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Evolution in Structured Populations

Gardner’s theory of multilevel selection: Where he goes wrong and why

Posted: January 28th, 2015 by Charles Goodnight

Two things have happened recently. First, Jonathan Pruitt and I (Pruitt and Goodnight 2014 Nature 514:359) have been asked to reply to a goodly number of letters to the editor concerning our paper on multilevel selection in Nature. These letters have made it clear to me that many people have a very basic misunderstanding of multilevel selection. Second, I was made aware of a recent paper by Andy Gardner (2015 Jour. Evol. Biol. doi: 10.1111/jeb.12566), which is impressive in the depths of  misunderstanding of multilevel selection that is in the paper. I have never met Andy, but I do know he is well established, and he can stand a little criticism from me. Thus, I thought his paper would be perfect for highlighting some of the very more serious misunderstandings people have about multilevel selection. There are so many problems with the Gardner paper that it will take me several weeks to work through them, so on that note, lets take his paper and start turning it into confetti. You have actually seen the opening salvo in my post last week about Fisher’s fundamental theorem. What brought that up was Gardner suggesting that the fundamental theorem was somehow special, or that it applied only to a specific subset of biological entities.

What I want to talk about this week is an idea that Gardner puts out nicely in the first sentence of the abstract: “The theory of multilevel selection (MLS) is beset with conceptual difficulties.” The truth is that MLS is in fact a mature theory. One that, at this point, has very few conceptual difficulties. We know group selection works, we know why it is so effective, we know how to extend quantitative genetics along several different pathways to incorporate the interesting results of group selection experiments, and we know how to measure MLS in the field. Finally, MLS methods are widely used in agriculture – your breakfast this morning may well have been dependent on MLS theory. Eggs, bacon (hogs) and toast (wheat) are commonly or exclusively selected using MLS methods. It is a mature settled theory, sure there is much to be done, but isn’t that true of all science?

So, why is Gardner so wrong? Well that can be seen in the first sentence of his introduction (Do you start to see why this might take a few weeks!): “Recent years have seen a resurgence of interest in the theory of multilevel selection (MLS: Price, 1972a; Hamilton, 1975; Sober & Wilson, 1998; Keller, 1999; Okasha, 2006; Wilson & Wilson, 2007; West et al., 2008; Gardner & Grafen, 2009; Leigh, 2010; Nowak et al., 2010; Lion et al., 2011; Marshall, 2011; Frank, 2012a, 2013).” What you should notice is that there are no serious multilevel selection experimentalists on this list, nor is there anybody on that list who I would call a true MLS theorist. I will not go through the list of why these people are inappropriate, other than to say that some are very old, many are philosophers, and many are advocates of kin selection, or for other reasons really should not be considered authorities on multilevel selection. One has to question where are (to list only modern authors) Wade (1977 Evolution 31:134, Wade et al 2010 Nature 463:E8), Bijma (Bijma et al 2007 Genetics 175:277), Muir (1996 Poultry Science 75:447), Eldakar (Eldakar et al 2010 Evolution 64:3183) Simon (Simon et al 2013 Evolution 67:1561), Ratcliff (2012 PNAS 109:5), Travisano (2004 Trends Microbiol. 12:72), Driscoll (Driscoll and Pepper 2010 Evolution 64:2682) or dare I suggest myself? These are people who understand multilevel selection. I should point out that it is not just this sentence where he fails to cite the relevant literature. With the exception of one vacuous (it will come up later) reference to a paper of mine, none of these authors appear in the literature cited.

This is a fundamental problem that I am seeing. Gardner, not to mention the authors of the letters to nature that we have been fielding, appear to be completely ignoring the MLS literature. I will admit my own failings in this matter. It is not infrequent that I will glance over an abstract and decide it is not important to what I am writing about. However, when writing outside my field (and yes, Gardner is working outside his field) I really do try to ask colleagues if they know of anything I have missed. In this case there is plenty that Gardner missed. As an example, the model he develops in his paper is totally incompatible with the results of Goodnight and Stevens (1997 Am Nat 150:S59). Nobody but Andy knows the real reason ignored the body of MLS literature. Hopefully it won’t happen in the future.

With these weak foundations, Gardener then goes on to list a series of things that he believes to be difficulties. These include:

  • the “precise meaning of group trait” – A group trait is either a trait measured on the group itself, or a composite of measures taken on the group members. Both can be appropriate. Like all studies of selection an understanding of the underlying biology is needed to identify relevant traits. Bottom line: experimentalists need to actually measure these “group” (really contextual) traits. As you might expect, those who measure them know what they are.
  • The “precise meaning of group fitness” – I have to give you that. However, the reason for this is that it is not relevant to the study of MLS. The relevant issue here is measuring selection in the field, and for this the appropriate approach is contextual analysis, which does not use “group fitness” (see Taylor, Wild and Gardner 2007 J. Evol Biol 20:301 for a demonstration that direct fitness, which is the same thing as contextual analysis, is an appropriate metric, Snideness aside, also look at Goodnight 2013 Evolution 67:1539).
  • There is “ambiguity as the focal level in a MLS analysis”. Here he is complaining about the distinction between multlevel selection 1 and 2. I do not like this language, and I am not the best to comment on it. The term was coined over 25 years ago, can we give it a rest? The basic problem is the level at which you assign fitness. Sadly he again shows his ignorance because the actually most relevant paper that gives a relatively simple explanation for this non-controversy is the one paper of mine he cites: Goodnight 2013 (pp 37-53 in: From Groups to Individuals). Sadly, while he did cite this chapter, it was not in the context of this problem, and when he did cite it, it is to make an invalid point.
  • Finally, he makes a big deal about MLS theory does not adequately able to handle class structured populations. First, off, there actually is a nice old paper on multilevel selection in age-structured populations (Mertz et al 1984 Evolution 38:560), although it really isn’t very useful in this context. More relevant, the reason that nobody has developed a method to study MLS in a class-structured population is that nobody has bothered – Most ant people these days are kin selectionists. The basic approach is actually conceptually quite simple:   I would follow Lande’s lead on analyzing sexual dimorphism (Lande 1980 Evolution 34:292) and phenotypic plasticity (Via and Lande 1985 Evolution 39:305). I would describe a separate trait for each cast, plus one or several contextual traits to describe the overall composition of the colony. Each individual would express only one of the individual traits, but of course, all would experience the contextual traits. Then it would be a small matter to modify the methods of Lande and Via and Lande to use them in this system. It actually isn’t that different from the approach Gardner advocates, but it is far more elegant, and it is far more consistent with the existing methodologies for related problems.

