The correlated character scenario shows how a character-fitness covari-ance can be a by-product of direct selection on another character at the same hierarchical level. But what we are really interested in, vis-a-vis the levels of selection, is the idea that a character-fitness covariance might be a by-product of direct selection at a different level. This section and the next examine cross-level by-products in hierarchically organized systems.

For simplicity we assume two hierarchical levels, with strict nesting and a fixed number of particles per collective, as in Chapter 2. It may be useful to keep an abstract representation of hierarchical organization in mind; this is depicted in Figure 3.3.

Intuitively it is easy to see how cross-level by-products in the particles collective direction might arise. For since a collective's character and fitness usually depend on, and are sometimes defined in terms of, the characters and fitnesses of its constituent particles, selection at the particle level might have effects that 'filter up' the hierarchy, generating a spurious character-fitness covariance at the collective level. The challenge is to spell out this intuitive argument in precise terms. Precisely how must character and fitness at the two levels be related

Figure 3.3. Particles nested within collectives in order for such 'filtering up' to occur? And what about cross-level by-products running in the opposite direction? These questions needed to be addressed separately for MLS1 and MLS2, given that 'collective fitness' means something different in each; I consider them in turn.

Recall the essential features ofMLSl: the particles are the 'focal' units, and collective fitness is defined as average particle fitness. Evolutionary change is judged to have occurred when the frequency of different particle-types in the global population of particles has changed. As we saw in Chapter 2, the evolutionary dynamics can be described by an equation in which characters and fitnesses are attributed to both particles and collectives. This was illustrated by the Price decomposition:

Collective-level selection Particle-level selection wAz = Cov (W, Z) + E (Covk(w, z)) (3.4)

where Cov (W, Z) is the covariance between collective fitness and character; and E(Covk (w, z)) is the average of the within-collective covariances between particle fitness and character. Recall that for this version of Price's equation to hold true, collective character Z must be defined as average particle character, and collective fitness W must be defined as average particle fitness. Note also that equation (3.4) presumes zero transmission bias at the particle level.

An example much discussed in the group selection literature of the 1980s shows how cross-level effects can arise in MLS1 (cf. Sober 1984; Grafen 1985; Nunney 1985a; Heisler and Damuth 1987). Apopulation of organisms is subdivided into groups. (So the collectives are groups and the particles are organisms.) The fitness of any organism depends only on its own phenotype and not which group it is in—there are no 'group effects' on organismic fitness. As Heisler and Damuth (1987) note, most biologists would say that no group selection is occurring in this situation, for the evolution of the system can be predicted without taking group structure into account. Nonetheless, there may well be a covariance between group fitness and group character. Some groups may be fitter than others simply because they contain a higher proportion of fit organisms.

Sober (1984) illustrated this point with an example in which an organism's fitness depends only on its own height—any two organisms of the same height have the same fitness, whatever group they are in. A group composed mainly of tall organisms, so with a high average height, will be fitter than a group composed mainly of short organisms; thus group fitness and group character covary positively. Though Sober's discussion was not framed in terms of the Price equation, his example neatly illustrates how cross-level by-products may complicate the Price approach to MLS1. In terms of equation (3.4), Sober has described a case where a non-zero value of Cov (W, Z) is a by-product of selection at the particle level.

One consequence of this is that the Price approach to MLS1 is called into question; for it detects a component of collective-level selection where intuitively there is none. This suggests that equation (3.4) may not in fact partition the total change into components corresponding to the two levels of selection, despite what Price and Hamilton thought. This important issue is tackled in Section 3.4.

As a way of analysing cross-level by-products in MLS1, Heisler and Damuth (1987) advocate a statistical technique from the social sciences called 'contextual analysis'.9 The basic idea is to regard a collective's character as a 'contextual' character of each particle in the collective. So in Sober's example, average group height becomes a contextual character of every organism in the group. Each organism in the global population is therefore assigned two characters: an individual character (height), and a contextual character (average height of its group), both of which may affect its fitness. The crucial question is then: does the fitness of a particle depend only its own character, or does it also depend on its contextual, that is, collective, character?

It is important to see why this is the crucial question. The reason is that when a particle's fitness depends only on its own character, as in Sober's example, any character-fitness covariance at the collective level must be a by-product of selection at the particle level. (And arguably, the Price equation will mislead about the true levels of selection.) Conversely, where particle fitness is affected by collective character, at least some of the collective-level covariance is due to processes occurring at the collective level itself; not all of it is a side effect of lower-level selection. So determining whether particle fitness is affected by collective character provides an important clue about whether a cross-level by-product is in play.

9 Boyd and Iversen (1979) provide a thorough introduction to contextual analysis, with some discussion of conceptual and philosophical issues. For applications of contextual analysis to levels-of-selection problems in biology, see Heisler and Damuth (1987), Goodnight et al. (1992), Tsuji (1995), Pederson and Tuomi (1995), Getty (l999), Okasha (2004b), and Aspi et al. (2003).

