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Evolution in Structured Populations

A quick review of the phenotypic perspective, pt 1

Posted: August 11th, 2015 by Charles Goodnight

Some of the recent comments I have received have made me realize that maybe I should re-emphasize some of the very early points I made on this blog. The point of this blog is to blatantly promote a phenotypic view of evolution, and do try to dislodge the dominant paradigm of the gene as the center of evolution. In the discussion that follows it is convenient to use Dawkins as a straw man. My own feeling, based on no evidence, is that most evolutionary biologists accept Fisher as a brilliant founder of modern genetics, and accept his as a very genic view of evolution. Interestingly, Dawkins perspective, working through the lens of Williams, is the logical outcome of taking Fisher’s work to its extremes. So, just as I feel most evolutionary biologists accept Fisher, I feel that that they are deeply uncomfortable with Dawkins, but most of these biologists would have trouble articulating exactly why Dawkins is wrong. Somewhere between Fisher’s deeply mathematical prose and Dawkins polemics something has gone awry. My feelings are that where Dawkins goes astray is very fundamental, and goes all the way back to Fisher. Basically Fisher imagined a genetical world that was a reasonable abstraction for a world in which we had no idea what a gene was, we were at the very beginnings of our understanding of inheritance, and we lacked the computational machinery to do anything more than relatively simple analytical models.

Another reading of Fisher, however, is that quantitative genetics is fundamentally a phenotypic model. The average offspring is the mean of the parents, but the loss of variation due to averaging is recovered in the form of within family variation. We can interpret Fisher’s book is an example of the phenotypic view of evolution that is illustrated using a simple Mendelian model of genetics, but which can be expanded as necessary and as computational power allows. Viewed in this way the phenotypic perspective I am advocating may be more of a descendent of Fisher’s legacy than the more classical genic view.

Matt Foley

Just a bit of self promotion, and maybe a bit of motivational speaking (http://gallery4share.com/c/chris-farley-snl-matt-foley.html)

So here are some of the relevant points:

1) Phenotypes create new phenotypes: At first blush this is just a change in perspective. At the risk of setting up a Dawkinsonian straw man, the classic genic view is that genes make copies of themselves, and use phenotypes as a mechanism to protect themselves, and help them survive to the next generation. This is why Dawkins refers to DNA as “immortal coils”. In the phenotypic perspective parent phenotypes create offspring phenotypes using “transition equations”. These transition equations are accepted to be impossibly complex, and so we accept at the outset that the best we can do are approximations. The simplest approximation to a transition equation is probably the heritability of quantitative genetics, or the simple Mendelian math of a Punnett square, however, in many situations it will be useful to add complications ranging from maternal inheritance and indirect genetic effects, to epigenetic effects, and all the way up to cultural effects.

What this change of perspective buys us is that genes are no longer the center of evolution. There are no such things as vehicles and replicators. These are the construct of a fevered mind that deeply misunderstands evolution. Instead, genes are relegated to being a prominent, but certainly not the only, contributor to the transition equation. This leaves the transition equation as an open ended construct that can incorporate new scientific findings. Rather than having to totally reconstruct our understanding of evolution every time we come up with a new mode of inheritance, we simply need to recognize that the transition equation was more complex than we had originally thought, and we need to modify that equation appropriately.

2) Some aspects of our understanding of evolution change with a shift to a phenotypic perspective, but our basic understanding remains remarkably similar. There have been many definitions of evolution, some of which have relied on a genic view. For example, a classic definition is that evolution is change in gene frequency. Re-framing our understanding to a phenotypic perspective demands a careful rethinking of what we mean by evolution. My own definition is evolution is the change in the distribution of phenotypes in a population due to the gain or loss of individuals. This definition is consistent with phenotypically oriented classic definitions, but ends up being more specific in many ways.

Classically there have been four forces of evolution  that have been identified: mutation, migration, selection and drift. These have mostly been defined in genetic terms. Thus, drift is often called “genetic drift”, mutation is discussed in terms of change in DNA structure.   However, these terms can be defined and discussed in phenotypic terms without reference to the specifics of the underlying mechanisms of inheritance. Clearly migration and selection do not need reference to genes, and our understanding of them really does not need to change at all. From a phenotypic perspective “mutation” need not be genetic change. It can be any change that randomly alters the phenotype of an individual, and that does not correlate with fitness. There is the interesting caveat here, however, that based on our definition of evolution, such random changes do not become “evolution” until they are passed on to offspring. Similarly, drift can be viewed as a change in phenotype frequencies due to the random gain or loss of individuals. With the phenotypic perspective, however, a fifth force must be recognized. This force is easily ignored in the genic world, but cannot be ignored in the phenotypic perspective. This is force is secular environmental change. A lasting change in the environment, such as global warming, can change the distribution of phenotypes directly, and in at least some cases it will be an intergenerational event. For example, global warming is changing sex ratios in some reptiles. If we assume an individuals sex is fixed at hatching, then indeed this change in the distribution of males and females is an evolutionary change by our definition.

I seem to have run into my self imposed thousand word limit, so I will continue this review next week.


Down the rabbit hole: More on multispecies organisms

Posted: July 23rd, 2015 by Charles Goodnight

I just tripped and fell down another rabbit hole. I was going to skip this week, but I would love input on this issue, so here it is. Earlier I argued that the organism was a multispecies entity. This makes perfect sense if we consider mitochondria to be symbiotic bacteria in a host cell, and we talk about the microbiome. Now here is the question: If you catch the flu, or get a bacterial infection (to keep it cellular), is that disease part of you as an organism?


Dang another rabbit hole.

There are two important points to remember. First, in the phenotypic view I am advocating considering the phenotype to be a vector through time, with every trait (a measured aspect of the phenotype) having a time element. Thus, it is not my weight, but my weight when I am 19001055824 seconds old (that is approximately how old I am while writing this). This means that even very temporary things such as whether you are inhaling or exhaling is technically a valid trait. Thus, if you have a fever of 104 degrees on a Saturday morning, that is the value of the trait “body temperature”  at that particular moment. The question is, do we make a distinction because that temperature is “caused” by a flu virus? The truth is I am beginning to believe we cannot make that distinction.

Taking a clearer example. Consider a person who chooses to dye their hair purple. This color comes out of a bottle, and it is no sense genetic or otherwise heritable (well, maybe in some odd cultural sense). That said, it is part of the phenotype. If you were to categorize people by the trait “hair color”. this person would go into the “purple”. Thus, it is a valid trait, and a valid part of their phenotype. How do we deal with this? I would argue that the best way would to consider the bottle of hair color to be a non-heritable or environmental influence on the phenotype. By analogy, I think it is perfectly reasonable to suggest your 104-degree fever is also part of your phenotype.

purple hair

This woman has a purple hair. It is certainly part of her phenotype, but probably not heritable.   (from http://darkuro.tumblr.com/)

So, your fever is part of your phenotype, but is the virus part of you as an organism? Certainly, we would not consider the bottle of hair color to be part of an organism. It is an external aspect of the environment that changes your hair color. Cold air temperatures may cause you to put on a coat (the coat wearing trait?), but it is certainly not part of your body. However, the virus differs here. It is IN your body, and in fact it is in your cells.