So basically what we see in Gardner’s paper (and by extension many of the letters to Nature) is a failure to be aware of and to understand the relevant literature. The problem is not failure to cite the relevant papers per say, rather the problem is that by they do not know the literature and understand the field. As a result the authors end up looking foolish for raising issues that do not exist, and suggesting methodologies that in this case are clumsy, but as we shall see in the next week or two, also methodologies that simply give the wrong answer. I am aware that it is often easy to miss important papers, but to paraphrase the old saying about the law: Ignorance of the literature is no excuse.


We all are guilty of not adequately reading the literature. Nevertheless, it is something to be avoided.  (From http://imgur.com/gallery/1DEYI)



A one line proof of Fishers fundamental theorem

Posted: January 21st, 2015 by Charles Goodnight

Lots of people get bent out of shape about Fisher’s Fundamental theorem, and spend lots of pages talking about it. The problem is that people tend to see the FFT as being magical. Theoreticians promote this because, well, the basic proof is so simple that you need to add some sort of complication to justify your existence (Ok, some of the complications are interesting). So without further adieu, here is my one line proof of Fishers Fundamental Theorem

Added in edit:  I completely forgot to tell you what Fisher’s Fundamental Theorem was!

Fisher (1930) stated that “The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.”.  I would strike the word “genetic” from that sentence.

First, define terms:

The relative fitness of the ith individual =Relative fitness

The frequency of the ith individual = pi (usually that will be 1/N, but the truth is general, all we need is a frequency, and in reality it should be particle, not individual in the definition)

The frequency of the ith individual after selection, or put another way, the fitness weighted frequency of the ith individual =freq after selection

The mean relative fitness isMean relative fitness

I use the symbol mean fitness after selectionto denote the mean fitness after selection


Some math you need to know:

12=1 (not sure what to call this other than a truism)

(A*B)*C = A*(B*C) (the associative property of multiplication)


SO the one line proof:

FFT proof



In a recent article, Andy Gardner (1015, J. of Evol. Biol, DOI:10.1111/jeb.12566)says:


“Today, disagreement still persists as to the correct interpretation of the fundamental theorem. For example, whereas Okasha and Ewens both regard the theorem as concerning the selection of genes, I regard it as concerning the selection of individuals.”


Are you serious? I don’t think it applies to Koalas. Of course FFT applies to genes, and to phenotypes, and yes, even to Koalas. Heck it applies to anything to which relative fitness can be applied. OK, maybe I will give this to Andy: there is really no meaningful way to assign fitnesses to genes, so maybe the fundamental theorem isn’t so useful in that situation.


This actually raises an important point. Yes, the FFT is a truism, but that doesn’t mean it is meaningful in all circumstances. When people refer to it as the “not so fundamental theorem” they are not complaining about the theorem itself, but about the application of that theorem to a particular biological situation.


Koalas don’t know or care about Fisher’s Fundamental Theorem. However, just because you don’t know or care about something doesn’t mean it doesn’t apply to you. (http://en.wikipedia.org/wiki/Koala)

Genes, society, sexism and racism

Posted: January 7th, 2015 by Charles Goodnight

James Watson has been in the news for more than just his efforts to sell some bullion. He has also been in the news for his completely outrageous racist and sexist comments. Two of the more famous ones are the time he told a reporter that he is “inherently gloomy about the prospect of Africa” because “all our social policies are based on the fact that their intelligence is the same as ours whereas all the testing says not really”; and, with regard increasing the number of women scientists, the time he wrote “I think having all these women around makes it more fun for the men but they’re probably less effective.”. It is tempting to dismiss this as Watson being a jerk, until we get Lawrence Summers more eloquently saying pretty much exactly the same thing. One suspects that these characters are simply the tip of an iceberg dominated by people who are better able to hold their tongue, but are nevertheless to some degree racist and sexist. The bottom line is that racism and sexism are everywhere. Indeed I would be lying if I claimed not to be racist, and I suspect most of my readers would be as well. The point of this is that if you consider yourself to be part of a particular group, be it a racial group, a sex, a sexual orientation, or a social or religious group, if there is a stereotype associated with that group, you will be reminded of it on a daily basis.

imposter syndrome

Stereotype threat: living up or down to the stereotype of the group you self associate with. http://web.stanford.edu/dept/CTL/cgi-bin/academicskillscoaching/its-not-you-its-stereotype-threat/

Here is where it gets interesting. It turns out that these stereotypes do affect your performance in many aspects of life. There is a group of psychology and education researchers who have been studying what is now called “stereotype threat”.   It turns out that if, before giving them a test, you remind students of a standard stereotype, it will affect their performance. This is very nicely described in a review article by Schmader and Croft (2011, How Stereotypes Stifle Performance Potential. Social and Personality Psychology Compass: 792–806):

In 1995, Stanford researchers Claude Steele and Joshua Aronson published a series of highly influential experiments. They reasoned that for those who are the targets of negative stereotypes of intellectual inferiority, even subtle reminders of these stereotypes can cue a concern with confirming them. . . . To demonstrate this phenomenon, they asked White and Black undergraduates to complete a set of verbal problems. For half of the sample, they described this task as a diagnostic measure of verbal intelligence. For the other half of the sample, it was merely described as a problem solving exercise. When students believed that their intelligence was being assessed, Black students performed more poorly than their White peers, replicating the typical racial gap in standardized test scores that is so often found. Remarkably, when the same task was described in a more neutral way – as a laboratory exercise – Black students performed significantly better and their performance was equivalent to that of their White peers after controlling for individual differences in past test performance.



When students were given a test in which they were made aware that it was an important test, (and one that blacks might stereotypically be thought to do poorly in) blacks, but not whites, were negatively influenced by that information. http://menghublog.wordpress.com/2012/12/06/race-and-iq-stereotype-threat-r-i-p/

Apparently you can do this with almost any group for which there is a stereotype. Tell the women in a class that this is a subject women are not good at and they will perform poorly, or convince any group that they are inferior, and they will perform poorly. Importantly, however, the students must believe the stereotype at some level. In the Steele and Aronson study Blacks, but not Whites, were affected by the doubt raised by the investigators. I could not find a study addressing it, but I would tend to doubt that stereotype threat would work for made-up stereotypes that were not part of our underlying cultural assumptions.

