Number of the Month

September  2005


Let’s face it! Religious zealots are not very nice people, but for sheer callous purblind prejudice German Environmental Minister Jürgen Trittin takes some beating. Blaming the appalling devastation in New Orleans on America’s failure to sign up to the doubly fraudulent Kyoto racket indicates why we have to borrow the word Schadenfreude from the German language. No wonder the reaction from America has been so strong! It would be just as much a mistake, however, to assume that all Germans are tarred with the same brush. That nation has its share of compassionate and rational people. It is just that they do not happen to be in charge at the moment.

Unfortunately, the disaster was inevitable, just as the San Andreas earthquake is. The trouble is we do not know whether it will be tomorrow or in a thousand years time. Here is the opening paragraph of a section of Sorry, wrong number!

The Nature of Risk

The days of man are but as grass: for he flourisheth as the flower of the field.

For as soon as the wind goeth over it, it is gone; and the place thereof shall know it no more.

Psalm 103

It is, of course, impossible to extract human emotion from the risk/reward calculation. Statistically, air travel is one of the safest forms, yet a large proportion of the population is terrified of it. I know people who will not go near a plane but drive their cars like suicidal maniacs. The chances of being killed in an air crash are about three in a million journeys. If this seems a big risk, put it in context by imagining that you fly every day of your life. You would have to do this for over 900 years before you reached an evens chance of being killed. In any given area of activity there is a tendency for the risk-effect product to remain constant. Thus, when the Americans built the levees, they exchanged a large risk of small floods for a small risk of large floods. Similar rules apply to energy production or transport. It is in the field of health, however, that the greatest amount of anxiety is generated, often deliberately.

When the horror of the human agony and loss has abated we will be left with the cultural loss. Those of us who always meant to go there and never did have lost the chance forever. Things man-made pass into oblivion. For some it is the work of their gods, while for others it is the second law of thermodynamics. 

Yet it is all down to the statistics of extremes. That strain again! It had a dying fall. The great American statistician E J Gumbel, among others, showed us how to calculate the return period of events such as giant hurricanes. What he could not do is tell us when the next one is going to happen.

How many preachers in churches, temples and mosques around the world are now attributing the agonies of New Orleans to various forms of misbehaviour by its residents? For them it is just another Sodom or Gomorrah. Sign up to our product or it will happen to you! The Greens are among the nastiest of them.

The rest of us can spare a moment to share human compassion with all those desperate people who have endured such unwonted terror. What those poor folk went through does not bear thinking about. Disaster and pain might be waiting round the corner for any of us. There is no comfort in the knowledge that some lamebrain will use it to promote some specious propaganda.

Chaos umpire sits,
And by decision more imbroiles the fray
By which he reigns: next him high arbiter
Chance governs all.
Paradise Lost

Apropos of nothing

For your edification there is a site called sourcewatch, which contains a reference to your bending author. The only authority quoted is our old cobber Tim Adhominem, ad nauseam. It is quite clearly a fine example of his oeuvre, worth preserving for posterior. What better example of his method and his accuracy could you find than:

Brignell claimed a relative risk of 1.5 was not statistically significant. The only evidence he could find to support this claim was a book written by...Brignell!




12/09/05 (The Times)

The camera does not lie

When you have been involved in an activity such as monitoring the frauds and errors involved in the abuse of numbers in the media and politics it is easy to convince yourself that you are unshockable. So it was with your bending author until he became involved in a BBC investigation of hand-held speed cameras. These devices have often been mentioned in Number Watch and they are probably the cause of more impassioned correspondence from all over the world than any other topic. We have observed before that they are a fruitful source of income for governments, and are protected as a random and secure form of taxation, but the degree of official obfuscation, Orwellian misrepresentation and sheer bureaucratic bullying involved in the UK is quite startling. The BBC programme demonstrated quite clearly that with a tiny judicial movement of the camera you could record a stationary car as going more than thirty mph. The Managing Director of the importer of the instruments, apparently the regular police expert witness in such cases, stated that such an error would not occur if the car was moving.

