Maternal death coincidence?

Two women admitted to the maternity unit of the Royal Hampshire hospital on December 21st 2007 died of Streptococcus A infections within two days, one on December 23rd and one a day later. In a BBC News article the hospital said "their deaths appear to be coincidental".
So what is the chance of such an event happening by chance alone?

Before we can answer this question, some general points apply:

1. When working out how 'surprising' an event is, it does not make sense to try to work out the chance of the specific event in all its precise details, as every exact combination of events is uniquely complex and unpredictable.
2. We can work out the likelihood of these events occuring by chance by first considering the narrow and very specific situation: e.g. two deaths from Strep A among mothers admitted on the same day to the same maternity unit (as was the situation in this case). We can then look at the chance of a rather wider range of events occuring: deaths from all causes, or deaths from the same cause within a day of each other. This way we can compare the 'chance' of the broader class of events occurring with that of the specific example.
3. We should pretend it is January 1st 2007, and ask ourselves - what is the chance of such an event happening somewhere in the UK this year?
4. We can only work out the chance of these two deaths happening by assuming they were unrelated. This is not the same as the chance that there was no common cause, given the fact that the two deaths occurred. There is a subtle but very important difference – see the end for explanation

Relevant statistics

The report Saving Mothers’ Lives, published by the Confidential Enquiry into Maternal and Child Health in December 2007, provides the basic historical data we need:

• There are around 650,000 births per year in the UK.
• There are around 100 maternal deaths per year from all causes.
• Between 2003 and 2005 (i.e. over three years), there were 8 maternal deaths from Streptococcus A infection.

This means that in the nearly 2 million births that occurred over these 3 years, there were only 8 deaths from Strep A, i.e. only 1 in 250,000 mothers died from Strep A infection.
This in turn means that the chance that either of these particular two mothers would die from Strep A infection is 1 in 250,000, and so the chance that both would die of the disease is 1 in 250000 x 250000, which is 1 in 63 billion.

But we need to consider the number of opportunities for such an event to occur to any two mothers admitted to hospital on the same day, given that the Royal Hampshire handles around 3000 births a year, and is only one of many units in the UK.

Two deaths from Strep A

This is the narrow, specific event that occurred: was it a coincidence? Was it 'chance'?

We first consider the event: two mothers admitted on the same day to the same maternity unit die from Strep A infection.

• It helps to think of the really surprising event - the second death. So first let's assume there has been a death from Strep A on a maternity unit the size of the Royal Hampshire. There are on average around 8 admissions a day, and so the chance that another of those admitted on the same day would die is around 1/250000 for each of the remaining 7 mothers, which is 7/250000 or about 1 in 35000.
• But as there were 8 deaths in 3 years, we can say that there are on average 8/3 Strep A deaths overall in the UK per year, so the chance that in any year there is a death followed by a further death = 8/3 * 1/35000, or around 1 in 13000.
• So we would expect, on average, that once in every 13000 years a UK maternity unit will have two mothers who were admitted on the same day die from Strep A.

So this is a very remarkable event, assuming it happened by chance alone. If we allow the second death to occur in a mother admitted within 1 day of the first death, this means that instead of 7 opportunities there are around 8+7+8=23 (adding together all the women admitted over the three day period - excluding the woman who has died already).

If we divide our figure for 2 deaths from Strep A in women admitted on the same day (1 in 13000 years) by 23/7 we get a figure close to 4000.

So we can expect, on average, that once every 4000 years a UK maternity unit will have two mothers die from Strep A who were admitted within a day of each other.

This would still be a very surprising event.

Two deaths from any cause

This is a broader class of events: what is the chance of this occurring?

Let's consider the event that two mothers admitted on the same day die from any cause (not necessarily the same one).

• There are around 100 maternal deaths a year, and so a 100/650000 = 1 in 6500 chance of death for an average mother
• After a death from any cause, the chance that a second death will occur in a woman admitted on the same day is around 7/6500 or 1 in 900
• But there are 100 deaths overall in the UK per year, so the chance that any of them is followed by a further death = 100 * 1/900 = 1 in 9
• So we would expect, on average, that once every 9 years a UK maternity unit will have two mothers who were admitted on the same day die from some cause or other

So the fact that two women admitted on the same day both die would not be such a remarkable coincidence, even though the maternal death rate is so low. It is the fact that in this case they both died of the same infection that is relevant.

Why does this calculation not express the chance that the deaths were a coincidence?

The explanation of the subtle difference mentioned previously goes something like this.

We have assessed the chance that such an event would occur, GIVEN that it was only a 'coincidence'.

People (quite reasonably) will want to know the chance that 'coincidence' was working, GIVEN the events that occurred.

These are two totally different questions, so have totally different answers (but if you have trouble immediately grasping this, you will be joining almost all lawyers and judges!).

To get at the second quantity ('Was coincidence at work?'), we can go back to the specific details of the case (Two women, two cases of Strep A, one day.) and no longer need to embed it in a broader class of events (Two women, two deaths by any cause, same day).

But the difficulty with getting this second quantity is that we have to look at so many other pre-existing factors before we can work out the likelihood of the actual event.
In this case, we have to look at factors such as:
(a) what is the chance that this particular event would occur, given it was not just coincidence, and
(b) what is the chance that there is an infection problem in the Royal Hampshire?

These quantities are much more difficult to work out than the probability of such an event occurring as a pure 'coincidence'.

To make a legal analogy, before calculating a probability of 'guilt' we need to consider both the probability of the evidence being true, and also the background level of suspicion against the accused, even before the direct evidence is taken into account.

So it is not straightforward to come up with a statement that 'there is a 1 in XXX chance that the deaths were unrelated' - that just can't be assessed without a lot more assumptions.

But you can say that: EITHER a very rare event has happened, OR there is some common factor.

And that's how almost all scientific investigations, including all drug trials for example, have to be reported.