So, what does this have to do with evolutionary biology? WELLLLLL, once long ago I was asked if I thought there were racially associated genes for intelligence. I answered Yes, I thought there were, and that they were exactly the same genes that cause the features we use to identify race. The next thing that happened is that I realized keeping my mouth shut was a good idea.

This is where the phenotypic view comes in. My argument is that because genes (broadly defined, since X chromosome number enters into this too), in addition to what we traditionally might assign to a gene, also have effects associated with them due to the social milieu in which they are found.   I seriously doubt that the loci that effect skin melanization also have a physiological effect on intelligence. But there is no doubt that an individual with sufficiently dark skin to be considered to be an African-American is treated differently from a White-American (great example of micro-aggression: What do you call a Caucasoid-American?). This differential treatment is part of their phenotype that, within the context of our society, is every bit as much a property of the loci in question as their effect on skin color.

This is an important aspect of the phenotypic approach. The phenotype is a construct of the patterning elements, including the non-heritable elements. Context means everything. Normally we think of this in terms of epistasis and the idea that gene expression depends on its interacting partners. However, it is more than that. The effect of a gene on the phenotype must take into account all of the forces affecting the formation of the phenotype. This is not to say that in many circumstances these can be ignored, but this racial bias is very emphatically a situation in which non-genetic cultural factors ARE influencing the expression and even the very function of a gene, and cannot be ignored.

So the bad news is yes, there are racially associated genes for intelligence; they are the ones that influence the phenotypes we associate with race. The good news is that we, as a society, made them intelligence genes, and we can unmake them. There are good solid strategies for minimizing stereotype threat. The best one is to get rid of the stereotype. That may be impossible, but as teachers we can also be aware of this and help our students. We can make them aware of stereotype threat. Just being aware that it exists will help a student recognize it and perhaps reduce their response to it. We can also work to encourage students to think of themselves as individuals and work to turn the negative stereotypes into personally positive messages. Finally, interestingly, stereotype threat apparently isn’t as powerful when a test is perceived as being not very important. Perhaps having more evaluations that are individually worth relatively less might help vulnerable students.

Genetic distance and FST

Posted: December 17th, 2014 by Charles Goodnight

First off, I did a search for papers that used contextual analysis in some form or another to analyze experimental data. This is the list I came up with. It seems pretty pitiful for a statistical method that (1) works and (2) with the exception of Heisler and Damuth using a very small data set to demonstrate the technique, has been wildly successful at detecting multilevel selection. I am hoping that I missed some important references. If you know of any that I missed, please let me know! If I didn’t miss anything, well, it looks like it is time for us to get to work!

Aspi, J., A. Jåkålåniemi, J. Tuomi and P. Siikamåki (2003). “Multilevel phenotypic selection on morphological characters in a metapopulation of Silene tatarica.” Evolution 57: 509-517.

Donohue, K. (2003). “The Influence of Neighbor Relatedness on Multilevel Selection in the Great Lakes Sea Rocket.” American Naturalist 162(1): 77-92.

Donohue, K. (2004). “Density-dependent multilevel selection in the great lakes sea rocket.” Ecology 85: 180-191.

Eldakar, O. T., D. S. Wilson, M. J. Dlugos and J. W. Pepper (2010). “The role of multilevel seleciton in the evolution of sexual conflict in the water strider Aquarius remigis.” Evolution 64(11): 3183-3189.

Heisler, L. and J. D. Damuth (1987). “A method for analyzing selection in hierarchically structured populations.” American Naturalist 130: 582-602.

Herbers, J. M. and V. S. Banschbach (1999). “Plasticity of social organization in a forest ant species.” Behavioral Ecology and Sociobiology 45: 451-465.

Laiolo, P. and J. R. Obeso (2012). “Multilevel Selection and Neighbourhood Effects from Individual to Metapopulation in a Wild Passerine.” PLoS ONE 7(6): e38526.

Moorad, J. A. (2013). “Multi-level sexual selection.” Individual and Family-level selection for mating success in a historical human population 67(6): 1635-1648.

Pruitt, J. N. and C. J. Goodnight (2014). “Site-specific group selection drives locally adapted group compositions.” Nature 514: 359-362.

Stevens, L., C. J. Goodnight and S. Kalisz (1995). “Multi–Level Selection in Natural Populations of Impatiens capensis.” American Naturalist 145: 513-526.

Tsuji, K. (1995). “Reproductive conflicts and levels of seleciton in the ant pristomyrmex pungens: contextual analysis and partitioning of covariance.” American Naturalist 146: 587-607.

Weinig, C., J. Johnston, C. G. Willis and J. N. Maloof (2007). “Antagonistic multilevel selection on size and architecture in variable density settings.” Evolution 61: 58-67.

The second thing I wanted to talk about was that I was asked about the relationship between inbreeding coefficients and genetic distance. I thought I would share my answer, in part to be told where I was wrong. My disclaimer is that all I know about genetic distance, is that it is something I rarely care about. . .

Consider a metapopulation with M alleles, with the mth allele having a frequency of pm in the overall metapopulation. We would like to calculate d, which from I got a formula cited by Smouse and Peakall (1999, Heredity 561-573) to be:

Screen Shot 2014-12-17 at 11.43.46 AM

Here the summation is over the M possible alleles, and yijm is the number of alleles of type m in individual i in the jth deme. This takes on a value of 0, 1, or 2.

If we are interested in the average genetic distance between deme j and deme l then we would calculate this as:

Screen Shot 2014-12-17 at 11.44.02 AM

We can now define dmax to be the maximum value that can take on. This will occur when the FST = 1. In an infinite metapopulation this means that every population will be fixed for an allele, and pm of the populations will be fixed for the mth allele.

If demes j and l are fixed for the same allele the genetic distance is 0. For allele m this occurs with probability (pm)2. If deme j and l are fixed for different alleles the genetic distance is:

Screen Shot 2014-12-17 at 11.44.11 AM

For alleles m and n this occurs with probability pmpn, thus:

Screen Shot 2014-12-17 at 11.44.20 AM

We want a measure that is a function of FIT and FST (I just figured out that I have never talked about FIS,  FST and FITTry this) that goes from zero to 1. When FST = 0, dij,kl = 0, and when FST = 1 dij,kl =1.

