Lucy Letby: Victim of Flawed Statistics?

Exploring the Texas Sharpshooter Fallacy

Further discussion of the flawed use of statistics in the Courtroom is available in Twisted Logic: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024, and also in Probability, Choice, and Reason, by the same author and publisher,

The Texas Sharpshooter Fallacy and the Lucy Letby Case: A Statistical Illusion?

The Texas Sharpshooter Fallacy is a cognitive bias where patterns are imposed on random data after the fact, creating the illusion of meaningful correlation. In criminal cases, this fallacy can lead to wrongful convictions when evidence is selectively framed to confirm a pre-existing hypothesis while ignoring contradictory data. In the case of Lucy Letby, did this fallacy play a significant role in shaping the prosecution’s argument?

Breaking Down the Fallacy: The “Barn Wall” of Hospital Deaths

Imagine a barn wall riddled with bullet holes.

• A skilled sharpshooter carefully aims at a pre-drawn target and hits the bullseye. This represents a genuine pattern, a case where evidence is gathered before forming a conclusion.
• A Texas sharpshooter, on the other hand, fires randomly at the barn, then paints a target around the densest cluster of bullet holes, claiming accuracy. This is a false pattern, created by selectively highlighting data that supports a conclusion while ignoring data that doesn’t.

The key mistake in the Texas Sharpshooter Fallacy is that the pattern is imposed after the data is already collected, rather than discovered through an objective analysis of all relevant information.

How This Applies to the Lucy Letby Case

1. The “Barn Wall” = All Neonatal Unit Deaths

• The neonatal unit at the Countess of Chester Hospital experienced multiple infant deaths and collapses over a specific period.
• The prosecution focused only on the subset of deaths and collapses that occurred during Letby’s shifts, effectively painting a target only after identifying her as a suspect.
• This ignores other infant deaths and medical complications that occurred during the same period when Letby was not present, much like ignoring other bullet holes on the barn wall.

2. Painting the Target Around Letby

• The prosecution used a chart showing that Letby was present at all the deaths/collapses for which she was charged.
• However, at least six other deaths during the same period were excluded from this analysis because Letby was not present for them.
• This selective focus creates a misleading illusion:
  • If Letby had been present for those deaths, they likely would have been included in the charges.
  • Because she was absent, they were ignored, despite potentially having the same medical causes as the deaths attributed to her.

This is a classic case of defining a pattern after seeing the data, rather than objectively analysing all neonatal deaths to determine if there was truly an unusual pattern.

Why This Statistical Error Matters

The Texas Sharpshooter Fallacy distorts the perception of probability and causation. In Letby’s case, it led to several key statistical misunderstandings:

1. Random Clustering Happens Naturally

• In any high-risk medical environment, adverse events will cluster randomly without intentional wrongdoing.
• Letby worked many shifts, increasing the likelihood that she would be present during multiple tragedies by chance alone.
• The prosecution failed to show whether other nurses, working similar hours, might also have appeared in clusters if all deaths had been analysed.

2. Base Rate Neglect: Ignoring the Expected Frequency of Nurse Presence

• The prosecution claimed that Letby’s presence at so many incidents was statistically improbable.
• But how often were other nurses present for multiple collapses?
  • If most nurses worked 40% of shifts, but Letby worked 60%, she would naturally be present for more deaths.
  • Without comparing her shift pattern to other nurses, the statistical claim that her presence was “too unlikely to be coincidence” is unsubstantiated.

3. Confirmation Bias: Interpreting Evidence Through a Guilt-Focused Lens

• Once Letby was identified as a suspect, investigators re-examined medical cases only from shifts she worked, looking for signs of wrongdoing.
• This ignores cases with similar medical outcomes that occurred when she was not present.
• If the same unexplained symptoms or medical complications were found in cases where Letby wasn’t working, the argument that she deliberately caused harm would weaken significantly.

4. The Prosecutor’s Fallacy: Misinterpreting Probability

• The jury was told that the probability of Letby being present for all these deaths by chance was “1 in 3.5 million”.
• This misleading argument makes two major mistakes:
  1. It assumes each death is an independent random event, when clusters happen naturally due to factors like seasonal infections, staffing levels, and equipment failures.
  2. It ignores alternative explanations, including poor hospital conditions and misdiagnosed medical complications, which might have been responsible for many of the deaths.

Expert Criticism: The Fallacy in Action

Several statisticians and medical experts have questioned the statistical reasoning behind Letby’s conviction:

• Dr. Richard Gill (Former Chair of Mathematical Statistics, Leiden University): Argued that the prosecution’s statistical argument was a “classic Texas Sharpshooter” mistake, cherry-picking data and excluding deaths where Letby wasn’t present.
• Prof. Jane Hutton (Professor of Statistics, Warwick University): Emphasised that all neonatal deaths should be analysed, not just a subset supporting the prosecution’s narrative.
• Medical Experts: Pointed out that the hospital’s mortality rate remained high even after Letby was removed from duty, suggesting systemic failures rather than the actions of a single nurse.

The Danger of the Texas Sharpshooter Fallacy in Criminal Justice

The Letby case is a textbook example of why cherry-picked statistics can create false narratives in the courtroom.

• Humans instinctively seek patterns, even in random data. When jurors see a chart where Letby’s name is the only one with multiple deaths, they may assume intent, even if the pattern is artificially constructed.
• In ambiguous medical cases, statistical manipulation can override weak physical evidence and lead to wrongful convictions.
• By focusing on Letby as a “bad actor”, the hospital avoids scrutiny over systemic failures in neonatal care, including understaffing, medical errors, and resource shortages.

The Bigger Picture: Does This Prove Letby’s Innocence?

The Texas Sharpshooter Fallacy does not prove Letby is innocent, but it does cast significant doubt on the prosecution’s statistical reasoning. When combined with:

• Disputed medical evidence (e.g. air embolism diagnoses contradicted by experts).
• No direct witnesses to wrongdoing.
• A struggling hospital with a high infant mortality rate, even after Letby’s departure.

…it suggests that the “pattern” of Letby’s presence at deaths may have been artificially constructed rather than genuinely significant.

In Justice, as in Statistics, Correlation ≠ Causation

If the jury was swayed by a pattern that was painted after the fact, then Letby may have been convicted not on solid proof, but on a fallacy. This case serves as a cautionary tale: when statistics are weaponised in courtrooms, they must be scrutinised rigorously, because mistaking correlation for causation can cost an innocent person their life.