ON MULTILEVEL SELECTION AND KIN SELECTION: CONTEXTUAL ANALYSIS MEETS DIRECT FITNESS

Authors


Abstract

When Hamilton defined the concept of inclusive fitness, he specifically was looking to define the fitness of an individual in terms of that individual's behavior, and the effects of its’ behavior on other related individuals. Although an intuitively attractive concept, issues of accounting for fitness, and correctly assigning it to the appropriate individual make this approach difficult to implement. The direct fitness approach has been suggested as a means of modeling kin selection while avoiding these issues. Whereas Hamilton's inclusive fitness approach assigns to the focal individual the fitness effects of its behavior on other related individuals, the direct fitness approach assigns the fitness effects of other actors to the focal individual. Contextual analysis was independently developed as a quantitative genetic approach for measuring multilevel selection in natural populations. Although the direct fitness approach and contextual analysis come from very different traditions, both methods rely on the same underlying equation, with the primary difference between the two approaches being that the direct fitness approach uses fitness optimization modeling, whereas with contextual analysis, the same equation is used to solve for the change in fitness associated with a change in phenotype when the population is away from the optimal phenotype.

In recent years there has been an increasing debate over whether the evolution of altruism is better understood using a kin selection, inclusive fitness approach, or a multilevel selection approach. It is a bit perplexing that this controversy has recently been growing because it has long been settled that the multilevel selection approach and the kin selection approach are mathematically similar, although several models point to difficulties with the Price equation that suggest that there are important differences between the two approaches (Simon et al. 2012; Van Veelen et al. 2012). Nevertheless, the choice of which provides better insights into the evolution of social behavior at some level appears to be an aesthetic choice. Indeed, as early as his 1964 paper, Hamilton recognized that there were two distinct approaches to studying social evolution. He briefly describes the “usual approach,” which is effectively a multilevel selection approach and goes on to describe his “inclusive fitness” approach as an equally valid, and potentially more tractable solution. In this essay I will discuss the different approaches, and highlight some of the equivalencies and differences between them.

A Difference in Traditions

When Hamilton defined his concept of inclusive fitness he specifically was looking for a measure of the fitness of an individual in terms of the individual's behavior, and the effects of this behavior on other individuals, and particularly on other related individuals. His conception is an optimality approach that is consistent with other related traditions, such as models of evolutionary stable strategies (ESSs) (Maynard-Smith 1974). In an ESS model, the perspective is to consider the fitness of an individual and ask if the fitness increases or decreases if the underlying genes or a behavior of the individual are changed. As a result the question of interest is how fitness changes as a function of phenotype. Mathematically, this class of models is interested in determining math formula, that is the change in fitness as a function of a change in the alleles at the gene of interest, or of the trait of interest. Because these are optimality models, the goal is to identify values of x such that math formula. Such points will either be fitness maxima or minima, and by inspection it is possible to find fitness maxima, or values of x that, under most, but not all, circumstances, are ESS solutions (Taylor 1989). In contrast, multilevel selection models have grown out of the field of quantitative genetics, and the one that I will be specifically discussing, contextual analysis, follows directly from the tradition of using regression to measure the change in phenotype due to selection (e.g., Lande and Arnold 1983; Arnold and Wade 1984). Mathematically, these models also examine rate of evolution but now use math formula, that is the change in relative fitness as a function of phenotype. Unlike the optimality models, these methods specifically focus on how relative fitness changes as phenotype changes when the population is away from the optimum. As such, the focus is on the current mean phenotype of the population and the selective forces acting on the population, and it is assumed that the population is not at a phenotypic optimum. For this reason, it is typically not of interest to solve for math formula. The difference in the goals of optimality models and evolutionary rates models means, for example, that on the one hand inclusive fitness oriented studies never report the strength of kin selection, and on the other hand multilevel selection studies typically do not identify the optimal phenotype.

