Gardner’s theory of multilevel selection: Parsing the Model

Continuing our discussion of Gardners paper on “the genetical theory of natural selection” (Gardner 2015 Jour. Evol. Biol. doi: 10.1111/jeb.12566) I want to turn from complaining about his failure to read the literature, and this week start talking about the model itself.

He starts the model with a discussion of Fishers fundamental theorem, which I have already shown is not particularly complex. Then he goes on to expand this using Robertson’s (1968. In: Population Biology and Evolution, R.C. Lewontin, ed.) result that the change in a trait is equal to the covariance between a trait and relative fitness.

Gardner 2 eq 1

It is worth mentioning that although it is usually presented the other way around, in fact, Fisher’s fundamental theorem is actually a special case of the response to selection on any trait. To see this just replace the trait, z, with relative fitness.

Next he goes on to express concern about selection in a class structured population. His approach actually works, as long as there is no multilevel selection. As I said last week, I think his approach is rather clumsy, and there is a much better way using standard quantitative genetic methods. So, my overall comment on that part of the paper is “meh”.


Gardner’s approach to evolution in stage structured populations? “Meh” (From )

Now we get to the meat of the issue. He then goes on to develop his genetical theory of multilevel selection. First off, he develops his theory in terms of breeding values. This, has a number of possible definitions. His definition is “. . . a weighted sum of the frequencies of the alleles that the individual carries, the weights being decided by linear regression analysis. This is strangely worded, but basically correct. It hides a HUGE problem that he is ignoring. To see this consider a more standard definition of breeding value: The sum of the average effects of the alleles that make up an individual. The average effect of an allele is basically the effect of that allele averaged across all possible genotypes. This works fine in Fisher’s imaginary world of infinite population size and random interactions. It does not work well when populations are structured, and interactions are not random. If you have multilevel selection then you have population structure.   If you have population structure average effects, and thus breeding values are not constant.

This is why this is so insidious: The assumption of constant breeding value appears reasonable, and it is consistent with all of the classic models. It is the central feature of his model, that there is population structure, that invalidates the assumption of constant breeding values.  It is so obvious that Gardner did not consider the possibility that breeding values might not stay constant, although quite entertainingly he did very clearly, if unknowingly, explain why they wouldn’t. On page 3 he writes:

“Fitness may be decomposed into its genetical and environmental components, that is vi = gi + ei, where ei captures nonadditive genotypic effects (such as dominance, epistasis, synergy and frequency dependence) as well as other more obviously environmental effects.”

Well, no, that is not true. That partitioning is done by least squares, and epistasis and dominance will shift between components as we move from group to group. However note that even here he is completely unaware that when genes interact it might have evolutionary implications. And that is where Gardner falls short: his model requires that breeding values stay constant. They do not. The correct subscripting should be gij, that is, the breeding value of the ith individual in the jth deme. Experimental (De Brito, et al. 2005. Evolution 59: 2333) and theoretical work shows that gij will vary in a way that is not predictable either from the individual nor the group measured in isolation. However, I am a generous man, so lets assume they are constants for the moment, and just keep in the back of our head that this is a fatal flaw in the underlying assumptions of his model.

He then goes on to use the two level Price equation to develop his “genetical model of multilevel selection”:

Gardner 2 eq 2

OK, I hate his notation. Here it is a form that doesn’t hurt my head:

Gardner 2 eq 3


Gardner 2 eq 4is the change in the mean breeding value due to selection

Gardner 2 eq 5is the between populations correlation between relative fitness and breeding value (and yes, I refuse to use v for relative fitness)

Gardner 2 eq 6is the average covariance between relative fitness and breeding value within populations

So what is wrong with this?

Well for starters its been published before. Wade, in his paper “Hard Selection, Soft Selection, Kin Selection, and Group Selection” (1985. Am Nat 125: 61) develops a model which has the following equation:

Gardner 2 eq 7

I won’t burden with telling you all of the details of what all the symbols mean, except to say the first term on the right hand side is the mean within population covariance, and the second term is the among populations covariance. I should also say that if you sum over the K loci, the result is the breeding value. In other words, with slightly different notation it is exactly the same equation that Gardner uses. One would think a proper citation would be in order.

