1. Introduction

Beginning with work in the 1920s, Sewall Wright (Reference Weir and Cockerham1921, Reference Weir and Hill1969) developed a set of mathematical concepts known as “F-statistics” or “fixation indexes,” which measure various kinds of departure of gene frequencies from Hardy-Weinberg equilibrium. F-statistics such as , , and and closely related concepts can be used to measure nonrandom patterns of genotype frequencies resulting from a variety of factors. Depending on context, F-statistics can be used to describe and provide evidence for various patterns of nonrandom mating, multilevel population subdivisions, degrees of isolation by distance, drift, migration, and natural selection (Weir Reference Walsh1996; Weir and Hill Reference Weir2002; Holsinger and Weir Reference Eagle2009; Ishida Reference Holsinger and Weir2009). Consequently, F-statistics have close connections to ideas central to evolutionary theory and continue to be used in a broad range of biological applications, both theoretical and practical, across diverse species, using genetic data of various kinds (Weir and Hill Reference Weir2002).

In this article, I focus on the use of , in empirical applications, to characterize genetic and other differences between subpopulations and to identify causes of those differences. Weir and Cockerham's (Reference Weir1984) popular approach to statistical estimation of is based on an assumption sometimes formulated as a claim that actual populations tested are sampled from an infinite set of (counterfactual) subpopulations, all derived from a single ancestral population. Claims about counterfactual populations appear in various contexts in evolutionary biology, but provides a particularly interesting context for examining implications of assumptions about them because of 's diverse empirical use and well-developed theory of estimation, because Cockerham and Weir's assumption is more substantive than many similar assumptions made in other contexts in population biology, and because in many applications the counterfactual populations assumption plays an essential role in allowing to distinguish the effects of drift from those of natural selection or migration. I will argue that Cockerham and Weir's approach to estimation of implies that selection and drift are distinct factors in evolution and that the way in which F-statistics and other statistical tests are applied to computer simulations of evolving populations shows that evolutionary processes involve objective, causal probabilities. The empirical success and theoretical justification of Cockerham and Weir's approach thereby provide support for a view of evolutionary theory as identifying evolutionary causes; this view is at odds with some recent philosophical accounts of evolutionary theory.

2. Goals

“Statisticalist” philosophers and biologists have argued for the following related theses (e.g., Pigliucci and Kaplan Reference Nei and Kumar2006; Walsh Reference Van Fraassen2007, Reference Voight, Kudaravalli, Wen and Pritchard2010; Matthen Reference Lewis2009; Matthen and Ariew Reference Matthen2009): (a) Natural selection and drift cannot be distinguished. (b) Neither natural selection, nor drift, nor any other so-called evolutionary force can be reasonably construed as a cause of evolutionary change. On this view, evolutionary change happens, and it has causes, but the only causes of change for a population are interactions between particular circumstances and organisms with particular genomes and physiologies. We can collect statistics about the effects of these underlying causes, but nothing general can be said about higher-level, general causes of evolution.

I will argue that a popular method for use of , a widely used mathematical tool for characterizing genetic differences between populations, (1) seems to be routinely able to distinguish the effects of different evolutionary “forces” in appropriate contexts and (2) requires that there be objective probabilities concerning the evolution of populations. I will also argue that these probabilities are causal, that is, that they capture a probabilistic causal relationship. Thus common techniques used—apparently successfully—by evolutionary biologists seem to be inconsistent with several statisticalist claims. My argument does not necessarily imply that the statisticalists are wrong, though. The fact that the work of many scientists depends on a given assumption does not guarantee that the assumption is correct. Other scientists may have different approaches (and that is true here). Moreover, in general, scientific claims and practices and their philosophical implications can reasonably be challenged by other scientists and by appropriately informed philosophers. However, widespread, long-standing, apparently successful practices in a science create prima facie justification for implications of those practices. Such success adds significantly to the burden of proof for those with contrary claims—a burden both to justify their claims and to explain why existing practices that seem to be extremely successful are carried out in the way that they are.

3. : Overview

As mentioned above, measures the degree to which populations of organisms (within the same species) differ in their genetic composition. It has been used in diverse biological applications, but my focus here is on its use to distinguish population-level factors such as selection and migration from mere drift.Footnote ¹ A simple example will help illustrate the core mathematical idea of . First we divide an overall population (or metapopulation) of organisms into groups (populations, subpopulations, demes, families, etc.). For one genetic locus (one gene) with two alleles A and a in a haploid species, characterizes the degree to which A and a are not randomly distributed between populations. In very simple cases,

where p is the frequency of A in all of the populations combined. For diploid genetics, we can think of as the degree to which the overall population departs from Hardy-Weinberg equilibrium because genes are sorted in a partially nonrandom manner into populations. In fact, there are a number of semantically distinct but mathematically equivalent ways of characterizing the meaning of ; some characterizations are more useful in some contexts than in others.