There are two distinct measures of fitness, absolute fitness, and relative fitness. Absolute fitness is typically an appropriate measure of fitness, such as lifetime reproductive success, measured on an arbitrary scale. Relative fitness is defined as the absolute fitness divided by population mean absolute fitness:

math image(1)

The mean relative fitness is one, with high fitness individuals having a relative fitness greater than 1, and low fitness individuals having a relative fitness less than 1. Kin selection models are typically vague about the scale on which “fitness” is measured. For example, Taylor and Frank 1996 never actually define fitness when they develop their model, although it appears that they are using absolute fitness. This vagueness is not an issue in kin selection models because the goal of such models is to identify the point at which math formula and the units in which fitness is measured does not affect the value of x at which this occurs. Changing the scale on in which W is measured will change the absolute value of the curve, but it will not change the location of the optimum. In contrast, quantitative genetics models are explicit that it is relative fitness that is used in the models. This attention to scale is important because quantitative genetics models are interested in the rate of a process, rather than the end point of evolution.

In most optimality models, the goal is to identify the genotype that will provide the optimal fitness. Thus, Hamilton 1964 develops his model around genotypic effects, and focuses on relatedness as a means of measuring the probability of shared genes. Similarly in Taylor and Franks 1996, model fitness is maximized as a function of x, which is given generically as “a variable”; however, the implication is that x is some measure of genotype or allelic value at a locus. Indeed, in one of their examples they refer to “the genic value x of the gene” (Taylor and Frank 1996). In their formulation the “recipients” phenotype, y, and the “actors” phenotype, z, are both functions of the underlying genetic variable x. In contrast, in the quantitative genetics formulation the response to selection is the result of two distinct components, as illustrated by the breeders equation (Lande 1979; Falconer and MacKay 1996),

display math(2)

P−1S is the selection gradient, which is a measure of the forces of selection acting on the phenotype with no reference to the genetic basis of those phenotypic traits. The genetic component of the breeder's equation is encompassed in the additive genetic covariance matrix, G. Because of the distinction that is made between selection and inheritance, studies of selection in natural populations are typically based on the regression of relative fitness on phenotype, and the genetic basis of the traits under selection is included later, often from measurements taken in a separate study. The genetic basis of the traits is incorporated when the G matrix and the selection gradient are multiplied to predict the response to selection.

Thus, in comparing the two models, inclusive fitness and direct fitness models have genes and genetic effects as an implicit part of their model, whereas quantitative genetic models have focused on the phenotype with genetics being added as a second element that is outside of the actual measure or modeling of how selection acts on the phenotype. This distinction shows up in models in that the direct fitness models are built around changes in genotype, expressed as dx, whereas selection models are built around changes in phenotype, expressed as dz.

INCLUSIVE FITNESS AND DIRECT FITNESS

Taylor, Wild and Gardner 2007 recently provided a detailed comparison of the direct fitness and inclusive fitness approach, so there is no need to present an overall outline of the comparison between these two approaches. Both of these methods lie well within the optimization approach favored by Hamilton, and used in ESS modeling. Thus, both seek to find a value of genotype that maximizes fitness. The principle difference is that Hamilton's inclusive fitness approach assigns to the focal individual the fitness effects of its behavior on other related individuals, whereas the direct fitness approach assigns the fitness effects of other actors to the focal individual. This is the distinction that Hamilton was making: direct fitness is what Hamilton referred to as the “usual” approach for incorporating “neighborhood modulated fitness,” and the inclusive fitness is the alternative approach that he advocated (Hamilton 1964).

As Taylor Wild and Gardner 2007 point out, the inclusive fitness approach can be useful in developing theoretical models because it is an intuitively natural way to think about the evolution of social behavior. That is, in the inclusive fitness approach the fitness effects of a variant genotype are considered in terms of the effect of that genotype on the individual and all of the relatives with whom it interacts. The inclusive fitness approach can also be useful in experimental settings in which interactions and relatedness can be experimentally manipulated (e.g., Komdeur 1992; Creel and Waser 1992; Krakauer 2005; Priest et al. 2008); however, there are limitations to Hamilton's inclusive fitness approach. In particular, the apparent fitness of an individual has to be partitioned between the fitness assigned to the individual due to its own actions, and a portion of that fitness that must be assigned to the inclusive fitness due to its actions on other individuals. To avoid double counting of fitness, the inclusive fitness of the focal individual must be subtracted from the fitness of the individuals with whom it is interacting. Although there is no theoretical or conceptual problem with this, it can, as a practical matter, become problematical, and as a result, the “close accounting of all fitness effects of a particular item of behavior, is often difficult to formulate” (Taylor and Frank 1996, see also Grafen 1982; Creel 1990; Lucas et al. 1996; Queller 1996).