The nice thing about Wade’s Price partitioning being published 30 years ago is that it has been around long enough, and we have known that it doesn’t work for 20 years, and we know why. As long ago as the 1990’s I was talking to Steve Frank about this (I am sure he doesn’t remember, so Steve, if you are reading this tell me if I am wrong) and he told me that he was well aware of the partitioning, but he never called the among group covariance group selection. I also know that Mike Wade, who originally published the Price covariance model 30 years ago, has come to realize that the Price equation is inadequate.

What is wrong with the Price equation is actually quite simple, and is really the same as William’s (1966, “Adaptation and Natural Selection”) famous distinction between a “fleet herd of deer” and a “herd of fleet deer”. The problem is that if there is only selection at the individual level, say the slowest deer get eaten, then there will be some herds that by chance have a large proportion of fast deer. The Price partitioning will identify this variation in group composition as a positive covariance between group mean fitness and group mean phenotype; however, it will be entirely due to individual selection and the fact that there is variation among groups in the proportion of fleet deer. In mathematical terms, we can divide the Price covariance at the group level into a partial covariance between group mean fitness and group mean phenotype independent of individual level effects, plus a residual covariance between group mean fitness and group mean phenotype that is caused by individual fitnesses and phenotypes.  Only the partial covariance holding individual effects constant should be considered “group selection”  the other portion is changes due to selection at the individual level:

Gardner 2 eq 8

The Price equation cannot make this separation.  It should come as no surprise that this partitioning is best done using contextual analysis. You can work out the math yourself if you want. The equations you need are in Goodnight et al. (1992 Am. Nat. 140:743).

However, there is a much more serious issue than something so minor as the model being fundamentally flawed at this high level. This is the problem I mentioned before, and that is that he is partitioning breeding values. In an additive world this should work, however, if there is one lesson that comes out of the experimental group selection literature it is that it does not work in the real world (Goodnight and Stevens 1997. Am. Nat. 150:S59). This is an important point I have made in the past, when theory and experiment disagree the theory is wrong.

Indeed, there is no theoretical justification in Fisher’s additive world for me saying it is wrong. The reason I know that you can’t do that partitioning is because I have done and read the experiments (e.g., Goodnight 1990 Evolution 44:1614 & 44:1625). The problem is that when individuals interact their interactions affect the phenotype. While it may not change breeding values at the individual level, it does change them at the group level. And this is exactly what we have found. Group selection experiments work way too well. When we have done experiments where the causes can be teased apart we know that the reason that group selection works so well is because it can act on the interactions among individuals. In other words interactions among individuals become part of the breeding value at the group level. The Price partitioning assumes you are partitioning a constant, however experiments show us that the breeding value at the group and individual levels are not the same thing.

In short, the only way to develop a “genetical theory of natural selection” is to go Full Monty multivariate quantitative genetics, and treat the group and individual traits as separate, but correlated traits. Contextual analysis does half of this, what remains to be done is to work out why the G matrix is the way it is. Fortunately, Bijma and friends have gone a long way in this direction (e.g., Bijma et al. 2007. Genetics 175: 277, Bijma 2014 Heredity 112:61).


You have to go Full Monty multivariate quantitative genetics if you want to have a chance at developing a genetical theory of multilevel selection. (hope the beefcake doesn’t offend.) (

So, thus we find that the basic model is flawed in several fundamental ways. First, it is a re-derivation that is, except for details of notation, identical to a model by Wade published in 1985 (it is clear he was unaware of Wade’s work so there is no possibility of plagiary here). Second, Wade’s model, and thus Gardner’s model, was shown to incorrectly partition group and individual selection, and third, based on experimental and theoretical work, it is clear that the basic underlying assumption of constancy of breeding values is fundamentally flawed. Efforts to partition breeding values into within and among group components using the Price equation are doomed to failure due to interactions among genes and individuals. Ignoring these issues, however, well, I guess the model is fine.

Next week will be the last on this paper.  Basically last week we covered the introduction, this week was the model.  Next week will be the discussion.  If I can’t cover it in three weeks it ain’t getting covered.

Added in postscript:  Andy:  I feel badly about so thoroughly trashing this paper.  If you would like to respond I will post your response with no edits other than a short paragraph at the beginning giving attribution.  (you might want to wait until next week after I discuss the implications of your model).


Leave a Reply