Working this out (the excel worksheet is available here: genetic distance work sheet)

Screen Shot 2014-12-17 at 11.44.34 AM

If we assume random mating within demes then FIT = FIS.

Screen Shot 2014-12-17 at 11.44.44 AM


Note that when FST=0, d = 0, and when FST=1, d = 4. The problem, of course is that we want to multiply this by dmax. For this to work we need the equation to go from 0 to 1. Thus, we divide by 4:

Screen Shot 2014-12-17 at 11.44.54 AM


Screen Shot 2014-12-17 at 11.45.03 AM

OK, A lot of algebraic noise. What this is telling us is that using Smouse and Peakall’s formula, there is a fairly direct relationship between FST and Screen Shot 2014-12-17 at 11.45.14 AM. Basically the difference is that genetic distance is based on identity by state, whereas F is based on identity by descent. If, at the start, every allele is unique then Screen Shot 2014-12-17 at 11.45.23 AM. If not, then dmax will be some number smaller than 4, and Screen Shot 2014-12-17 at 11.45.32 AM. If you care here is a graph of my equation:

Screen Shot 2014-12-17 at 11.47.04 AM

Genetic distance standardized to a maximum value of one as a function of FST.  If mating is non-random then  FIT will not equal FST and the results will be somewhat different.

Finally, I was asked about our fly collecting trip. Well do to a whole bunch of odd events we are understaffed to take care of a new batch of flies, so the trip has been postponed until January. The other question was about how I was going about bringing flies back to the US. The answer is I am not. I strongly recommend doing research in Brazil, but if you do get a Brazilian collaborator, and do your experimental work in Brazil, and leave your samples there. The reason is simple. We, as in the US and other first world countries, have been pillaging countries like Brazil for too long, and they are, unsurprisingly, sensitive about this. Doing research in Brazil is dead easy IF you have a Brazilian collaborator and you do the work in Brazil.

OH, and yes, I am slowing down my posts for a while, but I will still be occasionally posting as the occasion arises.

Hiatus announcement and group selection 1 and 2.

Posted: December 4th, 2014 by Charles Goodnight

The main piece of sad news this week is that I am just simply overwhelmed, and I am going to have to take a hiatus from writing. I will try to post occasionally, but look for once or twice a month rather than weekly. The reason is that I signed a book contract. I need to get writing on that, but before I do I have a chapter to complete, and I need to keep my research going – we are off to collect Flies in Southern Brazil next week. In some sense, it is good that I am slowing down. The logical progression of the blog would start to lead me into unpublished territory. The phenotypic approach has a lot to say about the evolution of sex, and the origin of life, for example. I think it makes sense to keep these a bit under my hat until they can get out in a citable peer reviewed format. The reason it is perhaps not so good is that there are a lot of ideas tearing at me that really do belong in the blog. Some subjects I should write about: It dawned on me that there is a difference between the contextual traits of contextual analysis on the one hand and indirect genetic effects on the other; I am once again confronted with philosophers of science talking about “group selection 1” and “group selection 2”, which are terms that I think confuse the issue and interfere with a nuanced understanding of multilevel selection. These are just issues that came up in the last week. Thus, I am not ready to abandon the blog yet, but I do think I need to slow down.

The book contract mentioned above is to write a book on exactly the topic of the blog, and as a result hopefully put it in a more permanent and citable form. The target date for a draft is a year from January, so it will be occupying a lot of my time for the next year.

So to rather randomly choose one of these topics. Lets talk about group selection 1 and group selection 2. This concept was introduced by Heisler and Damuth (1987, Am Nat 130:582), and recently popularized by Okasha (2006) (http://www.amazon.com/Evolution-Levels-Selection-Samir-Okasha/dp/0199556717/). The basic idea is that when you have individuals, or particles, you can do a multilevel analysis of selection on particles and selection on the collective, or you can just do an analysis of collective. Group selection 1 is an analysis in which fitness is measured at the particle level, and contextual effects of higher levels of organization are included. In other words, group selection 1 is what we think of when we think of group and individual selection acting simultaneously. In contrast, in group selection 2 we ignore the particles and focus on the collective. Thus, we might look at the fitness of bacterial colonies even though we know full well that these colonies are made up of individual cells.

This concept has gotten good traction in the philosophy world, and I will agree that it raises an important point. That is, it makes clear that results change depending on your point of reference. In the past there has been a lot of useless ink wasted when people were arguing about things like whether species selection was just the summed effects of individual selection, when in fact, they might be the same thing. That is, from a group selection 1 perspective in which the individual organisms are included in the analysis, it could indeed be that species selection is just the cumulative effects of selection on individuals. Even if this is true, however, if we take a group selection 2 perspective then indeed it is species selection, since we are only looking at the collective, or species.

So, this all sounds very positive, so what’s the problem? The problem is that every system is always group selection 1 and group selection 2 at the same time. Cells are made up of subcellular components, organisms are made of cells, groups are made of organisms. The levels need not be strictly hierarchical. For example, I “belong” to a number of different groups: My family, my department, the Evolution society, the blogosphere. These groups are in no sense hierarchical, and yet the do overlap to some extent. The group selection 1/2 perspective implies that there are really only two such levels, and basically enforces a false dichotomy. Question: are you working on particles or collectives? Answer: Yes.

So, rather than be destructionist I would like to offer a much better alternative. Let us clearly identify the level at which we assign fitness. This is Okasha’s particle, and my individual. Let me repeat that: The level at which we assign fitness is the individual. Then, rather than having the ambiguities of what exactly the levels are when we talk about group selection 1 and 2, we can instead clearly say that in this study the individual is the cell, whereas in that study the individual is the organism. The conclusions will, of course be different, but we don’t have to argue about them. We will know why they are different. They are different because they have different perspectives, they assign fitness at different levels.

Again, I apologize for taking a hiatus on my blogging. Hopefully I will be able to put up posts at a lower rate, and still keep this blog alive. One reason I will not be blogging next week is that we are going fly collecting. This will be an adventure, so don’t feel sorry for me!

Screen Shot 2014-12-04 at 1.59.48 PM


We will be driving south into wine country to sample Tephritid flies.  There are a number of interesting species complexes here.  Hopefully we will be able collect some of interesting species.