Because of the difficulty of applying Hamilton's inclusive fitness approach, the direct fitness approach (Taylor and Frank 1996) has gained increased acceptance and has become the preferred method of modeling kin selection (Taylor et al. 2007). The advantage of this approach is that in most circumstances it is easier to assign fitnesses and avoid the double counting problem. Although this is not technically considered an “inclusive fitness” approach, it is nevertheless a minor problem to recapture the inclusive fitness of an organism as (Taylor et al. 2007):

display math(3)

where Winc is the inclusive fitness, δwk is the effect of the focal individuals behavior on the fitness of the kth recipient, cov(ek,z) is the covariance between the genetic value of the kth individual and the phenotype of the focal individual, and cov(e,z) is the covariance between the genotype and the phenotype of the focal individual.

CONTEXTUAL ANALYSIS AND MULTILEVEL SELECTION

Contextual analysis is a method for analyzing multilevel selection that was adapted from research in the social sciences (Heisler and Damuth 1987; Goodnight et al. 1992). Following the methods for analyzing phenotypic selection developed by Lande and Arnold (1983; Arnold and Wade 1984), a multiple regression of relative fitness on phenotype is performed. Contextual analysis extends the methods of the Lande and Arnold approach by including “contextual traits,” that is traits measured on the group or neighborhood, in the multiple regression. Theoretical analysis confirms that this method does work, and can correctly identify the levels at which selection is acting for a number of classic models of selection (Goodnight et al. 1992). Importantly, contextual analysis is an outgrowth of standard quantitative genetic methods that are used to measure selection in field situations. As a result contextual analysis is a methodology that can, and is, regularly used in field studies of selection (e.g., Stevens et al. 1995; Tsuji 1995; Aspi et al. 2003; Weinig et al. 2007; Eldakar et al. 2010). It has the advantage of providing quantifiable measures of the strength of selection at multiple levels, and allowing a direct comparison of field measures of the strength of group and individual selection.

Although contextual analysis is a well-established technique, there are a number of misconceptions surrounding it, and multilevel selection in general. First, in recent years, the term multilevel selection has started to supplant the term group selection. This new terminology makes sense because group selection is mainly of interest when there is also selection acting at a lower level, that is, when selection is acting at multiple levels. Multilevel selection refers to situations in which selection is acting simultaneously at multiple levels, whereas group selection applies specifically to selection acting at the level of the group.

Second, the distinction between “old” and “new” group selection that is sometimes made (West et al. 2007) is a canard. Methods such as contextual analysis are entirely consistent with the “old” use of the term group selection, as evidenced by the finding that contextual analysis successfully identifies a classic group selection model as indeed being group selection (Goodnight et al. 1992). What has happened is that methods such as contextual analysis have made it possible to recognize that the severe limitations placed on the concept of group selection in the early days of the debate (e.g., Maynard-Smith 1976) are not justified.

A more logical distinction to be made is the distinction between what has been called the “adaptationist” school, and the “genetic” school (Goodnight and Stevens 1997). The adaptationist school focuses on existing adaptations, seeks to develop evolutionary explanations for those adaptations, and is generally associated with optimality modeling and thinking. Using this approach, traits of interest are studied in detail, and plausible explanations for the evolution of those traits are developed. For any given trait there are inevitably multiple explanations for how it could evolve, as a result rules are needed for deciding which is the most plausible explanation. One such rule is Williams’ “principle of parsimony,” in which Williams suggests that in explaining adaptations, the lowest level at which selection can act should be considered the most plausible. Other rules are of course possible; however, William's principle has been widely adopted as an appropriate guideline for the adaptationist school even though it has been criticized (Sober and Wilson 1998). The principle of parsimony stands as an obvious explanation for why the adaptationist school has such an emphasis on the evolution of altruism. In particular, using Williams’ principle, group selection will never be “invoked” unless it is opposed by individual selection. It is also worth emphasizing that the optimization models associated with kin selection primarily lie well within the adaptationist approach.

