A further important point is that Lucy Letby worked longer hours than many of the other nurses. In that case, she would be more likely to be present at any particular death or collapse. To confirm this fact, here is a quote from the book Baby Killer 2. Is Lucy Letby Innocent, by Stu Armstrong, ISBN 9798301688835, printed by Amazon. (I strongly recommend this book.) Pages 34 to 35:- Legal analyst (barrister) Mark Reynolds mentioned the high number of Letby’s shifts compared to her colleagues naturally increased her exposure to critical incidents.
ITEM 12. Considering this unit alone, the “spike” in deaths might seem like an improbable statistical anomaly. However, if you consider all 200 similar units in the country, the death rates would form a frequency distribution in the shape of the “normal distribution” or “bell shaped curve”, and you would expect at least SOME of the units to have similar “spikes” in the death rate, purely by chance. This point is made in the book Baby Killer 2. Is Lucy Letby Innocent, by Stu Armstrong, ISBN 9798301688835, printed by Amazon – as follows:- Pages 34 to 35:- Legal analyst (barrister) Mark Reynolds argued that “the statistical evidence - - - lacked sufficient context, such as the frequency of deaths and collapses in comparable neonatal units”.
ITEM 13. On a U-Tube video, it is stated that there was an apparent “spike” in deaths on the ward. When Lucy Letby left the ward, the deaths immediately stopped. This appeared to indicate her guilt. However, the ward was downgraded, so as to receive less severe cases AT EXACTLY THE SAME TIME as Lucy Letby was transferred. This explains the sudden drop in infant deaths. This did not come out at the trial. This being the case, the coincidence of a sudden drop in infant deaths does NOT indicate her guilt at all! To confirm this, go to the following web page:-
Lucy Letby: A Full Day At Thirlwall
https://www.youtube.com/watch?v=jbu-Rmk-ceU
Here is another quote from a U-Tube video:-
Lucy Letby: Trial's statistical evidence was a ‘scientific fake’
Feb 6, 2025
https://www.youtube.com/watch?v=GAOjeSUZeq8
“Statistical evidence helped convict Letby… and the main statistical constructs, a number of statisticians have said, were just a statistical abomination.” Statistician Peter Elston calls the statistical evidence given during Letby’s trial “a scientific fake.”
Here is another quote from a U-Tube video:-
Dr Richard Gill and According to Carl discuss similarities between Lucy Letby and Ben Geen cases.
https://www.youtube.com/watch?v=n7cHtiNtgOA
Statistician Doctor Richard Gill calls the Letby case “a farcical disaster”.
Here is another quote from a U-Tube video:-
“I’m Pretty Sure They Got It Wrong” Academic Casts Doubt On Baby Killer Lucy Letby’s Conviction.
https://www.youtube.com/watch?v=gIWKK8ZlqfM
“All of the evidence suggested that Lucy Letby was on her own, and it now turns out actually that was not the case”. Gill states that the unit was taking cases which were much too serious for the facilities which it had. Gill states that the investigation “broke all the rules of a sensible forensic investigation – it’s utterly biased”. Richard Gill is Emeritus Professor of Mathematical Statistics, and consultant in forensic science.
ITEM 14. Here are further quotes from another U-Tube video:-
The Untold 1 in 2 Trillion Anomaly in the Lucy Letby Stats
https://www.youtube.com/watch?v=98vdeBipbC8
Of the “suspicious” events, 8 involved single babies, and 9 involved twins or triplets. There was one set of triplets, and 9 babies who were twins. (My comment:- Twins and triplets are more vulnerable than single babies. Also, if one twin or triplet dies or collapses, their siblings are more likely to follow suit, simply because they have similar genetic make-up.)
ITEM 15. Here are further quotes from another U-Tube video:-
Lucy Letby: Medical Experts Speak Out
https://www.youtube.com/watch?v=U4s08JzfSKk
In the case of baby C, the x-ray used to indicate deliberate harm to the baby was taken on the day before he died. Lucy Letby was not on the ward before this x-ray was taken, and therefore could not be held responsible for the death. She was, however, on the ward on the day of the infant’s death, and got blamed for it.
