To all those wondering, what’s so amazing about this: The formula when graphed can visually reproduce itself!

Choose the following value for n:

Now plot the set (x, y – n) for x : [0, 106], y : [n, n +16] such that the inequality holds. The result is:

The effect results in the formula above being referred to as Tupper’s self-referential formula.

References:

1. Wikipedia page

At a children’s hospital in Netherlands in 2001, an unexpected infant death caused a review of past incidents. It was discovered that child nurse Lucia’s shift coincided with many of these incidents. Subsequently, these deaths till then thought unremarkable, were marked suspicious, and charges were pressed against Lucia for murder of her patients. The mainstay of the prosecution was their statistical argument: the probability of a nurse’s shift coinciding with so many incidents by mere chance was minute: 1 in 342 million! Lucia was sentenced to life imprisonment for 4 murders and 3 attempted murders.

A note of caution here: In two cases against Lucia, medical evidence was quoted as the basis of the guilty judgment. This evidence has since been contested by medical experts. It is not the point of this writing to convince you that Lucia was definitely innocent. I only point out the incongruity in the statistical argument, the miscalculation involved in obtaining 1 in 342 million, and the questionable relevance of this probability figure.

Based on preliminary analysis following the death of an infant during Lucia’s shift in September 2001, Lucia was suspected of being involved in as many as 30 incidents across hospitals she had worked in. Many of these could however not be linked to Lucia because of her absence during these. This set of incidents was no longer regarded as suspicious. This is the first and most primary of the statistical errors in this case. The list of suspicious incidents should have included all (and only) those deaths/near-deaths for which definite natural reasons could not be assigned. For instance, if there was a total of 50 incidents, of which 20 were unexplained, Lucia being present in (say) 10 of them, then the list should include all, and only, these 20 incidents; not only the 10 Lucia was present at. Lucia’s presence itself should not be criterion for classification. This is called confirmation bias: In the verification of a hypothesis, selecting relevant data again based on the same hypothesis. This obviously stacks the odds against Lucia!

This problem was further compounded during data collection. Initially, many of the ‘suspect’ incidents were classified as ‘natural’, including the last infant death. When inquiries were made so as to reclassify incidents as suspicious or natural, the people who were asked these questions very obviously knew they were being asked to ascertain Lucia’s involvement. Considering that she was already being reviled in the media, and the 342 million figure was so widely known, their responses have very little chance of being unbiased.

Selectiveness in assessing data has another major issue. A very relevant question that needs to be asked is: What was the trend in ‘suspicious’ deaths before nurse Lucia arrived at the scene? Lucia’s hospital unit, reported 6 unexplained deaths during her two years there. However, the same unit also reported 7 unexplained deaths for about the same length of time before she arrived at her hunting ground!

Another issue is the choice of statistical models for determining the probability value. In such real-life problems, various statistical methods can be applied depending on the extent of information available and the assumptions. Using different models, estimates of the same probability, computed as 1 in 342 million by the prosecution, have been calculated to be as high as 1 in 10 and 1 in 48 (after also making corrections for the other issues with this figure). I am not detailing here the more technical issues (like multiplication of p-values instead of Fischer combinations, and deliberations on which models – Bayesian/Epidemiological etc.) regarding the computation of the 1 in 342 million number, but you are welcome to see the references for more.

Not only is the figure of 1 in 342 million highly suspicious, the question arises: is this number relevant at all. And here again, is the prosecutor’s fallacy. Mark Buchanan states in his article in Nature:

“The court needs to weigh up two different explanations: murder or coincidence. The argument that the deaths were unlikely to have occurred by chance (whether 1 in 48 or 1 in 342 million) is not that meaningful on its own — for instance, the probability that ten murders would occur in the same hospital might be even more unlikely. What matters is the relative likelihood of the two explanations. However, the court was given an estimate for only the first scenario.”

It’s not the probability of an innocent nurse’s shifts coinciding with the incidents, but the probability of a nurse whose shift coincides with the incidents being innocent, which is important. (This I pointed out earlier too, in the first article in this thread.)

Following protests by statisticians across the world, a Dutch government committee was set up to deliberate on whether or not to reopen Lucia’s case. Other evidence was also called into question recently following emergence of new facts. The case was subsequently reopened in October 2008. It might be the case that the culprit is statisticide, and not Lucia.

