### Dr Bayes' medical marvels

The Bayesian approach to statistics is a fascinating subject, which I cover at some length in my book Dice World. What Bayes theorem enables you to do is to improve an estimate of the chances of something happening when you have additional information, and to use one set of probabilities to calculate another linked one.

This can be extremely useful and powerful when, for instance, calculating the effectiveness of disease screening tests, which can be very confusing due to wildly varying conditional probabilities. It's worth getting your head around a bit of probability symbology to get on top of this. In these simple formulae, the '|' sign is read as 'given'.

So, for instance if I have a test that will flag up the presence of a disease 90% of the time, which isn't too bad, I can write that as
P (Positive result | Disease ) = 90% - the probability of a positive result in the test, given the person has the disease, is 90%.

The problem comes, and sadly this has happened for real, when this is represented by the medical profession or (more often) the press office of a university or the press in general as the test being '90% accurate.' This is because it's perfectly possible with the same test for P (Disease | Positive Result ) to be, say, just 20%.

What that's saying is that the probability of a person who tested positive having the disease is only 20%. This, of course, is an important part of what people want to know after a test. I've just had the test and it came up positive. What's the chance that I actually have the disease? In this case it is surprisingly low, given the apparent 'accuracy' of the test.

The reason this can happen is that to work out the first figure, P (Positive result | Disease ) we are only considering the population of people who have the disease, which might be quite small. But for the second figure, P (Disease | Positive Result) we are looking at the population who had the test, which could be a much bigger number, overwhelming the number of correct positive result tests from the smaller population of sufferers with the false positive result tests from the larger population of tested people.

This makes those in the business who understand probability wary of mass screening programmes for relatively rare conditions - they result in tests that, even when very likely to get the result right on any particular individual, can come up with a distressing false positive more often than a true outcome, putting the patient through a time of horrible stress unnecessarily.

See David Colquhoun's blog for more detail on the risks of using these kinds of screening tests.

### Why I hate opera

If I'm honest, the title of this post is an exaggeration to make a point. I don't really hate opera. There are a couple of operas - notably Monteverdi's Incoranazione di Poppea and Purcell's Dido & Aeneas - that I quite like. But what I do find truly sickening is the reverence with which opera is treated, as if it were some particularly great art form. Nowhere was this more obvious than in ITV's recent gut-wrenchingly awful series Pop Star to Opera Star , where the likes of Alan Tichmarsh treated the real opera singers as if they were fragile pieces on Antiques Roadshow, and the music as if it were a gift of the gods. In my opinion - and I know not everyone agrees - opera is: Mediocre music Melodramatic plots Amateurishly hammy acting A forced and unpleasant singing style Ridiculously over-supported by public funds I won't even bother to go into any detail on the plots and the acting - this is just self-evident. But the other aspects need some ex

### Is 5x3 the same as 3x5?

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3. Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable. It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5.  However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign. Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only wor

### Which idiot came up with percentage-based gradient signs

Rant warning: the contents of this post could sound like something produced by UKIP. I wish to make it clear that I do not in any way support or endorse that political party. In fact it gives me the creeps. Once upon a time, the signs for a steep hill on British roads displayed the gradient in a simple, easy-to-understand form. If the hill went up, say, one yard for every three yards forward it said '1 in 3'. Then some bureaucrat came along and decided that it would be a good idea to state the slope as a percentage. So now the sign for (say) a 1 in 10 slope says 10% (I think). That 'I think' is because the percentage-based slope is so unnatural. There are two ways we conventionally measure slopes. Either on X/Y coordiates (as in 1 in 4) or using degrees - say at a 15° angle. We don't measure them in percentages. It's easy to visualize a 1 in 3 slope, or a 30 degree angle. Much less obvious what a 33.333 recurring percent slope is. And what's a 100% slope