Quantifiers

Scientists and Engineers

Not everyone in medicine or public health has strong quantitative skills.  Therefore, when matters of degree come into play, the task of informing decision makers is turned over, at least in part, to those that do.  These quantifiers have a wide variety of job titles; they may be statisticians, risk analysts, modelers, or computer programmers.  For those who make use of their skills, it is often thought that all quantifiers are more or less interchangeable – the good ones will all get the same right answer. Yet, that has never been true.  There are and always have been two very different species of quantifiers.

Theoretical modelers are, first and foremost, scientists.  The generally have some experience in producing data, and they know from personal experience that the data may not be what it is purported to be.   Protocols aren’t always followed correctly, corrections aren’t always correct, the assays may be flawed, the data may be misrecorded or even falsified.  As a result, doubting the data comes naturally.  If a model doesn’t quite fit the data, it is quite possibly because the data are not quite as advertised.  If the data can’t be completely trusted, then it also follows that the model can’t be completely trusted either.  So, getting a model that is approximately true is the most that one can realistically expect.

Mathematical modelers are, first and foremost, mathematicians or engineers.  In fact, they are generally much better at math than the theoreticians.  They also tend to believe the data completely and totally.  As a result, the only purpose of a model is to extract the essential truth from the data. Since models can’t really be trusted, it is even better to extract the truth without a model or theory.  Of course, this isn’t really possible, but if the model is quite simple, then perhaps no one will notice.

At the heart of this divide is the definition of probability.  A mathematical modeler will, of course, want a mathematical definition.  Whether it is a frequentist or Bayesian interpretation, mathematical probability gives an equation the necessary direct contact with the data that allows the conclusions to be inexorably deduced from the data and perhaps some necessary priors.  Inductive reasoning is not part of the equation.  Like Hume (1748), the theoretician recognizes the probability of chance that belies the mathematical definition of probability, but also uses the term in a manner that bears more similarity to the legal definition, i.e. a theory or model is to be preferred when there are arguments for it to be made.

The Interpretation of Quantum Mechanics

The great modeling divide is not unique to biology.  It can also be found in physics.  In one camp, you have Albert Einstein (Quoted in J R Newman, 1956):

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality

Yep, that’s just what a theoretician would say.  In the math camp, you have Heisenberg and the other authors of the Copenhagen interpretation of quantum mechanics (QM), where uncertainty is held to be as essential part of the fabric of the universe.   The theoretical modelers of physics have adopted a different view known as the ensemble or frequency interpretation, which asserts that quantum mechanics is statistical model that describes variation (Buchanan, 1999).

Popper (1982) noted that the problem of interpretation came about precisely because quantum mechanics transformed physics into a statistical problem.    He also suggested that the introduction of the concept of ‘propensity’ would solve the interpretation problem. However, the distinction between the Copenhagen and Ensemble interpretations is arguably just a matter of semantics or perhaps grammar.  On this view, no modification of quantum mechanics is really necessary; the Uncertainty Principle is pretty much the same with either interpretation, but an ensemble theorist would prefer something more like the Variability Principle as a name for it.   But, if nothing else, switching up the terminology solves the Schrödinger’s Cat in the box paradox.  Any given cat is either alive or dead inside the box.  In a series of cat boxes, QM will predict the number of dead cats.  That’s because QM really is not about just a cat – it’s about cats:

Copenhagen Interpretation: One cat, half alive-half dead

Ensemble Interpretation: Two cats; one alive, one dead

Things to Avoid

In epidemiology and quantitative risk assessment, the services of a good statistician or engineer are more or less indispensable.  But, some caution is warranted.  The best bet is to try to get them to stick to the math and computer programming.  In any case, there are two things to watch out for:

• Uncertainty and Variability.  Do not depend on an engineer to keep these two very different concepts straight.  They may, but it is far more likely that they will botch it up.  The cat test is useful here: If they see more than one cat, they are probably OK.
• Picking the Model.  Don’t let them do it.  There is no telling what you will get.  Wild implausibility is a distinct possibility.  And, if they try to tell you that no model is needed, don’t believe them.  It’s in there, and it is probably describing a very complex system with a very simple equation.

References

Buchanan M (1999).  The end of uncertainty.  New Scientist (6 March 1999).

Hume. D. (1739). A Treatise of Human Nature, EC Mossner (ed.), London: Penguin Books.

Newman, JR (1956).  The World of Mathematics (New York 1956).

Popper, KR (1982).  Quantum Theory and the Schism in Physics.  Routledge, New York.

Official Post Soundtrack

Martha and the Muffins (1980).  Paint By Number Heart.  In: Metro Music, Track 2.

Post Notes

This is thesis post #5.  This is a major theme post which has two themes that will be reinforced over and over again.  The first is that inductive reasoning is a critical part of any quantitative analysis. The second is that conflation of uncertainty with frequency is a serious problem.