Consider our microbiome. There certainly are aspects of the microbiome that are acquired from our parents, either at birth, or because we live next to them as infants, and many of these we will pass on to our children.   Thus, they are heritable from the phenotypic perspective. However, others are picked up late in life, perhaps when we temporarily change our diet, and then lost again, perhaps when we revert to our old diet, and are not heritable. I think a strong argument can be made that this microbiome should be considered part of the multispecies organism: Selection acts on the whole organism; outside of perhaps prokaryotes, single species organisms don’t exist; as far as I know, animals cannot survive without their symbionts. From an experimental perspective, it is difficult or impossible to separate symbionts that are heritable from those that are non-heritable, and perhaps more important both can have significant effects on our phenotypes in ways that can affect our fitness. Thus, I think it can be argued that all aspects of the microbiome, whether heritable or not, should be considered part of the organism. Nor does it makes sense to me to argue that there is a minimum residence time before a symbiont or disease should be considered part of the organism. Such a waiting time is necessarily arbitrary, and as a result there will always be situations that are ambiguous.

Now comes the question: Should we make the distinction between the bacteria that we picked up on vacation that makes it easier to digest shrimp from another bacteria that gives us diarrhea? I cannot think of a criterion that does not require special pleading that incorporates the former, but not the latter into the organism.

One final caveat is that it is important to remember that the most appropriate unit to assign fitness depends on the trait being investigated. Thus, the colony might be the appropriate unit if we are examining colony defense, the organism if we are examining foraging behavior, and the cell if we are examining cancer. Perhaps the organism is best thought of as being equally fluid. A flu infection is an assault on our bodies, thus if our trait is immune response, maybe the organism is everything but the flu virus, whereas if we are looking at body temperature the organism is everything including the flu virus. This is a bit of a conundrum for me, and I am happy to get any feedback that anybody else may have.


Some thoughts on aging and the phenotype

Posted: July 16th, 2015 by Charles Goodnight

I have been gone a while. Something of a creative meltdown after the Evolution meetings. Perhaps one to many Caparinha, at what might have been the best party ever at an Evolution meeting. Leave it to the Brazilians to throw a party with enough food and liquor and the wackiest live music ever. In any case, I am back in Vermont, eating kale and other healthy things that us aging hippies do, and thinking about the pain of having to ply my profession as a teacher. Time to get back to work.


Admittedly a terrible picture, but the party was one of the best.   If you missed the Evolution meetings this year you made a mistake.

Jake Moorad started a discussion with me about how aging affects individuality. My first thought was that I have no idea. As usual, such an answer means that it is a really interesting question. After giving it more thought I have come to realize that it has no effect at all. After all, an individual is the level at which we assign fitness, which is potentially quite arbitrary. In most cases the “individual” will be an organism and its associated symbionts. Thus, despite the fact that an individual changes as it ages, I think it should have little influence on what we call an individual. What it does change, however, is it complicates what we think of as the phenotype. The fact that the phenotype changes over time is not a trivial issue, and it is one that needs to be given some attention.

How to view the phenotype as a vector through time is a topic I have discussed before, and one that is a general issue for the phenotypic view of evolution. My solution is to treat the phenotype of an individual as a vector through time that begins at formation of the individual, and ends at their dissolution. There are a couple of interesting things in that sentence. Note that I am specifying the “individual”. It seems to me that, although perhaps not essential, it makes sense to assign phenotypes at the same level at which we assign fitness. I could be argued out of that, but for the moment it seems right (which is not terribly convincing to me, let alone anybody else). Second, I speak of the “formation of the individual” and “dissolution” of the individual. If we assign fitness at the level of the organism, this will be at conception or birth, depending on your perspective, and ends when the organism dies. But remember two things: First, if we assign fitness at a level other than the organism, “birth” and death may not be an appropriate terms. However formation and dissolution will always be appropriate, since our definition of evolution involves the gain and loss of evolution, individuals must of necessity have a beginning and an ending if they are to be considered to evolve.

If we take a classic genic perspective of the individual as a single species organism it is easy to ignore the time dependent aspects of the phenotype. Most importantly under a genic perspective genes are the only important heritable effects on the phenotype. These all enter at the time of formation (birth) and are unchanging through time (this last ignores somatic mutations of course).   This is not true from a phenotypic perspective. Culture is a prime example. The language you speak whether you are most comfortable with a fork and knife or with chopsticks are all heritable aspects of phenotype that are added after birth. Similarly, your cultural parents will not necessarily be the same as your genetic parents. Children learn their earliest language from their parents, but the vast majority of their language comes from peers and children that are slightly older than they are. This acquisition of heritable elements is not limited to culture, of course. We acquire much of our microbiome from our parents, and other individuals with which we live, and in many social insects, such as termites, trophallaxis is essential for their survival.

Screen Shot 2015-04-20 at 12.25.47 PM

The phenotype as a vector through time. Effects entering from the top are possibly heritable inputs, effects leaving below the lines are products of the phenotype. Note that the phenotype ends at death. This is why products such as beaver dams and human produced things like books are part of the phenotype, and not part of the distended (or is it extended, I always get it wrong) phenotype.

However, it is not just the acquisition of heritable elements that affect the need to consider the time element in the phenotype. Traits also change over time.   At a simple level every trait will have a time element. One way that this is handled is to simply to measure traits at a time when they are stabilized. For example, most vertebrates have targeted growth. Thus, there is a period between when adulthood is reached, and before senescent decline that traits are stable enough that we can effectively ignore the time element. In reality, of course, we should always include the age component, thus, it should be considered weight at age X, not adult body weight. However, speculating about how things ought to be done is different than doing things, and well, I for one will not be angry at people who simply measure adult body weight.

This does raise one additional interesting point. That is we can measure the time element of a trait to whatever precision we choose. Thus, in principle we could measure a trait such as whether an animal is inhaling or exhaling, or whether their heart is in systole or diastole. From an evolutionary perspective it would be silly to measure such highly time dependent transient traits, nevertheless it emphasizes the point that traits are aspects of the phenotype we choose to measure, and as such can be measured to whatever precision is appropriate.

More interesting, however, is the expression of traits with non-genetic inheritance. In some instances traits might not be expressed at all until the causal elements are acquired. For example, the trait of speaking a language cannot be expressed until the language is learned. Language is not acquired at birth, and if it is not the mother tongue, it may be acquired quite late in life. Further, if this person goes on to teach others their newly acquired language we can say that they have a heritable trait (the language understanding) that they acquired late in life.