Who is this guy? What are his academic qualifications? Are they relevant? What institutions have admitted him to their fellowship?  Apart from making money out of importing boxed instruments, what does he actually know? Is he likely to give a disinterested opinion?

To a simpleton such as your bending author, zero is just as valid a speed as any other. At what speed does the transition occur between a thirty mph error and a zero error? Does this managing director actually understand the algorithms that are implemented in his goods? Is he aware of the problems of fitting a straight line to a data sequence? How does he decide whether a sequence is linear or non-linear? Incidentally, if you too are confused at this point, the instrument does not work on the Doppler principle, but calculates a sequence of distances by determination of time of flight of the infra-red beam, over a period of about one third of a second. The data are only rejected if they are not in a straight line. Who decides what is a straight line? How straight is straight?

Some poor victim is going to lose his licence, possibly even his livelihood, because the programmer of a bit of microprocessor firmware writes a few glib lines of code. Does that programmer understand the theory of quantisation noise, the z transform, the limitations of the method of least squares, the degree of tremor in the human hand, the propensity of  human subconscious to achieve the result that is expected and desired or the ruthlessness of the Establishment in obtaining funding for its overweening  bureaucracy?

The behaviour of the Home Office in this saga is quite extraordinary and menacing. Perhaps, when one has spent twenty years being a professor of industrial instrumentation (possibly the only professor of industrial instrumentation), one might have developed an exaggerated respect for one’s own appreciation of the basic rules of the subject. Nevertheless, it would not seem unreasonable to propose that, in evaluating an instrument, one of the essential elements is to determine and investigate the possible failure modes. Not only did the Home Office fail to do this, but they have assiduously prevented anyone else from doing it either. Furthermore, according to the coda of the recent BBC programme, the Home Office have threatened that anyone who uses the information coming from that programme in his defence is likely to receive a higher fine.

There was a time when the level of fines was determined by the judiciary. What have we come to when you are punished for daring to defend yourself against a dubious charge? Turning the police into random tax collectors causes a great deal of resentment and damage to their ability to tackle real crime. Police in North Wales, the fiefdom of the egregious Richard Brunstrom (who generated our Number of the Month for June, 2005) are given targets for the number of arrests of motorists, regardless of the actual number of offences. It can be shown that people are able to learn subconsciously to achieve desired results in measurement. All it needs in the case of the hand-held speed camera is a movement of the front of the camera of the order the thickness of a human hair. With a little acquired skill you can beat the algorithm that flags an error for non-linearity and become the constabulary champion of North Wales. Among others, Jones the Burglar will love you for it. Time was when Home Office scientists were fiercely independent of politics and widely admired, while chief constables were appointed for their skills as thief takers, not their political correctness.

Testing an instrument is like testing a scientific theory – you try to break it. It is clear that the Home Office only tried to demonstrate that the device worked in favourable circumstances. It is evident that the whole concept of the instrument is seriously flawed. Rather than exploit the Doppler Effect, it takes a series of readings over a period of about a third of a second. The speed is calculated by fitting a straight line to the data sequence. Only if the data do not form a straight line is an error flagged. Just fitting a straight line is fraught with difficulties; determining whether the line is actually straight is a whole new ball game. The major potential source of falsity is “slip error” in which the beam rakes the side of the vehicle.

All individuals to some extent experience normal hand tremor. The frequency is reduced but the amplitude increased by carrying a mass (such as a speed camera). In addition, the process of aiming is a feedback process involving vision, nerves and muscles, resulting in behaviour such as lag and overshoot. If this seems nit-picking, note that the distance between the number plate and the edge of a car subtends an angle of a fraction of a degree at the distances usually involved. This means considerably less than 1 mm of motion of the front of the camera relative to the back. Try focussing a pair of powerful binoculars on a moving number plate a couple of hundred metres away, or try taking a photograph of a moving object with the shutter speed set at a third of a second.

There is no reason to suppose that slip error will necessarily produce a non-straight line of data. The only reasonable conclusion you could come to in view of all these caveats is that thousands of motorists have been wrongly convicted, some of them losing their livelihood as a result.