What can be done with ? Once we have estimates from genetic data, then, for example, we can compare the estimate of at one genetic locus of interest—for example, one that codes for a known phenotypic effect—with the pattern of estimates of values at other loci. This would allow us to determine whether at the locus of interest is unusually large, perhaps by using additional statistical methods such as bootstrapping. If the difference between the value at the locus of interest and values at other loci is statistically significant, that can provide evidence that the genetic differences between populations at this locus are the result of natural selection.

4. Estimation, Statistical Sampling, and Genetic Sampling

The F-statistic (and others: , , etc.) is usually defined in terms of distributions of alleles in populations. Generally we do not have complete data about these distributions. Therefore, we take random samples from actual populations. Call this statistical sampling (Weir Reference Walsh1996; Holsinger and Weir Reference Eagle2009). We then compute a function of the distribution in the samples, where this function would approach the true value of as we collected more samples. The value of such a function is an estimate of . In statistical terminology, this means that is a parameter of groups of populations; it is a property of the overall metapopulation. Parameters are contrasted with estimates, which are statistics—properties of our data. (The term “F-statistics” is thus misleading.)

In addition to statistical sampling, we can also think of a population at time as sampling genes from a population at an earlier time . Descendant populations usually will not have the same distribution of alleles at a locus as an ancestor population. This is a result of randomness in the processes leading to later generations (drift) and other factors such as selection, which bias whether some alleles make it into subsequent generations. Call this genetic sampling (Weir Reference Walsh1996; Holsinger and Weir Reference Eagle2009). This extends the common notion of single-generation sampling of gametes or zygotes and so forth to multiple generations since the descendant populations containing sampled alleles can be the result of many generations of genetic sampling processes. Note that members of the same species typically have common ancestors, and more generally, all living organisms have common ancestors. Therefore, for any set of populations, there is some common ancestral population from which they all have descended. (Mutation of course generates some of the alleles in descendant populations as well. However, since mutation rates are very low, for many purposes we can ignore this fact.)

There are thus two levels of sampling relevant to . First, natural processes leading from an ancestral population to contemporary populations sample sets of alleles from those in the ancestral population. Then researchers sample alleles from contemporary populations.

5. Counterfactual Populations

We have seen that distinct current populations can be viewed as containing samples of alleles from the nearest common ancestral population. Alternatively, we can think of a current population as itself sampled from the set of all possible populations that could have descended from a single ancestral population. More precisely, this is the set of possible combinations of genotype frequencies using alleles sampled from the ancestral population. Weir and Holsinger (Weir Reference Walsh1996; Holsinger and Weir Reference Eagle2009) put this point by saying that actual populations are sampled from hypothetical “replicate populations.” This phrasing is common in evolutionary biology. Philosophers will naturally want to ask questions such as these about so-called replicate populations: (a) What is the ontological status of these counterfactual populations? (b) What sort of possible worlds do they involve? However, trying to answer these questions directly would lead us into irrelevant, poorly specified issues. It is better to begin by asking other questions: (1) What role do the replicate populations play in the equations that define and its estimators? (2) How do we count replicate populations and weight them in these equations? I will begin by summarizing the answer to question 2 and then elucidate this answer by giving a detailed answer to the first. This will lead to a clearer understanding of the nature of the fictional “replicate populations.”

The answer to question 2 is that the “replicate populations” are not equally weighted: there is a probability distribution over them. In fact, though talk of hypothetical populations is a common, heuristically useful way of talking about certain probabilities in evolutionary biology, what is essential is only the assumption that there are such probabilities (cf. Abrams Reference Abrams2006). It is probabilities, not counterfactual populations, that actually enter into the mathematical models. More specifically, we assume that there is a probability distribution over possible allele frequencies for loci found in current populations. In the present context this distribution is defined in terms of probabilities of sampling alleles from an ancestral population. That is all that talk of possible replicate populations amounts to.

Of course, it is possible to view probabilities as involving possible worlds (e.g., Van Fraassen Reference Strevens1980), but since there will usually be uncountably many possible worlds (Lewis Reference Huff1973), we generally cannot just count worlds, and probabilities may need to be treated as playing a more fundamental theoretical role than sets of possible worlds (Abrams Reference Abrams2006). Similarly, talk of frequencies in all possible evolutionary histories (Matthen and Ariew Reference Lynch and Walsh2002, Reference Matthen2009) or all possible lives of an organism (Ramsey Reference Northcott2006) usually does not make sense on its own because there are infinite numbers of histories or lives (Abrams Reference Abrams2009). In the case of genetic sampling for , it is possible to classify counterfactual populations so that there is a finite number of them, since all of the replicate populations are finite in size. Nevertheless, populations cannot simply be counted because they will not be weighted equally.Footnote ² I turn now to a more detailed look at the role of genetic sampling in Cockerham and Weir's approach to estimating .

