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Lecture 8 Hypothesis testing: SIDS Hypothesis testing 1 Testing alternative hypotheses • Suppose we want to compare how a given piece of evidence (e) bears on the probabilities of H1 and H2. p ( H 1) p (e / H 1) p ( H 1 / e) p (e) p ( H 2) p (e / H 2) p ( H 2 / e) p (e) p ( H 1) p (e / H 1) p ( H 1 / e) p (e) p ( H 1) p (e / H 1) p ( H 2 / e) p ( H 2) p (e / H 2) p ( H 2) p (e / H 2) p (e) • The last formula is very useful and worth examining. Hypothesis testing 2 The ratio of probabilities • Let us look at the ratio of posterior probabilities of H1 and H1: p( H 1 / e) p( H 1) p(e / H 1) p( H 2 / e) p( H 2) p(e / H 2) • The posterior probability of H1 is higher than the posterior probability of H2 if and only if the product of prior probability and likelihood is higher for H1 than for H2. • The formula gives the ratio of probabilities of the two theories. If these two theories are the only possibilities (one of them must be true), then we can immediately obtain the probabilities themselves. • For example, if p(H1│e) is four times higher than p(H2│e), then p(H1│e) must be 0.8, and p(H2│e) must be 0.2 (under the assumption that either H1 or H2 is true). Hypothesis testing 3 The Sally Clark case • November 1996: Christopher Clark (11 weeks old) dies in the presence of his mother. • December 1997: Harry Clark (8 weeks old) dies, and again only the mother was present. • 1999: Sally Clark convicted of double murder. • 2001: First appeal against the sentence (unsuccessful). • 2003: Second appeal (successful). • An expert witness for the prosecution relied on a probabilistic argument that created an uproar among the statisticians. • Intervened: the Royal Statistical Society, the President of the RSS, two professors of statistics (Oxford and UCL), the President of the Mathematical Society, the Governor of the Bank of England... Hypothesis testing 4 SIDS and Meadow’s law • Sudden infant death syndrome (SIDS) is “the sudden death of an infant under 1 year of age, which remains unexplained after a thorough case investigation, including performance of a complete autopsy, examination of the death scene, and review of the clinical history.” • SIDS is an infant death, due to unknown natural causes. • SIDS happens very rarely, so the repetition of SIDS in the same family must happen even more rarely. • Meadow’s law: “One case of SIDS in a family is a tragedy, two cases is suspicious, and three cases is a murder until proven otherwise.” (Goldfinger’s rule!) • Applied to the Sally Clark case: the chances of one SIDS: 1 in 8,500. The chances of two SIDS: 1 in 73 million. Hypothesis testing 5 Bayes’s theorem (the odds form) p(2S / E) p(2S) p(E / 2S) (1) p(2M / E) p(2M) p(E / 2M) (2) p(2S) p(S) p(S2 / S) (3) p(2M) p(M) p(M2 / M) p(2S/E) p(S) p(S2/S) p(E/2S) (4) p(2M/E) p(M) p(M2/M) p(E/2M) The ratio of posterior probabilities of 2S and 2M depends on: 1. The ratio of prior probabilities of S and M. 2. The ratio of repetition probabilities of S and M. 3. The ratio of likelihoods of 2S and 2M. Hypothesis testing 6 Prior probabilities of a single S and a single M • SIDS happens more frequently than infant murder, and so p(S)/p(M) is significantly greater than 1. • But the difference is sometimes exaggerated. • An unspecified proportion of officially declared SIDS are not really SIDS but murders. • True, some officially declared murders are also not really murders but SIDS, but for two reasons the mistakes are far more frequent in the former direction. • First, the crucial witness usually declares the case to be SIDS and denies the murder hypothesis. • Second, the case is classified as murder only if it is proved beyond reasonable doubt, whereas the SIDS classification is based largely on ignorance. Hypothesis testing 7 To square or not to square, that is the question • Meadow claimed that p (S & S2) = p (S) x p (S). • In general, however, p (S & S2) = p (S) x p (S2│S). • So, Meadow’s claim entails that p (S2│S) = p (S), which can be called the independence hypothesis (IH). • There are two possibilities: 1. Meadow was not aware that his claim entails IH, and he committed an elementary probability mistake. 