Finally, some traits we may be interested in might be rate of change over time. Such traits might be the slope of the decline in fertility with age since puberty. To be honest I am not sure of the language to use to describe such traits, since such traits are explicitly incorporating the time element, and do not fit well with my phenotype as vector metaphor. If anybody has any ideas I would be pleased to hear them.

The point of this is that the evolution of aging remains an important issue. However, I am inclined to think it will have at most a small effect on our understanding of individuality. Where I think it will have its big impact will be on how we think about the phenotype, and the necessity to think of traits as being age dependent.



Why there is no Genic Selection

Posted: June 17th, 2015 by Charles Goodnight

This is the week before the Evolution meetings, so the big question of the day is what can I post that I believe to be true, and will rile enough people up to get a good discussion going. I decided Sam Scheiner was a good target – we were graduate students together, he is a good guy, and a great scientist at NSF. BUT, one day on an online discussion, I don’t remember where, he took umbrage at my statement that there is no such thing as genic selection. So, that is today’s thesis: There is no selection on genes. I will actually soften that a little, and bring up a special case that is indeed genic selection.

Sam Scheiner

Sam Schiener – Currently at NSF, long ago a fellow graduate student with me (Sadly, all my incriminating photos are in Vermont). We spent long hours hanging out and working in the Barnes Greenhouses, which have since been torn down. (left: https://www.researchgate.net/profile/Samuel_Scheiner right: http://ian.umces.edu/blog/2011/02/19/more-randy-alberte-memories/ )

First, an important philosophical point. As I have probably said more times than necessary, quantitative geneticists divide evolution by natural selection into selection and the response to selection.   Selection is an ecological process that has no reference to whether or not a trait is heritable. As an extreme example, consider a situation in which one person is painting random numbers on the backs of turtles, and a second investigator, seeing the numbers decides to select for those turtles with the largest numbers on their backs. This is selection, even though it obviously has no genetic basis. More realistic traits can have a heritability anywhere from 0 to 1, so there can be no logical cutoff where we say it is no longer selection.   This is of more than philosophical importance. It is of practical importance. The ecological process of selection really is only studied in nature.   That is, we, as evolutionary biologists, are mainly interested in the selective forces acting in natural natural populations in natural settings. Further, the appropriate way of studying selection in nature is to use the regression approaches of Lande, Arnold and Wade (Lande & Arnold. 1983. Evolution 37: 1210-1226; Arnold & Wade. 1984. Evolution 38: 709-718). In contrast, the response to selection is a function of genetics. From a practical perspective heritabilities are measured using breeding designs and statistically comparing relatives.  In these designs, the mating structure is forced, and comparisons are most easily done in the laboratory (e.g., Falconer and Mac Kay 1996, introduction to quantitative genetics). Finally, there will be numerous situations where we will be interested the measurement of selection acting on interesting organisms or in interesting situations, but we have no knowledge of the underlying heritability of the traits.  The distinction between selection and the response to selection tells us that such studies are interesting even without simultaneously doing genetic studies.

Thus, for both philosophical practical reasons it is reasonable to separate selection from the response to selection. So, what is the point of this? Genic selectionists are arguing that we can act AS IF selection were acting on genes. As I have argued in the past doing this is fraught with dangers. But more importantly, we have to ask the question do we want to play “as if” games if we are scientists. It would be great for the molecular biologist to be able to assign fitnesses to individual alleles, but they are interested in the genes, not in how selection is working. Such reductionism is very suspect, and, if it does work, probably working for the wrong reasons, so even if your idea of understanding evolution is ignoring the actual process, and just focusing on the change of gene frequencies you are likely to be disappointed.

If on the other hand, we are interested in how selection is acting, then “as if” doesn’t cut it. We want to know where selection is acting and such reductionism tells us nothing about the ecology of how selection actually works. For this we need to study selection, not its consequences.  When we talk about selection it is best to always identify an “among” and a “within”. The “among” is what level of organization selection is acting on, and “within” is what the range, or scope, of selection is. Thus we can have selection among groups within a metapopulation, selection among organisms within a population, selection among cells within an organism. The point is, to say that even if selection at different levels causes the same change in gene frequency (as I keep saying, they don’t) is to say they have the same consequence, not to say that they are the same thing.

So what about genic selection? Well, first off we need an among and a within. The among is pretty obvious, it is alleles. But what is the among within? Except in special cases I will get to, the among within cannot be the organism. Loci and thus alleles are always grouped into genomes and genomes are properties of cells (ignoring viruses here). Further, mitosis stops within genome selection. If you are heterozygous today, you will be heterozygous tomorrow. This is easily handled by contextual analysis. Simply put, because of mitosis, there is no variation in fitness among genes within the genome. Even if we assign fitness at the level of the allele, there can be no variation in fitness among the alleles within the genome, and indeed the lowest level at which it is possible for there to be variation in fitness is at the level of the cell. Thus the lowest level at which it is even theoretically possible for selection to act is at the level of the cell.


A Genic selection. Which is the best Gene? All are good, but the correct answer is Gene Kelly. He is well known as a jumping Gene. Top row (left to right) Gene Wilder, Gene Hackman, Gene Simmons. Bottom row (left to right) Gene Kelly, Gene Autry, Gene Tierney.

Of course there is an exception to this, and that is transposable elements. Now you can have selection among transposons within the genome. That is because each insertion site can be thought of as a bit of habitat for the transposon to insert into, and for most transposons a transposition event is a form of reproduction. Thus, selection among transposons within the genome is a special case example of genic selection. Lest you want to revel in having finally justified the term genic selection be aware that having high transposition rates is usually not good for the organismal phenotype, so there is selection at the level of the organism against jumping genes. Not only is this an example of genic selection, it is also an example of multilevel selection in which the gene is the individual, and the organism is the group. And, McClintock help us, genes that “choose” not to jump are being altruists (uggh.  Hate that term).



Why reductionism DOES work: Individuals to genes

Posted: June 4th, 2015 by Charles Goodnight

In the last couple of posts I have suggested that reductionism is for chumps. Two weeks ago I argued that gene interactions made average effects wonder around all over the place, and last week I argued that indirect genetic effects mucked up the works if there was population structure. This would seem to imply that quantitative genetics doesn’t work. Tell that to anybody who works in the agricultural breeding industry and they will laugh at you. Possibly more than any other field you can take quantitative genetics to the bank. You want lean pork, more marbling in your steak, more lysine in your corn? Quantitative genetics will do it for you. Furthermore, heritability estimates are technically only valid for the generation in which they are measured, however, the reality is that the common rule of thumb is that they are generally usable for 10 or more generations, and often appear to be pretty close after 100 generations. So, if reductionism doesn’t work why does quantitative genetics work so well? I will argue that as may be true of many complex systems, it works for the wrong reasons.