How should we determine this? Simply looking for a correlation between particle fitness and collective character is insufficient, as Heisler and Damuth stress. Even if a particle's fitness is not affected by its collective character, the two will still be correlated so long as particle fitness is affected by particle character. This is because particle character and collective character are themselves correlated—since the latter is defined as average particle character. (In Sober's example, tall organisms are more likely to be found in groups with high average height, obviously, so an individual's height will tend to be correlated with the average height of its group.) So we need to check whether there is a correlation between particle fitness and collective character that is not due to the correlation between particle fitness and particle character.

Contextual analysis addresses this question using a standard linear regression model:

where particle fitness wis the response variable, and the two independent variables are particle character z, and collective character Z.10 (Think of Z as a relational property of each particle in the collective.) pi and p are the partial regression coefficients. So pi measures the direct effect of particle character on particle fitness, controlling for collective character; while p2 measures the direct effect of collective character on particle fitness, controlling for particle character. For simplicity we again assume linearity, zero interaction, and the absence of unmeasured influences on particle fitness; so the partial regression coefficients can be interpreted as measures of direct causal influence.n

Heisler and Damuth argue that selection at the collective level requires p2 to be non-zero. This means that information about the collective to which a particle belongs is relevant to predicting the particle's fitness, over and above information about the particle's own character, that is, it signals a 'collective effect' on particle fitness. In Sober's example, where an organism's fitness depends only on its own height, p2 is zero—once

10 Equation (3.5) could be formulated more perspicuously using indices, i.e. wjk = Pizjk + ^Zk + ejk, where wjk and zjk denote the fitness and character value of the particle in the kth collective, Zk is the character value of the kth collective, and ejk is the residual. The unindexed form is used in the text for ease of reading.

11 The most thorough exposition of contextual analysis, due to Boyd and Iversen (1979), explicitly allows the possibility of interaction between individual and contextual characters, by adding an interaction term to the regression model. This complication, though important for any actual empirical study, will not be examined here.

you know an organism's height, further information about its average group height does not help predict its fitness.12 Of course, if you did not know the organism's height, then learning the average height of its group would help you predict its fitness. But collective character is not a predictor of particle fitness once particle character has been taken into account. That is the key point. More generally, contextual analysis tells us that if p2 is zero, then a non-zero value of Cov (W, Z) must be a by-product of particle-level selection. 13

Notice that contextual analysis is simply a special case of the Lande—Arnold model of selection on correlated characters (Heisler and Damuth 1987). Indicative of this is that equation (3.5), the contextual regression model, has precisely the same form as equation (3.2). The only difference is that in the contextual model, one of the characters whose effect on particle fitness we are interested in is a 'contextual' character. As we saw, the Lande—Arnold model is meant to help tell whether a given character-fitness covariance is a by-product of selection on a correlated character. The same is true of contextual analysis. But since the two correlated characters, in the contextual model, are at different hierarchical levels, this means that the by-product in question is a cross-level by-product. Thanks to the expedient of treating a collective's character as a relational property of its constituent particles, contextual analysis succeeds in reducing cross-level by-products to the single-level by-products that the Lande—Arnold model analyses.

It may not be clear how contextual analysis achieves this reduction, for the following reason. Cross-level by-products, as defined earlier, occur where direct selection at one level produces a character-fitness covariance at another level. But contextual analysis deals exclusively with particle fitness—it says nothing about collective fitness. So while it is clear how contextual analysis could help determine whether a covariance between collective character and particle fitness is a by-product of direct selection at the particle level, this seems to be changing the concept of a cross-level by-product. The original question was whether a covariance between collective character and collective fitness might be a by-product of particle-level selection, but contextual analysis does not seem to address that question.

12 The condition ft2 = 0 is closely related to Sober's (1984) idea that group selection requires group membership to be a 'positive causal factor' in the determination of individual fitness; see Okasha (2004c) for further discussion.

13 Modulo the simplifying assumption that all of the causal influences on fitness have been captured in the contextual regression model, of course.

In fact this objection is incorrect, for the two covariances in question are identical. The covariance between collective character and collective fitness, Cov (W, Z), is equal to the covariance between collective character and particle fitness, Cov (w, Z), given that collective fitness W is defined as the average fitness of the particles in the collective.14 So contextual analysis does address the issue of cross-level by-products as we defined them. From equation (3.5), and from the fact that Cov (W, Z) = Cov (w, Z), it follows that:

Collective-level Covariance By-product of Selection on z Direct Selection on Z

Equation (3.6) is a useful decomposition for understanding byproducts running in the particles collective direction. The LHS term is the character-fitness covariance at the collective level. The first RHS term, ^iVar(Z), measures the cross-level by-product arising from direct selection on z. The second RHS term, ^2Var(Z), measures direct selection on the collective character itself. Equation (3.6) shows that the collective-level covariance is actually made up of two parts, one due to selection at the collective level itself, the other 'caused from below', by selection at the particle level.

Contextual analysis helps illuminate cross-level by-products of the sort illustrated by Sober's height example. But does it offer a complete solution to the problem of cross-level by-products in MLS1? (Contextual analysis does not apply to MLS2, since it treats the particles as the 'focal' units.) And what are its general implications for the levels-of-selection debate? These questions are tackled next.

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