Variance in a structured world

Posted: November 19th, 2014 by Charles Goodnight

So far I have been writing about things that in some sense I fancy I know something about the answer. Today, all I have is conundrum. The conundrum I have is this: one of the unspoken themes of this blog is that the “mean field approximation” is inadequate, and yet I am failing to provide an alternative.   Sadly, this is the comment from my physicist friend for which I had no answer.

Consider one of the most basic mean field statistics, variance. Variance is fundamental to our understanding of selection. For example, the selection vector, S, is the covariance between a trait and relative fitness, which in turn is a product of the standard deviations of the trait, relative fitness, and the correlation between the two:

Variance eq 5

Now, lets think about what this means. As far as selection is concerned it means that we are effectively lining everybody up, and choosing those with the most favorable value of the trait, and discarding those that don’t measure up.   In effect this is very much like the start of a race: Everybody lines up at the start line, the gun goes off, and the runners can be ranked based on the time it takes them to cross the finish line.

Race start

The start of a race is a situation in which the variance in a trait, such as muscle mass, can be used with some confidence to predict the outcome the race. The important point is that all are competing equally at the same time and under the same conditions. (http://www.dailymail.co.uk/travel/article-2176034/London-2012-Olympics-Things-London-Olympics.html)

Unfortunately, most of the world is not like this. Even in the world of sports it becomes more complicated. Consider the NCAA basketball tournament. In this contest only pairwise contests are possible. Thus, a team is never competing against the entire field, but is instead competing against a single competitor. This means that for any given contest the performance of teams not in the contest are (at least for the moment) irrelevant. Thus, a measure of the variance in some trait, say team mean free-throw percentage, is irrelevant, whereas the difference in the means of the trait between the two teams is highly relevant. This has some interesting consequences. For example, it is not unusual for a low ranked team to put everything it has into the opening game and defeat a top ranked team only to get trounced in the next game because they don’t have the depth to continue at that level of play.


The results of the 2014 NCAA basketball tournament. Note that the championship was played between 7th ranked Connecticut and 8th ranked Kentucky. Also there are a number of interesting upsets, such as 3rd ranked Duke being defeated by 14th ranked Mercer. These anomylies indicate that summary statistics, such as variance, are not always predictive of the final results. (http://www.printyourbrackets.com/ncaa-march-madness-results-2014.html)

Turning to nature, it is the same thing. As far as our favorite gazelle is concerned it doesn’t matter how fast cheetahs run on average, or how fast the fastest cheetah runs. What matters is how fast the cheetah that is chasing it can run. In a large panmictic population with random interactions mean field summary statistics such as variance are indeed appropriate for predicting the response to selection it is very much like the race example I started with, and we are justified in calculating S as the covariance between relative fitness and the trait value. But what do we do when selection is taking the form of a tournament or interactions are local?

The easy solution that I have used is to assume that the population is structured using an island model of migration. This is a metapopulation in which each subpopulation has random mating and random interactions. In addition, migration among subpopulations is random, with no effect of distance on probability of migration. This is in effect a two level mean field approximation in which we can use variances to describe selection among individuals within subpopulations, and another set of variances to describe selection among subpopulations within the metapopulation. This is fine, and probably often a good approximation, but it is at least conceptually unsatisfying in continuous populations with localized interactions.

Another solution is to use the recently very popular network approach. I am honestly not sure how this works, so I will leave it to others rather than embarrassing myself. That said, I have concerns about this approach in the practical world of measuring plants and animals in natural populations where measuring connections may be difficult or impossible.

So, what is the answer here? The simple truth is that I don’t have one, but I do have some ideas. My thought is that we do something along the lines of a weighted mean and variance, and that we weight the variance by the probability that the interaction will occur. For example, if we have a continuous population the standard manner for calculating the variance in a particular trait in the population would be:

Variance eq 4

Two things to note: (1) yup, I am using the MLE formulation of variance, not the BLUE (Best Linear Unbiased Estimator) one. It may be biased, but from a theoretical perspective it is cleaner. And (2) pi can be thought of as the frequency of the ith type. My thought is that the pi can be any number, as long as it sums to one. Thus, we can replace pi with another value, say qi as long as it also sums to one. I suggest that we define a number, say kji = the probability that individual j interacts with individual i, and:

Variance eq 3

Thus, for each individual we would get a separate “local” mean and variance:

Variance eq 1


Variance eq 2

I will admit at this point I am stuck and running out of space, however, my thought is that we could similarly calculate a local variance for relative fitness and a local covariance between local relative fitness and local phenotype. Summing across individuals (S Zj) may well give a more meaningful estimate of the selection differential in the population. Estimating kji might be difficult, but perhaps reasonable estimates could be obtained using home range distributions, or other behavioral measures.

Actually, there are people who are much more adept at such things than I am so I am sure there is a better solution, but I just don’t know what it is.



On partitioning fitness

Posted: November 6th, 2014 by Charles Goodnight

I realized last week that I should probably explain more clearly what I meant by temporal components of fitness. As a result, a discussion of variance will have to wait until next week (I actually am flying to California for a National Academies Keck Futures Initiative meeting next week, so the next update may be two weeks).

What I am using is Arnold and Wade’s (1984, Evolution 38:709) episodes of selection paper. In this paper they partition episodes of selection using a very simple idea. They start with the multivariate breeders equation, but in reality there is no need for that. Lets forgo the opportunity to be cool and use matrices, and instead use the simple univariate breeders equation:

Screen Shot 2014-11-06 at 6.01.31 PM

Where (Yea, you know the drill): z̄’ is the mean phenotype between generations, VA is the additive genetic variance, VP is the phenotypic variance, and z̄* is the mean phenotype within generations due to selection.

Another point (yea, I know: boring) but one that bears repeating, is that z̄ etc. are real quantities with units attached to them. For example, the average weight of men in the United States is z̄ = 195.5 pounds. It is important to remember that, all equations aside, we really are talking about the real world!

From our perspective the important part of this equation is the selection differential S:

Screen Shot 2014-11-06 at 6.02.43 PM

What Arnold and Wade pointed out was that you could easily divide this selection vector into multiple episodes of selection by simply adding and subtracting the mean phenotype at various moments in the history of the organism:

Screen Shot 2014-11-06 at 6.02.53 PM

In other words, it is quite easy to divide an overall selection differential into multiple episodes of selection. So, that tells us about selection, but does not tell us about fitness. However, that is easy if we further expand our equation. In particular, the mean of the population after selection, but before reproduction, is found by taking the fitness-weighted sum of the traits:

Screen Shot 2014-11-06 at 6.03.04 PM

What can be seen from this is that indeed each episode of selection is associated with an episode specific relative fitness.