2. Meadow was aware that his claim entails IH, but he did not see it as a problem because he believed that IH is true. • RSS did not consider option 2 at all, but immediately embraced 1, the fallacy scenario. • Two reasons: (a) Meadow gave no justification for IH, and (b) RSS thought that there are strong a priori reasons against IH. Hypothesis testing 8 The probability of a second SIDS • “This approach *the squaring of the single SIDS probability] is, in general, statistically invalid. It would only be valid if SIDS cases arose independently within families, an assumption that would need to be justified empirically. Not only was no such empirical justification provided in the case, but there are very strong a priori reasons for supposing that the assumption will be false. There may well be unknown genetic or environmental factors that predispose families to SIDS, so that a second case within the family becomes much more likely.” (RSS 2001). • …or perhaps less likely? • Isn’t this an empirical issue, not to be decided by “strong a priori reasons,”, i.e. speculation? Hypothesis testing 9 Should p(M) be squared too? • Dawid’s “equivalence argument”: if p(S) is squared, then the same thing could be done with p(M) with equal legitimacy. • The final result: p(2M) >> p(2S). • The equivalence argument is wrong. • Even if squaring of p(S) were dubious, the same procedure with p(M) could be, and would be, much more dubious. • There is a strong reason to believe that the probability of a second infanticide in the family would be substantially higher than the probability of the first infanticide — if there is no knowledge that the first case was infanticide. • In “Beyond Reasonable Doubt,” Helen Joyce works with the assumption that p(M2│M) = 0.1. Hypothesis testing 10 Figure 1a: Prior probabilities of S and M (SIDS independence) 1.00 0.80 0.60 M Probability 2M S 0.40 2S 0.20 0.00 0% 5% 10% 15% 20% Proportion of misdiagnosed SIDS Hypothesis testing 11 Figure 1b: Prior probabilities of S and M (SIDS dependence) 1.00 0.80 M Probability 0.60 2M S 0.40 2S 0.20 0.00 0% 5% 10% 15% 20% Proportion of misdiagnosed SIDS Hypothesis testing 12 Likelihoods of double SIDS and double murder • What are p(E│2S) and p(E│2M)? • What is E in the Sally Clark case? • Is E merely the fact that both children died? • But then, p(E│2S) = p(E│2M)= 1. • Was Clark really convicted just on the basis of prior probabilities, as Helen Joyce suggests? (“The lightning does not strike twice.”) • The judge’s explicit instruction to the jury: “I should I think, members of the jury, just sound a note of caution about the statistics. However compelling you may find those statistics to be, we do not convict people in these courts on statistics. It would be a terrible day if that were so.” Hypothesis testing 13 Why is this evidence “worrying”? • The judge in the first appeal: “Young, immobile infants do not sustain injury without the carer having a credible history as to how the injury was caused.” • “We and others have gone through the movements of resuscitation on cadavers and have found that it is extremely difficult to fracture ribs in an infant by pressing on the chest or by any of the usual methods of artificial respiration. Fractures of the ribs, however, can be relatively easily produced by abnormal grasping of the child’s thorax. The presence of fractures in any site in a child younger than 1 year should be considered as caused by abuse unless proven otherwise.” (John Emery) Hypothesis testing 14 Why is this evidence “worrying”? • The judge in the first appeal: “Young, immobile infants do not sustain injury without the carer having a credible history as to how the injury was caused.” • “We and others have gone through the movements of resuscitation on cadavers and have found that it is extremely difficult to fracture ribs in an infant by pressing on the chest or by any of the usual methods of artificial respiration. Fractures of the ribs, however, can be relatively easily produced by abnormal grasping of the child’s thorax. The presence of fractures in any site in a child younger than 1 year should be considered as caused by abuse unless proven otherwise.” (John Emery) p(2S/E) p(S) p(S2/S) p(E/2S) 5.6 0.0069 0.2 0.008 p(2M/E) p(M) p(M2/M) p(E/2M) Hypothesis testing 15 Figure 2A Double SIDS or double murder: posterior probabilities (I) 1.00 2M 0.80 Probability 0.60 0.40 0.20 2 SIDS 0.00 0% 5% 10% 15% 20% Proportion of misdiagnosed SIDS Hypothesis testing 16 Figure 2B Double SIDS or double murder: posterior probabilities (D) 1.00 0.80 2M Probability 0.60 0.40 0.20 2 SIDS 0.00 0% 5% 10% 15% 20% Proportion of misdiagnosed SIDS Hypothesis testing 17

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posted: | 10/25/2011 |

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