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July 08, 2004

Defining Group Selection: Price's Equation

In an earlier discussion of group selection (here) I said that much of the argument about the importance of group selection was a disagreement over semantics rather than facts.

In re-reading the earlier post I think I could have usefully mentioned George Price’s covariance formula, as explained by W. D. Hamilton, Narrow Roads of Gene Land, vol. 1, p. 332....

The following, unless otherwise stated, deals only with the selection of genes, not of memes, cultural traits, etc.

Suppose we wish to consider the change of frequency of a gene in a population from one generation to the next. Suppose also that the population is divided into a number of groups, either in some natural way (e.g. as discrete geographical sub-populations), or by a decision of the investigator. Price’s equation provides a means of analysing any overall change of frequency of the gene into two components.

One component (the ’intragroup’ component) takes account of the change of frequency of the gene within each group, so that the total contribution of these changes is simply the average of the intragroup changes in frequency, weighted by the size of each group and its own fitness (rate of growth or decline).

The other component (‘intergroup’) is more subtle, and takes account of any covariance between the fitness of the groups and the initial frequency of the gene within them. Thus, if groups which have a high initial frequency of the gene tend to have higher (or lower) fitness than others, this will affect the frequency of the gene in the total population, but it is not taken into account in the intragroup component. (Incidentally, in Price’s formula the intergroup component comes first, but that is just a matter of presentation.)

The covariance (a statistical measure of the association of two variables) can also be expressed as the product of the regression of group fitness on gene frequency, times the variance in the initial gene frequency itself. An important consequence of this is that if there is no variance between groups in the initial gene frequency, then the intergroup component of the Price formula must be zero.

The beauty of Price’s formulation is that it provides a general, mathematically watertight, and relatively simple framework for identifying the contributions of different levels of selection. (In principle it can be extended to more than two levels, but I won’t consider that. For the formula itself, see the Technical Note below.)

It is naturally tempting to propose that the intragroup component should be identified with individual selection, while the intergroup component represents group selection. I think there are two problems with this suggestion. One is that some processes potentially affecting the intragroup component, such as reciprocal altruism and help given to relatives, are regarded by some biologists as group-selection phenomena, while some processes affecting the intergroup component, such as the geographical association of relatives, would be regarded by other biologists as contributing to kin selection rather than group selection. So the proposal is unlikely to be acceptable to either side in the debate. More seriously, whether a process falls within or between groups may depend on how the population is divided up. So it would be difficult to say categorically whether a process was ’individual’ or ’group’ selection without considering every possible way of dividing the population.

Nevertheless, even if the formula does not provide a knock-down solution to the group selection controversy, it does offer a means of clarifying the issues. For any process which is postulated as a form of group selection, it should be possible to specify whether it contributes to the intragroup or intergroup component of the Price formula.

If it contributes to the intergroup component (assuming that this is non-zero), it should be possible further to specify what gives rise to the covariance between gene frequency and group fitness. Crucially, there cannot be non-zero covariance unless there is also variance in each variable. An important part of the analysis is therefore to explain how the initial variance in gene frequency arises, i. e. how some groups come to have a higher frequency of the gene in question than others. So far as I am aware, there are five main ways in which such variance may arise:

a. Pure chance: the random assortment of individuals leading to differences in gene frequency between different groups. As Hamilton shows, the random assortment of individuals in a single generation does not create enough between-group variance for an ’altruistic’ gene to increase in frequency through the intergroup component of Price’s formula, if the fitness effects of the gene are simply additive. However, if the fitness effects are synergistic (i. e. increase disproportionately as the number of altruists increases) this objection does not necessarily apply.

b. Genetic drift over several generations in isolated groups could lead to a divergence of gene frequencies between the groups.

c. Individuals with relevant traits may associate together for behavioural reasons.

d. Individuals living close together geographically are also likely to be genetically related, and therefore to have a higher frequency of certain genes than the population average. This is especially true in small local populations with little migration. Such populations are also liable to inbreeding and genetic drift due to the chance loss of alleles from the local gene pool, so this source of variance tends to overlap with (b).

