Will Rogers, Clever Kiwis and Medical Magic

Sir Robert Muldoon was a memorable, pugnacious prime minister of New Zealand. In a typically diplomatic moment, "Piggy" Muldoon commented on his many emigrating countrymen: "New Zealanders who leave for Australia raise the IQ of both countries". 

Is that possible? Well, a New Zealander with IQ lower than New Zealand's average would raise that average by leaving. And, if this same New Zealander had an IQ higher than Australia's average, their arrival would also raise Australia's average. 

So, Muldoon's is a very clever double-joke, simultaneously having a go at both Australians and emigrating New Zealanders. Unsurprisingly, Muldoon's joke has been used in many other contexts, and unsurprisingly it's not really Muldoon's joke. 

It seems that the joke originated with the great American humorist Will Rogers. Born in Oklahoma, Rogers was describing his fellow "Okies" emigrating to California during the Dust Bowl of the 1930s.

However, our simple analysis above demonstrates that there is also a serious principle underlying Rogers' excellent joke. Given two collections of objects, it may be possible to raise (or lower) the averages of both collections simply by switching some of the objects from one collection to the other.

This principle, now known as the Will Rogers phenomenon, has some important and surprising applications. For example, in the diagnosis of diseases such as cancer, patients are often categorized into different "stages". Then, advances such as improved diagnostic techniques can alter the manner in which patients are categorised, resulting in stage migration. In turn, this stage migration can lead to an illusory improvement in recovery rates.

To illustrate, suppose we have a number of patients suffering a particular illness. Imagine that a third of our patients are very ill and even with prolonged treatment only 30% will fully recover. Another third of the patients are not very ill and 70% of them would make a full recovery with little or no treatment. Finally, there is a middle, moderately ill group, 50% of whom would recover with or without treatment.

Now, suppose our diagnostic techniques are poor, and that we cannot distinguish the slightly ill and moderately ill patients. In that case, we'd put them altogether in one group, with a combined recovery rate of 60%.

But then our diagnostics improve, and we can now determine the symptoms of the moderately ill patients. We decide these patients should receive the same treatment as the seriously ill patients. The recovery rate of those "serious" patients will then rise from 30% to 40%, and simultaneously the recovery rate of the "slightly ill" patients will have risen from 60% to 70%. Magic!

Now, if you're a doctor witnessing your patients at all stages doing better on average, you may innocently but erroneously attribute this "improvement" to your brilliant doctoring. Much less innocently, the manager of a hospital might "improve" the performance of the hospital's intensive care unit, and of the rest of the hospital, by simply weakening the admission criteria for the intensive care unit. The opportunities for error and abuse are endless.

The Will Rogers phenomenon is just one more example of how averages and other statistical quantities can so easily confuse and mislead. We’ve written about such trickery time and time and time again. So, maybe we've written enough, and perhaps it's time to stop? 

Last week, Prime Minister Gillard delivered a very important speech. Her speech began:

We are a nation of 23 million people, whose median age is 37, an average which continues to rise.

No, it's probably too soon to stop.

 

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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