In contrast to the adaptationist school, the genetic school is generally associated with multilevel selection methods, and does not rely on rules for deciding among possible evolutionary explanations. Instead, students of this approach seek to examine the process of adaptation as it occurs, either experimentally in the laboratory or in field studies. It is this focus on the ongoing processes of selection that makes rules such as Williams’ principle of parsimony unnecessary. In the laboratory this approach has been used to demonstrate that populations respond rapidly to experimentally imposed group selection, and that indirect genetic effects are primarily responsible for the surprising effectiveness of group selection experiments (Goodnight and Stevens 1997). Field studies using contextual analysis have demonstrated that in nature, multilevel selection is far more common than previously anticipated (Stevens et al. 1995; Tsuji 1995; Aspi et al. 2003; Weinig et al. 2007; Eldakar et al. 2010).

Equally important, because selection is observed as it is occurring there is much less emphasis on the evolution of altruism. In the genetic school, group selection is equally easy to detect whether it is acting in concert or in opposition to individual selection. A reading of classic group selection experiments makes this distinction apparent. Wade 1977 performed group selection on population size, Goodnight 1985 performed group and individual selection on leaf area, and Craig 1982 performed group selection on migration rate. None of these traits is obviously an “altruistic” trait, and importantly, in Goodnight 1985 there were several treatments in which group and individual selection were applied in the same direction.

Neither the adaptationist nor the genetic approach should be considered “better.” The adaptationist school has a history of examining traits in which we are vitally interested, but which may be resistant to experimental manipulation. In contrast, the genetic school tends to focus on traits that are easily measured and susceptible to experimental manipulation, even if they are less interesting as adaptations (Goodnight and Stevens 1997).

CONTEXTUAL ANALYSIS AND DIRECT FITNESS

Contextual analysis and direct fitness come from very different approaches, nevertheless, if we make reasonable modifications of the two approaches it is possible to show that they are based on the same equation. To do this, start with the direct fitness formulation suggested by Taylor and Frank 1996. Taylor and Frank point out that using an inclusive fitness approach that we want to find the value of x such that math formula, where W is fitness measured on an arbitrary scale, and x is genotypic value, or some similar measure of interest. They then use the chain rule to examine the change in fitness as a function of the individual trait (y), and the group trait (z). Note that this is a redefinition of “z,” but it is done here to be consistent with Taylor and Frank's notation.

display math(4)

In words, the change in fitness as a function of genotype is due to the change in fitness as a function of individual phenotype times the relationship between individual phenotype and genotype plus the change in fitness as a function of interacting group's phenotype times the relationship between the group phenotype and the focal individuals genotype.

The group trait, z, can potentially be any measure that is made at the group level. In Taylor and Frank's model it is the relatedness weighted mean phenotype of interacting individuals. In contextual analysis it may be summary statistics such as the group mean phenotype, or the size of the largest or smallest individual. Alternatively, it may be a “contextual trait,” such as population density, that is solely a property of the group and has no individual level analogue. The generality of what qualifies as a “group trait” is an advantage in that these methods can easily generalized beyond the evolution of altruism, and can be used to study a wide range of systems. As an interesting example, Molofsky et al. 1999 used a modification of contextual analysis to measure the relative influence of genetic and community level environmental factors on the invasibility of different genotypes of reed canary grass. However, this generality means that these methods cannot be applied without a solid understanding of the biology of the system being investigated.

Contextual analysis differs from direct fitness in that we are interested in the change in fitness as a function of a change in phenotype, rather than identifying the optimum. To equate the two equations, I assume that Taylor and Frank's “W” is relative fitness. From equation (1) it is clear that relative fitness is equal to absolute fitness divided by a constant (math formula) and thus this transformation will not effect the results of the direct fitness approach.