It is a bit involved, but Arnold and Wade show that absolute fitness can be partitioned easily as long as the absolute fitnesses are multiplicative. That is:

Screen Shot 2014-11-06 at 6.03.16 PM

where capital Wi etc. is the absolute fitness of the ith individual. Absolute fitness is the fitness of an individual on an arbitrary scale – for example, it might be the number of eggs a frog lays ranging from zero to the maximum clutch size possible – whereas the breeders equation uses relative fitness, which has a mean of one. For each episode of selection the relative fitness must be calculated as Screen Shot 2014-11-06 at 6.03.25 PM .

What it means for fitnesses to be multiplicative is that they need to be conditioned on previous events. Examples of appropriate conditional fitnesses would be:

W1 = the probability of being born alive

W2 = the probability of surviving to adulthood given that the individual was born alive

W3 = the probability of successfully mating given that the individual survived to adulthood.

W4 = the number of eggs the individual lays given it successfully mated

Note that in each case the fitness is a function of previous events. Thus, survival to adulthood is only measured on those individuals that are born alive, etc.

The beauty of this is that as I pointed out last week it is probably impossible to measure true fitness. What this paper tells us is that even if we can’t actually measure total fitness, as long as we are honest about what temporal component we measure, and the temporal component of fitness is multiplicative (e.g., we measure survival to adulthood only on live born individuals) we are safe. We simply need to make sure that we are aware that other episodes of selection, and other temporal fitness components affect the evolution of the trait. For example, Arnold and Wade measured selection for mating success in male frogs. They found that mating selection favored larger males. One could immediately ask why frogs aren’t huge. A reasonable answer would be that there are selection episodes other than mating that keeps the males size in check.

The net result is that we can define “fitness” in conceptual terms that cannot be realistically measured, and then when we actually need a definition that can be used in actual research or modeling just use a valid temporal fitness component and call it good. Last week I defined fitness as the probability of founding a lineage that persists until some arbitrary point (such as a speciation event) in the distant future. First off, there is an obvious problem with this definition for sexual species, so that was definitely not the last word in fitness definitions. But for the purposes of this discussion it is also a useless definition from a practical perspective. However, using this idea of partitioning fitness it is easy to see we can have this useless conceptual definition, and at the same time use a temporal component of fitness that Can be measured. For example, to the list above we can add an additional component of fitness:

W5 = the probability an egg gives rise to a lineage that survives to the arbitrary point in the future times the number of eggs laid.

Now we have our unworkable conceptual definition of fitness, and we can still, say focus on sexual selection and W3, the probability of successfully mating given that the individual survived to adulthood. In short, we can have our cake and eat it too. We can (although apparently I can’t) develop a comprehensive definition of fitness that will satisfy the most demanding critic, and still develop a temporal component of fitness that is usable in experimental settings.

A conversation with a physicist: Some thoughts on fitness

Posted: October 29th, 2014 by Charles Goodnight

This past week I went over to the university at a nearby city, and talked to some physicists interested in complex systems, and among other things, biology. As seems to be the nature of physicists turned biologists, I was impressed with some of their ideas, but also impressed with their lack of knowledge of biology, and, to be blunt, their view that biologists were basically inept when it came to theoretical issues. Perhaps the most impressive simple example was one (note no names, but an American, not a Brazilian) wanted to publish a paper showing that the Fisher’s fundamental theorem was not universal. Now that is a paper that would be met with a large yawn. Aside: I find FFT to be a fun mathematical truism, given a set of assumptions, but I seriously doubt anybody has taken its universality seriously in a very long time.

FFT equation 7

Fisher’s fundamental theorem is a mathematical truism, but only if the underlying assumptions are met. It is well known that these assumptions will only rarely be exactly met in real systems, and that this theorem should be used to guide intuition, rather than to make quantitative predictions.

However, there were two issues that came up that I want to cover more seriously. Today I will talk about fitness, and next week I will talk about variance in evolution.

I was telling this physicist about some of my ideas, and in the process talking about “fitness”. In the course of this discussion I was repeatedly told that unless we really understood terms we couldn’t proceed. So as a result I kept being more and more specific, sadly, about everything but fitness. It was only later that it dawned on me that he didn’t think I knew what I meant by fitness. I actually think that this is one of the problems that physicists have with biologists. They tend to come up with wacky examples of where a naïve concept they attribute to biologists doesn’t apply, and then say that because this definition does not apply in this case we have no idea what we are talking about. This is, of course, a bit disturbing for somebody who has published models in which naïve definitions of fitness don’t apply (e.g., Goodnight et al. 2008. Complexity 13(5): 23-44). In short, I think I have a pretty danged good idea of what I mean by fitness.


I might just know something about fitness!      bankai.    (from http://pt.wikipedia.org/wiki/Ichigo_Kurosaki)

What this physicist was missing is that when I talk about “fitness” it is in fact shorthand for “temporal component of fitness”. Please bear with me on this, since I have never attempted to define fitness in a way that I would apply to all situations. But if I were to attempt to define fitness it would define it to be something along the lines of:

Fitness is the probability that an organism will start a lineage that will persist for an arbitrary period into the future.

I think a good boundary would be speciation, thus, I might narrow the definition of fitness to be:

Fitness is the probability that an organism will start a lineage that will eventually participate in the founding of a new species.

Obviously, this is an unworkable definition, but still, there are reasons why a truly general definition would need to be something along these lines. For example in the paper cited above (Goodnight et al. 2008. Complexity 13(5): 23-44) we examined the dynamics of a predator that was subject to mutations with how aggressive it was. What we found was that the optimal aggressiveness was a balance between growing quickly, but not so quickly that a lineage exhausted its prey and went extinct. The interesting thing was that in the short term these “optimal” lineages were always invasible by a more aggressive mutation, but in the long term these more aggressive lineages always went extinct. The point is that very aggressive lineages had an apparent high fitness over the course of a few generations, but a low fitness over a longer term. A true definition of fitness would have to incorporate this.


long term pred prey study

τ is a measure of aggressiveness, or how quickly the predator consumes the prey. Note that in this example a highly aggressive (cyan and dark blue) predator appears, but eventually burns out and goes extinct. (Figure from Goodnight et al. 2008. Complexity 13(5): 23-44.)

