1. Introduction

In this article I recount some empirical and associated statistical difficulties confronting going accounts of group selection in respect of the conceptions of ‘group’ employed by them. In brief, most such theories require real groups but define the reality of groups in ways that make it impossible to test for the reality of the groups employed by a population model. If those definitions are to be taken seriously, no group selection model can ever be employed on real observational data in any reliable fashion; even the diagnosis of the presence or absence of group selection is statistical nonsense given these accounts. There are alternatives, but they require a nominalism about groups that most theorists (although rather fewer practitioners) abjure.

Models of group selection require groups: some mapping of individuals to collections of individuals. On some theories of group selection, those collections comprise merely nominal groups; that is, any mapping will do, so long as every member of the population is mapped to some collection. Consider, for example, models employing neighborhood variables. Each unit, each oak tree on a hillside perhaps, is mapped to the collection of oak trees that are within, say, 10 meters of the focal unit. Each oak tree has its own neighborhood, so every member of the population is mapped to some (possibly empty) collection. But no oak tree is in its own neighborhood, and any given oak tree might be in the neighborhood of several others. Further, the neighborhoods—the collections of oak trees—need not themselves be in any interesting sense biologically united, need not comprise a ‘real’ group, apart from the fact that they happen to be all the oak trees within 10 meters of some other oak tree. The group is formed by appeal to a mere Cambridge property, as it were.

Some (e.g., Glymour and French Reference Glymour and French2009) are content with neighborhoods or other merely nominal groups, and I will call these positions ‘nominalist’. But I do not mean by nominalism to require that groups are not real in any biological sense; nominalism merely says that reality, and so reality in any particular sense, is irrelevant to questions about whether group selection is operating. Neighborhood and contextual analysis, for example, are nominalist methods because they do not require that group members be united by physical, social, or biological relations. But neither do these methods exclude such real, unified, groups. Thus for nominalists, as I mean the term, the members of the groups may, but need not, be united by some set of physical, social, or biological relations, and, if they are united, the relevant relations may, but need not, be known to obtain.

At least some biologists, especially those engaged with actually modeling wild populations (see sec. 4) adopt practices that are consistent with nominalism. But many orthodox accounts of group selection are less permissive: it is thought that the collections to which individuals are mapped ought not be mere neighborhoods but rather groups that are real in the sense that members’ evolutionary fates must be tied to one another in some interesting and important way. There are roughly three strategies for defining groups so as to induce such partitions: appeal to expert knowledge, appeal some observable social or biological relation among individuals, or appeal to fitness-affecting interactions among individuals. The last is by far the most common strategy, and accordingly we will begin with it.

2. Fitness-Defined Groups

On this method of inducing groups, two individuals are held to be members of a common group (if and) only if they affect one another’s fitness. As Sober and Wilson put it, “a group is defined as a set of individuals that influence each other’s fitness with respect to a certain trait but not the fitness of those outside the group” (Reference Sober and Wilson1999, 92), which definition they take, plausibly, to summarize the essential feature of groups as understood by Darwin (Reference Darwin1871), Haldane (Reference Haldane1932), Wright (Reference Wright1945), Williams and Williams (Reference Williams and Williams1957), and Hamilton (Reference Hamilton1975). Godfrey-Smith (Reference Godfrey-Smith2008) endorses a similar requirement in at least some circumstances and cites in support Uyenoyama and Feldman (Reference Uyenoyama and Feldman1980), Wilson (Reference Wilson1980), Michod (Reference Michod1982), and Wade (Reference Wade1985). Others endorse some version of the requirement but add further restrictions or allow exceptions of one kind or another. Okasha (Reference Okasha2006), for example, distinguishes between ecological and genealogical groups and requires only the former to satisfy the requirement; he also requires that ecological groups be capable of “free living.” Sterelny (Reference Sterelny1996), Maynard Smith (Reference Maynard Smith1998), and Nunney (Reference Nunney1998) arguably endorse some version of the requirement and also, as Okasha notes, require a richer functional organization among the elements of the group. Thus, for many theorists, symmetric, fitness-affecting causal interactions between (nearly all) members of the group and the absence of such interactions between (nearly all) members of different groups is a necessary feature of group structure: given a mapping G from the domain of individuals to subsets thereof, if the ‘groups’ identified by the mapping systematically include individuals that do not influence one another’s fitness, then the mapping does not identify real groups, and there is no group selection acting on that population with respect to those collections of individuals. I say that theories including this kind of constraint on group structure employ fitness-defined groups.

