# Levels of selection & the full Price Equation

In the post below on the Price Equation I stayed true to George Price's original notation in his 1970 paper where he introduced his formalism. But here is a more conventional form, the "Full Price Equation," which introduces a second element on the right-side.

Δz = Cov(w, z) / w

One can specifically reformulate this verbally for a biological context:

Change in trait = Change due to selection on individuals + Change due to individual transmission

The first element on the right-side is explicable as selection upon a heritable trait. w is the conventional letter used for "fitness," so w is population mean fitness, and serves to normalize the relation. "z" is the trait. The term "individual" can mean any set of entities. The straightforward plain interpretation may be that "individual" means a bounded physical entity, so that the covariance is measuring selection across individuals within a population conditional upon a correlation between trait value and fitness.

What then is the second element? The "E" represents expectation, just as "Cov" represents covariance. Purely abstract statistical concepts which can be drafted to various ends. In the frame I presented above, it is transmission bias from the individual to their offspring. In a deterministic system without stochasticity this is often just 0, so it is omitted from original Price Equation, but, it can be understood genetically as meiotric drive, mutation, random drift or biases introduced through Mendelian segregation. In other words, the covariance is measuring the change across the whole population due to processes which apply on the level of the population, while the expectation is simply tracking parent-offspring dynamics independent of that covariance.

But "individuals" need not be conceived of as physical individuals. One could imagine individuals being cells within a multicellular organism. The application of this in terms of the spread of cancers is obvious. Or, one could move "up a level," and conceive of the individuals as a collection of individuals, groups. Then, the second element, the expectation, could be transmission bias within the groups. So the verbal form of the equation would be:

Change in trait = Change due to selection on groups + Change due to group transmission

"Change due to group transmission" simply refers to within group selection. In the context of what I've been talking about the past week that refers to selection against altruism within groups. There will be a bias, all things equal, to favor cheaters and selfish strategies within groups. "Change due to selection on groups" simply refers to group fitness conditional upon the frequency of altruists. The more altruists, the more likely that the group is to be selected.

Here is the full Price Equation expanded to show within and between group dynamics (assume "population mean fitness" = 1, so omit the denominator):

Δz = Cov(wi, zi) + {Covj(wji, zji) + Ej(wjiΔzji)}

The subscript refer to:

i = group
j = individual

Though really they're simply referring to levels of organization or structure. The following would be acceptable:

i = species
j = group

i = individual
j = cell

i = culture
j = subculture

i = religion
j = sect within religion

(and of course, you could continue to "expand" across levels of organization)

In concrete terms, let's imagine that "z" is an allele. A gene variant. Also, let's focus on group & individual scales. Again, the first element, Cov(wi, zi), refers to the covariance between fitness of the groups and frequency of genes within those groups. The second element is more complex now, as a covariance term is nested within the expectation. The expectation is evaluated over all the groups, as you have to assess transmission bias on a group by group basis. The within group covariance is now evaluating evolutionary dynamics in terms of relative fitness of individuals within the group, with specific individuals being referred to by the subscript "j." The more individuals within the group, the greater the weight of this covariance. This is important, because you need to weight the effect within the groups by the sizes of the groups. Additionally, there is still the issue of transmission bias, the expectation of change from parent-to-offspring which isn't a function of the covariance between the trait (gene) and fitness.

In sum:

1) The existence of a formalism does not entail that it is empirically ubiquitous. Because it can be does not mean it is.

2) For a less agnostic and more verbal treatment, see David Sloan Wilson.

3) Much of the above is based on Steve Frank's review (PDF) of the Price Equation.

Citation: George Price's contributions to evolutionary genetics, S. Frank,
Journal of Theoretical Biology, Vol. 175, No. 3. (07 August 1995), 10.1006/jtbi.1995.0148

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The price equation has no predictive power, it is just an identity and as such it is often misused. Unfortunately this message has not been spread. Check this: doi:10.1016/j.jtbi.2005.04.026

By Julian Garcia (not verified) on 11 Nov 2009 #permalink