1. Within populations genetic effects will tend to be additive for statistical reasons.

In other words, selection and drift make gene interactions go away. Yes, genetic drift and selection can cause the additive genetic variance to increase, but it happens by statistically depleting the epistatic variance. After only a few generations of small population size the population can be treated as if there was no epistasis. In other words, as long as you stay within populations reductionism often provides a fairly accurate picture of the world. But you need to be careful. Another population may also act additively, but it will be a different additivity with alleles having different effects on the phenotype.

VA by generation

Twenty five generations of brother sister mating starting with equal amounts of AXA epistasis and additive genetic variance. The small population size increases the additive genetic variance, but also decreases the epistatic genetic variance, which is the difference between the green and red lines. After only a few generations of small population size there is very little epistatic variance available.

2. In a well connected metapopulation you may not see much differentiation for local average effects.

The way to detect epistasis is to look among populations either by examining the variance in local average effects (Goodnight 2000. Heredity 84:587), or the variance in local breeding values (Goodnight 1995, Evolution 49:502). This runs into two problems. First, nobody ever listens to me, so this experiment has been exactly twice (plus one in progress unpublished experiment) (De Brito, et al. 2005. Evolution 59: 2333, Drury & Wade 2011. JEB 24:168), and second, the variance in local breeding values is a function of the migration rate among subpopulations within the metapopulation.

I have not published this work, so I am violating my personal rule to not put unpublished results on my blog, but I think this is relevant, and it is part of a much larger model on speciation. Lets just say there is more than one paper coming out of this model, I am coming off of sabbatical, and well, it might be a year before this part gets written up. In any case If we look at a single metapopulation with an infinite number of demes. By the way, infinite demes is an assumption that is very suspect. For example, the approximation that equation works pretty well if there are infinite demes, but falls apart in finite metapopulations (yea, that is another paper that will out of this model. . . ).

With that in mind if we look at the variance in local breeding values as a function of Nm it becomes apparent that in order to get a significant variance in local breeding values migration rates need to be relatively low.


The effect of migration on the variance in local breeding values. Gene interaction is much more detectable among populations than within populations.   The appropriate measure being the variance in local breeding values or the variance in local average effects. Shown here is the variance in local breeding values as a function of Nm, or the number of migrants entering a deme per generation.   Note that migration rates must be below one migrant per generation before variance in local breeding values will be statistically detectable. Green dots (upper left corner) is zero migration, red dots are migration rates of 0.005. Scatter is due to different deme sizes. VAA = 1, VA = 0, generation 30,000.

To see the interplay between migration rate and deme size a three dimensional graph may help:

3d Graph

3d graph of the variance in local breeding values as a function of migration rate (M) and deme size (N). Unfortunately, JMP does not render surfaces exactly correctly. The graph should reach up to a value of 2 for zero migration.

There are a few caveats. First this is a drift model. There is no selection. If selection were to be added (good luck with that) I would speculate that selection against migrants offspring (e.g., hybrids) would mean much higher levels of population differentiation. Second, this model uses island model migration with infinite number of demes. Isolation by distance would dramatically increase the population differentiation and allow detectable gene interactions at much higher migration rates.

So this puts us in a relatively interesting situation. The models of drift and selection within demes tells that epistasis will be difficult to detect, and models of migration among demes tells us that migration rates above about 1 migrant per generation will also make gene interactions difficult to detect. Thus migration has the effect of tying the population together, and as a result preserving a lot of alleles.   The larger the metapopulations and the more the migration the more overall number alleles that will be preserved. However, such situations are ripe to explode if migration is ever restricted, or two metapopulations are separated. The variation is there, and thus no measurable epistasis, but once the populations are separated those interactions will pop out of hiding and show up again between species where migration rates are lower or non-existent. This would argue that again, the additive model is working for the wrong reasons. Just because gene interaction is statistically hard to detect doesn’t mean it isn’t there. It may simply mean that the conditions are such that it is hidden.


Not a whole lot of ice from my view from the ships deck! (http://blogdasa.com/2012/12/27/5-documentarios-que-me-tornaram-uma-pessoa-mais-bacana/)

One last thought. This also argues that it is reasonable to speculate that Dobzhansky and Muller are wrong. You don’t need two, or even any mutations for speciation to occur, just a barrier to gene flow that can be anything from isolation by distance to a road to disruptive selection.Barriers to reproduction will naturally arise.

Next time:  I will NOT talk about why you cannot reduce group selection to individual selection.  I say this for one simple reason:  I got nothin.  As far as I can tell indirect genetic effects are so powerful that any attempt to reduce group selection to individual selection is destined to end in tears.





Why reductionism doesn’t work, Part 2: Groups to individuals

Posted: May 26th, 2015 by Charles Goodnight

Williams (1966) famously wrote “In explaining adaptation, one should assume the adequacy of the simplest form of natural selection, that of alternative alleles in Mendelian populations, unless the evidence clearly shows that this theory does not suffice.” This principle of parsimony makes two interesting points. The first phrase “In explaining adaptation” makes the point that Williams was interested in examining patterns, and then using those patterns to infer how selection acted in the past. This is very different than modern MLS approaches in which the process of selection is examined. This is why parsimons (A bit of artistic license with the spelling) are so unimportant in modern MLS theory: such rules are not necessary if you are studying the process rather than inferring the process from standing patterns. More importantly, this principle implies that group selection can in many cases be reduced to individual selection, or even genic selection. The only thing that stands in the way of doing this is the ecology. Unless the trait is “altruism”, and thus impossible to evolve at a lower level, there is no reason not to act as if it was one of these lower levels of selection.


The principle of persimmony: persimmons come from a persimmonious tree (https://www.flickr.com/photos/giagir/5185254421).

But is this really true?   Last week I discussed why individual selection can’t be reduced to genic selection. It turns out that the situation is worse trying to reduce group to selection on the underlying individuals. So with that long-winded introduction out of the way, the main reason that group selection cannot be reduced to individual selection is indirect genetic effects (IGEs). Indirect genetic effects occur when genes in one individual affect the phenotype of another individual.

This is an effect that has been seen time and time again. The most aggressive chickens lay the most eggs, but also suppress the egg laying of their cage mates (Muir 1996, Poultry Science 75:447), crop plants aggressively interact such that the highest producing plants most strongly suppress their neighbors (Griffing 1977 in: Proceedings of the International Congress on Quantitative Genetics, August 16-21, 1976.) and many more examples. The important thing is that interactions that are internal to the unit of selection can contribute to the response to selection, whereas if they are external to the unit they cannot. Thus group selection can act on IGEs, but individual selection cannot.

To see this it is easiest to use the Price equation. The Price equation divides the covariance between a trait and relative fitness into within and between group components. It is easy and convenient to use this partitioning to make the point I want to make, but it is important to emphasize that the Price partitioning should never be equated with group and individual selection (Are you listening West and Gardner?).

