Tuesday, 27 September 2011

Bowel Cancer Statistics - a funnel plot

I'm grateful to David Spiegelhalter of Understanding Uncertainty for the suggestion that I should display these data in a funnel plot.  (Click on the plot for a full-screen version.)

For a given population size, and assuming that the death rate is uniform across each authority, an authority has the same probability of falling outside the funnel lines whatever its size - the funnel is narrower at the right-hand end of the plot where the authority sizes are larger and the standard deviation of the distribution is smaller relative to its mean.

Under the uniform distribution assumption, the probability of a point falling above the upper dotted line is 2.5%, as is the probability of a point falling below the lower dotted line.  There are 378 points, so typically nine or ten of them would be above and below the dotted lines.  For the dashed line, the probabilities are 0.2646%, which I chose so that one point would typically be above and below the lines.

What stands out from the plot is that Glasgow City's poor result is fantastically unlikely to be random.  There may be a meaningful pattern too in the figures for North Lanarkshire and Falkirk, which cover the area from Glasgow east-north-east to the Firth of Forth above Edinburgh.

Only Westminster falls below the lower dashed line.  Since we expect one point there at random, this is perhaps not worthy of much attention.  However, there are a lot of points - 21 - below the 2.5% line, most of them in south-east England (one of them - Stirling - is in Scotland).

I suspect that there are genuine regional variations in outcome, Glasgow aside.  But the data need looking at over regions larger than most local authorities.

(Wales, which is presented as a single, very large, region is a long way off the right-hand end of my plot.  But it falls comfortably within the funnel.)

[This post is a follow-on to my previous analysis of the same data]

Monday, 26 September 2011

Blaming the Bankers

Bankers here means Investment Bankers.  No one is blaming the cashiers in High Street banks, I hope.

Daniel at Crooked Timber has written a piece to the effect that bankers are not all that bad, inspired by Joris Luyendijk, an anthropologist who's writing a blog about bankers for the Guardian.  The mainly leftish commentators on those blogs are almost all unpersuaded.  I've been meaning to write a piece myself on a similar theme: here it is.

1) Bankers are Greedy.

Yes, bankers like money.  But that's the basis of capitalism - people would rather have more money than less.  Footballers, musicians, doctors, academics, they all conform.

The real grievance isn't that bankers like money.  It's that they get it.

2) Bankers' wealth has come in the last three years from making the rest of us poor.

This is wrong.  Bankers (other than those working directly for the government) do not have the power to make huge sums of money disappear.  It's rather the opposite: by securitization of loans they made huge sums of money appear, or, if you prefer to count it thus, they made huge sums of money circulate faster.  The effect was to make some people - in particular those who sold property - richer, and to make lots of people who owned property feel richer.  The money ultimately went to pay for imported goods which people enjoyed having, hence the increasing balance of trade deficits in the UK and USA since 1998.  With the massive losses the banks inflicted on themselves through their mortgage-backed bond holdings, this source of money has dried up.

Governments outside the euro zone could, if they chose, restore their people's purchasing power by printing (electronically) more money - Quantitative Easing - to make good the lost circulation.  They don't do it on anything like the scale they'd need to because of frightening analogies with Zimbabwe or the Weimar Republic.  Arguably they shouldn't do it because international trade needs rebalancing before China owns everything.  But this rebalancing would be needed even if there had been no banking crisis.

In short, you can fairly blame the bankers for making us think we had more money than we did.  But not for the fact that we didn't really have it.

3) Bankers are useless

Investment banking activities fall into roughly the following categories.
i) Efficient allocation of capital.  This is necessary, and a lot of banking activity either promotes liquid markets - even short selling of shares - or creates the securities needed for corporates to achieve a desirable capital structure.
ii) Recirculation of money.  This is unnecessary: whatever the velocity of circulation the government is equally able (or unable) to adjust the money supply to give the value it wants to the left-hand-side of the Equation of Exchange.
iii) Tax avoidance.  This is unnecessary: governments could greatly reduce it by implementing flatter tax rules in which income would be taxed at the same rate whether it came as salary, dividends, debt interest, or capital gains.
(One might add to this category other tricks to change the appearance of accounts, such as the swaps Goldman Sachs used to help the Greek government conceal its true debt.)