This is one of those cases in which it is difficult to understand why the fundamental design decision was made. Laser Doppler velocimetry  has been successfully applied to the measurement of speeds of everything from blood corpuscles to rolled steel. Why throw it aside for a mechanism with such obvious contradictions?

Salt of wisdom

After climate, alcohol, tobacco and fat, salt comes pretty high up the nonsense league table. This piece of numerical wizardry from the Daily Telegraph’s dietary advice will be appreciated by collectors:

How much should I eat?

The health authorities advise eating no more than 6g per day. This includes processed foods so check the ingredients lists on labels.

Note that sodium (often noted on labels in place of salt) is more than twice the strength of salt. So 2.5g sodium equals 1g salt. It is a ruse by the industry to make you think you are eating less salt.

Always taste food before adding salt because it may not need it. Be aware that salt is "hidden" in or added to many everyday foods, including breakfast cereals, biscuits, stock cubes, soup, ready-cooked meals (especially those containing meat), crisps and other snack foods.


Vioxx and the power of placebos

The wrong shape
Title of a Father Brown story by G K Chesterton

Someone is mad, and it is possibly your bending author.

When you have spent many of your early professional years examining time-to-failure curves and have the equations for the general case written down in your yellowing forty year old PhD thesis, it is easy to develop the delusion that you understand them. Seeing something as inexplicable as the placebo plot quoted last month, finding that it has produced drastic effects, yet seems to have remained unchallenged, tend to produce a disturbance of sleep patterns and dereliction of duty.

If it is taken at face value, there are two things you can deduce from the plot:

(a)    Once on a placebo, only two percent of patients will suffer a cardio-vascular incident.

(b)   If they have not had an incident within three years, and continue the treatment, they will never have one.

When you have examined dozens of such plots, you come to realise that localised, often bizarre, deviations in the statistical behaviour are not unusual. When you have only one plot available, and that produced by enormous effort of many people, it can be invested with a significance that is not merited.

There is an element of chartmanship in the use of a cumulative plot. For example, the little flurry of incidents at eighteen months opens up a gap that is preserved for the rest of the three-year span. In a histogram it would just be an isolated outlier. The important observation is that the slope is constant. In other words, the drug group behave just as though there were no effect and particularly no response time (unless it is improbably instantaneous).

The behaviour of the placebo group in the final year of the trial, however, is nothing short of bizarre. It is not an exponential rising to saturation, because the proportions are all wrong. If it were an exponential, it would represent a time constant of about eighteen months. This is absurd, as there is no possible physical mechanism by which starting to take a placebo could result in immunity from any disease in such a short time. There are two reasons that it cannot be related to age. First, if anything, the likelihood of such an event will increase with age. Second, any such short term effect would be smeared out (convolved) by the spread of ages in the group. Most remarkable of all is that there appear to be only two events in the whole third year of recording. Even allowing for the fact that there is marked attrition of the cohort, especially with the premature termination of the trial, you would expect about ten.

The rate of events, as indicated by the initial slope of both plots, is about 1.4% per year, which corresponds to a time constant of about 70 years. As most people have experienced a CV event by this age, the order of magnitude would seem to be reasonable , though it depends on the assumption that no one is actually immune. If this calculation is roughly correct, the plots over a period of only three years would be expected to be straight lines.

Incidentally, the reason we have to deal with percentages rather than numbers is that the plots are actually estimates of the distribution of times to event, F(t), by means of an estimator (Kaplan-Meier) designed to compensate for the attrition of numbers in the study. In fact, only about one third of the patients make it to the end.

The jealous gods of randomness are perverse. In a physical experiment you get a result that induces all sorts of hypotheses and dreams of glory. Then you painfully manufacture new samples, rerun the tests, and the theories fade like the mirages they are. In a physics lab the research student producing such a result would be told to go back and do it again.

As the old saying goes – the devil is in the detail. In this case, there is not only the outlandish behaviour of the placebo group, but to add salt to the illusion there is another of those little random above-trend flurries in the drug group at the end of the observation period.