Imagine we have a metapopulation in which individuals interact within groups but not between groups. The individuals interact in some manner that affects all individuals in the group in the same way. That is, perhaps they release waste products into their environment and everybody gets equally poisoned, or on a more positive note, perhaps they release some chemical public good. Further imagine that we have a trait, z, that is influenced by direct genetic effects (DGE), indirect genetic effects (IGE) and environmental effects. Thus, the trait value of the ith individual in the jth deme is:

Zij = DGEij + IGE.j + eij

Further imagine that the fitness of the ijth individual relative to the metapopulation mean fitness is wij, and the correlation between environmental effects and fitness is zero (just to get them out of the way).

To bring this back to my posts on Gardner, if I was following his model, at this point what I would want to do is partition the “total breeding value” so I could compare it with his partitioning of Fisherian breeding values. “Breeding value” is defined by Fisher (1930, Falconer and MacKay 1996) to be the average value of an individual’s offspring measured as a deviation from the population mean. This breeding value assumes that there is no population structure and that offspring interact randomly with other individuals in the population. Because they ignore population structure Fisherian breeding values cannot be partitioned. Bijma and Wade (2008. JEB 21: 1175-1188) solved this by defining “Total Breeding Value” to be the average value of an individual’s offspring measured in their native social environment as a deviation from the metapopulation mean. Unlike Fisherian breeding values, total breeding values can be partitioned. If you prefer to partition total breeding values replace “z” with total breeding value in the equation below, and replace DGE’s and IGE’s with their additive genetic equivalent.

If we put all this together, using the Price equation to partition the covariance between total breeding value and relative fitness we get an algebraic explosion!

Equation 1

Or much more simply:

equation 2

So, in words, this simply tells us that the within demes covariance between phenotype and relative fitness (red in the equation) includes ONLY direct genetic effects, whereas the between demes covariance between phenotype and relative fitness (blue in the equation) includes both direct and indirect genetic effects. This is shown graphically in the following figure:

Screen Shot 2015-03-25 at 2.04.15 PM

The sources of variation for a trait and the group mean of the trait. For clarity I have left the total variance proportions the same for the group mean trait, even though in most situations the direct genetic effects and the environmental effects would be reduced due to averaging. Although the genetic components underlying the trait are unchanged by taking the average, the heritable component does change. For the individual trait only the direct effects are heritable, whereas for the group mean trait both the direct and indirect genetic effects are heritable.

What this is saying is that from an evolutionary perspective a trait and the group mean of a trait are actually different traits. Because group selection can act on both direct and indirect effects it can produce genetic changes that are qualitatively different than selection acting on the individual level. As I have pointed out numerous times this is not a minor theoretical issue that experimentalists can ignore. Indirect genetic effects have shown up as major factors in the response to group selection in every situation where it has been possible to infer there presence, including both experiments specifically designed to detect them (e.g., Goodnight 1990 Evolution 44:1625), or where it was obvious even though the experiment did not have explicit treatments to detect them (e.g., Muir 1996).

Next week, as promised for this week, but not delivered:  Why reductionism does work.



Why reductionism doesn’t work; Part 1, Individuals to genes

Posted: May 18th, 2015 by Charles Goodnight

One thing that often used to happen, perhaps not so much any more, is that people will say that we don’t need to worry about levels of selection because all selection can be reduced to selection acting directly on genes. George Williams perhaps put this view best, first with his principle of parsimony, which argues that reductionism is the right perspective:

“In explaining adaptation, one should assume the adequacy of the simplest form of natural selection, that of alternative alleles in Mendelian populations, unless the evidence clearly shows that this theory does not suffice”

and in the same book, and more explicitly, which says that reductionism is works:

“No matter how functionally dependent a gene may be, and no matter how complicated its interactions with other genes and environmental factors, it must always be true that a given gene substitution will have an arithmetic mean effect on fitness in any population.”

All I can say to this is GAHHHH!

Brave 2 frustrated

Merida expresses her opinion on genetic reductionism (taken from http://giphy.com)

I think a lot of people know that you cannot think of selection as acting on genes, but a lot of people also can’t articulate why it doesn’t work. So, if anybody asks you, the simple answer is that reductionism doesn’t work because of interactions. At the individual level this will primarily be gene interactions of dominance and epistasis.

In a fully additive system there would be no problem, and this IS the problem.  Our intuition about genetics was developed using simple additive models.  In an additive system, knowing at what level selection was acting would be nice information, but the fitness of the phenotype can always be algebraically reduced to fitness effects on individual loci.   In other words, in additive systems, how the genes are packaged really doesn’t affect the effect of genes on the phenotype. To see this consider a phenotype affected by a single locus additive trait:

Genotype A1A1 A1A2 A2A2
Frequency p2 2pq q2
Fitness 1 1-Z/2 1-Z

(I use Z to emphasize that we are not talking about fitness. Selection will be affected by the packaging for the simple reason that some of the selection is on heterozygotes). We can calculate the average effect of the A1 allele on the phenotype we would discover that it is:

Original genotype genotype after substitution probability change
A1A1 A1A1 p2 0
A1A2 A1A2 ½ 2pq 0
A1A1 ½ 2pq Z/2
A2A2 A1A2 q2 Z/2

So, the average effect of the A1 allele is:

Screen Shot 2015-05-16 at 12.37.15 PM

Now consider a haploid system

Genotype A1 A2
Frequency p q
Fitness 1 1-Z/2

The average effect with the same phenotypic effects (adjusted for ploidy). Now the local average effect of the A1 allele is:

Original genotype genotype after substitution probability change
A1 A1 p 0
A2 A1 q Z/2

So, the average effect of the A1 allele is: you guessed it:

Screen Shot 2015-05-16 at 12.37.24 PM

The effect of the allele on the phenotype is not affected by the packaging.

Now lets do the same thing with a dominant system:

Genotype A1A1 A1A2 A2A2
Frequency p2 2pq q2
Fitness 1 1 1-Z

Now the average effect of the A1 allele on the phenotype becomes:

Original genotype genotype after substitution probability change
A1A1 A1A1 p2 0
A1A2 A1A2 ½ 2pq 0
A1A1 ½ 2pq 0
A2A2 A1A2 q2 Z

So, the average effect of the A1 allele is:

Screen Shot 2015-05-16 at 12.37.34 PM

turning to the haploid system

Genotype A1 A2
Frequency p q
Fitness 1 1-Z/2

Now the local average effect of the A1 allele is:

Original genotype genotype after substitution probability change
A1 A1 p 0
A2 A1 q Z/2

The average effect in the haploid system is now different than in the diploid system,.

