Contextual analysis and Hamilton’s rule

After another brief hiatus to talk about religion, lets return to contextual analysis.  As I have pointed out several times contextual analysis uses the same equation as the direct fitness approach of kin selection.  This implies that we can re-capture Hamilton’s rule using contextual analysis.

Hamilton’s rule is the center of kin selection.  In one of its standard forms it is argued that an altruistic trait will evolve whenever:

Ham rule equation 1

Hamilton’s rule applies specifically to the evolution of altruism, which is a special case of multilevel selection in which group selection and individual selection are acting in opposition.  In fact, that is my definition of altruism:  An altruistic trait is a trait that is favored by group selection, but opposed by individual selection.  Thus, we can say that altruism will increase (within a generation, more on that later) whenever the strength of group selection is greater than the strength of individual selection.  Thus we need to satisfy the condition:

The strength of group selection > The strength of individual selection

or

Ham rule equation 2

Where the vertical lines mean absolute value.  In words the absolute value of the partial covariance between relative fitness and the contextual trait must be greater than the partial covariance between relative fitness and the individual trait.

If you square both sides and do a bunch of algebra you eventually get:

Ham rule equation 3

or

Ham rule equation 4

Where  Ham rule equation 5etc. is the correlation between relative fitness and the contextual or individual trait,  Ham rule equation 6etc. represent the partial correlations, and Ham rule equation Ais the correlation between the individual trait and the group mean of the trait.  When these correlations are squared they are equal to the fraction of the variance in the first variable that is “explained” by the second.  Thus:

Ham rule equation 7= the fraction of the variance in relative fitness that is attributable to variation in the contextual trait holding the individual trait constant = the strength of group selection
Ham rule equation 8 = the fraction of the variance in relative fitness that is attributable to variation in the individual trait holding group mean constant = the strength of individual selection

and perhaps surprisingly

Ham rule equation 9= the fraction of the variance that is among groups.

Thus, this gives us a very nice equation:

Strength of Individual Selection   < the fraction of variance among groups
Strength of Group selection

Now, in Hamilton’s hyper-additive gene centric world it turns out that the fraction of variance among groups is exactly equal to the relatedness within groups. Thus in Hamilton’s world Ham rule equation 9 = “r”, or relatedness.

Similarly, we can equate the strength of individual selection with Hamilton’s “cost”, and the strength of group selection with Hamilton’s “benefit”.  Thus in additivity land we have exactly recaptured Hamilton’s rule.

Howeverrrrrrrrr, it is not exactly Hamilton’s rule.  Hamilton was working strictly from optimality theory, and with the genic view.  Turning first to the optimality issue, the problem is that an optimality approach finds the place where the forces are exactly in balance.  But notice it is the FORCES.  This makes an implicit assumption about the underlying genetic nature of the traits, and the population structure.  In particular, it assumes that even though kin interact, when it comes to mating they nevertheless mate at random.  Consider what would happen if kin interacted because the population was structured, and they therefore also had to mate with each other.   What happens is that the variance within kin groups goes down, and the variance among kin groups goes up.  In the extreme you would end up with two kinds of groups (remember, kin selection typically assumes a “gene” for altruism).  Some groups would be homozygous for the altruistic gene, some would be homozygous for the cheater gene.  Guess what:  The altruistic gene would win every time.  In other words, balancing the forces is not the whole story.  The partitioning of heritable variance to the variation within and among groups also matters.

In contrast the contextual analysis version of Hamilton’s rule is actually a competing rates solution.  That is the forces balance out where the intensity of selection and heritabilities at the two levels balance out.  This brings us to our second difference, the idea that Hamilton’s original perspective used a “genes eye” view, and assumed that the group mean altruism was a simple function of the average frequency of an altruism gene.  On the other hand contextual analysis treats the individual trait and the group trait as separate traits that may have different heritabilities.  This gives rise to a very interesting point.  If we are interested not just in the within generation balance of forces, but also the response to selection then we need to add in the heritabilities.  In a paper (Goodnight 2005, Popul Ecol 47:3) I derive the following equation:

Ham rule equation 11
where h2grp and h2ind are the heritabilities at the group and individual levels, VA(*) is the additive genetic variance for the trait, and rA is the additive genetic correlation between the two traits.

Man that is ugly.  Its ugly because it includes heritability, plus the effects of both direct and indirect selection.  If we ignore indirect selection for the moment (I am not trying to pull a fast one, just trying to make the equation pretty) and do some rearranging we get:

Ham rule equation 12(plus correlated responses to selection)

Or

Ham rule equation 13

This is actually pretty profound.  What it is saying is that Hamilton’s “r” is really the ratio of the heritabilities of the group trait to the individual trait.  The heritability is affected both by the genetic structure AND the breeding structure.  Consider, for example, our bodies.  In this case because the cells in our bodies are the product of mitosis, and genetically nearly identical h2ind will be very nearly zero.  This same process that makes the cells within an individual more similar makes different individuals more different.  As a result in this case the ratio of group heritability to individual heritability will be very nearly infinity.  Compare this to Hamilton’s rule where r maxes out at 1 (which I emphasize is somewhat smaller than infinity).  Yes, when you take heritability and breeding structure into account under many circumstances the evolution of “altruism” becomes hardly surprising at all.

inigo

Hmmm, relatedness close to infinity.  Inconceivable! (image taken from http://software-carpentry.org/blog/2013/10/you-keep-using-that-word.html)

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