Here are further quotes from another U-Tube video:-
ITEM 16. Lucy Letby's conviction was 'unsafe' due to 'misleading data'
https://www.youtube.com/watch?v=7j19jiU95pk
Statistician Jane Hutton of The University of Warwick states that “there were multiple lines in the chart presented by the prosecution, involving double counting”. Twins and triplets are not marked in the chart (If one dies, the other is more likely to die) No information as to when the medical rounds were done. Also no clear definition of an “unexplained event”. (This means that Dr Evans can call events unexplained once he knows that Letby was in the ward at the time. If he had not known her whereabouts, he might not have called it an unexplained event.) The ward had very serious failings relating to the medical doctors and the lack of advanced nurse practitioners. The door swipe data was incorrect. The police thought nurses were leaving the unit when in fact they were returning. This means that Letby was NOT alone on some occasions when the police thought that she WAS alone with the babies. In other words, occasions when the police thought that she might have been alone with an opportunity to harm a baby, in fact she was not alone with the baby, and therefore did NOT have the opportunity to harm the baby. Question:- Can two people enter the unit at the same time? Ward round of consultants 2 or 3 times a week. They should have been 2 or 3 times a day!
ITEM 17. Here are further quotes from another U-Tube video:-
There was an apparent “spike” in deaths on the ward. When Lucy Letby left the ward, the deaths immediately stopped. This appeared to indicate her guilt. However, the ward was downgraded, so as to receive less severe cases AT EXACTLY THE SAME TIME as Lucy Letby was transferred. This explains the sudden drop in infant deaths. This did not come out at the trial. This being the case, the coincidence of sudden drop in infant deaths does NOT indicate her guilt at all!
To confirm the above, go to the following U-Tube video:-
Defence didn't tell Jury this - But Why?
https://www.youtube.com/watch?v=jbu-Rmk-ceU
ITEM 18. Here is another quote from the internet:-
There were deaths on the ward when she was NOT on the ward., but the jury did not hear about these deaths.
To confirm the above, go to the following U-Tube video:-
Defence didn't tell Jury this - But Why?
https://www.youtube.com/watch?v=GyBr06bKjpY
Here are some quotes from Private Eye Special Report. The Lessons of The Lucy Letby Case, by Doctor Phil Hammond. MD. (available online.)
Private Eye Report, Part 1:- “A survey found two thirds of the country’s neonatal units did not have enough medical and nursing staff”.
ITEM 19. Private Eye Report, Part 1:- “The excess deaths and harms did indeed return to expected levels when Letby was removed from the unit, but this also coincided with the unit being downgraded - - - - (to) deal with babies who did not need intensive care”.
ITEM 20. Private Eye report, Part 6:- “The jury didn’t hear about all the other deaths that occurred on the failing unit”.
ITEM 21. Private Eye report, Part 8:- The Mail on Sunday reports a new audit of baby deaths at the hospital which shows that many rapid deteriorations of babies occurred when Lucy Letby was NOT on duty!
ITEM 22. Private Eye Report, Part 6:- “The jury - - - didn’t know that the swipe card data was wrong, and that Letby might not have been alone with babies when she was alleged to be”.
ITEM 23. Private Eye Report, Part 7:- Jane Hutton, a professor of statistics at Warwick University was engaged by the Cheshire Police to do a statistical analysis of the Letby allegations. She told them that she needed full information regarding all deaths at Chester, and staff rotas, and details of deaths at other units. The prosecutor then instructed the Cheshire Police “not to pursue this avenue”. Subsequently, Professor Hutton has turned into “a very vocal critic of the way statistics were used in the trial”. (My comment:- Obviously the Crown Prosecution Service did not want the jury to realise how weak the statistical evidence in the case really was.)
ITEM 24. Private Eye Report, Part 7:- the report mentions “failure to redact staff names - - - - from the clinical records before handing them to Dewi Evans” (My comment:- He SHOULD have seen these clinical records “blind”, ie:- NOT knowing which babies had been under the care of Lucy Letby. The fact that he DID know which babies were in her care allowed him to classify deaths and collapses as “suspicious” or NOT “suspicious” entirely depending on Lucy Letby’s presence or absence.)
The following quotes are from the article in The New Yorker (issue for May 20th, 2024) by Rachel Aviv, entitled – A Reporter at Large – Conviction.
ITEM 25. Page 40:- The Royal College of Paediatricians and Child Health noted that the spike in deaths in the unit in 2015 coincided with a spike in still-births on the maternity ward. (My comment:- This suggests infection in the hospital that was transferred from the maternity ward to the neonatal unit.)