References:

• Mark Buchanan’s article in Nature
• Oxford Journal publication detailing statistical issues in the case
• Wikipedia page on Lucia de Berk

Sally Cark’s first son died a few weeks after his birth in 1996. She gave birth to another child in 1997, who also died subsequently, at the age of eight weeks. The latter death aroused suspicion, and Sally was charged with the murder of her two kids. While there was no definite evidence of Sally’s having murdered her children, the prosecution’s case rested on the claim that two natural cot deaths in the same family are extremely unlikely. There was evidence of physical trauma in both cases. However, this could have been related to Sally’s efforts at resuscitation.

The prosecution’s expert witness, Sir Roy Meadow, former Professor of Paediatrics, University of Leeds, stated that “one sudden infant death in a family is a tragedy, two is suspicious and three is murder unless proven otherwise”. He further deduced that for a family like the Clarks (affluent, non-smoking), the probability of a single cot death was 1 in 8,543, so the probability of two cot deaths in the same family, was around “1 in 73 million” (1 / 8543 × 8543, obtained by the product rule for probability of independent events). This small likelihood of an innocent mother facing two cot deaths, seems to have caused the jury to have considered Sally the murderer beyond reasonable doubt. Sally was convicted and sentenced to life imprisonment. (I can discuss the media reviling the killer mother and hostility in prison, but that is not my point here).

In this case, its not merely an issue of transposed conditionals (discussed in my last post), but also sheer miscalculation of the probability figure to obtain ’1 in 73 million’. It was only in 2002, that the Royal Statistical Society expressed its concerns about Prof. Roy’s calculation.

The first lacunae pointed out, was that the assumption of two cot deaths in a family being independent events, was ill conceived. It could be the case that genetic or environmental factors are involved and the events are highly correlated. The second concern was related to the fallacy of transposed conditionals. The RSS stated that the ’1 in 73 million’ figure should not have been interpreted as the probability of Clark’s innocence. What was required of the jury was to weigh up the relative likelihood of two competing explanations for the deaths: double cot death is very rare, but double murder is likely to be rarer still, so the probability of Clark’s innocence was quite high in absence of any other available evidence! This is also referred to as prosecutor’s fallacy.

Another objection raised, was regarding the probability figure for a single cot death also not being 1 / 8543. This figure ignored the male gender of the Clark babies. Among males this probability is about 1 / 1300.

The prosecution’s statistical argument contains all the lacunae school students are guarded against in probability classes: wrong base probabilities, unfounded assumptions of independence, and misinterpretation of conditional probability. Further, misrepresenting the facts to an equally naive jury, compounded matters. It took two appeals, intervention by elite statisticians, and three years in prison, for Sally Clark to undo, to a limited extent, the damage statisticide caused her.

References:

1. Sally Clark: Home Page
2. Wikipedia page on Sally Clark
3. Plus Mathematics magazine article on the case
4. RSS public statement about the case

Prof. Paulos emphasizes the distinction between:

• probability of an innocent person having an array of evidence against him; and
• probability of a person with an array of evidence against him being innocent.

To the naive, this might look like mere word-play, but the distinction is as stark as between innocent and guilty. The former probability has often been abused by chest-thumping prosecution lawyers presenting bewildering probability numbers to jurors. The slick prosecution attorney will say something like – “the probability of someone innocent having A, B, C pieces of evidence against him is 1 in a million!” (also notice his mouth-left-wide-open look for added effect). And there goes half the jury – already decided on a conviction.

I will steal Prof. Paulos’ example to illustrate the vast difference this misrepresentation/misinterpretation makes:

Consider a murder trial in a town of 1 million people. It is agreed by the prosecution and defence that the person who committed the crime is a resident of the town and has a certain type of moustache. Now, it so happens that only 2 (X and Y) of the 1 million people have such a moustache. The prosecution presents its case against X claiming that the probability of an innocent having X’s moustache is 2 in a million! This is not a lie, but the jury interprets it as a strong indication of X’s guilt. The fact however, is that the same statement can be made for Y. It follows that the probability that X, given that the murderer has his kind of moustache, is innocent is not 2 in a million, but in fact 1 in 2 in absence of any other evidence!

The above ruse (called fallacy of the transposed conditional) has probably been pulled by lawyers in innumerable cases with substantial impact on jurors.

I found instances of statisticide of above nature and others, in several trials: the O.J. Simpson trial, the Sally Clark trial and the Nurse Lucia trial being very public examples. I will detail each of these cases in separate posts soon.