Reporter's evidence note · R v Clark [2000] EWCA Crim 54 · as of
The figure counts how often two infants in one family with three stated characteristics would both die of SIDS if the deaths were independent. It does not count how likely it is that two deaths which did occur were natural, and this record contains no measured comparison that could turn it into a probability of guilt.
Decision: publish no numeric probability of guilt or innocence.
when one is looking ex post at whether two deaths were natural or unnatural, the 1:73 million figure is no help(para 162).
The appeal against conviction on each count is therefore dismissed.(para 258). The two infants who died are not named in this record and this note does not name them.
approximately one in 73 million(para 122). Rounding adds 17,151 to the denominator, about 0.02 per cent.
does not take account of possible familial incidence of factors other than those included in Table 3.58(para 122).
a chance of 1 in 73 million live birthsand added that there are
about, say, 700,000 live births a year, concluding
it is saying by chance that happening will occur once every hundred years. The division is 73,000,000 ÷ 700,000 = 104.3 years. It divides a per-family figure by a count of all live births of every kind, so it mixes a per-birth denominator with a per-family one. A per-year restatement would need the annual number of qualifying families with two live births, which the record does not give.
ball parkestimate was, in the court's words, made
not unreasonably(para 139).
emphasises the possibility that there might be factors predisposing a family with one cot death to another(para 139).
is not an exercise the courts would perform(para 160).
tells us nothing whatsoever as to the guilt or innocence of the appellant(para 141).
Write F for a family with two live births and the three stated characteristics. Write r for the chance that the second infant dies of SIDS given that the first did, and h for the chance that both infants in such a family die by maternal homicide. Neither r nor h has a sourced measurement in this record.
(1)P(two SIDS | F) = P(first SIDS | F) × P(second SIDS | first SIDS, F) = (1 ÷ 8,543) × r
(2)The reported figure sets r = 1 ÷ 8,543, giving 1 in 72,982,849.
(3)Bounds: 0 ≤ r ≤ 1, so 0 ≤ P(two SIDS | F) ≤ 1 in 8,543, that is 117.1 per million qualifying two-birth families. A joint chance cannot exceed the first-child chance; the independence point, 0.0137 per million, sits inside this range but nothing in the record singles it out.
The figure is P(two SIDS | F). It is not P(two SIDS | two deaths, F), which equals P(two SIDS | F) ÷ P(two deaths | F), and the second term adds up every cause of two infant deaths. The appellant's argument at para 160 makes the related point that the figure has no significance without the likelihood of a competing possibility.
(4)O = P(two SIDS | F) ÷ P(two maternal homicides | F) = (r ÷ 8,543) ÷ h
With the numerator somewhere between zero and 1 in 8,543 and h unknown, O is zero when r is zero and grows without limit as h approaches zero. The record therefore places no bound on O. Even if the numerator were fixed at the reported figure, O would depend entirely on h, and no source gives h.
(5)Posterior odds (two SIDS to two maternal homicides, given E) = O × [P(E | two SIDS, F) ÷ P(E | two maternal homicides, F)]
E is the actual physical and medical evidence. The ratio's numerator is the likelihood under two SIDS deaths, matching the numerator of O; the reciprocal would give odds the other way round and must not be combined with O. The excerpts quantify no finding, so this ratio is unknown. An absence of an identified natural cause is part of E: its weight is the ratio of how likely that absence is under each account, which the record does not measure, and it is not proof of homicide.
(6)P(two SIDS | E, F, and one of only these two accounts) = posterior odds ÷ (1 + posterior odds)
Even that would be conditional on only two accounts being possible. They are not exhaustive, so a probability of guilt would need a prior and a likelihood for every account. None of the terms in (4), (5) or (6) has a value in this record.
The control below changes only r, an assumption. It shows what formula (1) returns for each value; it does not estimate r, and the homicide comparator does not change because nothing measures it. With scripts off, the table after the chart lists every value the control offers.
| Hypothetical r | Two SIDS deaths per qualifying two-birth family | Per million such families | Two maternal homicides comparator |
|---|---|---|---|
| 0 (lower bound) | zero | 0 | Unknown |
| 1 in 100,000 | 1 in 854,300,000 | 0.00117 | Unknown |
| 1 in 8,543 (independence) | 1 in 72,982,849 | 0.0137 | Unknown |
| 1 in 1,000 | 1 in 8,543,000 | 0.117 | Unknown |
| 1 in 100 | 1 in 854,300 | 1.17 | Unknown |
| 1 in 20 | 1 in 170,860 | 5.85 | Unknown |
| 1 in 10 | 1 in 85,430 | 11.7 | Unknown |
| 1 in 4 | 1 in 34,172 | 29.3 | Unknown |
| 1 in 2 | 1 in 17,086 | 58.5 | Unknown |
| 3 in 4 | about 1 in 11,391 | 87.8 | Unknown |
| 1 (upper bound) | 1 in 8,543 | 117 | Unknown |
The highlighted row is the reported figure's assumption. Every row is a hypothetical input to formula (1); the comparator column reads "Unknown" in every row because no value of r supplies it.
Missing-data conclusionFor every value of r, the two maternal homicides comparator is Unknown, so the prior odds cannot be computed and no probability of guilt follows. The record has no sourced point estimate for r; the first-child estimate is the only measured input.
witnessTestimony and pre-trial notice, as reported by the court
The notice tied the estimate to family circumstances: The likelihood of SIDS varies with social circumstances.
