Contextual Analysis Versus The Neighbour Approach

Recall how contextual analysis works. It uses a multiple regression model to assess whether an individual's fitness is affected by individual character, group character, or both:

w = ftiz + ftZ + e where w is individual fitness, z is individual character, and Z is group character, defined for the moment as average individual character. fti and ft are the partial regressions of fitness on individual and group character respectively. Presuming the individual character is transmitted perfectly, this permits the overall change to be written as:

Group selection Individual selection

Contextual partition-. wAz = ft Cov (z, Z) + ft Var (z)

which constitutes a rival to the Price equation partition. The opposition between the contextual and Price approaches to MLSi was examined in Chapter 3, Section 3.4.

To apply the contextual partition to the population depicted in Figure 6.1 above, assume that individuals reproduce asexually and that types breed true. We define z = 1 if an individual is A, 0 if it is B; so Z is the proportion of A types in a group. So for example, an A individual in an AABB group has z = 1 and Z = i/2- Obviously, z is the overall frequency of the A type, and Az the change in frequency over one generation.

Though Nunney (1985a) employs a different formalism, his approach is a close relative of contextual analysis, but with one difference. Whereas the contextual approach looks for 'group effects' on individual fitness, Nunney looks for 'neighbourhood effects' on individual fitness. In effect, Nunney is using a variant of the contextual regression model in which the second independent variable is neighbourhood character X, rather than group character Z. An individual's neighbourhood character is the average character of its neighbours, that is, of all the individuals in the group except itself.21 So in the example above, an A individual in an {AABB} group has X = 1/3, while a B individual in the same group has X = 2/3. The regression model is:

w = ^3z + ^4X + e where fa and fa are the partial regressions of fitness on individual and neighbourhood character respectively. Let us call this the neighbour approach to group selection. The overall evolutionary change can then be expressed as:

Group election Individual selection

Neighbour partition : wAz = faCov(z,X) + &Var(z)

which constitutes a rival to both the Price and the contextual partitions. Of course, the neighbour, contextual, and Price partitions are all correct as statistical decompositions of the overall change, but at most one of them can be correct as a causal decomposition.

In motivation, the contextual and neighbour approaches are similar— both are intended as correctives to the Price approach—but they are subtly different. Consider the component of change that each attributes to group selection. On the neighbour approach, this component is fa Cov (z, X); on the contextual approach, it is fa Cov (z, Z). Crucially, Cov (z, Z) = Var (Z) so is always non-negative. But the term Cov (z, X) is very different. This term is the covariance between individual character and neighbourhood character, and may take on any value. If groups are formed at random, then Cov (z, X) will equal zero—there will be no correlation between an individual's character value and that of its neighbours. If groups are formed assortatively, then Cov (z, X) will be positive—an individual's character will correlate with that of its neighbours. Therefore, Cov (z, X) measures departures from randomness in the formation of groups.22

21 Note that there is a simple relation between an individual's neighbourhood character X, its individual character z, and its group character Z, i.e. X = 1/(n-1) (nZ — z). If follows from this that ft = n/(n-1)ft, so there can only be 'group effects' on individual fitness if there are 'neighbourhood effects' on individual fitness and vice versa

22 The contextual and neighbour approaches also embody different criteria for individual selection. On the contextual approach, individual selection requires fitness differences between individuals with the same group character, i.e. in the same group. But

This explains why Nunney regards non-random group formation as a prerequisite for group selection. If groups are formed randomly, then even if an individual's fitness is affected by which neighbours it has, that is, even if is non-zero, there will be no component of group selection, since ^4 Cov (z, X) will equal zero. By contrast, on the contextual approach, there will be a component of group selection, since Cov (z, Z) will be non-zero. Put differently, the contextual approach defines 'group selection' as direct selection on an individual's group character; while the neighbour approach defines 'group selection' as direct selection on an individual's neighbourhood character. In general, direct selection on a character y only affects the evolution of a character x if x and y are correlated. There is an intrinsic correlation between an individual's character and its group character (given how the latter is being defined); but there is only a correlation between an individual's character and its neighbourhood character if groups are formed assortatively. Hence Nunney's thesis that assortative grouping is necessary for group selection.