Sunday, 25 September 2011

On Capital Punishment

Capital punishment is wrong.  Not because all murderers deserve to live, but because it diminishes us all when the state kills on our behalf.

In the developed world, only the USA carries out judicial executions (the vast majority of the world's executions are in China).  I'm writing about this now because of the attention focussed on the execution of Troy Davis in Georgia on the night of 21st September.

Much of what has been written about it has been assertions of Davis' guilt or innocence of the capital crime he was convicted of.  The crime was committed in 1989 in Savannah, Georgia, at about 1am outside a Burger King restaurant.  One of a group of three young men got into an argument with a homeless man who refused to give him a beer.  Another of the group struck the man with a gun.  A security guard at the restaurant, who happened to be a moonlighting policeman, went to the victim's aid and was shot dead by the assailant.  The identities of the group of three men are not in dispute; the only question is which of them committed the assault and murder.

Perhaps unsurprisingly, polemicists on both sides have wildly exaggerated the strength of their cases.  At the direction of the US Supreme Court, the District Court in Savannah (where the crime was committed) re-examined the evidence in June last year, 21 years after the murder.  This at least makes available a reasonably objective account of what evidence there was, in two parts.

The court account shows that there is no forensic evidence against Davis, and in particular that claims are false that the dead man's blood was found on a pair of   his shorts which were excluded from evidence in Davis's original trial because they had been seized illegally by police.

It shows equally that claims are wildly exaggerated that seven out of nine key witnesses have recanted their testimony.  The two most important witnesses were the homeless man, Larry Young, and his girlfriend who was drinking with him, Harriet Murray.  Young gave an affidavit to the defence saying that his original evidence had been whatever the police wanted.  But the defence declined to call him in person at the 2010 hearing, despite his being present, and despite being warned by the court that his affidavit would carry much less weight if he were not called.  Murray gave a statement to the defence, which she declined to have notarized, largely repeating her original evidence.  There were discrepancies, but nothing approaching a recantation.  She has since died.

Identification evidence is notoriously unreliable.  But the identification in this case is unusual in that it is needed to determine which of the group of young men was which, not to determine who was in the group.  It seems to me that the reliability in general of this sort of identification would be quite easy to test, and that if either side were interested in the truth they would have done the research at some time in the last 22 years.

The Savannah Court's conclusion was "After careful consideration and an in-depth review of twenty years of evidence, the Court is left with the firm conviction that while the State's case may not be ironclad, most reasonable jurors would again vote to convict Mr. Davis of Officer MacPhail's murder".  And it explained at considerable length that it was not the Court's business to conduct a retrial, but to determine whether the original verdict could not reasonably have been arrived at in the light of new evidence.

I don't know what the difference is between "ironclad" and the "beyond reasonable doubt" standard for convictions used in the USA as in the UK.  But I suspect that an expensive legal team has got a lot to do with it.

Tuesday, 20 September 2011

Having worked for UBS equities for many years, I ought to be writing something intelligent and well-informed about the \$2.3bn lost by a rogue trader at the bank.  Reportedly, the trader, Kweku Adoboli, lost the money by buying equity index futures while booking fictitious ETF trades, so that the futures appeared to be hedges for the ETFs.

In fact I am baffled.  First, Adoboli should not have been able to book fictitious client trades (nor genuine ones).  He should certainly not have retained systems access from his previous role in back office.  Second, fictitious trades should have been detected by reconciliation procedures - according to the charges he faces he started doing this in 2008, so there's been ample time to catch up with him.  And third, the Treasury desk will have had to post margin, in cash or securities, on his losing futures positions.  \$2bn or so is not a trivial amount (no, really) and they'll have been looking for offsetting margin from the purported client trades.  This should have shown up very quickly.

Which goes to show what I know, since evidently he got round all these obstacles for an extended period...

Monday, 19 September 2011

Bowel Cancer Statistics - update

I've revised the post below, having done some work on the data.