6. and the Method of Moments

Cockerham (Reference Wright1969, Reference Wright1973) showed that (which he called ) can be derived from sample frequencies using the method of moments, which is a generalization of the statistical method known as analysis of variance or ANOVA. I will give one component of a derivation of using this approach, for haploid organisms in isolated populations with negligible mutation. This discussion will involve some mathematical detail, but the mathematics required are relatively minimal. I will follow the convention, from statistics, of using “mean” for an unweighted average defined over properties in a sample and “expectation” for a probability-weighted average over the items from which the sample is taken. My presentation, based on Weir (Reference Walsh1996, chap. 5) and Lynch and Walsh (Reference Ishida1998, chap. 18), will involve various abbreviations, which I will define as we proceed.

I begin with an overview of the strategy for estimating . First, we define expressions and for mean squares involving between-population and within-population averages, respectively, of sample data. We also define expressions and for expectations (˙) of those quantities. These expressions have the following properties:

1. 1. Expectations of and of do not mention sample data but only

   • • a few constants,

   • • a value for (which is initially unknown), and

   • • frequencies in an ancestral population for the contemporary populations under study. (These are also unknown.)

2. 2. Expectations and can be combined in an expression equal to :

   
   where c is a constant defined in terms of number of populations and sample sizes (see below). It turns out that the ancestral frequencies do not appear in the result of evaluating the expression on the right; they will get divided out.

We can then use a similar expression involving and as an estimate of :

Since, on average, and equal and (the latter are expectations of the former), we can use this last expression as an estimate of .Footnote ³

In order to illustrate the role of “replicate populations” in the estimation of , it will be sufficient to examine one component of the derivation of the estimator (2). Take the expectation operator to represent a probability-weighted average over possible populations genetically sampled from an ancestral population. Let

• • be the frequency of a given allele in the sample from population i,

• • be the frequency of the same allele in the combined sample from all populations,

• • p be the frequency of the allele in the ancestral population,

• • c be a constant depending on the number of populations and their sample sizes:

  
  where is the sample size in population i, and #pops is the number of populations. The derivation of this expression is not important here, but note that it is indeed a constant since the number of contemporary populations and their sizes can be taken to be fixed.

One key step in the derivation of the estimator for is to calculate a simple expression for . In equation (3), is the expression within parentheses in the center term, and a bit of manipulation produces the result on the right:Footnote ⁴

Note that appears on the right-hand side of equation (3). This ultimately allows estimating in equation (2). Note also that the middle term of equation (3) depends only on data that we can readily determine: we know the number of populations with which we are dealing, we know our sample sizes from each population, and we can presumably determine the frequency of an allele of interest in our sample from the ith population, as well as its frequency in all of our samples combined. The right-hand side of equation (3), though, refers to the frequency p of the chosen allele in the ancestral population. We do not actually know what that frequency is since we have only samples from current populations descended from this unknown ancestral population. However, it turns out that a similar equation for also includes as a factor, so that is divided out in equation (1). This means that the values of ancestral frequencies can be ignored in the end.

Thus ancestral allele frequencies do not appear in the final steps of the derivation of the estimator for . Nevertheless, the idea that there are probabilities of allele frequencies in possible populations is essential to the derivation of equation (3). Of course, these probabilities are unknown, too. So equation (3) and other components of the derivation of an estimator for depend on both frequencies and probabilities that are unknown and that would be extremely difficult to determine at best. However, the assumption that these probabilities exist and provide the weights for calculating the expectation in equation (3) is what determines the form of the right-hand side of equation (3). Without this assumption, the right-hand side of the equation would have a different form:Footnote ⁵

where q is the frequency of the allele in current population i, and is the average frequency in all such populations. The relevant formulas involve assumptions about what kind of statistical process produced the current data and differ depending on those assumptions.

To sum up, in Cockerham and Weir's method of estimating , equation (3) and other similar expressions, which together are used to derive formulas for estimating , involve first treating the sample frequencies as derived from the actual frequencies in each population (statistical sampling) and then treating those actual frequencies in populations as derived from an earlier parent population (genetic sampling). The assumption that frequencies in current, actual populations are probabilistic effects of processes leading from an ancestral population to contemporary populations is therefore part of what justifies equation (3) and similar expressions. This assumption is essential to Cockerham and Weir's way of estimating for the purpose of determining significant differences between contemporary populations. For example, if we estimate in order to get evidence that there have been differences in natural selection between populations, we are depending on this assumption.