Page 45. William C. Thompson, Professor of Criminology at The University of California stated that medical murder cases are particularly prone to errors in statistical reasoning. Daniel Kahneman, winner of the Nobel Prize in economics states that people do not have good intuitions when it comes to the basic principles of statistics. Burkhard Schafer, a Law professor at The University of Edinburgh stated that the prosecution (in The Lucy Letby trial) “seems to have ignored statistics”.
ITEM 26. Page 45. For one baby the diagram showed Letby working a night shift, but she was working day shifts at the time.
The following quotes are from the book Lucy is Innocent, by Paul Bamford, SECOND EDITION, ISBN number 9798326484130 (I strongly recommend this book.)
ITEM 27. Page 27 to 28:- States that Lucy Letby worked more overtime than many, working nights, working the whole year round, with predilection for the most vulnerable babies.
ITEM 28. Page 28:- 18 deaths in the unit that year, Letby only charged with seven of them. There were 60 incidents that year (ie:- deaths or collapses).
ITEM 29. Page 67:- In 6 of 7 deaths, the coroner found no evidence of foul play.
ITEM 30. Page 69:- The hospital started admitting very vulnerable babies which COINCIDED with Lucy Letby starting as a neonatal, nurse.
ITEM 31. Page 72:- Professor Kendrick (an accredited authority on neonatal pathophysiology – to see his credentials see pages 25, and 42, and 356 of the book) calculates the correlations of deaths with Lucy Letby’s presence, factoring in the following factors:- (A). She worked extra shifts. (B). She was unlucky encountering triplets who are extra vulnerable. His conclusion is that deaths on Lucy Letby’s shifts “could all have been down to chance”.
ITEM 32. Page 82:- Baby N. Frequent apnoea episodes were ascribed to Lucy Letby’s attacks. However, these episodes continued after the baby went to another hospital.
ITEM 33. Page 138:- Many times, when incidents occur, Lucy Letby was the only one with qualifications to look after a child of high risk.
ITEM 34. Page 138:- Doctor Evans stated that Baby L was the 60th incident that he was examining. In that case, there were 60 incidents, whittled down to 25 incidents.
ITEM 35. Pages 150 to 156:- The author points out that Lucy Letby did a lot of night shifts; and the lack of cover by medics at night made these shifts more “risky” for her (ie:- more at “risk” of being present at a death or collapse).
ITEM 36. Page 156:- The deaths jumped from 2.7 per annum to 18 per annum. Only 7 deaths could be attributed to Lucy Letby. The other 11 deaths MUST have been due to sub-optimal care. In that case, the 7 deaths attributed to Lucy Letby might also have been due to sub-optimal care.
ITEM 37. Page 158:- Statistician Professor Richard Gill states that the police counted incidents (ie:- deaths and collapses) per nurse, but not “hours worked per nurse”. Then he makes an important statement. “We now know that the incidents were called “suspicious” because Lucy was there”.
ITEM 38. Page 321:- Professor Kendrick prepares a table which shows that 14 out of 40 collapses were on night shifts. Also, for 13 of these collapses, Letby was NOT alone on the unit.
ITEM 39. Page 341:- Professor Kendrick states that a nearby hospital suffered a similar spike in deaths.
ITEM 40. Page 342:- There were three spikes in collapses and deaths on the unit. June 2015 and January 2016 and June 2016. Lucy Letby was not associated with the babies in the January spike.
Page 343:- Professor Kendrick states that the share of deaths that fell on Lucy’s shifts were about average for her work roster.
ITEM 41. Page 427:- There were some alleged attacks on babies by Lucy Letby when she was, in fact, on another unit at the time.
To substantiate the above comment, here is a quote from a U-Tube video:-
Was there ever a crime?
Episode 4: Star Witness
https://www.youtube.com/watch?v=24dj9O-K33o
Baby C. Doctor Evans stated that Letby injected air into his stomach and killed him. However, it turns out that she was not on the ward at the critical time. This only came out AFTER her conviction.
ITEM 42. THE LUCIA DE BERK CASE.
There is another case very similar to The Lucy Letby case. This is the Lucia de Berk case. There are some interesting and illuminating parallels between the two cases. The two cases involved exactly the same egregious error of statistical reasoning, and this error was clearly and definitively demonstrated in the de Berk case, so that she was set free, after six years in prison. Lucia de Berk was accused of killing various patients. Just as in The Lucy Letby case, the prosecution classified deaths on the ward as “suspicious” or NOT “suspicious” entirely depending on Lucia’s presence.