Fitness-defined groups introduce a decidedly intractable discovery problem for those who would model populations using real data. If group selection occurs only over real rather than nominal groups, we can diagnose the occurrence of group selection only given a prior diagnosis of which partitions of the population yield real groups. And to do that one must specify a test for the existence of the group-defining relations, here fitness-affecting interactions. Such tests are complicated by a necessary vagary. Godfrey-Smith (Reference Godfrey-Smith2008) points out, correctly, that often no partition of the population will yield collections of organisms, cells in the partition, each of which strictly satisfies the constraints on fitness-defined groups. But he and others suppose, as will I, that often enough there are partitions on which the resulting collections approximately satisfy the relevant constraints, hence the ‘nearly all’ in the above framing. I will gloss ‘approximately satisfy’ thusly: we require a partition of the population such that the network of fitness-affecting interactions is relatively dense within groups and relatively sparse between groups. This is vague in respect of what counts as ‘dense’ and ‘sparse’, and there is clearly a disputable boundary: there will be cases in which it is unclear whether there are fitness-defined groups because there are only a few fitness-affecting interactions and such as do occur are only slightly more frequent within than between groups. But also I assume that there are clear limiting cases, that is, cases in which fitness-affecting interactions occur but are so sparse even within groups that on any reasonable interpretation of those who require fitness-defined groups no such groups are present.

Developing tests for the satisfaction of vague boundary conditions is of course problematic. But as it turns out precision here is not to the point. Minimally, we want a procedure that, when given a partition, reliably determines whether the collections generated by that partition do or do not satisfy the (now vague) constraints on fitness-defined groups. No procedure of this sort is now possible, exactly because there are no reliable methods for determining the existence of fitness-affecting interactions between individuals.Footnote ¹ Such interactions involve cross-unit causal dependencies, which turn out to be empirically inaccessible. More precisely, on any reasonable precisification of ‘dense’ and ‘sparse’, available methods are not appropriately sensitive to the frequency or distribution of cross-unit fitness-affecting interactions. In fact, there is in the literature just one general strategy for identifying the presence of the relevant fitness-affecting interactions.Footnote ² Before elaborating the challenges it faces, it will be useful to have a clear view of what makes it so difficult to discover cross-unit causal dependencies.

2.1. Inferring Cross-Unit Causation

The fitness-affecting interactions employed to define real biological groups are a species of cross-unit causation. Cross-unit causation occurs whenever the traits of one individual, or unit, causally influence the traits of another. It is simple enough to define such cross-unit causal dependencies. For instance, on an interventionist conception of causation (Pearl Reference Pearl2000; Spirtes, Glymour, and Scheines Reference Spirtes, Glymour and Scheines2000; Woodward Reference Woodward2003) it is perfectly reasonable to say than Anya’s income causally influences Boris’s education and that this is so if and only if there is some intervention on Anya’s income that changes the probability density over Boris’s education. But discovering the truth about such causal hypotheses is not so simple at all.