e. Different selection pressures may apply in different places, leading to geographical variation in gene frequencies and consequently to variance in gene frequency between groups living in different places. Since the variation in gene frequencies is a straightforward consequence of individual natural selection, this is seldom considered relevant in the context of group selection, though in principle it should be, since it might lead to a non-zero intergroup component in the Price formula. However, an intergroup component will not arise merely because each group is specially adapted to its local conditions. This will produce variance in gene frequency between groups, but in itself it will not produce covariance between gene frequency and group fitness.

Provided the investigator clearly specifies how the population is divided into groups, and how any covariance between group fitness and gene-frequency arises, it is a matter of linguistic convenience whether a process is described as group selection, individual selection, kin selection, or whatever. Hamilton’s own suggestions (op. cit., p. 337) seem quite balanced and sensible: “if we insist that group selection is different from kin selection the term should be restricted to situations definitely not involving kin. But it seems on the whole preferable to retain a more flexible use of terms; to use group selection where groups are clearly in evidence and to qualify with mention of ’kin’ (as in the ’kin-group’ selection referred to by Brown), ’relatedness’ or ’low migration’ (which is often the cause of relatedness in groups), or else ‘assortation’, as appropriate. The term ’kin selection’ appeals most where pedigrees tend to be unbounded and interwoven, as is so often the case with humans”.

Oddly, Price’s equation has been taken by enthusiasts as somehow vindicating the importance of group selection. But quite apart from the point that Price’s formula does not automatically resolve the semantic dispute, it remains entirely a matter of fact whether or not the intergroup component of the formula is important in nature. I am not aware of much evidence that it is - if there were, enthusiasts for group selection would presumably bring it forward.

To mention briefly the group selection of memes, cultural traits, etc., in principle the Price-Hamilton approach could be applied, but we have a much less clear idea of what the units of replication are, and how to measure their frequency and fitness, than in the case of genes. At present I don’t think there is any alternative to giving a detailed ad hoc explanation of what is meant by ‘group selection’ in any particular case of cultural evolution.

Technical Note (for masochists only)

Price’s equation applies to selection within a population of ‘particles‘. It may help to imagine these as genes at a single locus in a haploid population, so that the number of particles equals the number of individual organisms. (This is my suggestion, just to help the interpretation. I have also modified the notation to avoid using subscripts and Greek letters.) If the number of the total population in the first generation is N, suppose that this is divided in some way into groups. We denote the number in each group by lower-case n (not necessarily the same for each group). We use the letter S to denote summation over all groups, so Sn = N, as the total population number must be the sum of the group numbers. We are interested in genes of a particular type, A. We designate the frequency of A within the population as Q, and within each group as q (not necessarily the same for each group). QN is the total number of A genes in the population, and qn is the number in each group, therefore Sqn = QN, since the total population is just the sum of its groups. (NB: when the product of two variables appears within the scope of a summation, as in Sqn, this indicates that the value of one variable is to be multiplied by the corresponding value of the other variable before summing.)

So far we have been dealing only with the first generation. We assume that there are discrete generations. For each value in the first generation, the corresponding value in the second generation may be designated by an asterisk. Thus the total population in the second generation is N*, the population of a group is n*, the frequency of A in the second generation is Q* for the whole population and q* for each group. The total number of A genes in the second generation is Q*N* for the whole population and q*n* for a group.

We can now define a concept of fitness, denoted by W = N*/N for the whole population and w = n*/n for a group. W (or w) therefore measures the rate of growth or decline of a population from one generation to the next. Note that NW = N* and nw = n*, so Snw = Sn* = N* = NW.

Finally, we may denote the change in frequency of A as dQ = (Q* - Q) for the whole population and dq = (q* - q) for a group.

With this notation Price’s equation may be expressed as follows (adapted from Hamilton‘s version):

WdQ = Snw(q - Q)/N + Snwdq/N.

