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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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