The reason for this is that causal dependencies are evidenced by the statistical associations they generate in the data, and (more important) the absence of those causal dependencies is evidenced by the absence of associations in the data. But to find associations in the data, or their absences, variable values must be paired. Sample covariance, for example, is defined as the mean product of paired deviations from the mean. Consider Boris and Anya again. If one wanted to test the theory that income causes education, one might look to see whether there is a sample covariance between the variables: where i indexes individuals and n is the number of individuals in the sample, is

$\frac{1}{n-1}\sum _1^n\left(\mathrm{Income}\left(i\right)-\overline{\mathrm{Income}}\right)\left(\mathrm{Education}\left(i\right)-\overline{\mathrm{Education}}\right)$

nonzero? If so, Income and Education are associated and so potentially causally related; if not, then neither causes the other, and, more, they share no common cause. But calculating the covariance requires that observations of income and education be paired, which is what the index i is doing. Normally, i just indexes subjects, so we pair the observation of Anya’s income with the observation of Anya’s education and the observation of Boris’s income with the observation of Boris’s education, and so on. But clearly that will not do to test for cross-unit causation: that Anya’s income influences Anya’s education and Boris’s income influences Boris’s education implies nothing about whether Anya’s income influences Boris’s education. Hence, any test of cross-unit causation requires some other way of pairing observations.

Nor will it do to simply pair Anya’s income with Boris’s education, for that gives us a sample of one, from which nothing can be inferred with any reliability (e.g., the sample covariance is not even defined on a sample of one). One could simply consider every possible pair, but it is not clear what a nonzero sample statistic from such a data set would mean, and, not unrelatedly, for samples of any size at all, the signal generated by the causal dependency would be swamped. The problem is really twofold. First, observations must be paired or otherwise grouped, unit a with unit b, unit c with d, and so on (a matching problem). Second, within the resulting groups the observations must be ordered, potential cause–potential effect, a’s education with b’s income, c’s education with d’s income, and so on (an ordering problem).

There are methods for avoiding the ordering problem, for example, the use of intraclass correlations or demographic variables, but their values provide at best deeply ambiguous evidence for the existence of pairwise causal dependencies.Footnote ³ Some of the relevant problems are simply endemic to causal inference. For example, we could match observations of Education and Income by pairing off our population into families; thus, if Anya and Boris are related as spouse to spouse, they would share a common value for an index variable Family. We do not actually need to arbitrarily order spouses to determine whether units within a given family are likely to share similar education values, and we can test for that using intraclass correlations. If the units with a given family are especially likely to have similar Education values, then Family is associated with Education. If so, then either Family influences Education, or they share some common cause, for example, Income or City of Residence. Similarly, if we map units to groups with a function G and the intraclass correlation of unit fitnesses within a group is nonzero, we have reason to think either the units influence one another’s fitness or there is some common cause of their fitnesses. The latter possibility is worrisome if we are committed to fitness-defined groups, but the risk of confounder bias is ever present and unavoidable if we wish to learn from observational data. If this were the only worry, we could proceed apace. Unfortunately, it is not. The substantive worry is best illustrated as it arises in the use of demographic variables, which method is by far the most common in discussions of group selection.

3. Groups Otherwise Defined

Groups need not be fitness defined. One might instead define groups by some other real physical, biological, or social relation and so without explicit reference to fitness at all. In particular, if one has some prior commitment to some particular mapping, the worries about proper specification may reduce to worries about whether demographic variables defined on that mapping are causally relevant to the effect of interest. Suppose, for example, that Anya is Boris’s mother, and one is in particular interested in whether one’s mother’s income or one’s father’s income has a greater influence on education. Then one might simply define the variable Mother’s Income, measured on individuals. If one can for each individual identify a mother and her income, one can proceed again by using the index i over units to pair observations in order to calculate relevant sample statistics.