So, how do we deal with this obviously unworkable definition of fitness? The answer that my colleague did not understand is that we work with components of fitness. As Arnold and Wade (1984. Evolution 38: 709-718)pointed out so long ago, as long as episodes of selection are described in a multiplicative manner (that is conditional probabilities) it is valid to study a component of selection. In the predator prey example, it is perfectly valid to define a component of fitness that is lifetime reproductive success. If you did this you would discover that for this fitness component, the “fitness” of an aggressive mutant was very high. A biologist who chose their words very carefully would acknowledge that there may well be other later selective events that would counter the effects of this fitness component, but that would not invalidate conclusions about differences within lifetime fitnesses.

Since we are being careful, there are caveats even here. That is, technically the selection events have to be independent, and that will often not be the case. I think that this is another issue between biologists and physicists, however. That is, that in theoretical physics there is this concept of an exact solution. I seriously doubt that any theoretical biologist believes that we could develop a model that gave an exact solution in a living system. It is simply too complicated. Thus, as long as different episodes of selection, and thus temporal components of fitness, are not too highly correlated the assumption of independence will not hurt the ears of biologists too badly.


Unless you are really sensitive, a small amount of non-independence among selection episodes should not hurt your ears. (http://factsaboutbirds.blogspot.com.br/2010/08/desert-animals-list.html)

So, in sum, is life time reproductive success “fitness”? The answer is clearly no, since multigenerational processes can also enter into the equation. Instead, we need to think of it as a temporal fitness component. Am I apologetic about frequently referring to life time reproductive success as fitness, even though I know it is technically incorrect? No, I am not. Referring to it as something else is cumbersome, and honestly, in nearly every experimental system I am aware of, it is the major component of true fitness. It is basically a convenient shorthand and something that is usually pretty close to the truth.


In “truth” as in horseshoes sometimes close is good enough.  (http://lrossentertainment.wordpress.com/tag/outdoor-games-wedding-ideas/ photo credit, Sam Beebe, Ecotrust)

Spider Group Selection

Posted: October 22nd, 2014 by Charles Goodnight

I could keep going on what an individual is, but at least at this point I have put out the main points I have been thinking about. It may be a subject we will return to in the future. What I really should talk about a bit is our recent paper, “Site-specific group selection drives locally adapted group compositions” (Pruitt and Goodnight, Nature 514:359–362).

I am actually a bit surprised that it has gotten so much positive press with so little backlash. But with a very few exceptions we are not being dismissed. I guess times have changed. I am actually not so much going to describe the study so much as talk about what we demonstrated and why it is in actuality a fairly small step.

First a disclaimer: This is John Pruitt’s project. I was honored when he asked me to participate in the analysis of the data, and sometimes I am not sure I did a lot more than provide cover so that he could talk about multilevel selection.

As with all science, this study is but a small step in a long line of studies.


Two very apt adages “the cutting edge of science is dull” and more positively “if I can see so far it is because I stand on the shoulders of giants.” I have also heard this second being “if I can see so far it is because I stand on a mound of midgets”. It seems to me that all three are apt. The first because science proceeds in small steps. The giant and midget analogies are correct because there are giants (Darwin, Fisher Wright and the like) but advancing science also depends on midgets like you and I. (left: http://www.telegraph.co.uk/travel/travelnews/10088510/Bland-reaches-out-to-Dull-and-Boring.html, right: http://www.stellabooks.com/articles/dr_seuss.php )

The reason for that long-winded preamble is that in reality we did a relatively small thing. First some history. It has long been demonstrated that there were serious flaws in the reasoning of theoreticians dismissing group selection (Wade 1978, Quart. Rev. Biol. 53:101-114 – this paper is remarkable in how prescient it is given that it was published nearly 40 years ago). At the same time Wade was also doing the first studies showing that group selection worked in the lab (Wade 1977 Evolution 31:134-153). Later studies would demonstrate that indeed Wade had been correct in his quarterly review article: group selection could act on interactions among individuals in a way that simply was not available to individual selection (see Goodnight and Stevens 1997 Amer. Natur. 150:S59-S79), and a series of selection experiments led to the wide-spread adoption of group selection as a means of live stock improvement (e.g., Muir 1996 75:447-458). Thus, by this time we really have answered the early questions: In the lab group selection works, and it works so well because it can act on interactions among individuals. The next question was whether group selection IS acting in nature. The methodology for this was brilliantly provided by Heisler and Damuth (1987 Amer. Natur. 130:582-602), and promptly ignored. Fortunately, in recent years this method has been rehabilitated, and shown to work both in theory and in action. The results of these efforts is that, although not enough cases have been examined, it is pretty clear that group selection is at least not uncommon. To give you some idea about how common it might be, (1) recognize that theoretically we can show that soft selection is a mixture of group and individual selection (Goodnight, Schwartz and Stevens 1992, Amer. Natur. 140:743-761), (2) the constant yield law (Weiner and Freckleton 2010 Ann. Rev. Ecol. Evol., & Syst. 41:173-192) is almost universal in plants and (3) the constant yield law is a form of soft selection. Thus, group selection may indeed be extremely common in nature.

So, in some sense the main questions are answered: The models were wrong, group selection works, and it may be very common in nature. What was missing is that up to this point we did not actually have an example of an adaptation that was unequivocally a result of group selection. That is what Pruitt and Goodnight provided. This is no small feat. Consider how many solid examples we have of adaptation due to individual selection. There are a lot of traits that almost certainly are, but have never been demonstrated to be individual level adaptations. Examples of these are things like our hearts. Clearly an adaptation, but has it ever REALLY been demonstrated to be so? The examples we do have are few and far between, and all represent a huge amount of work on somebodies part. I am thinking of examples like the evolution of lead tolerance in grasses (Antonovics and Bradshaw 1970 Heredity 25:349-362) peppered moths (Kettlewell and later Majerus), and beak size in Galapagos finches (Peter and Rosemary Grant).


Social spiders have a group level adaptation that is the proportion of the colony that is aggressive vs. the proportion that is more docile. Populations have evolved to display a mix of these two proportions that is apparently optimal for their local environment. When the distribution is experimentally adjusted it always returns to the evolved ratio regardless of the environment in which they are raised. By the way, this strikes me as grasshopper hell. (From http://www.readcube.com/articles/10.1038/nature13755 — this is a nice writeup by Tim Linksvayer. I recommend it if you have not already seen it.)

