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# The Price Equation

Gene ExpressionBy Razib KhanNov 10, 2009 8:18 PM

In the comments below I referred to the "Price Equation." Here is what William D. Hamilton had to say about George Price's formalism in Narrow Roads of Gene Land:

A manuscript did eventually come from him but what I found set out was not any sort of new derivation or correction of my 'kin selection' but rather a strange new formalism that was applicable to every kind of natural selection. Central to Price's approach was a covariance formula the like of which I had never seen...Price had not like the rest of us looked up the work of the pioneers when he first became interested in selection; instead he had worked out everything for himself. In doing so he had found himself on a new road and amid startling landscapes....

In Selection and Covariance, a short letter to Nature in which he introduces his eponymous equation, Price concludes:

...it seems surprising that so simple a relation as equation 1 has not (to my knowledge) been recognized before. Probably this is because selection mathematics has largely been limited to genetical selection in diploid species, where covariance takes so simple a form that its implicit presence is hard to recognize (whereas if man were tretraploid, covariance would have been recognized long ago); and because, instead of using subscripts as "names" of individuals (as I have done), the usual practice in gene frequency equations is to use subscripts only as names of gene or genotype types, which make the mathematics seem quite different. Recognition of covariance (or regression or correlation) is of no advantage for numerical calculation, but of much advantange for evolutionary reasoning and mathematical model building

The "equation 1" mentioned above is: ΔQ = Cov(z,q)/z In other words, the change in "Q" equals the covariance of "z" & "q" divided by the mean of "z". It can be rewritten as: ΔQ = &betaz&sigma^2/z "ΔQ" is naturally the change in frequency of "Q". One can think of "z" above as fitness, and "q" as the proportion of a trait (or gene frequency). The second equation shows "β", the regression coefficient of "z" on "q", and the variance of "q." One recalls from simple evolutionary logic that selection is conditional on variation in the trait which is under selection, and, a correlation of that variation with genetic variation. Much of what we know conceptually is densely packed into the Price Equation. Wikipedia's entry for the Price Equation is fine as it goes, but I would point you to David B's Defining Group Selection: Price's Equation. The verbal treatment has a particular focus obviously, but it is very lucid. The Darwin Wars & Defenders of the Truth have more much more on George Price, both his science and peculiar biography. This formalism is also the basis under which Peter Richerson and Robert Boyd have developed their research program on the evolution of culture, outlined in Not by Genes Alone.

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