This method has several advantages. Since groups are no longer fitness defined, we need not employ tests for an association between demographic variables and fitness as (unreliable) tests for fitness-affecting interactions but rather as (at least sometimes reliable) tests for existence of a causal connection between the demographic variable and fitness (i.e., as tests for group selection). Further, the range of methods by which to identify groups is considerably expanded. For example, some causal interactions, generally social but sometimes biological, are directly observable: grooming, mating, feeding, and so on are pairwise interactions that can be seen. Even though whether the interaction in turn effects fitness cannot simply be observed, the fact of the interaction can be. Collections of such interactions constitute a social or biological network with various structural features that can be used to partition a population into groups. Differently, one might, as with mothers or more generally with families, identify groups simply by adverting to reasonably well understood features of social life. One could with equal ease appeal to more narrowly held expert knowledge of particular collections—for example, one could identify baboon troops or lion prides by appeal to the expert knowledge of field biologists observing the baboons or lions; certainly the identification of nests, colonies, clutches, and like quite often proceeds by appeal to such expert knowledge. Differently again, one could represent observations of mating, feeding, grooming, or the like with a graphical model and then deploy some clustering algorithm on that model to produce a partition of the population. Herbers and Banschbach (Reference Herbers and Banschbach1999), for example, employ several of these strategies when they individuate ‘nests’ and ‘colonies’ by appeal to pairwise behavioral tests, spatial location, and genetic data.

On any of these ways of partitioning a population into groups, one can then sensibly seek to test whether demographic variables defined on that partition causally influence reproductive success, although it should be said that the statistics here are seriously nontrivial and clustering is more a matter of art than science. We should recognize that even setting aside statistical concerns this strategy has certain disadvantages. First, the resulting groups are not fitness defined. Groups are rather defined by whatever physical, social, or biological relation is employed individuating them, which relations will themselves often be unknown when one appeals to expert knowledge. Second, while there is nothing essentially wrong with an appeal either to expert knowledge or widely shared common knowledge, the best defense of such everyday or expert identification of groups invites a nominalism about groups.

4. Nominalism Again

Suppose we defend a particular partition of baboons into troops by appeal to the fact that the experts, the field biologists who spend time actually observing the baboons, recognize just those groups. Even if we did not bother to build a graphical model of the grooming, mating, display, and so on, behaviors or bother to use a clustering algorithm to induce a partition of the population into troops, we could reasonably rest content with the resulting partition of the population. This is because we can be reasonably sure that any good clustering algorithm applied to such a model would generate the same groups as the field-workers recognize. And that is so because if a clustering algorithm did not routinely return just the clusters recognized by the experts, we would reject the clustering algorithm as bad news: whatever it is that such algorithms are identifying, it is not the groups of interest to us, however objective or real they might be. But if this is the proper defense of an appeal to expert knowledge, and that appeal is legitimate, nominalism about groups seems entirely appropriate. The right groups to consider are whatever groups happen to interest us at the moment.

This of course implies that diagnoses of the presence or absence of group selection are necessarily relative to an arbitrary, although not unmotivated, partition of the population into groups. But at least this aspect of nominalism seems not only harmless but correct: if there are in fact cross-unit fitness-affecting causal dependencies, it will be true that on some partitions of the population demographic variables influence fitness; this will be true, for example, for those partitions that yield collections satisfying the constraints of fitness-defined groups. But there often will be other partitions on which demographic variables do not cause fitnesses; this will be true when the collections generated by the partition never include individuals that affect one another’s fitness. Insofar as MLS1 versions of group selection are, at least in part, a matter of demographic variables causally influencing individual reproductive success, judgments about whether group selection is actually occurring really ought be relative to the partition of the population employed. That the implicit relativization is made explicit by nominalism then looks to be a feature rather than a bug.

Nominalism, as I am here using the term, is anathema to many, perhaps most, of the standard classical discussions of group selection and much of the theoretically motivated commentary thereon. In that tradition, groups must be real, and reality is a matter of fitness-affecting interactions. Individuals come as units of a group, it is thought, when, and only when, to some appreciable extent the evolutionary fate of each individual is bound up with that of the others. But experimental and observational work, especially that in the traditions following Heisler and Damuth on the one hand or Wade on the other, is often less demanding. Some discussions seem to presuppose the importance of fitness-affecting interactions. For example, Pruitt and Goodnight, writing in defense of an earlier paper (Pruitt and Goodnight Reference Pruitt and Goodnight2014) in which they claim to have demonstrated group selection in a social spider, seem to think it is important that individuals in a colony succeed or fail together, that is, that the behavioral or social interactions used to identify groups also constitute fitness-affecting interactions. They write, “Our case study is clear because both the target and agent of selection are above the level of the individual: the target of selection (group composition) is a trait that an individual cannot have, and the agent of selection (extinction) is the textbook example of strong group selection. We showed that A. studiosus colonies live or die as a unit” (Pruitt and Goodnight Reference Pruitt and Goodnight2015, E4).