(para 118). At trial the answer described the squared figure as the risk of two infants dying of SIDS in that family by chance alone
(para 114).
appellantArgument, as quoted by the court
Relying on the fresh evidence of Dr Evett and Professor Dawid, the appellant argued that without reference to the likelihood of a competing possibility, the figure has no relevance or significance at all
(para 160).
expertAs reported by the court
The court reports that Professor Dawid goes on to question the appropriateness of squaring 1 in 8,543 in order to produce the figure of 1 in 73 million.
(para 139). The same paragraph reports that he emphasises the possibility that there might be factors predisposing a family with one cot death to another.
(para 139).
courtIts own conclusions
On the ball park description: We do not imagine that the Crown would demur from that suggestion.
(para 139). On the comparison: legally speaking the exercise is not realistic
(para 160). On the figure: It is merely a distraction.
(para 162). On the appeal: dismissed (para 258).
Read fairly, the court accepted the statistical criticism and still dismissed the appeal. The excerpts in this record do not include the court's treatment of the medical evidence, so this note does not characterise it. The court also recorded that the Crown had no statistical expert either
(para 120).
The call stays at "publish no number" unless every item below is supplied from a dated, citable source.
Each passage is quoted exactly from source SC1. The tag shows whose words they are: court, witness, expert notice, study text, fresh expert evidence, or the appellant's argument.
witnessWitness testimony at trial, as reported by the court. Speaker: A trial witness answering a question (not named in this excerpt).
Q Later on in the paper does it go on to work out the risk of two infants dying of SIDS in that family by chance alone?
A Yes, you have to multiply one in 8,543 times 1 in 8,543 and I think it gives that in the penultimate paragraph, its points out that it's approximately a chance of 1 in 73 million .... It gives a chance of 1 in 73 million live births and in England, Wales and Scotland there are about, say, 700,000 live births a year, so it is saying by chance that happening will occur once every hundred years.
expert noticeExpert's pre-trial notice, as quoted by the court. Speaker: The expert's notice (author not named in this excerpt).
The likelihood of SIDS varies with social circumstances. The most recent estimation of the incidence in England is that for a family in which the parents do not smoke, in which at least one has a waged income and in which the mother is over the age of 26 years, the risk is 1:8,543 live births.
expert noticeExpert's pre-trial notice, as quoted by the court. Speaker: The expert's notice (author not named in this excerpt).
Thus the chances of two infant deaths within such a family being SIDS is 1:73,000,000.
courtThe court's account of the trial. Speaker: Court of Appeal.
The trial therefore started without the defence having the assistance of a medical statistician, and in the event, no such witness was called. But the Crown had no statistical expert either.
study textCESDI study text, as quoted in the judgment. Speaker: CESDI study.
Since the factors will generally remain the same (with the possible exception of maternal age below 27 years) the risk of SIDS to a subsequent child in a family in which one infant has already died will range from one in 214 to one on 8,543. this does not take account of possible familial incidence of factors other than those included in Table 3.58.
study textCESDI study text, as quoted in the judgment. Speaker: CESDI study.
For a family with none of these three factors, the risk of two infants dying as SIDS by chance alone will thus be one in (8,543 x 8,543), ie approximately one in 73 million.
fresh evidenceFresh expert evidence, as reported by the court. Speaker: Professor Dawid, reported by the court.
Professor Dawid goes on to question the appropriateness of squaring 1 in 8,543 in order to produce the figure of 1 in 73 million.
fresh evidenceFresh expert evidence, as reported by the court. Speaker: Professor Dawid, reported by the court.
emphasises the possibility that there might be factors predisposing a family with one cot death to another.
fresh evidenceFresh expert evidence; the words 'not unreasonably' are the court's assessment. Speaker: Professor Dawid, with the court's assessment.
He suggests, not unreasonably, that the 1:73 million must be regarded as a "ball park" estimate.
courtThe court's own response. Speaker: Court of Appeal.
We do not imagine that the Crown would demur from that suggestion.
fresh evidenceFresh expert evidence, as reported by the court; 'obvious' is the court's word. Speaker: An expert giving fresh evidence, referred to as 'He' (not named within this paragraph's excerpt), with the court's assessment.
He also makes the obvious point that the evidential material in Table 3.58 tells us nothing whatsoever as to the guilt or innocence of the appellant.
appellantThe appellant's argument, as quoted by the court. Speaker: The appellant's case on appeal.
The fresh evidence of Dr Evett and Professor Dawid demonstrates that the prosecution invited the jury to adopt the figure of 73 million as having a significance in itself when in truth, without reference to the likelihood of a competing possibility, the figure has no relevance or significance at all
courtThe court's comment on the missing comparison. Speaker: Court of Appeal.
The competing possibility identified is a double infant murder by a mother. That may be capable of being expressed in terms of a statistical probability, but legally speaking the exercise is not realistic - see R -v- Denis Adams (No 2) [1998] 1 Cr App R 377, which shows that it is not an exercise the courts would perform.
courtThe court's stated conclusion on the statistic. Speaker: Court of Appeal.
Therefore we accept that when one is looking ex post at whether two deaths were natural or unnatural, the 1:73 million figure is no help. It is merely a distraction.
courtThe court's disposition of the appeal. Speaker: Court of Appeal.
The appeal against conviction on each count is therefore dismissed.
Every number on this page was parsed from the quoted passages and recomputed independently. Nothing published after the as-of date was used.