Contextual analysis is a multivariate approach focusing on phenotype rather than genotype, thus we remove the genotype from the equation by multiplying through by math formula to get:

display math(5)

and math formula to get:

display math(6)

This can be converted to the multivariate regression form of contextual analysis:

math image(7)

or

display math(8)

With contextual analysis, we are interested in the change in multivariate phenotype

display math

which we can get by multiplying by the phenotypic variances:

display math(9)

This is the familiar Lande and Arnold 1983 formula for the effects of selection, except that it is applied to a system with group and individual traits rather than two correlated individual level traits. In words, this formulation divides the total change in Z due to selection, math formula, into the direct effects of selection, math formula, and the indirect effects of selection due to phenotypic correlations among traits, math formula. The change in math formula, math formula, is the change in the multivariate mean phenotype within a generation, providing a link between the rate of change in phenotype and the sometimes rather esoteric concept of partial regressions of fitness on phenotype.

The contextual analysis approach as it is developed here uses the relationship between fitness and phenotype, math formula, and ignores the relationship between fitness and genotype. This is in keeping with traditional selection analysis, which treats within generation selection as an ecological process independent of the response to selection (Lande and Arnold 1983; Arnold and Wade 1984). In the context of selection analyses, including contextual analysis, genetics can be included by completing the breeders equation, by multiplying the “S” vector, math formula, by GP−1, where G is the genetic covariance matrix, and P−1 is the inverse of the phenotypic covariance matrix. In the case of a single individual trait and a single group trait. The genetic covariance matrix would be:

display math(10)

where VarA(y) and VarA(z) are the additive genetic variances for the individual and group traits, respectively, and CovA(y,z) are the additive genetic covariances between the group and individual traits. This genetic covariance matrix in effect converts the phenotypic based contextual analysis model into the genotypic model of the direct fitness models. The advantage of using the G matrix rather than the genotypic value “x” is that it is a richer metric that does not specify a simple genetic relationship between the individual trait and the group trait. Experimental studies clearly demonstrate that the response to group selection is much greater than would be predicted based on simple genetic models. Importantly, there is also clear evidence that group selection acts on indirect genetic effects (Goodnight 1990a, 1990b; Goodnight and Stevens 1997). This implies that an individual trait and a group level trait should be considered separate traits with different genetic bases. Thus, the G matrix may be more appropriate than trying to describe the genetics of selection acting at multiple levels than is the single genotypic value of x that is used in traditional kin selection theory.

The problem with using the G matrix is that the methods for measuring the genetic basis of group level, or contextual, traits have not been developed. Measuring the heritability of contextual traits is complicated by the fact that in most interesting cases indirect genetic effects will contribute to their heritability. As a result standard randomization methods used for calculating the heritability of individual level traits may not be appropriate. Some work on this has been done in the context of agricultural methods (Griffing 1977; Muir 2005; Bijma et al. 2007; Bijma and Wade 2008), but much remains to be done in this area.

From this it can be seen that contextual analysis and direct fitness both use identical equations. This makes sense because this regression approach is the logical way of relating changes in fitness to changes in phenotype; however, the choice of how this equation is used depends on the question asked. In the direct fitness approach the question is what is the optimal genotype that maximizes fitness. For this it is appropriate to solve for math formula = 0 because this can be used to find maxima and minima. On the other hand, while the rate of change approach can be used to identify optimal phenotypes (e.g., Preziosi and Fairbairn 2000), in many cases this will be impractical. Thus, for theoretical questions focusing on the optimal solution to a problem, or identifying an ESS the direct fitness approach will often be more appropriate, whereas contextual analysis will tend to be less helpful.

In contrast, if the goal is to detect multilevel selection in the field, or to see whether there is selection acting on a trait, contextual analysis is more appropriate. This method directly addresses the problem of how selection is acting, and it does so using the well-developed methods of quantitative genetics as it has been applied to selection in natural populations. One of the strengths of contextual analysis in this regard is that it works well regardless of how the contextual traits are defined. Thus, it will work equally well with classic group selection (e.g., Maynard Smith 1976) in which the contextual traits are properties of the isolated group, classic kin selection in which the contextual traits are properties of the kin group, or even continuous populations with neighborhood interactions (Stevens et al. 1995).

Because inclusive fitness and the direct fitness approaches do not provide insights into how selection is currently acting, it is of no use in detecting multilevel selection in the field. Because of this it is perhaps not surprising that searching the literature reveals that while there have been several studies based in the quantitative genetics tradition (e.g., Wade 1980; Silk 1984; Breden and Wade 1989; Muir et al. 2013) there have been no studies using an inclusive fitness approach that report the strength of kin selection, or compare its strength to that of other evolutionary forces, in the nearly 50 years since Hamilton 1964 first introduced it.

