A theoretical test of contextual analysis

After two weeks off – week one was the fun review of Allen et al. week two was grading – it is time back to my main agenda of wandering through the things the ideas that have made me question the genic approach.

When I was first introduced to contextual analysis I was convinced that it didn’t work.  As an aside, this was distressing because it was introduced to biology by two of my former room mates in graduate school, and I really didn’t like being at odds with them!  So, we set out to prove them wrong.  We failed.  BUT we did get a paper out of it for our efforts (Goodnight, Schwartz and Stevens 1992 Am. Nat. 140:743-761)!

What we did was imagined we had a metapopulation that consisted of a large number of subpopulations, and then imagined that there was a trait, Z, and for each subpopulation a group mean trait Z̄  .

Metapopulation structure

We then assigned fitnesses based on three classic models of selection: group selection, hard selection, and soft selection.

Models of Selection

Group selection:  The fitness of an individual is determined entirely by the group mean phenotype and the individual phenotype has no effect on fitness.  Hard Selection:  The fitness of an individual is determined entirely by its individual phenotype, and the group mean phenotype has no effect on fitness.  Soft selection:  The fitness of an individual is a function of its phenotype relative to the group mean phenotype.  Top:  Degree of shading indicates fitness.  Bottom:  Red lines shows the relationship between individual fitness and individual phenotype, blue line shows the relationship between group mean phenotype and fitness.

Intuitively we would like contextual analysis to detect the “group selection” model as group selection, the “hard selection” model as individual selection, and the “soft selection” model as, well um, not really sure.  The Soft selection model is a classic frequency dependent selection model.  Consider an individual of intermediate phenotype.  This individual will have a high fitness in groups that have on average low phenotypes, and a low fitness in groups that have on average high phenotypes.  Thus, it clearly has a component of individual selection, but the fitness of an individual is, at least in part, a function of the characteristics the group they belong to.  This sounds a lot like group selection.

What we need to have to analyze these are the simple regressions of fitness on Z and Z̄:

relative fitness = intercept + slope * Z  (or Z̄ )

and what we are really interested in are the slopes.  The other thing we need are the slopes of the partial regressions of relative fitness on Z holding Z̄ constant, , and the partial regression of relative fitness on Z̄ holding Z constant.

In the rarified world of theory regressions are best expressed as covariances.  To scare off the mathematically uninitiated I will spell the first equation out as summations, but because that ends up cluttering up the work space, I will skip the summations in the later equations.  In any case our four regressions are:

Covariance Equations

Key to terms:  Cov(X,Y) = covariance between X and Y, Var(X) = variance of X, Corr(X,Y) = the correlation between X and Y. Cov(X,Y.Z) etc. is the partial covariance between X and Y holding Z constant etc.

The point of that ugly math is that regressions and partial regressions are different things.  If you believe me, feel free to ignore the math!

So now that we have these we can actually work out our different models of fitness.  Turning first to group selection, in contextual analysis we use partial regressions.  Without going into the rather gruesome math (and ignoring some very cute tricks) it turns out that:

partial corr equal 0

and thus,

bwzzbar equal zero

On the other hand,

bwzzbar equal b

In words, when we analyze a model of group selection, we discover that contextual analysis can be used to identify that group selection, but not individual selection is acting.

This seems like a trivial point:  If we apply group selection we detect group selection, but it is an important step in confirming that this is a valid approach.  Does the model provide the correct answer in a simple well understood system.

In contrast to the group selection model, the hard selection model intuitively should be individual selection with no group selection.  After all, the fitness of an individual is determined entirely by its own phenotype, with no influence from the group mean phenotype.  Again using the same reasoning as before, and a couple of clever tricks – We wrote this paper a long time ago and every time I go back to it, it throws me for a loop because of all of the clever tricks in it.  In the hard selection case the clever trick was figuring out that the simple regressions of fitness on phenotype and fitness on group mean phenotype were parallel. In any case, it is possible to show that there is individual selection:

Hard Sel ind component

but no group selection:

Hard sel group component

As an aside, this is in contrast with the Price equation, which would “detect” group selection, but that is for a future post!

Finally, if we apply contextual analysis to the soft selection model you find that there is individual selection acting, but the slope is not quite what you would expect:

Soft Sel ind component

and it becomes clear that individual selection is not the whole story.  Turning to the partial regression of fitness on group phenotype, it can be seen that there is group selection that is equal and opposite in strength to the individual selection:

Soft Sel group component

This result has generated a lot of confusion and senseless hot air.  This is because in a pure soft selection model there is no among group variance in fitness, and thus it seems reasonable that there should be no group selection.  However, the truth is that in the pure individual selection (hard selection) situation there IS variance among groups, due to indirect selection.  In the soft selection case group selection is PREVENTING there from being any variance among groups. This is seen in the simple regressions of relative fitness on group phenotype:

Hard Soft simple regs

In other words, there must be group selection acting if you are going to have individual selection with no variation in fitness among groups.  This should come as no surprise, since this result is completely non-controversial when talking about selection on correlated traits at the individual level, as I discussed in my blog on selection on multiple traits .

As a result of doing these analyses I was convinced that contextual analysis was in fact the correct way to analyze multilevel selection.  Several years after the original (to biology) publication of the contextual analysis approach (Heisler and Damuth 1987 Am. Nat. 130: 582-602) the kin selection folks re-discovered it and called it the direct fitness approach (Taylor  and Frank 1996 J. Theor. Biol. 180: 27-37).  It was satisfying to see the efficacy of this approach independently confirmed.  Sadly, it also confirmed that many “kin selectionists” were bound and determined to pointedly ignore the multilevel selection people.  A point that was driven home several years later when Taylor, Wild and Gardner (2007 J. Evol. Biol. 20: 301-309) confirmed the efficacy of the “direct fitness approach”, and then West, Griffin and Gardner (2008 J. Evol. Biol. 21:374-385, Gardner, A. and A. Grafen 2009 J. Evol. Biol. 22: 659-671) subsequently dismissed contextual analysis as being incorrect.

Ignorance is Bliss 1 quarter

(http://www.gocomics.com/calvinandhobbes/2012/05/20#.UrHXro2kBVs).  I thought the whole comic was a bit more catty than I wanted to be.

2 Responses to “A theoretical test of contextual analysis”

  1. Lorraine;

    Thanks for the nice comments. It means a lot coming from you. I think you are right about the zealotry, but I like to think it has an intellectual basis. That is, I came to contextual analysis as a skeptic, and I studied it, and I was convinced by the data. I think that having done that I have a fairly deep appreciation for its power.

    Don’t get too cocky, however. I think you will find that I don’t agree with you on everything. In particular, I am not enamored of “group selection 1″ and “group selection 2″. I would argue instead for identifying the level at which fitness is assigned, which I would call the level at which we assign individuality.

  2. Lorraine Heisler says:

    Great blog, Charles. The whole series. I haven’t kept up with the MLS and CA literature, but I’ve noticed the approach gaining traction in recent years, in no small measure due to you. It’s no surprise that the kin selection crowd is working hard to reclaim lost ground. Theoreticians’ egos hang by thin threads. It’s so common (and from a distance amusing) to see modelers rearrange the terms of the same model, give them new names, and claim this proves that a different arrangement of the terms is wrong. As if nature cares about how we parse it.

    “When I was first introduced to contextual analysis I was convinced that it didn’t work.” I remember those arguments, fondly. I guess it confirms the old saw that there’s nothing like the zeal of a convert ;)

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