But many other studies seem to employ partitions in which groups are defined by appeal to expert knowledge. Moorad (Reference Moorad2013), for example, employs a variable measuring the number of mates for an individual’s father, which father is in turn identified from birth records rather than genetic data; thus, an individual’s male parent is identified by appeal to records of common-knowledge identifications. Tsuji (Reference Tsuji1995), for another example, identifies ant colonies (Pristomyrmex pungens) with single nest sites, claiming that colonies were monodomous (i.e., had but one nest site), but offers no data in support of that claim (i.e., colonies have been individuated on the basis of Tsuji’s expert knowledge). Similarly, Breden and Wade (Reference Breden and Wade1989) consider egg clutches, which clutches are distinguished one from another not by data on which an analysis is performed but by expert knowledge. This use of expert knowledge need not be epistemically fraught, but neither are the resulting groups constructed on the basis of fitness-affecting interactions.

Yet other studies are based on physical or biological relations other than fitness (e.g., Herbers and Banschbach Reference Herbers and Banschbach1999; Laiolo and Obeso Reference Laiolo and Obeso2012). Laiolo and Obeso partition a metapopulation of Dupont’s lark into local populations following Vögeli et al. (Reference Vögeli, Serrano, Pacios and Tella2010), who divide the study location into patches on the basis of bird movement, census data, and habitat. Still other studies individuate groups in apparently arbitrary ways. Aspi and coauthors, studying selection on patches of Tatar catchfly, identify a patch with “a group of individuals within a maximum (arbitrary) distance between individuals of five meters” (Reference Aspi, Jäkäläniemi, Tuomi and Siikamäki2003, 510). Eldakar et al. (Reference Eldakar, Dlugos, Holt, Wilson and Pepper2010), in a study of water striders, identify pools and pool regions on an ephemeral stream bed, the latter being a major pool with its “immediately connected” minor pools. No criteria are given for distinguishing ‘immediately connected’ pools; whatever criteria were used were apparently unmotivated to details of water strider behavior or biology. Weinig and coauthors (Reference Weinig, Johnston, Willis and Maloof2007) designed experimental ‘patches’ of A. thaliana by planting seeds in pots; within each pot seeds were planted on a 3 × 3 grid, with 1 centimeter between grid locations. Pots are treated as patches, but no test for within-pot interactions of any kind are offered in justification of the choice of 1 centimeter distances between seeds. Again, this is not problematic, unless we require that groups be fitness defined.

Discussions in empirical papers often underdetermine the authors’ considered views about the nature of the groups required for group selection. On the one hand, all of these studies, excepting Pruitt and Goodnight (Reference Pruitt and Goodnight2015), are consistent with nominalism; in none of them is group identification explicitly made definitionally dependent on fitness-affecting interactions. Perhaps, then, the authors are perfectly satisfied with groups defined by real physical, social, or biological relations, even if those relations are not pairwise fitness-affecting interactions. Or perhaps the authors simply care about the groups they use, whether or not they have some underlying biological reality. On the other hand, it might be that the authors are laboring under the mistaken assumption that a causal connection between demographic variables and fitness suffices to establish that the groups on which the variable is measured are unified by dense networks of fitness-affecting interactions. Indeed, even pieces like Pruitt and Goodnight (Reference Pruitt and Goodnight2015) are open to alternative interpretations. Perhaps Pruitt and Goodnight are committed to fitness-affecting interactions. But perhaps they are not and are instead simply responding to a critic who is. But at least this much can be said. The methods employed in these studies do not in fact demonstrate that the studied groups satisfy the conditions on fitness-defined groups. To the extent that the fitness-defined groups are a necessary condition of group selection, the studies simply do not demonstrate group selection. Hence, to the extent that the observational, experimental, and inferential practices therein are acceptable, fitness-defined groups are not necessary conditions of group selection, and, what is more, nominalism about groups looks to be perfectly acceptable.