CONTEXTUAL ANALYSIS AND THE PRICE EQUATION

Remembering that y is the individual trait, and z is the group trait, in its simplest form the Price equation as applied to change in phenotype for multilevel selection is (Wade 1985, see Frank 1997 for a full derivation)

display math(11)

or

display math(12)

which is another form of equation 6. In past studies Δy has been equated with individual selection and Δz has been equated with group selection (e.g., Wade 1985). However, the relationship between the Price equation and contextual analysis shows that both of these quantities have two components, that is y can change either as a result of individual selection acting directly on the individual trait, or as an indirect effect of selection acting at the group level. Similarly, z can change either as a result of group selection or as a result of individual selection leading to a change in the mean phenotypes of the groups.

This issue of equating the change in mean phenotype with group selection has led to considerable confusion. For example, Wade 1985 used the Price equation to examine the case of “hard selection.” In this model, the fitness of an individual is solely a function of its phenotype. Intuitively, there should be no group selection in this model; however, there are group-level effects of selection acting at the individual level. That is, individual selection within groups will lead to variation in the mean fitness among groups. Using the Price equation, this variation in group-mean fitness is equated with group selection. This reasoning lead West et al. 2007 criticize multilevel selection theory because “Even a nonsocial trait can be ascribed a group selection component simply because groups containing fitter individuals are themselves favoured by selection!” This criticism is indeed true if the change in group means as expressed by the Price equation is equated with group selection; however, using contextual analysis the change in the group means due to indirect effects of selection within groups are appropriately attributed to individual selection (Goodnight et al. 1992).

A more illustrative example is given by soft selection. In soft selection within group individual selection is performed; however, exactly the same number of individuals are selected from each group. Because there is no variation among groups there can be no covariance between group phenotype and group mean fitness, and based on the Price equation no group selection is acting (Wade 1985). However, this does not make intuitive sense, because the fitness of an individual is a function of group membership. For example, an individual of intermediate phenotype would have high fitness in a group with a low mean phenotype, and a low fitness in a group with a high mean phenotype. In contrast, contextual analysis indicates that the lack of variation among groups is maintained by group selection acting in opposition to the indirect effects of individual selection (Goodnight et al. 1992). This apparently counterintuitive result actually makes sense from a quantitative genetics perspective. To see this, consider two correlated individual level traits, for example, snout-vent length and body weight in a lizard. If length and weight have a positive phenotypic correlation then longer lizards will also tend to be heavier. If we were to select for longer lizards while keeping body weight constant we would have to not only select for longer lizards, but also select for lighter animals to counteract the indirect effects of selection for length on weight. It is the same with soft selection. Variation among groups will inevitably result in some groups having a greater number of large individuals. As a result with pure individual selection these groups of larger individuals would be overrepresented in the selected population. This is an indirect effect of individual selection on the group mean of the population. Group selection counteracts this indirect selection, and equalizes the representation among groups.

CONTEXTUAL ANALYSIS, INCLUSIVE FITNESS, AND HAMILTON'S RULE

Because contextual analysis is closely related to direct fitness, it is not surprising to find that the change in inclusive relative fitness can be calculated from contextual analysis. To see this recognize that using the direct fitness approach the change in inclusive fitness is calculated as (Taylor and Frank 1996):

display math(13)

It follows from equation 9 that

display math(14)

is also the change in inclusive fitness.

Hamilton's rule is typically addressed as an optimality problem, however, it can also be considered a competing rates problem. Treated as an optimality problem it can be solved using equation 13 for math formula, which is the standard method of treating Hamilton's rule using inclusive fitness. In contrast, when treated as a competing rates problem, contextual analysis is more appropriate. Goodnight et al. (1992) solved Hamilton's rule by equating benefit with group selection, cost with individual selection. They then solved for the situation in which the strength of group selection was greater than the strength of individual selection, and showed that:

display math(15)

where math formula is the squared partial correlation between the individual trait and relative fitness, math formula is the squared partial correlation between the group trait and relative fitness, and math formula is the squared correlation between the group and individual trait. In words, math formula, the cost, is the strength of individual selection, math formula, the benefit, is the strength of group selection, and math formula is the fraction of the total variance that is among groups.

