Evaluating Price’s Equation

I have previously written several posts on the work of George Price. This one contained general reflections on Price’s reputation; this one attempted to explain how he arrived at the first (1970) form of his famous equation; this one did the same for the later (1972) version of the equation; and this one one attempted to give a more intuitively satisfying account of the ‘covariance’ term in the equation.

I said I would conclude with some general comments and criticisms. This post will comment on aspects of the equation except its application to the problem of group selection, while the final post in the series will cover that.

The full equation

Price’s 1970 paper contained two versions of the equation, a short form:

[1] dQ = cov(z,q)/Z

and the full version:

[2] dQ = cov(z,q)/Z + Sz[i]dq[i]/NZ.

(For an explanation of the notation see the previous posts.)

Dealing first with the full version, its main virtue is to analyse the change in a gene frequency (or other properties of a population) rigorously into two components: one of which depends on the covariance between the fitness of individuals in the ‘parent’ generation and their own possession of the gene in question, and the other of which depends on any change of frequency of the gene among the offspring as compared with their own parents.

I think this is a genuinely important contribution to our understanding of selective processes. As Price himself pointed out, it is in principle applicable to changes other than those of gene frequencies, or even outside biology altogether. (These applications require some reinterpretation of the terms involved.) For example, if we wish to analyse an increase in average height in a population (excluding migration), this can be broken down into any covariance between the height of individuals and their number of offspring, and any tendency for offspring to be taller than their own parents. The two components may work in the same direction or in opposite directions, as would be the case if tall parents have fewer offspring than short parents, but offspring are taller than their parents. Price’s Equation provides a clear way of analysing such issues.

A particularly interesting application of the equation has been made by Steven A. Frank. Suppose we take the property of interest (represented by q in the equation) to be fitness itself. The equation can then take the form:

[3] dZ = cov(z,z)/Z + Sz[i]dz[i]/NZ.

The first term on the RHS now looks distinctly odd. What on earth is the covariance of fitness with itself? Odd though it may seem, it is mathematically quite legitimate, and if the term cov(z,z) is expanded using the definition of covariance it will be seen to equal S(z[i] – Z)(z[i] – Z)/N. But this is the same as the variance of z. Equation [3] may therefore be interpreted as showing that the change in average fitness between two generations depends on two quantities: the variance in fitness among the parent generation, and the change (if any) in average fitness between parents and their offspring. Some readers will have noticed a similarity in this proposition to R. A. Fisher’s Fundamental Theorem of Natural Selection. Steven A. Frank has shown that with an appropriate interpretation of the FTNS, it can be simply and elegantly derived from Price’s Equation [Frank p.20-25]. Anything that makes this possible must be regarded as a notable contribution to biological theory.

My only reservation about this contribution is that the conceptual breakdown of the change between the two components does not necessarily correspond to different causal processes, because the same process may sometimes contribute to both components. I fell into this trap myself in an earlier post when I said that the full equation ‘makes a sharp distinction between the effect of a gene on the fitness of the individual organism and any changes in gene frequency occurring solely during the transmission of genes from parents to offspring. These two types of effect can therefore be divided neatly between the two components in the right hand side of the equation.’ To see that this is misleading, consider the case of segregation distortion. This may happen because in individuals who are heterozygous for the relevant gene, the ‘male-producing’ allele manages to disable gametes from the same parent which carry a ‘female-producing’ allele. In this case the gene may have two effects: it will certainly increase the proportion of males among the offspring, but it may also reduce the overall number of offspring, since fewer viable gametes are produced. (See Ridley p. 172 for an actual example.) The same causal process (the sabotage of ‘female-producing’ gametes) therefore affects both components in the equation simultaneously.

The short form

Turning to the short form of the equation, dQ = cov(z,q)/Z, this is strictly valid when there is no change in gene frequency between individual parents and their own offspring, and a good approximation if such changes are confined to small random fluctuations in a large population. Whether it is actually useful, either for practical or conceptual purposes, is more debatable. Price himself, in his 1970 paper, said that ‘Recognition of covariance… is of no advantage for numerical calculation, but of much advantage for evolutionary reasoning and mathematical model building’. Biologists have differed in their opinions on this. Few paid the equation much attention until the mid-1980s, which is a long time for a supposedly valuable tool to go unused. This may be contrasted with the rapid and widespread use of evolutionary game theory, which Price also helped create in the early 1970s. Some very distinguished evolutionary biologists, including W. D. Hamilton, Alan Grafen, and D. C. Queller, claim to have found the ‘covariance’ approach to selection enlightening, but on the other side John Maynard Smith, in the interview mentioned in my last post, had no time for it. This may be, as JMS himself suggested, because scientists’ minds work in different ways.