In Section 6.6 we noted that the evolution of strong altruism requires assortative grouping, while weak altruism does not. The neighbour partition yields a simple proof of this fact. Suppose type A is strongly altruistic. The first term of the neighbour partition, ^Va^z), must then be negative. For measures the change in individual fitness if an individual with fixed neighbours has its z-value increased by one unit, that is, if a selfish individual converts to altruism and stays in the same group. By the definition of strong altruism, such an individual suffers a fitness loss, hence ^3 <0. So if strong altruism is to evolve, that is, if Az >0, then the second term of the neighbour partition, ft4 Cov (z, X), must be positive. Since random group formation implies Cov (z, X) = 0, it follows that strong altruism cannot evolve with random group formation. This conclusion does not go through for weak altruism, however, for weak altruism implies ¿83 >0. So for weak altruism, Az > 0 is compatible with Cov (z, X) = 0.

In Chapter 3, we argued that the contextual approach is theoretically superior to the Price approach, as the latter detects group selection even when individual fitness depends only on individual character. However, on the neighbour approach, individual selection requires fitness differences between individuals with the same neighbourhood character; such individuals cannot be in the same group. A consequence of this is that on the neighbour approach, individual selection can operate even if there is no within-group variance in fitness, something that is impossible on both the contextual and Price approaches. See Okasha (2004b) for discussion.

this consideration does not support the contextual approach over the neighbour approach; for they both yield the 'correct' answer in the case that the Price approach gets wrong.

How then should we choose between the contextual and neighbour approaches? One possible argument for the latter stems from a general consideration about the likely causal influences on individual fitness. Since an individual interacts directly with its neighbours, not with its neighbours-plus-itself, neighbourhood character is the sort of character that could directly influence an individual's fitness, while group character is not. If an individual's fitness is affected by interactions with its neighbours, we should expect a correlation between individual fitness and neighbourhood character, and thus between individual fitness and group character;23 but arguably only the former correlation is causal. So the neighbour approach yields a more satisfactory causal decomposition.24

This argument in favour of the neighbour approach seems right, given that group character is defined as average individual character. There seems no way that this group character could affect individual fitness directly, rather than via neighbourhood character. But as we saw in Chapter 3, the contextual approach is also applicable to 'emergent' group characters, which are not the average of any individual character. Where emergent characters are at issue, it is less clear that their causal influence on individual fitness must be indirect, mediated by neighbourhood character. For the set of neighbours of a given individual is not a 'real' biological unit the way a group is, so any given emergent character may not even be well-defined on this set. Therefore, where individual fitness is affected by emergent group characters, the contextual approach provides the better causal decomposition; the neighbour approach may not even be applicable.

This suggests that the neighbour approach is most plausible during the early stages of the evolution of cooperation/altruism, where cohesive,

23 Here it is important to remember that ^2 and always have the same sign; see note 21.

24 Importantly, there can be no purely statistical solution to the problem of whether group character Z or neighbourhood character X causally affects individual fitness. One might think that this question could be resolved by including both Z and X, along with z, as independent variables in a regression analysis, with fitness as response variable. But this will not work. Since any one of the variables {z, Z, X} is a linear function of the other two, the relevant partial regression coefficients will not be well-defined. (This is known as 'the problem of perfect collinearity'.) So extra-statistical considerations concerning the actual mechanisms by which neighbourhood and group characters might causally affect fitness are the only way of choosing between the approaches.

integrated groups have not yet evolved, but where individuals do engage in social interactions. During such stages, an individual's fitness may well be affected by factors such as the proportion of altruists in its neighbourhood. This ties in with a point stressed by Nunney (1985a), that the neighbour approach is perfectly applicable even if discrete, non-overlapping groups do not exist in the population. But where cohesive groups have evolved, with discrete boundaries and emergent properties of their own, the contextual approach makes more sense, for it is likely that individual fitness will be affected by emergent group characters. Still later, groups may be so cohesive that we wish to treat them as the focal units, that is, to move to an MLS2 framework. The idea of a transition between MLS1 and MLS2 is discussed in Chapter 8.

What then of Nunney's thesis about non-random group formation, and the correlative thesis that weak altruism is really a form of selfishness? I think Nunney is partially right. There are situations, such as those described by the simplest models for the evolution of altruism of the sort used above, where the neighbour partition seems 'better' than the contextual or Price partitions, in the sense of yielding a more accurate causal decomposition of the overall change. And if the neighbour partition is taken to define individual and group selection, Nunney's thesis follows immediately. However, there are other situations, typically involving cohesive groups rather than just interacting individuals, where the contextual approach seems superior; and modulo that approach, group selection does not require non-random group formation. If this is correct, then a general verdict on Nunney's thesis is not possible; it needs to be assessed on a case-by-case basis.

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