The point that math formula is the fraction of variance that is among groups needs to be explored in more depth. In particular, if all variation is “additive” in the sense that the group trait is the mean of the individual traits within the group, then math formula is equal to Hamilton's original conception of r as the genetic relatedness among interacting individuals. If there are interactions, such as dominance or epistasis, or if there are social interactions, such as indirect genetic effects, or policing, then math formula will tend to be larger than Hamilton's r. Thus, the fraction of variance that is among groups is the more general concept that reduces to Hamilton's relatedness in cases where there are only additive effects.

Goodnight 2005 extended this equation to include genetic effects on the group and individual level traits.

display math(16)

where math formula is the regression of the individual level additive effects on the group level additive effects, and math formula is the regression of the group level additive effects on the individual level additive effects.

Both the inclusive fitness approach and the multilevel selection approach work well for solving Hamilton's rule. They both work with different assumptions, and thus may be appropriate in different settings. The inclusive fitness method uses the approach of optimizing fitness, and thus is appropriate when the details of the evolutionary rates are either equal or unknown. The multilevel selection approach attacks the evolution of altruism as a competing rates problem. This approach will often be best in experimental situations because the strength of group and individual selection can be directly measured using contextual analysis, and the problem can be solved even if the “optimal” phenotype is not experimentally knowable. Also importantly, following Goodnight 2005 the heritability at the group and individual level can be incorporated and the group and individual traits treated as separate genetically correlated traits with different heritabilities.

An advantage of the multilevel selection approach comes from the competing rates perspective. This is important because in dynamic systems the final phenotype is determined by the strength of selection and the genetic variation for the trait rather than the optimum. Thus, for example, inbreeding and policing can act to change the heritability at the individual and group levels. Importantly, standard inclusive fitness theory does not take the effect of inbreeding on heritable variance into account. In effect, kin selection models typically assume that interactions are among relatives, but reproduction occurs at random so that the degree of inbreeding equals zero. To see the importance of this consider the extreme case of cells in a typical animal. They are all the result of mitosis from a single fertilized egg. Thus, the only source of genetic variation among cells within an organism is somatic mutation. In this case “additive genetic variance” at the cell level (VarA(y)) is nearly zero, “organismal” level additive genetic variance (VarA(z)) is likely nonzero, and math formula is much larger than 1, and approaches infinity. Thus, it is not surprising that the cells within our bodies behave “altruistically,” and that in this case altruism may be much easier to evolve than indicated by the traditional interpretation of Hamilton's rule that puts a maximum value r = 1. In this case, with math formula, the population will evolve solely due to organismal selection, with selection at the cell level having no evolutionary consequence.

This distinction between inclusive fitness models being optimality models, and contextual analysis being a competing rates model indicates that while these two models are mathematical identical, the conceptual differences are substantial. Inclusive fitness approaches are methods that extrapolate from existing conditions to develop models to predict the equilibrium point for the population. When there are genetic or demographic complications that cause the costs and benefits to change as the population evolves, the predictions of such models will frequently be violated (Simon et al. 2012; Van Veelen et al. 2012). In contrast, the multilevel selection contextual analysis approach is a description of the current direction of the population at the current point in time. Speculation as to the evolutionary consequences of the measured selection is frequently done, but underling assumptions, such as heritabilities remaining constant, used in such extrapolation lies outside of the statistical method per se.

Inclusive fitness models find the “optimal” solution under a restrictive set of circumstances, and a particular set of assumptions. Three typically unstated assumptions are, first, that the heritabilities of the traits are constants unaffected by mating structure, second, that the genetics are additive, and third, that the correlation between the group and individual trait is negative one. The contextual analysis “competing rates” model provides a means of relaxing all of these assumptions. That is, genetic variance components at the group and individual level are typically measured values. It is known that additive genetic variance can change in a complicated manner with changes in population structure (e.g., Bryant et al. 1986a,b; Goodnight 1988, 2004; Willis and Orr 1993). The additive situation, where VarA(z) is a simple function of FST is a special case. If there are social interactions, and especially policing behaviors enforcing uniformity within groups it will often be the case that VarA(z) is much greater than predicted by the additive model. Also, because the group and individual level traits are not necessarily the same, there is no assumption that the correlation between the traits at the two levels will be negative one, or even very high. This has the important possibility that in some cases the “costly” individual level trait can evolve to minimize that cost without affecting benefit to the group.