In order to verify this, I will now provide some quotes from the book Math on Trial – How Numbers Get Used and Abused in The Courtroom, by mathematicians Leila Schneps and Coralie Colmez, published by Basic Books, 2013, pages 121 to 145:- “The - - - deaths at which Lucia had been present were all RECLASSIFIED AS UNNATURAL; THEY HAD ALL BEEN DECLARED NATURAL WHEN THEY OCCURRED.” (My capitals and highlighting.) - - - - “THEY HAD ALL BEEN DECLARED NATURAL DEATHS UNTIL IT HAD BEEN NOTICED THAT THEY HAPPENED ON LUCIA’S WATCH. - - - There were two deaths on the list - - - at which Lucia had NOT been present (as subsequently discovered) - - - (but) no one tried to calculate the difference they (these absences) made to the damning probabilities - - - - as murders, there was no medical evidence to justify that claim (Then a person with some training in statistics gave odds against chance occurrence for de Berk’s presence at these deaths as 1 chance in 342 million). - - - If Lucia was not on shift, deaths were considered natural, but if she had been present, they were murders – the status of each death seemed to hinge on whether or not Lucia had been present. - - - - - deaths were termed suspicious only in hindsight”. (My capitals and highlighting.)
Finally, Tom Derksen, a professor at the Faculty of Philosophy of the Radboud University in Nijmegen investigated the various deaths, with the help of various professional statisticians. He analysed the incidents, and DEMONSTRATED CONCLUSIVELY that the “suspicious” deaths had been “cherry picked” specifically to coincide with Lucia’s presence on the ward (either deliberately, or by unconscious “confirmation bias”). Lucia was released, and completely exonerated.
To see comments from statistician Professor Richard Gill, in which he states that the “suspicious” incidents were “cherry picked” specifically so as to incriminate Lucia, and that incidents that merited being called “suspicious” were deemed NOT “suspicious” if Lucia was NOT present, go to the following U-Tube video
The statistical side of the case of Lucia de Berk - comparison with Lucy Letby
https://www.youtube.com/watch?v=bY1yKwXiIrE
ITEM 43. STATISTICS IS A DIFFICULT SUBJECT.
Statistics and probability theory is beyond the capability of anyone but a trained statistician, and even they have difficulty! The adjectives that I would apply to the study of statistics and probability theory are as follows:- Abstruse, mysterious, ambiguous, arcane, confusing, cryptic, enigmatic, esoteric, impenetrable, opaque, recondite, unfathomable, counter-intuitive, deceptive, treacherous, and “slippery”. Now I will provide a few quotes from highly qualified sources showing that statistics is a difficult subject, and that statistical “blunders” are only to be expected unless a qualified statistician is employed.
Here are some quotes from the book Statistical and Data Handling Skills in Biology by Roland Ennos, published by Pearson, 2012. On page 3 the author asks:- “Why is statistical logic so strange?”. On the same page, he states “The logic of hypothesis testing is rather counterintuitive.” (An understatement, in my opinion!)
Here is a quote from the book Math on Trial – How Numbers Get Used and Abused in The Courtroom, by mathematicians Leila Schneps and Coralie Colmez, published by Basic Books, page 61:- “Probability is a delicate subject, because it can often run contrary to elementary intuition.” (Again, an understatement, in my opinion!)
Here is a quote from The Private Eye Special Report – The Lessons of The Lucy Letby Case, by Doctor Phil Hammond, MD. (available on the internet.), Part 13:- “Statistician Professor David Spiegelhalter told the (Thirwell) enquiry (that) humans are not very good at judging data - - - (they) try to find patterns that may not actually exist”.
The following quotes are from the article in The New Yorker (issue for May 20th, 2024) by Rachel Aviv, entitled – A Reporter at Large – Conviction.
Page 45. William C. Thompson, Professor of Criminology at The University of California stated that medical murder cases are particularly prone to errors in statistical reasoning. Daniel Kahneman, winner of the Nobel Prize in economics states that people do not have good intuitions when it comes to the basic principles of statistics. Burkhard Schafer, a Law professor at The University of Edinburgh stated that the prosecution (in The Lucy Letby trial) “seems to have ignored statistics”.
