Statistics and the Law The case of the negligent nurse Willem R. van Zwet, University of Leiden Bahadur lecture Chicago 2005

k=8 incidents (i.e. cases of unexplained death or re-animation); N=27 nurses; S=1029 eight-hour shifts; s=142 shifts of Lucia; F: the fact that Lucia is present during all 8 incidents; LI: hypothesis Lucia innocent; LG: hypothesis Lucia guilty.

Elffers (for prosecution). Suppose LI (Lucia innocent). Urn model: S = 1029 balls (shifts) ; s = 142 black balls (Lucia's shifts); S – s = 887 white balls. Draw at random k = 8 balls (shifts with incidents). F (Lucia present during incidents): all balls drawn are black. P(F  LI) = {s! (S-k)!} /{(s-k)! S!} = 1.1 × (+ correction & other facts) (× 27 × 0.001) = 3 × This is not due to chance! The rest is up to the court!

Judge combines for each incident  Statements of witnesses  Toxicological evidence  Statistical argument and concludes that 8 (attempted) murders have been committed by Lucia and hands down a life sentence. Judge did not separate the questions: (1) were the incidents (attempted) murders ? (2) if so did Lucia commit them ?

Appellate court (Gerechtshof Den Haag) Meester, Van Lambalgen (for defence) "Elffers' analysis has little relevance“  Attention was caught because something very improbable happened in a particular ward of a particular hospital. But there are many wards in many hospitals. So correcting by multiplying by N=27 is nonsense. Multiply by # nurses in the Hague, the Netherlands, the world, …?? Lottery example.  Urn model is too special. E.g. different probability of incident during night and day, or for certain nurses? Or Lucia more serious cases?

Appellate court completely muddled. Forget about statistics. She's guilty. Life sentence confirmed. Rule 1 of statistical consulting: Discuss with client what the problem is, what the roles of the client and the statistician are and what the statistician can and cannot do. Judge's question: Can it be due to chance that Lucia was present during all of these incidents? Wrong question (see Meester).

Preliminary question: Were these incidents (attempted) murders ?? This is up to the judge to decide on the basis of Statements by witnesses Forensic (toxicological) evidence but statistical analysis of the data F cannot possibly help answer this. Judge should decide whether incidents are (attempted) murders on the basis of other evidence. If not: go home. If so: start thinking. Judge should also decide who are possible suspects, in this case N nurses.

M= event that 8 (attempted) murders took place. Statistician may assume that the event M took place and the murders were committed by one of the N=27 nurses. Now Elffers' analysis is convincing !! It is the classical statistical test to solve this problem. This hospital is not just one of many but the one where the murders occurred and Lucia is one of 27 suspects Differences between night and day, between nurses, etc refer to natural deaths and these are murders. So if judge decides these are 8 (attempted) murders, then a statistical test tells us Lucia is guilty.

Interview of De Vos in leading newspaper. Aggressive Bayes! Two possible hypotheses LI and LG with a priori probabilities P(LI) and P(LG) F = facts (8 incidents with Lucia present) Bayes rule: P(LI  F) = P(LI,F) : P(F) = P(LI).P(F  LI) : {P(LI).P(F  LI) + P(LG).P(F  LG)}.

P(LG)= P(random nurse starts murdering) = (1/4) ; So P(LI) = 1- (1/4) Elffers computes P(F  M,LI)=1.1x and De Vos interprets this as P(F  LI) = 1.1 x and takes P(F  LG) = 1/2 ??!! Bayes: P(LI  F)  2.2 x x {P(LI) / P(LG)} = By some further changes you can raise this even more, so you can't convict! Big newspaper scoop !

My conditional Bayesian analysis. We know M took place so we do our calculations conditionally on M. Elffers: P(F  M,LI) = 1.1x10 -7 P(F  M,LG) = ½ ?!! Prior probabilities: P(LG  M) = N -1 = 1/27, P(LI  M) = (N-1)/N = 26/27.

Conditional Bayes: P(LI  F,M)  2.2 x10 -7 { P(LI  M) / P(LG  M) } = (N-1) x 2.2 x (cf. frequentist result: P = N x 1.1 x ). De Vos: 2.2 x x {P(LI) / P(LG)} vZ: 2.2 x x {P(LI  M) / P(LG  M)} Once murders have been established P(LI) and P(LG) are irrelevant (cf. O.J. Simpson murder trial). De Vos’ analysis is another attempt to achieve the impossible..

Conclusions: Elffers was a bit too vague when saying "the rest is up to you". The "client" has a right to more guidance. Murder needs to be proved first. In this context Meester objections were valid, but his claim that Elffers' analysis was irrelevant was not. The guilty verdict can be defended if and only if the non-statistical evidence proves that the incidents were attempted murders. Elffers analysis then proves the guilt of Lucia. Conditional Bayes gives the same result. De Vos’ analysis is just another attempt to achieve the impossible. The case goes to the Supreme Court.

PS Only 5 (attempted) murders ???