Finally, if we accept the inclusive fitness assumption that the correlation between group and individual traits is necessarily negative one, then it becomes apparent that the equilibrium will be a function of the relative strengths of selection, and the relative amounts of additive genetic variance at the two levels. There is no reason that the equilibrium between these two opposing forces will, in general, equal the “optimal” solution.

All of this suggests that an obvious modification of the direct fitness approach would be to follow the quantitative genetics lead and explicitly incorporate the G matrix into the calculation of the optimal phenotype. This would allow much greater flexibility in modeling the relationship between direct and indirect fitness effects, and potentially solve much of the problem the potential mismatch between the optimum predicted from inclusive fitness models and the equilibrium predicted from contextual analysis problems.

PUTTING KIN SELECTION AND MULTILEVEL SELECTION INTO A SINGLE RESEARCH AGENDA

Kin selection, including both Hamilton's inclusive fitness approach and the direct fitness approach, and multilevel selection differ primarily in the types of questions being addressed. Kin selection methods aim at identifying character states that maximize fitness, whereas multilevel selection methods aim at examining the effects of selection on trait changes. Far from being alternative approaches to the same problem these two approaches are complementary methods that can be used together to obtain a complete understanding of the evolution of a social behavior system.

Consider a classic inclusive fitness problem, such as alarm calling by female prairie dogs (Hoogland 1981, 1983). In this example it is observed that upon seeing a predator, prairie dogs will frequently sound an alarm call that warns neighbors, most of whom are relatives. This is an example of a behavior that appears to have evolved to maximize the inclusive fitness of the alarm caller because it minimizes the chance of the predator being able to capture individuals to whom they are related. To study this it would be important to determine if this behavior does indeed increase the inclusive fitness of the calling female. For this Hamilton's inclusive fitness approach would be useful because the number and relatedness of the neighbors could be measured and the change in inclusive fitness of calling versus not calling could be calculated. Because the question is whether calling affects inclusive fitness of an individual organism the double counting problem will not affect the qualitative result. A second question would be determining the optimal behavior for the caller, for example, under what circumstances they should call. For this fitness optimization involving the use of the direct fitness approach would likely be the best strategy for addressing this question because double counting could have an important effect on the final optimum. Next, it would be of interest to determine whether there was ongoing kin selection acting on the population, and if so how strong is the selection favoring alarm calling (group selection), versus the strength of selection favoring remaining silent (individual selection). For this, contextual analysis, which examines the change in traits as a function of selection, is the appropriate approach. Finally, if the population is not at the optimum behavior as determined by the direct fitness approach, and there is ongoing selection, as determined by contextual analysis then a quantitative genetic study to determine the heritability of the trait may be needed to determine why the population is not on the optimum.

Thus, in summary I have shown that both the kin selection, in the form of the direct fitness approach, and contextual analysis use the same equation, and this sense the kin selection approach and the kin selection approach are identical. Although they both use the same equation the two approaches differ in how the equations are used. Kin selection models are used to extrapolate from the current situation and identify the optimal or equilibrium solution, but they cannot be used to examine the process by which the population will achieve that optimum. In contrast, the multilevel selection methods are used to describe the processes that are acting on the population in its current state, but do not provide insights into the equilibrium state that the population will evolve toward. The clear message from this is that kin selection and multilevel selection should not be considered the same, nor should they be considered competing paradigms. Rather they should be considered complementary approaches that when used together give a clearer picture of social evolution than either one can when used in isolation.

ACKNOWLEDGMENTS

This work benefited from discussions with many people, particularly H. Axen, S. Cahan, L. Higgins, J. Schall, and M. Wade. I am particularly grateful to B. Muir who made insightful comments on an early draft of this manuscript.

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