I make one last observation. Insofar as one is willing to be nominalist about groups (i.e., to hold that the groups identified in a MLS1 model of a population need be no more real than that the individuals assigned to groups really do exist), much of the motivation for preferring group to neighborhood variables vanishes, and there is rather less reason to object to the use of demographic variables defined over neighborhoods rather than groups. Of course, for some that will be sufficient reason to reject nominalism. But those who do owe a fuller, more careful story about just how real groups are to be identified and why reality, so understood, is essential.

5. Implications

The situation then appears to be this. If we reject nominalism, we require real groups. The standard account of such groups is that any two individuals in a population belong to the same group if and only if there are fitness-affecting causal dependencies between them; that is, for some trait T, either T(a) causes W(b) or T(b) causes W(a). Such dependencies, it turns out, can be tested for, given current methods, only by testing for a causal dependence between individual fitness and a demographic variable $D\left(i\right)=\overline{T}\left(G\left(i\right)\right)$, where G maps individuals to groups. That method is seriously unreliable. Causal dependencies between demographic variables and fitness can, indeed will, hold even when most members of most groups do not influence one another’s fitness, and the associations that are evidence for the causal dependence between demographic variables and fitness can hold even when the mapping G(i) generates groups that are not characterized by a high density of pairwise fitness-affecting interactions.

There are alternatives that preserve something of the flavor of the initial proposal. We might define groups by means of causal dependencies between variables other than reproductive success and then ask whether these real groups are such that demographic variables defined over those groups causally influence the reproductive success of the members. When the relevant interactions are themselves directly observable, methods for using the network of such relations to partition the population are available, and, depending on the way in which group structure is induced, raise only manageable statistical problems. But in adopting these methods we must relinquish the two most important features of groups used to argue against nominalism about groups. First, the idea that groups should include only individuals that affect one another’s fitness must go, for even when it is true that demographic variables influence the reproductive success of individuals, it will not in general be true that the trait value of any one group member influences the reproductive success of any other; indeed it may well be the case that for most groups most members do not influence one another’s fitness. Second, although in some sense the group structure is understood to be generated by some underlying network of biological or social relations, we will often not infer a group structure from some representation of the underlying social or biological relations. Indeed, we will often neither represent the relations nor infer them from data but instead directly induce the group structure by appeal to either expert (e.g., baboon troops) or common (e.g., families) knowledge. But the best justification for such appeals invites nominalism insofar as it relativizes judgments about group selection to partitions of the population into groups and justifies the use of one rather than another partition by appeal to our interest in the resulting groups.

Finally, there remains nominalism, which offers a ready justification both for appeal to expert knowledge in inducing group structure and for the use of social or biological causal relations apart from those in which fitness enters as an effect variable. Groups are where you find them, and if your interest is in just these groups, either because those are the groups you care about after having watched Anya and Boris or the baboons or the oak trees, for many years, or because the observed patterns of social interaction generate just those clusters, or for whatever other reason, then the group selection hypotheses of interest will be hypotheses about the influence of demographic or aggregate variables measured on just those groups. As a nominalist who prefers neighborhoods to groups, I recommend this strategy, although the reader’s tastes may differ. But whatever one’s tastes, if we are to take inference seriously then neither definitions of nor tests for group selection may require that groups be defined by fitness-affecting causal dependencies among group members: to do so places the reality of groups, and hence the reality of group selection, beyond the reach of even our best methods and so also beyond the epistemic pale.