Notice that in the cases where we have good evidence of evolution by natural selection there is environmental variation. In the case of lead tolerance there are mine tailings such that soils with high lead content are adjacent to pristine land lacking the metal. In the case of the moths and birds there is sufficient data over a long enough period of time that we can see adaptation occurring as the environment changes. Pruitt’s spiders were closer to the mine tailings scenario. That is there were different local environments that had different optimal mixtures of spider behaviors, and we were able to show that the spiders are locally adapted, and regardless of the environment we placed them in, they always adjusted their colony to reflect the adaptive mixture of behaviors in the environment they evolved in. Thus, Antonovics and Bradshaw were able to show that “normal” plants were unable to grow well in lead poisoned soil, we showed that spiders are unable to adjust their mix of behaviors when placed in the “wrong” local environment.

In reality, then, that is all we really did. We showed that these spiders have a group level adaptation to local conditions. Since it is a property of the group that cannot be meaningfully measured on individuals, it really is a group level adaptation, and since it is non-plastic it is a heritable change. This is the last of the qualitative issues to be addressed about group selection. Group selection works in theory, group selection works in the lab and in agriculture, we see group selection in nature, and now we know that group selection leads to adaptive change in nature. What is left is to work out the quantitative questions, such as how common is group selection relative to lower levels of selection, under what circumstances is it sufficiently powerful (and traits sufficiently heritable) that it leads to adaptations that can be attributed to evolution by group selection.

Individuals as Multispecies Entities

Posted: October 15th, 2014 by Charles Goodnight

I am feeling a bit schizophrenic these days, splitting my time between developing models of species differentiation, experimental design to measure the genetics of species differentiation – yes, it can be done, and blogging. The last is a bit confusing because it starts me wondering, when did I become a philosopher.   And lets just say, there is a reason that they give PhDs for philosophy, and a reason I don’t have one.

Any way on the philosophy issue, today I want to clean up and finish up my discussion of individuality. Hopefully this will be short and sweet.

Here is the issue: I have defined individuality in terms of selection and evolution. I stand by those definitions, but it occurs to me that this does not fit well with our concept of organism. Consider the point I raised before which is that if we choose selection as defining the individual, in many, if not most cases, we will logically assign fitness at what we commonly call the organism. However, what we colloquially call organisms, it is now becoming apparent, are not a single species.



Revisiting an issue raised a few weeks ago, when studying selection on running speed in cheetahs it makes sense to assign fitness at the level of the organism. In this case the “organism” is a multispecies assembly. (http://theruniverse.com/2012/07/running-tips-from-cheetahs/)

It may be that there are a bunch of commensals and parasites inside that organism, but it is still the cheetah that eats or doesn’t eat, and reproduces or doesn’t reproduce. For the rest of this post I will use commensals to mean all of the species associated with a metazoan, including parasites and symbionts.  Remember, selection is blind as to the causes of the phenotype.   If our cheetah cannot run fast because it has a bad mix of gut bacteria (and a belly ache?) it still doesn’t get to eat, and it still starves. Thus, our selection definition of fitness is in this case identifying the assembly of species that is a cheetah as the “individual”.

Fine, you may argue, but it is the cheetah that survives and reproduces, and as such we ignore the rest as environment. The philosophical problem with this is that it is exactly the same issue with the gene-centered view. The gut bacteria may be along for the ride, but they are there. To ignore them is the same as in the gene-centered view ignoring other loci even though they are interacting with the locus you are focusing on. We know that doesn’t work. I would argue that the reason we know the reductionism to only the colony of “cheetah” cells doesn’t work is because nobody has ever bothered to ask the question.

There is, however, another issue. I would also argue that many of these commensals ARE heritable. Remember using the phenotypic view I consider “heritability” referring only to genes to be far too narrow. Basically many of the commensals ARE heritable. At one extreme, we can consider mitochondria to be “commensals”. The only reason that we normally don’t is that they are strictly vertically transmitted, and they have coevolved with their host cells to the point that neither can survive without the other (except, the case of chloroplasts and nudibranchs )


The emerald green sea slug, Elysia chlorotica, isolates cholorplasts from algae and incorporates them into their own cells, using them for photosynthesis. (from http://scienceblogs.com/notrocketscience/2008/12/28/solarpowered-green-sea-slug-steals-ability-to-photosynthesis/)

The case of mitochondria is not really so different from that of Wolbachia, except that insects cured of Wolbachia typically survive, whereas “curing” animals of mitochondria is probably not a good idea! (OK, our cheetah neither uses chloroplasts, nor has a wolbachia infection, hopefully you get the point).

Other commensals have a looser association, nevertheless, in the context of the phenotypic approach they have to be considered heritable. For example, an old study (Bettelheim, Breadon, Fairs, O’Farell and Shooter 1974, J. Hyg. Camb 72:67) found that in most cases mothers and their babies had the same serotypes of E. coli. One can imagine similar “inheritance” of even ectoparasites such as fleas (mother cheetahs infecting their offspring). Thus, yes, many of these commensals WILL be part of the patterning node. Others will be more opportunistic infections and thus nonheritable, but that does not negate the fact that many are indeed heritable.

So, that leaves us with the reality that in many cases “individuals” will be multispecies entities. You are not just you, you are you and your pets. Lest this bother you, please remember that commensals CAN affect your behavior. I have a friend that swears that an amoeba infection he got caused him to become depressed, and toxoplasmosis can apparently make people more reckless (and make the cat litter box less offensive). So, yes, parasites do affect the way you think, and no, even your behavior is not just your own.

So, here is my proposal. I think we do need a name for a colony of cells of the same species. I suggest we define “organism” as a colony of cells derived from a single cell and physically connected to each other.   In this view the cheetah “organism” would be the set of cells derived from the fertilized egg from the mother cheetah. On the other hand, the cheetah “individual” would be the actual creature that includes the organism and its commensals. I am not sure that this will ultimately end up being a good idea, but hopefully it is a good place to get the discussion started.


No, not that cheetah, I am talking about the one that can run fast. (from http://www.lavantis.com/2011/12/cheetah-chimp-of-tarzan-dies-at-80/)
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