Screen Shot 2015-05-16 at 12.37.24 PM

In other words, if we add the simplest possible form of nonadditivity the packaging does matter. Trust me it gets worse. I am way to lazy to put up tables for average effects in epistatic systems, but I have talked about this before. It turns out that the variance in local average effects is a measure of how the average effects of alleles are to genetic background. I have talked about these before, but it bears re-posting the relevant figure:

Drift and epistasis LAE graph

The important point is that the variance in local average effects is zero in additive systems, but non-zero when there are any sort of interactions. This means that the reducability of fitness effects on to genes is a reasonable exercise in additive system, but simply is not meaningful in epistatically interacting systems. To see how bad this can be, consider long-term directional selection in a system with AXA epistasis. Depending on the starting gene frequencies the average effect of an allele can actually reverse signs.  For what it is worth, the dashed lines are the local average effects for an additive system, and the solid lines are the local average effects for AXA epistasis.  This shows the contrast between additive systems and epistatic systems.  For the additive system, if you were to evaluate the fitness effects in generation zero they would provide a pretty good estimate of the fitness at the end (in this deterministic system an exact estimate).  On the other hand, for the epistatic system, estimates of allelic effects made in generation zero rapidly become useless, and by the time fixation is reached they are exactly wrong.

figure 12 AXA LAE

In one sense, Williams is absolutely correct. At any given instant it is certainly possible, in principle, to do a least squares regression analysis and assign fitness effects to individual loci. However in an epistatically interacting system those fitness assignments are ONLY good for the moment, or perhaps the generation, in which the assignment is done. Those effects will change as gene frequencies change, and not just gene frequencies at the locus under study, but gene frequencies at any other loci as well. So, my point is not that the assignment cannot be done, but rather that the assignment carries no information that is useful beyond the moment.

Next time I talk about why reductionism does work!

Epistasis in Monkey Flowers, and some general thoughts on epistasis

Posted: May 8th, 2015 by Charles Goodnight

So, at least my twitterverse has been on fire suddenly with the appearance of a new article in PLoS by Patrick Monnahan and John Kelly “Epistasis Is a Major Determinant of the Additive Genetic Variance in Mimulus guttatus”  .

It really is a nice study in which they identified 11 quantitative trait loci (QTL) in a single population of monkey flower, then used these to estimate the functional (also known as physiological) direct effects, and all of the two locus epistatic interactions. They then used these estimates to estimate additive genetic variances and total genetic variances in the population.

What is nice about this study is that they use actual data from a QTL analysis of a natural population, and then use the resulting analyses to estimate bi-allelic functional epistasis for each of the pairs of QTL. In fact it would be a great teaching tool to have access to some of those two locus genotypic values for teaching purposes! I would also love to have the actual allele frequencies, so that we could in fact estimate the standing statistical variance components in the natural populations. This also brings up a very important point: all of the models to date have put in fixed values for the genotypic values (or avoided the issue entirely using inbreeding coefficients). In the real world we collect organisms, identify genes, and phenotype them. There is ample room for error at every step. So the one thing we know for sure is that any QTL measures or assignment of phenotype to genotype is an estimate. This really is the first attempt to couple field estimates of genotypic values to variance components.

One other thing that is nice about this paper is that they bring up both the Kempthorne/Cokerham variance components and the more recent terminology of “positive”, “negative” and “sign” epistasis. Nicely, Hanson (2013 Evolution 67: 3501-3511) provided two locus examples of these types of epistasis. It turns out that if we set the gene frequencies to 0.5, and do the appropriate regressions we can directly relate these molecular concepts of epistasis to the quantitative genetic components. It also turns out that this is critical, for while functional epistasis is loads of fun, it is only the quantitative genetic variance components that tell us how phenotypic evolution works.

Anyway, from Hanson (2013) these different types of functional epistasis are:

Hansen figure

Using the JMP program shown below it is easy to show that positive epistasis is a hodgepodge of variance components (89% additive variance, 3.6% AXA epistasis, 3.6% AXD epistasis, and 3.6% DXD epistasis), negative and sign epistasis is additive variance and AXA epistasis (negative epistasis: 80% additive variance, 20% AXA epistasis, sign epistasis: 50% additive variance, 50% AXA epistasis). Maybe its because I am a curmudgeon, but I am happier with the old fart Kempthorne partitioning, because it relates directly to variance components, and can be much more easily converted to statistical genetic components.

Now here is the critical point. These variance components are a function of gene frequency, thus the variance components will change as gene frequencies change. Using the example of positive epistasis above I can now tell you the additive genetic variance for any two locus gene frequency:

va Plot cropped

Graph of the additive genetic variance for two locus two allele positive epistasis as described by Hansen (2013). A JMP program to calculate VA for a single gene frequency is listed below. Note that I rotated the graph to best show the shape of the surface. The highest additive genetic variance occurs when both the A2 and B2 alleles are at low frequency (around 0.2).

Finally, I know it is impolite to promote your own work, but well, it’s my blog and I will do what I want. My ego was a bit hurt by the fact that that my work on epistasis and additive genetic variance was not cited, in particular, my paper on average effects and additive variance (Goodnight. 2000. Heredity 84: 587-598.), which was quite relevant. That and my earlier paper using breeding values (Goodnight,1988. Evolution 42: 441-454) were the first papers to describe the conversion of epistasis in to VA, and they have historical significance if nothing else. I have long been fighting a bit of a rear-guard action to keep those papers from falling into the obscurity of common knowledge. There is actually another reason that they could have benefited from citing those papers. One of the things that comes out of those papers is that if you can write down the functional values for the 9 genotypes of a pair of interacting two allele loci you can use regression to calculate the additive genetic variance for any given gene frequency. I do actually know why they might have missed my paper. They use the Falconer partitioning that was first pioneered by Cheverud and Routman (1995. 139: p. 1455–1461) which is enough different that my paper really didn’t need to be cited, so it is hard to get too mad at them.

my party

Its my blog and I will whine if I want to. You would whine to if it happened to you. (picture from (http://www.amazon.com/Its-My-Party-Mercury-Anthology/dp/B000VHKHZA )

If you have JMP and are savvy in its use, the files that I use for calculating the additive genetic variance can be found here (variance regressions). I fixed it by changing the file extension to .txt.  It is still a .jmp, so after you download it please change the txt to jmp, then it should work.

Basically you add your own dependent variables, add the allele frequencies of your choice (I put it in as a formula, so use the get column info route to change those), and the linkage disequilibrium. Then run the script in the upper left hand corner. Finally, if the gene frequencies are other than 0.5 and in linkage equilibrium use sequential (type 1) sums of squares. Type 3 sums of squares will give you the wrong answer. If you have any questions feel free to ask me.  OK, if you want the program I need to send it to you under a separate cover, so email me if you would like it.  If I ever figure it out I will fix tings.




Matrix comparisons: Random skewers and selection skewers

Posted: April 25th, 2015 by Charles Goodnight

A week late and a dollar short, but lets continue comparing matrices. Continuing on with my blatant endorsement of statistical methods attached to my name. . .  Last time I talked about the “Rank”/“Signed Bartlett”/”Modified Mantel” tests for comparing the dimension size and shape of a pair of matrices. This is only one of several ways of comparing matrices. This set of tests has the advantage that it is basically non-parametric, and makes very few assumptions about the actual matrices. It is also useful because it directly compares matrices for easily interpretable differences. The problem with the these tests is that in most cases we don’t so much care about whether or not a pair of matrices are the same or different as whether they have the same or different effects on the evolution of the organism.