Thursday, 15 September 2011

'Three-fold variation' in UK bowel cancer death rates

[I rewrote this post on 19th September, having done further analysis of the data.  The overall conclusion is the same, but better supported by the analysis.  I corrected the penultimate paragraph on 4th October.]

I've copied the title from this BBC story.  The story is an uncritical account of a press release by the charity Beating Bowel Cancer.  "Beating Bowel Cancer calculates that over 5,000 lives could be saved every year".

The press release announces an on-line Bowel Cancer Map which allows one to find the (age-standardized) bowel cancer incidence and mortality for each local authority in England and Scotland.  This is based on 2008 figures provided by UKCIS.  (The raw data are available from UKCIS to registered users only.)

The headline finding is that death rates vary from 9.16 deaths per 100,000 in the semi-rural district of Rossendale (in Lancashire) to 31.09 deaths per 100,000 in the city of Glasgow.  I suppose that the calculation that over 5000 lives a year could be saved is based on reducing the death rate nationally from 17.68 to 9.16 - for a population of 61 million that would save 5,197 lives each year (17.68 is my calculation of the UK-wide death rate, using their data.  They give 17.27 for the death rate in England, and higher figures for Scotland, Wales and Northern Ireland).

But this statistical analysis is completely wrong.  Beating Bowel Cancer didn't reply to a request I sent them for a spreadsheet containing the numbers numbers behind the map, so I've scraped them out by semi-automated postcode query.  The total annual deaths in the data I've got is 15,936 compared with 16,259 reported for 2008 by Cancer Research UK, so I think I've been successful enough in extracting the data.  For a given expected death rate, the actual number of deaths in each district will be a random number sampled from a Poisson Distribution with mean equal to the expected number of deaths in that district.  I assumed that the expected death rate is the same throughout the country, and simulated on a spreadsheet numbers of deaths for every district in the UK. After each simulation, I found the district with the lowest actual death rate in that simulation, and the one with the highest.  The result was that on average the district with the lowest death rate has about 7 deaths per 100,000.  The district with the highest death rate has about 32 deaths per 100,000.  So the observed range - 9 to 31 deaths per 100,000 - is in fact slightly (not significantly) smaller than one would expect if the variation were purely random.  There's nothing in that range to suggest that the expectation is any different from one district to another.

What is happening here is that the analysis has been done over districts that are small enough for random variation to swamp any systematic variation.  Taking this to extremes, one might calculate deaths per household, and find that in the best households no one at all died that year of any cause.  If we could duplicate that for all households, we could all live for ever.

It's worth saying a bit more about the population sizes in each area.  The Bowel Cancer Map doesn't give these numbers directly, but it reports numbers of deaths and "age-standardised" deaths per 100,000.
Age-standardisation adjusts rates to take into account how many old or young people are in the population being looked at. When rates are age-standardised, you know that differences between the local authority areas do not simply reflect variations in the age structure of the populations...
From the data given it is simple to calculate the age-standardized population used for each area.  On this basis, Rossendale has 76,000 people and City of Glasgow has 675,000.  It is not surprising that the lowest death rate is in one of the smaller areas - they are the ones with the greatest random variation.  But if the results are simply random it is surprising that the highest death rate is in a large area.  So I do think that the high death rate in City of Glasgow is not purely random - the Cancer Research UK data confirm that death rates are significantly higher in Scotland.

One more thing.  Adding up the populations for each area, calculated from the deaths and death rates, I get a UK population of 89.75 million.  Since the true figure for 2008 was about 61 million, that's rather surprising.  Doing the same calculation on the data for number of cases, which the map also gives, I get a UK population of 83 million.  The Cancer Research UK table is using a population of just over 61 million, and therefore gets a noticeably higher death rate for about the same number of deaths.  [Update: I find that the CRUK table offers age-standardized rates also: they are very similar to the Beating Bowel Cancer rates.  The heading in the table reads "Age-standardised rate (European)...", so it seems that the standardization is to a European-average age distribution.]

I'm not sure whether I should be railing against the use of stupid statistics in a good cause...