The moral of the story is – when you see a statement such as RR=1.96 (C.I. 1.20-3.19) thereby hangs a tale. Could there be a better illustration of why real scientists are so circumspect about such claims? Often, especially when there is a political motivation, as with for example tobacco zealots, you only get the Trojan Number and the claim. In a case like this, the consequences are awe-inspiring. A valuable drug is withdrawn, millions of dollars fall into the pockets of lawyers without their having to chase a single ambulance, the loss is made up out of the pockets of the customers and lots of newspapers are sold.

By choosing the C.I. of 95%, the authors of the report have entered the one-in-twenty lottery. Have there been more than twenty studies on such a basis? If so, which ones were wrong, even on their own terms? How big was the cost? Then there is the question of premature termination. However correct it was ethically, mathematically it was a disaster. Only a third of the cohort made it to the end of the three year term of the trial.

Applied physicists have been making this sort of measurement for years (e.g. von Laue on electric breakdown in gases, 1925). They do not make claims of significance, but put in standard error bars to let the reader judge. The most extraordinary thing about this report is that the authors blandly observe (twice) that “the difference between rofecoxib and placebo appears to reflect a relative flattening of the placebo curve after 18 months compared with the preceding 18 months.” Did they not think to ask whether this behaviour was reasonable before basing such a drastic claim on it? Have they no scientific curiosity?

A clear account of the KM estimator contains the following comment (emphasis added):

Apart from the usual assumption of independence between the survival times. There is one crucial assumption behind the KM estimator, namely that the censoring is independent of the event of interest. This means in practice that the fact that we have seen a censoring must not make us change our belief about the survival of the individual.

When the censoring is dependent on the event of interest there are no ways of correcting for this!

Several days of searching this report for an iota of justification for withdrawing Vioxx lead to the conclusion that there is none. In addition there is no justification for the premature termination of the trial and the added obscurity that it engenders. At the time of the Tamoxifen fiasco, the European participants professed to be scandalised. Now they are all at it. The most trivial outcome is described as staggering and used as an excuse to stop the trial before it goes wrong.

Oh dear!

  28/09/0 5 

Number of the month – 150 million

Your bending author can hardly complain about the flu scare, when the following appeared in Sorry, wrong number! five years ago:

If you really cannot manage without something to worry about, may I suggest our familiar friend, the ’flu. It gets the headlines during a particularly severe outbreak, but is largely ignored, even by the most prolific scaremongers. Even in a good year it will kill more of the population than, say, road accidents, which receive a great deal more propaganda. The remarkable and fearful thing about this virus is its built in capacity to mutate. Each year new strains pop up somewhere in the world, an example of evolution in action. We rely on a modest collection of medical teams to isolate each new strain and develop a vaccine in time to give some protection to the most vulnerable members of the race. Every year they look at the odds and make their bet on our behalf, a gigantic gamble for very high stakes. Every now and then a really potent form emerges and spreads devastation around the world. In the Great War of 1914-18, the most mortal in history, the “Spanish” ’flu killed more Europeans than the war itself. Fit young men were dead within three or four days of contracting it. Even as recently as 1996 the death rate in Britain reached 5,000 in one week. Yet where are the SIFs, the high profile professors and the billions of dollars of taxpayers’ money they command?

It is likely that ’flu epidemics originate with and are spread by birds, but the amount of human travel since 1914 has increased beyond all bounds. A particularly virulent form could spread around the world in days. My contribution to the scare industry is to predict that the next disastrous world epidemic will be some form of influenza.

Then, of course, flu had largely gone out of fashion among scaremongers, but these things tend to come round again in the cycles of the scare industry. The point is that putting a number to the death toll is in the province of fools, knaves and those who make a living out of it. Then there’s the BBC:

"The range of deaths could be anything between 5m and 150m," the UN's new co-ordinator for avian and human influenza said in his BBC interview.

Regular number watchers will like the next bit: 

Dr Nabarro said he stood by the figure drawn from the work of epidemiologists around the world. 

And, of course, they don’t forget the religious message:

"It's like a combination of global warming and HIV/Aids 10 times faster than it's running at the moment," Dr Nabarro told the BBC.

Just imagine what it would be like if the BBC became as unpredictable as the objects of its scare stories.

Immediate footnote: but then there's this.



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