Obviously the size shape and dimension of a covariance matrix will be related to the ability to respond to selection, but the relationship may not be perfect. Two other approachs that has been developed are “random skewers” (Cheverud 1996 J. Evol. Biol. 9:5-42; Cheverud and Marroig 2007 Genet. Mol. Biol. 30:461-469; Revell 2007 Evolution 61:1857-1872) and “selection skewers” (Calsbeek and Goodnight 2009. Evolution 63:2627-2635). To see what a random “skewer” is consider that in a multivariate selection experiment the response to selection is given by:

R = GP-1S = Gβ

The β is a vector that describes the direct effects of selection on the different traits. The G matrix is sometimes thought of as a “rotation matrix” in that, while what it does from a biologists perspective is tell us what the R vector or response to selection, from a mathematicians perspective what it does is rotate and warp the β vector. Thus, if we take any arbitrary β vector and multiply it by two different G matrices the two matrices will rotate and stretch the β vector in different ways producing two different R vectors. We can use this because if the two matrices are identical the two rotated vectors will be identical, whereas if the matrices are different the two rotated vectors will also be different. These can be compared by calculating the vector correlation between the two vectors. In linear algebra terms this is (I am SO sorry I am doing this to you!)

Screen Shot 2015-04-25 at 9.07.06 AM

For the non-linear algebraic adept (he said raising his hand), the numerator is really just a means of calculating a covariance between the two vectors, and the denominator is the square root of the product of the two covariance matrices from the vectors.

So, with the random vectors approach what you do is generate a large number (1000 or more) random unit vectors. These represent a set of selection gradients in random directions. For each gradient you calculate the resulting R vector using your two matrices, and calculate the vector correlation. If the average correlation is close to one, then they are the same, whereas if it is less than one the two matrices are different.

The question, of course, is how close to one is close enough. Here again the bootstrap comes in. Following the approach I outlined last time, we generate a large number of pairs of matrices that are estimated from bootstrap samples of the same data set. Because they are estimated from the same data set there can be no true difference, so if we calculate the average correlation between these two matrices this will give us a distribution of the correlation when the null hypothesis is true. It is then a simple matter to compare the actual correlation with the bootstrap correlations. If the actual correlation is less than 95% (or what ever) of the bootstrap correlations then we can say that the two matrices are significantly different from each other.

This is an interesting point. Here we are using the null hypothesis that the two matrices are identical. Thus, we set up the bootstrap such that the null hypothesis was true, and compared our actual correlation with the bootstrap correlation. In the original random skewers approach the opposite was the case. The null hypothesis was that the two matrices were uncorrelated, and thus those papers use a different approach to significance testing. I googled hard for a joke about getting null hypotheses backwards, but apparently this is too subtle for the online community.

The selection skewers is similar to random skewers, with a few important changes. This analysis is appropriate if you are specifically interested in comparing how two populations will respond to a particular selection pressure. For example, you may have two recently diverged populations and want to determine whether the two populations will respond in the same manner to a particular selection pressure. In most cases you will likely have a known S vector, which is the raw selection differential. This is what I assume in the program I provided. In this case you first need to generate the b = P-1S vector. Then as with the random skewers you calculate the vector correlation, and compare the actual correlation to the correlation in the bootstrap data sets when the true null hypothesis is zero

The nice thing about both the random skewers and the selection skewers is that they give a real world idea of what changes in shape can do. The random skewers is agnostic as to how selection actually works, whereas the selection skewers tests a specific selection regime. This later is particularly interesting, since it is entirely possible for two matrices to have very different structures (as determined say by the rank/Bartlett’s/Mantel tests), and yet have this structural difference have very little actual effect on the response to selection. On the down side, however, the random and selection skewers lump a lot of information together. For example, it can be hard to determine whether a difference in response between to matrices is due to a difference in the total amount of available variation, or due to changes in the correlation structure leading to negative genetic correlations.

I guess the real lesson from all this is that there is no one best statistical test. Which is best depends on the question you ask. If you want detailed insights into the actual covariance matrices the rank/Bartlett’s/Mantel test may be best. If you want a summary of the difference in the ability to respond to selection random skewers may be a good choice, and if you have a clear a prior selection hypothesis to test the selection skewers is clearly the best.

To remind you I have an R script that performs these tests and can be relatively easily modified for different data sets and circumstances.

Here is the program: 

Writeup on how to use the program:  Matrix comparison writeup

The program:Bootstrap command

Relevant example data sets:

balanced stock females

stock data female

population 3 females

Statistical tests for comparing matrices

Posted: April 8th, 2015 by Charles Goodnight

I have been remiss. Quite a few years ago I found myself in the position of wanting to compare two genetic covariance matrices. At the time it was before the Flury hierarchy had been suggested by Pat Phillips (Phillips & Arnold 1999. Evolution 53: 1506-1515), so I found myself in a position of needing to invent my own. Later, apparently along with others I decided I wasn’t particularly enamored with the Flury hierarchy. This resulted in two publications (Goodnight & Schwartz 1997 Biometrics 53: 1026-1039; Calsbeek &. Goodnight 2009 Evolution 63: 2627-2635), the first of which is not particularly well known. The first publication also suffered from not having a good software implementation. With the appearance of R this has now been rectified. In any case I would like to remind people of these statistical methods for comparing covariance matrices.

First off, there is nothing wrong with the Flury hierarchy, I just don’t particularly find it intuitively useful. As I understand it the Flury hierarchy is a model selection approach, whereas the methods I will discuss are parametric statistical tests. I recommend you read Philips and Arnold’s papers and make your own decision. So enough preamble.

We had just done an experiment in which we sent a population a population bottleneck, and we had measured several traits. We wanted to know if the derived population and the ancestral population had the same genetic structure, aka, the same genetic covariance matrices. For a single trait we know exactly how to do this. You “simply” measure the additive genetic variance in the two populations and do an F test to see if they are the same or different. I put simply in quotations because measuring additive variance is never easy.

When we get to a multivariate settings things become more complicated. Again, we will likely use a MANOVA to measure an additive genetic covariance matrix for each population. We would then like to compare these to see if they are the same or different. The good news is that genetic covariance matrices are square and generally easy to work with. The bad news is that when we go multivariate there are several ways that matrices can be different. In Goodnight and Schwartz (1998) we decided there are three ways of interest. The matrices can be of different dimension, they can be of different size, and they can be of different shape. These are really independent ways of being different, so it makes sense to develop three tests. The way we tested these was using bootstrapping.