Hamilton remarks that with such notation the equation is ‘easy to derive’, and leaves it for his readers to do so for themselves. Well, I found it bloody difficult, so for the benefit of others I will give my own working:

WdQ = (N*/N)dQ = (N*/N)(Q* - Q) = Snw(Q* - Q)/N = SnwQ*/N - SnwQ/N .

So far, fairly mechanical. The non-mechanical part now is to recognise that SnwQ* = Snwq*. To me, at least, this is not obvious. But reflect that the total number of A genes in the second generation can be arrived at in two ways: either by taking the total of the second generation population and then multiplying it by Q*, or by calculating the number for each group in turn, and then summing the numbers. The first way gives N*Q* = SnwQ*, while the second gives n*q* = nwq* for each group, which is summed to get Snwq*. Since the total must be the same, whichever way we reach it, we have SnwQ* = Snwq*. (For the same basic reason, Snq = SnQ, and Snw = SnW.) We may therefore substitute Snwq* for SnwQ* in the expression SnwQ*/N - SnwQ/N, giving Snwq*/N - SnwQ/N.

Since q* = q + dq, we can rewrite this as Snw(q + dq)/N - SnwQ/N.

This may be rearranged as Snw(q - Q)/N + Snwdq/N, which is the right hand side of the desired equation WdQ = Snw(q - Q)/N + Snwdq/N. QED.

The first term on the right hand side is usually described as the ’covariance’ term. It measures the association between the fitness of the group and the extent to which the frequency q in the group diverges from the frequency Q in the whole population. But as a covariance it has several puzzling features, which took me quite a while to fathom. If it were a covariance between properties of groups, we might expect it to take the form S(w - Mw)(q - Mq)/G, where G is the number of groups, Mw is the mean of the w’s for the groups (i.e. the sum of the w’s divided by G) and Mq is the mean of the q’s for the groups (the sum of the q’s divided by G) So why is the denominator N and not G, why does the term n appear in the numerator, why is the mean of the q terms Q and not Mq (which is not necessarily the same as Q unless the size of each group is equal), and what has happened to the mean of the w’s?

The answer to the first three questions is essentially the same: the covariance is not a covariance between properties of groups as such but between certain properties of individuals defined by reference to their groups. The product sum consists of the sum of N terms, one for each individual, arrived at by multiplying the w and q of that individual’s group. The total number of ‘observations’ is therefore the total number of individuals, N, and not the number of groups, which explains why N is the denominator. It also explains why the term n enters into the numerator, so that each w and q is multiplied by the appropriate number of individuals, and why the mean of the q‘s is Q (the mean for all individuals) and not Mq. As to the absence of any term for the mean of the w’s, it would be legitimate to express the numerator in the form Sn(w -W)(q - Q), but it will be found that the terms involving W cancel out, as SnqW = SnQW, so that the expression may be reduced to Snw(q - Q). (Alternatively, one could cancel out the terms in Q, since SnwQ = SnWQ, and reduce the expression to Snq(w - W). I have not seen it in this form. The fuller form Sn(w -W)(q - Q) is perhaps preferable, as it makes it clear that variation in both w and q is needed before there can be an ‘intergroup’ component.)

Despite its odd appearance, Snw(q - Q)/N is therefore an ordinary covariance, provided it is understood not as a covariance between properties of groups but between properties of individuals defined by reference to their groups. It may be described in words as the covariance between the property of being a member of a group with fitness w and the property of being a member of a group with frequency q. The difference is subtle but important, because Price’s equation is only valid if the number of individuals in each group is taken into account. Similarly, in expressing the covariance as Variance x Regression, it is strictly the variance and regression of properties of individuals that needs to be measured, even though the properties inescapably involve their groups. This is a somewhat peculiar situation, and it is not very clearly explained by Hamilton, so it is perhaps not surprising that the formula has taken a long time to catch on.

Posted by David B at 06:15 AM