I am hardly qualified to take sides in this debate, but I will. As I pointed out in my last post, there is nothing inherently surprising about finding some connection between covariance and selection. If one analyses the concept of covariance itself more closely, the equation is dangerously close to triviality. It is neat, and ingenious, but does it give any insight into selective processes that we could not get more easily in other ways? Price himself thought that the equation could be especially useful when the covariance is expressed as ‘regression times variance’. In these terms the short form of the equation can be expressed as:

[4] dQ = reg(z,q)Vq/Z

where reg(z,q) is the regression of z on q, and Vq is the variance of q.

The change in gene frequency resulting from selection is therefore proportional to the regression of z on q. This can be represented diagramatically as the slope of a line through a scatter plot, where fitness is measured along one axis and the gene frequency of individuals along the other. Price claimed that ‘at any stage in constructing hypotheses about evolution by natural selection, one can visualize such a diagram and consider whether the slope really would be appreciably non-zero under the assumptions of the theory’. But this seems a very roundabout way of achieving something that could usually be reached more directly by considering the relative fitness of genotypes (which must be known or assumed for such a diagram to be constructed). I just don’t see any real advantage in Price’s approach.

Applications to inclusive fitness

Another claim about the equation in its short form is that it helps clarify, or even supersedes, W. D. Hamilton’s inclusive fitness theory. Oren Harman’s biography of Price implies that Hamilton got some important ideas for developing the theory from Price. There is some basis for this view, as Hamilton admired and championed Price’s work, and made some use of the short form of Price’s Equation in his 1970 paper on ‘selfish and spiteful behaviour’. (Hamilton later made use of the ‘multilevel’ version of the equation in an attempt to clarify the issues raised by group selection, but that is not covered in the present post.)

It is therefore undeniable that the equation can be used in formulating the theory of inclusive fitness. But one may still ask the historical question how far the equation actually helped Hamilton develop the theory; and the practical question whether the equation is useful for understanding inclusive fitness.

After his initial presentation of inclusive fitness theory in 1963 and 1964, Hamilton made four notable corrections or extensions of the theory:

1. He recognised that in the case of haplodiploids his original measure of relatedness was incorrect. He corrected it in a short paper published in 1971, after he became aware of Price’s work, but he did not refer to Price and there is nothing to suggest that Price’s Equation had any influence on this particular point.

2. He recognised that in cases where one individual is inbred and the other is not, or where one is more inbred than the other, then the appropriate measure of relatedness for the purpose of Hamilton’s Rule must be asymmetrical, and is technically a regression rather than a correlation coefficient. Hamilton introduced a revised coefficient in the 1970 paper which first made use of Price’s Equation [Narrow Roads, 177-82], and this may have led some to suppose that it was Price’s work which prompted the change (see e.g. Harman’s biography of Price, p 208). But Hamilton has said elsewhere (Hamilton, Narrow Roads, vol. 2, p.99) that he recognised the need for a correction ‘a year or two after’ his 1964 paper, which was long before he came in contact with Price. Indeed, he had recognised even in his first (1963) paper on altruistic behaviour that in principle a regression coefficient is needed [Narrow Roads, vol. 1, p.7], but he considered at that time that Sewall Wright’s measure of relationship, which is a correlation coefficient, was a good enough approximation in most circumstances.

3. Hamilton’s original work did not allow for ‘spite’, that is, for behaviour which is harmful to the individual’s fitness but even more harmful to others. His 1970 paper rectified this. Again, it is tempting to suppose that Price’s work influenced this development, and Harman (p.223) implies this, but Hamilton’s own 1970 paper says that the two men reached their conclusions independently. The basic point about ‘spite’ can be made with a very simple example: a gene which caused an individual to kill some of its own offspring, but all the other members of the population, might at a stroke greatly increase its frequency in the population. The point can also be made within the framework of the original version of Hamilton’s Rule, provided one is aware that Wright’s coefficients of relationship can be negative. (See my post here.) If we put the Rule in the form:

[5] br – c > 0

where b is the ‘benefit’ to the recipient of the action, c is the cost to the actor, and r is the measure of relationship between them, it is evident that if r is negative then b, the ‘benefit’, may also be negative (i.e. spiteful) and still satisfy the Rule.

