# Probability of a cancer cluster: 1 in a million

By Murray Bourne, 14 Aug 2007

In a recent cancer scare at Australia’s national broadcaster, 13 women developed breast cancer while working in the ABC’s (Australian Broadcasting Corporation) Brisbane studios.

In *612 studios are moving* (link no longer available), Professor Bruce Armstrong from the Sydney Cancer Centre at the University of Sydney says:

"What we’ve observed is about a 6.25-fold increase in breast cancer in women who are employed here at the ABC in Brisbane, compared with the general Queensland population of women," he explained.

"The probability the that increase is due simply to chance is about one in a million so we are looking at something that is almost certainly a real increase in risk.

A recent update, "14th cancer linked to ABC studio" [no longer available], describes the plight of the latest victim and adds:

However, no cause has been found, despite an extensive investigation.

We live in a world where the amount of radiation increases daily. I work almost directly under a wireless access point (for wireless broadband). The manufacturer will always claim that the equipment is safe. However, I imagine this was the same assurance given to the ABC women. There was clearly something amiss in the ABC’s Brisbane studios.

We never know what harm radiation is doing to us - until it is too late.

Risk assessment is a very important field of applied mathematics and statistics. I imagine any woman that worked in that Brisbane studio will find great difficulty in obtaining health insurance.

See the 5 Comments below.

15 Aug 2007 at 11:46 am [Comment permalink]

Hi Alan and thanks for your extended discussion on this topic.

Yes, I was puzzled where his "1 in a million" figure came from.

In an interview before he conducted the study (no longer available), Armstrong says:

By "calculus" I take it he means "calculation".

So does his figure mean that this cluster occurred 1 million times more than usual considering cases of similar-age women over a 10 year period?

15 Aug 2007 at 8:41 am [Comment permalink]

[...] this post at squareCircleZ, Professor Bruce Armstrong from the Sydney Cancer Centre at the University of [...]

15 Aug 2007 at 8:55 am [Comment permalink]

The statement by Bruce Armstrong is almost certainly not quite correct since the probability that something is due simply to chance is not computable and probably not even meaningful and is certainly not at all the same as the probability of its happening in a randomly chosen situation from a well defined population of cases (which is meaningful and often computable and is used by statisticians to define the "confidence level" or "statistical significance" of an experimental result.

This may seem like a mere quibble but it is is actually relevant to the story about cancer clusters since, in a world with several million observed groups of a hundred or so people, if the chance of a cluster happening given only random factors is one in a million then we may expect to see several such clusters occurring just by chance.

That's not to say that members of a cluster group shouldn't be worried - just that it's much more likely than Anderson suggests that they actually aren't at any higher risk than anyone else.

15 Sep 2007 at 9:43 pm [Comment permalink]

I agree that Armstrong's more extended quote is still not very clear - (and at the end he makes the same error I identified in the news story so I am guessing that he was not misquoted).

When he says before the study that "that number will be greater than the number expected. The key issue is how much greater and how unlikely is it that that number will have occurred simply by chance", I don't think his subsequent "million" is intended to represent the amount of increase in the frequency of cases but rather the number of groups you would have to look at in order to have a reasonable expectation of seeing that increase.

(For example if I toss six dice, the expected number of sixes is one, but the chance of increasing that number by a factor of six is just one in six to the sixth)

More precisely, I think that what the figure means is that if we took many equally sized random samples of similarly aged women and observed them all for ten years, then on average only one in a million of those samples would be expected to have as many or more cases of breast cancer showing up as occurred in the ABC Brisbane group. This is what I and most statisticians would call "the probability of having this number of cases occur by chance in a random sample", and it really can be computed on the basis of known, estimated, or guessed data about the population as a whole.

My original point was that it is quite wrong to describe this as the "probability the that increase is due simply to chance". If something of low probability (like me winning the lottery) happens, that doesn't mean that it is equally unlikely to have actually happened just by chance (ie me winning wouldn't make it almost certain that I cheated!)

More briefly: The probability of something happening by chance is not the same as the probability, once it happens, of its having been due simply to chance.

Or in symbols P(Event|onlyChance)!=P(onlyChance|Event)

16 Sep 2007 at 6:59 am [Comment permalink]

Thanks for your reply, Alan.

I wrote to Armstrong and invited him to comment on the issues raised in this post, but I have not heard from him.