This quote comes from a U-Tube video:-
Jane Hutton, a statistician at The University of Warwick (who specialises in MEDICAL STATISTICS) stated that almost all barristers, judges, and solicitors that she has spoken to say that they are not confident with, or are frightened of, or not competent in statistics. To confirm this, go to the following u-tube video:-
Lucy Letby: Statisticians have ‘serious concerns’ over data used in conviction
https://www.youtube.com/watch?v=IwELT-O0org
I myself have done various courses in statistics and probability theory, and I am the author of the book Binomial Probability Theory: A Comprehensive Guide, by Roger Elliott (available on Amazon). In that case, I am well aware of the possible pitfalls.
Let me tell you another story. I wrote a “paper” on statistical analysis, and sent it to a Professor of Statistics for a (paid) referee’s report. He sent back his report, together with his calculations. His calculations were incorrect. I notified him of this error. He wrote back “Sorry! I forgot that you have to multiply by two!” This was an elementary calculation, and a PROFESSOR OF STATISTICS got it wrong! Are you now beginning to see how “treacherous” and “slippery” and “mysterious” the subject of statistics and probability theory actually is? Are you now beginning to see that the jury in the Lucy Letby trial had absolutely ZERO chance of appreciating and evaluating the gigantic and egregious flaws in the complex statistical arguments put forward by the prosecution?
Let me now provide a further instance showing that even a trained scientist can be totally out of their depth when it comes to statistics and probability theory.
Let’s look at the case of Sally Clarke. She had three children. Two of them died of SIDS. (Sudden infant death syndrome.) She was accused that she had killed these two children by violently shaking them. The autopsy showed no actual evidence of violent shaking. Nevertheless, a supposed “expert” witness (Professor Roy Meadows, whose name deserves to live forever in infamy!) gave evidence in the court. He insisted that the statistical odds against BOTH babies dying from SIDS were 1 chance in 73 million. Now what baffles me is that this ridiculous evidence was accepted by the jury. Let me explain why this evidence was based on an insane fallacy of statistical reasoning. Suppose you choose a person at random from some absolutely randomly chosen part of the world. What is the probability that that person would have BLUE EYES? Let’s say that the odds would be 1 chance in 20. Now take another case. A married couple produce two children. One of them has BLUE EYES. Now what are the odds that the OTHER child will ALSO have BLUE EYES? It is NOT 1 chance in 20. Why not? Because the two children have the same parents, and therefore somewhat similar genetic makeup. The other child will almost certainly have BLUE EYES also. Now suppose that one of the children dies of SIDS. In that case, how likely is it that the OTHER child will ALSO die of SIDS? The obvious answer is PRETTY LIKELY! Why? Because both children, being from the same parents, will tend to have very similar genetic makeup. There are many illnesses that have a genetic origin. SIDS may well be one such illness. In that case, these odds of 1 chance in 73 million is completely incorrect. Sally Clarke spent years in prison before the antediluvian justice system realized that an error of statistical reasoning (of “schoolboy howler” proportions!) had occurred, and released her. If a (so called) “scientific “expert” could be so totally deficient in statistical reasoning skills, then what chance has the jury in the Lucy Letby trial got of understanding complex statistical evidence – especially when the data has been (deliberately?) “reverse engineered”, and some of the crucial data deliberately (?) hidden by the prosecution?
Let me now consider a further case. The Post Office introduced a new software system. Shortly after this, some of the postmasters of sub-post offices were accused of embezzling funds, and were forced to repay to the Post Office large sums of money, which they absolutely denied having misappropriated. There were soon well over 100 of these supposedly miscreant postmasters. Eventually the postmasters were all exonerated. The point of this story is that, after the first TEN miscreant postmasters, it should have been obvious that the software was at fault. A simple application of statistics and probability theory should have made this clear. The chances of ONE postmaster, taken at random, going “rogue” are (let’s say) 1 chance in 20. The chances of TWO postmasters going “rogue” are 1 chance in (20 x 20) = 1 chance in 400. For TEN postmasters to go “rogue”, the odds are 1 chance in (20 x 20 x 20 x 20 x 20 x 20 x 20 x 20 x 20 x 20) = 1 chance in many millions. This uses a statistical method known as The Law of Compound Probability. Actually, it is not quite as simple as that. You have to factor in the total number of postmasters, and then use binomial probability. Nevertheless, a simple statistical analysis would have made it clear that the software MUST be at fault. Unfortunately, no one carried out this statistical analysis, and now The Post Office is having to pay out millions of pounds in compensation. This story shows how totally incompetent humans are at statistics and probability theory. Our brains are just not built for these kind of calculations!