4. In a paper published in 1975 (based on a lecture given in 1973) Hamilton pointed out that altruism can spread if there is a positive association of altruists in the population, whether this is due to genetic relatedness (common ancestry) or to genetic similarity arising from other reasons (such as habitat preference). This is in principle an important extension of inclusive fitness theory. The 1975 paper is strongly influenced by Price’s Equation in its ‘multi-level’ form of 1972, but the basic point could also be made using the simple covariance formula. Anything that causes a positive covariance between fitness and the frequency of a gene in groups of a population could increase the frequency of a gene, and it is not difficult to see that factors other than common ancestry could in principle give rise to such a covariance. Price’s work may well have given Hamilton the inspiration for his treatment of this point in the early 1970s, but it is worth noting that Hamilton’s 1964 paper already recognised the possibility of extending the concept of inclusive fitness to cover genetic similarity for reasons other than common ancestry: ‘if some sort of attraction between likes [i.e. between genetically similar individuals] for purposes of co-operation can occur, the limits to the evolution of altruism… would be very much extended’ [Narrow Roads, vol 1, p54]. This is the origin of the ‘Green Beard’ concept.

Overall, I do not think that the short version of Price’s Equation played a major part in the development of inclusive fitness theory by Hamilton in the 1970s. However, it may still be argued that is useful for further work on that theory. Alan Grafen, one of the most brilliant recent evolutionary theorists, has made much use of Price’s work, and in 1985 showed how Price’s Equation could be used to derive a version of Hamilton’s Rule itself [Grafen 1985]. Grafen defines a certain quantity r in such a way that the frequency of the relevant gene will increase if:

[6] br – c > 0

This is identical with [5] above. Should we therefore say that Hamilton’s Rule can be derived from Price’s Equation? Well, not really. The term r represents different concepts in [5] and [6]. In [5] it stands for a measure of genetic relatedness, while in [6] it stands for an altogether more abstract quantity. It involves a ratio between two covariances, one of which measures the connection between possession of a gene and the frequency with which a given individual performs a costly social act, and the other of which measures the connection between possession of that gene and the frequency with which the same individual receives the benefit of such acts. The underlying model of the selective process is quite different from that in Hamilton’s Rule. The terms of Hamilton’s Rule represent the costs and benefits of a particular social act. The Rule states the condition under which that act will tend to increase the frequency of a particular gene which gives rise to the act. In contrast, Grafen’s derivation of [6] considers the totality of social interactions in a population, and states a condition under which the possessors of a particular gene will have an average fitness higher than that of non-possessors. This condition has no obvious connection with genetic relatedness. It is true that, as Grafen shows, Hamilton’s Rule in the ordinary sense complies with the condition. If individuals, on average, are sufficiently closely related by ancestry to the individuals with whom they interact for Hamilton’s Rule to be satisfied, then they will also meet the condition stated in Grafen’s model. But this cannot be deduced from Price’s Equation alone: it requires an elaborate analysis of the nature and measurement of relatedness. It is highly unlikely that anyone armed only with Price’s Equation would ever have hit on the expression [6] unless they had already arrived at something equivalent to Hamilton’s Rule by more direct means. It is arguable that Grafen’s version is both more rigorous and more general than Hamilton’s Rule in its usual interpretation (more rigorous because it is ultimately a mathematical tautology; more general because it allows for covariance due to factors other than common ancestry), but unlike Hamilton’s Rule in the ordinary sense it cannot directly be applied to any actual biological situation.

Conclusions

Summing up my subjective opinion on the value of Price’s Equation (apart from its use in ‘multilevel’ selection theory, and in areas outside biology), I think that the full version of the equation does make a notable contribution to evolutionary theory, but I am not convinced that the short version is as valuable as the enthusiasts claim. It has so far been used mainly by theorists with a taste for abstraction and mathematical elegance. I think that mathematical elegance in itself is less useful in biology than in physics, where it is often a clue to underlying laws of nature or to new phenomena, as for example Maxwell’s field equations led to the discovery of electromagnetic waves, and Dirac’s Equation led to the discovery of the positron. There is probably nothing comparable to this in biology; for example, Fisher’s comparison of his Fundamental Theorem of Natural Selection to the Second Law of Thermodynamics is quite misleading. The FTNS, when properly interpreted, is a mathematical tautology; whereas the Second Law of Thermodynamics is a fact of nature: we can easily imagine a world in which it would be false. So far as I know, Price’s Equation in its short form has not yet helped solve any hitherto unsolved problems in biology or led to any new empirical findings. Perhaps it will in future, but until then I don’t think we should get too excited about it.