The bootstrap: Bootstrapping is an interesting statistical procedure that was popularized in the 80s by Brad Efron (Efron 1979, The Annals of Statistics 7:1-26) (I took a workshop he offered somewhere around 1985). The basic idea is that if you have a data set you can create new pseudo data sets by randomly sampling with replacement from the original data. If enough of these bootstrap data sets are generated they will actually provide a distribution for the data. This at first seems counter intuitive, but as long as your data set is relatively large it works very well. To use this as a statistical test you need to decide what your null hypothesis is, and then figure out a random sampling scheme that makes that null hypothesis true. For example with a t-test, the null hypothesis is that the two populations have the same mean. You can make that null hypothesis true in several ways. You could simply combine the data from the two populations. Then randomly assign them back to the two populations without regard to original source. As a result there will be no true difference between the populations. If you randomly create several thousand of these pairs of populations you will get a distribution of observed differences in the means when you know the true difference is actually zero. You can then take the actual difference between the two populations and simply ask what percentage of the bootstrap differences that are more extreme than the difference in the actual data. That percentage is your probability of the observed difference occurring by chance. There are more sophisticated approaches, but this gives the idea.

In our particular test we had an ancestral population and a population derived from two generations of brother sister mating. We wanted to see if the two populations were the same or different. Our null hypothesis was that their covariance matrices were the same (this is important!), and we decided to use data from the ancestral population as our source for the bootstrap data.

Dimension: A genetic covariance matrix can be thought of as enclosing a space. Thus a univariate “matrix” is a single vector of a length that is equal to the variance. A two-trait covariance matrix defines a plane, a three trait matrix a cube, and so on.

Screen Shot 2015-04-08 at 5.47.22 PM

Figure 1; a one dimensional vector, and two and three dimensional matrices.

There are two things that can happen to the additive genetic variance after a population goes through a bottleneck. First it can disappear, that is, it can go to zero. Second, it can become so highly correlated with other traits that it becomes a linear combination of these traits. In graphic terms, in the three-trait case, that would be the equivalent of one of the vectors lying exactly in the plane of the other two vectors.

3 colinear vectors

Figure 2: in this matrix trait z is a linear combination of traits y and x. As a result all three lie in a single plane, and the resulting matrix is a two dimensional matrix.

Consider trying to compare two matrices with three variances. One is like the three dimensional matrix in figure 1, and the second has only two dimensions as in figure 2. It won’t work to compare these. As an analogy it is like asking which is bigger, a box or a sheet of paper. The three dimensional matrix has an extra dimension along which it can evolve that is qualitatively different from the two dimensional structure.

The way we tested this was to find the largest sub-matrix that had valid variances that were not linear combinations of other vectors. We then tested the absolute value in the difference in rank ( |RpopA-RpopB|) as our test statistic measured against the bootstrap populations were there was no true difference in rank. In this data set the difference in rank was not significant.

Screen Shot 2015-04-07 at 12.19.17 PM

Difference in size: As mentioned above, matrices can be considered to be planes, volumes or hyper volumes. It turns out that the determinant is a measure of the space enclosed by the matrix. For example, in a two-trait matrix the determinant is the area of the matrix, in a three trait matrix it is the volume, etc. Thus two matrices of the same dimension, regardless of shape, can be compared by comparing the determinants. The analogy is having two oddly shaped vases. We can compare them by asking how much water they hold. In this case shape is of no consequence, only the size of the space enclosed.

The important caveat is that they must be the same dimension. Again, the same question: which is larger the volume of a box, or the area of a sheet of paper. And again it is a meaningless question. We chose to resolve this by doing an “orthogonal projection” of the larger dimension matrix on the smaller dimension matrix. That is, we searched the matix pairs for a set of traits that had valid variances in both matrices. We did the analysis on this pair of sub matrices.

The next question, is how to compare the two determinants. It turns out that there is a good test, the multivariate Bartlett’s test that can be used. Bartlett’s test, has two problems. First, it is very sensitive to the assumption of multivariate normality, and second, it is not structured for use with MANOVA derived data. Still we can use the basic statistic and combine it with the bootstrap data, and it works perfectly well. One of the very useful features of bootstrap tests is that they make no assumptions about the distribution of the data. Also, if properly designed, they work well with virtually any experimental design. Interestingly, since, the standard test was not developed for use with MANOVA the parametric multivariate Barlett’s test was way to optimistic and the bootstrap ended up doing a much better job. A final modification is that we had an a priori interest in whether the derived genetic covariance matrix was significantly larger than that in the ancestral population. Thus, we multiplied the Bartletts statistic by 1 if the derived population was larger than the ancestral population and -1 if it was smaller, giving us the signed Bootstrap Bartlett’s test that allowed for both one tailed and two tailed tests.

Screen Shot 2015-04-08 at 6.05.37 PM

Shape: For shape we decided to go with a test similar to the Mantel test. Many rightly complain about the classic Mantel test for numerous reasons. However, the basic idea is useful. The idea is that you calculate a correlation between the pairwise elements of the two matrices. That is you pair up the elements of the two matrices and simply calculate the correlation among them. The problems with the traditional Mantel test for this application are three fold. First, the traditional Mantel has a null hypothesis that the two matrices are independent, whereas our null hypothesis is that the two matrices are identical. The bootstrap solves this by allowing us to generate a distribution of Mantel correlations among pairs of matrices that have a true correlation of 1.

Second, the Mantel test is meant to compare correlation matrices, which have 1s on the diagonal, whereas this is not true for a covariance matrix. In the classic Mantel test this diagonal is excluded, whereas in ours it is not. Third, all of the elements of a correlation matrix are between -1 and 1, whereas covariance matrices can have vastly different variances for different traits, which can inappropriately skew the results. This last we solved by standardizing the elements to the average of the diagonals of the two matrices. The final equation is somewhat ugly, so I refer you to the paper if you want the details. The results indicate that females, but not males, have a significant change in the shape of their covariance matrix. That is the population bottleneck significantly changed some of the variances and covariances among traits in the two populations, even though it did not change the total amount of additive genetic variance.


So, the point of this is simply to suggest one possible way to compare genetic covariance matrices. One of the reasons I really enjoy multivariate math (I can’t believe I said that) is that very simple ideas, like the variance of a trait, suddenly become so much richer, and can change in so many more ways as we move into a multivariate setting. Obviously simple multivariate math in a pale comparison with the real world, but this only serves to make the diversity of the real world even more easily understood.

The other reason I wanted to put this up is that I have an R program that does these analyses, along with random skewers and selection skewers, which I will talk about next time. I am not an R developer, so I would be more than pleased if somebody were to take this script and turn it into something that didn’t actually need to be adjusted for the needs of every data set. If you do choose to finish developing this, please let me know!

Here is the program: 

Writeup on how to use the program:  Matrix comparison writeup

The program:Bootstrap command

Relevant example data sets:

balanced stock females

stock data female

population 3 females


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