References
G. R. Price, ‘Selection and covariance’, Nature, 227, 1970, 520-21.
G. R. Price, ‘Extension of covariance selection mathematics’, Annals of Human Genetics, 35, 1972, 485-90.

Steven A. Frank, Foundations of Social Evolution, 1998.
Alan Grafen: ‘A geometric view of relatedness’, Oxford Surveys in Evolutionary Biology, vol. 2, 1985, p.28-89.
W. D. Hamilton: Narrow Roads of Gene Land, vol. 1, 1996; vol. 2, 2001.
Oren Harman: The Price of Altruism: George Price and the Search for the Origins of Kindness, 2010.
Mark Ridley: Mendel’s Demon: Gene Justice and the Complexity of Life, 2000 (UK paperback edition.)

1. I have often heard that the second law of thermodynamics is just a result of math (more specifically, statistics). Eliezer Yudkowsky would frame it in Bayesian terms as reflecting our ignorance about the microstates of component particles. I suppose the fact about the universe is that its initial state is uncommonly orderly (Julian Barbour instead says the initial state is initial by virtue of its orderliness).

2. “I have often heard that the second law of thermodynamics is just a result of math”

Only in the sense that it follows mathematically from certain empirical facts, such as the quasi-random behaviour of gas molecules. These empirical facts are not logically necessary: e.g. we could perfectly well imagine a world in which certain bodies were ‘heat attractors’ which absorbed heat from surrounding bodies even though these were colder, which would violate the SLT.

3. “two components: one of which depends on the covariance between the fitness of individuals in the ‘parent’ generation and their own possession of the gene in question, and the other of which depends on any change of frequency of the gene among the offspring as compared with their own parents.”

Would you restate (elaborate) that again in different words? thanks

4. Can I restate it in different words? Not sure that I can do it any better, but let me try a slightly different approach.

The frequency of a gene in a population can change for two reasons.

First, individuals who have the gene in question (or maybe more than one copy of it) may have more or fewer offspring than those who do not. Technically, this means there is a covariance (positive or negative) between the number of copies of the gene possessed by an individual and their number of offspring. If the covariance is positive, then other things being equal the frequency of the gene in the population will increase; if it is negative it will decrease.

Second, the frequency of the gene may change between parents and their own offspring. E.g. the parents may each have one copy of the gene but their offspring may have two, or they may have none.

Price’s Equation provides a framework for quantifying these two ways in which gene frequency in the population may change.

5. It’s this second point that I don’t quite get. How does it differ from the first? Give a simple example. thanks,

6. Let’s set up the simplest possible numerical example.

Haploid asexual organisms, 2 individuals (A and B) in the population. The gene of interest is marked X.

Case 1: Individual A has the gene X and has 2 offspring which also have the gene X. Individual B does not have the gene X and has no offspring. Result: frequency of the gene X increases from 50% to 100%.

Case 2: Individual A has the gene X and has 1 offspring with the gene X. Individual B does not have the gene X but has 1 offspring with the gene X (due to a mutation). Result: frequency of the gene X increases from 50% to 100%.

In case 1 the increase in the frequency of X is entirely due to the ‘covariance’ term of the equation. In case 2 it is entirely due to the other term.

7. Oh, a mutation! But that’s pretty rare. Are there other examples? BTW, we are talking alleles not genes I suppose? Can you give a good not a-sexual example? Does it work better if you work with sub-populations instead of individual couples?

8. David, excuse me, but let me amend my question: Can you give a simple non-a-sexual example that does not involve mutation? Does “transmission bias” refer to mutation or something else? I couldn’t find a definition for transmission bias on the web.

9. Mutation is indeed rare in genetic inheritance, but remember that Price wasn’t just interested in that.

Apart from mutation, anything that causes an offspring to differ from its parents would affect the ‘transmission’ term of the equation. Segregation distortion would be one example. In segregation distortion the offspring do not get a random sample of the parents’ genes. But even with random sampling the offspring dont’t always get exactly the same proportions of genes as their parents, so for example two AB parents may have offspring who are all AA.

In a large population these fluctuations of random sampling tend to cancel out, which is why the ‘transmission’ term of the equation is often negligible.

10. I see. Thanks. Does this relate to the fact that under the Price Equation group selection is (or so I read) theoretically possible but highly unlikely?

What about a proxies for inclusive fitness, such as “we grew up in the same small hunter/gatherer band together”? That would correlate with kinship. Would the result be considered group selection in the non-kin sense? Seems like it would.

11. I will be commenting on group selection in my final post.

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