Importantly, Cockerham and Weir's method has been enormously popular in biological applications. Web of Science lists over 6,700 citations of Weir and Cockerham (Reference Weir1984) in diverse applications (cf. Weir and Hill Reference Weir2002; Weir Reference Walsh2012). There are other methods for estimating that do not depend on the same assumptions, notably Nei's methods (Nei Reference Matthen and Ariew1987; Nei and Kumar Reference Matthen and Ariew2000), but they are much less widely used. The widespread usefulness of Cockerham and Weir's method creates a prima facie reason to think that the assumptions on which this method is based are justified and at least approximately correct in general.Footnote ⁶

7. Causal Probabilities

I will say that there is a “causal probability” for a set of event types when these events usually appear in systematic patterns of frequencies, and there is a set of factors in the world that can be used to manipulate these frequencies.Footnote ⁷ For example, adding weight to some sides of a die is a way of manipulating frequencies of outcomes from tosses of that die. This loose notion of a causal probability will be sufficient to make the points I want to make. What it captures is that some frequencies in the world can be readily controlled, or at least biased, by manipulating some physical factors. I say that causal probabilities govern these frequencies and anything defined in terms of them.

Note now that the probabilities involved in genetic sampling, that is, in the processes that produce contemporary populations from an ancestral population, must incorporate probabilities concerning selection, migration, nonrandom mating, and so forth. Thus they are probabilities of effects that give rise to the current populations, given an ancestral population state. Moreover, as I will argue below, these probabilities can in theory be manipulated by manipulating the ancestral population or by manipulating the conditions experienced by populations derived from it. Thus it appears that the probabilities that underlie the notion of genetic sampling should be understood as causal probabilities. (Some authors might argue that these probabilities must be propensities—indeterministic dispositions. I think that is implausible [Eagle Reference Cockerham2004; Abrams Reference Abrams2007, Reference Abrams and Kincaid2012a]. My suggestion is that they are my “mechanistic probabilities” [Abrams Reference Abrams and Kincaid2012a, Reference Abrams2012b], Strevens's [Reference Shapiro, Sober, Wolters and Machamer2011] “microconstant probabilities,” or Rosenthal's [Reference Ramsey2010] “natural range” probabilities.)

The claim that genetic sampling can be manipulated by manipulating past conditions seems intuitively obvious, but it can be bolstered by considering a common method for making inferences about current populations using computer simulations (e.g., Voight et al. Reference Strevens, Beisbart and Hartmann2006; Scheinfeldt et al. Reference Reisman and Forber2011; Huff et al. Reference Endler2012).Footnote ⁸ In these simulations, a population-genetic model is used to produce a sequence of simulated generations. At each point in time, one generation of a simulated population gives rise to a subsequent generation, which in turn gives rise to a third generation, and so on, usually for thousands of generations. The probability that an allele is inherited in any given generation can be affected by various modeled factors such as selection, mutation, mating patterns, population size, and so forth. (Sometimes the simulations use coalescent methods, which in effect run the generations in reverse; this does not fundamentally affect the points I make.)

One then lets such simulated populations evolve repeatedly. Numerous runs are performed at each of several parameter settings (different combinations of fitness values, population size at various times, recombination rates, etc.). Frequencies of results of such repeated simulation runs can then be used to estimate how probable it is that statistical tests on a sample from a real population would have resulted in the measured values, given similar values for parameters in a real-world population. For example, one can estimate the probability that a current population would have the state it has, or that a sample from it would have the state it has, in the presence or absence of past natural selection within the population. The important point is that by performing numerous simulation runs at each setting of parameter values, so as to measure frequencies of specified outcomes, one is manipulating these frequencies. That is, such a use of simulations involves causal probabilities.

In applying the results of such simulations to real populations, we are assuming that real populations behave in a way that is similar to the simulated populations. We decide what kind of past events (selection, pure drift, population bottlenecks, etc.) are likely to have produced observed data on current populations by in effect assuming that the past was subject to manipulations like those done in the simulation. That is, we are assuming that probabilities of what we observe today would have varied depending on how past conditions had varied. Thus this use of simulations investigates the relationships between past conditions and such probabilities and does so by manipulating the probabilities in ways analogous to ways that real-world probabilities could have been manipulated. Therefore, to the extent that such simulation methods used in applications of population genetics are appropriate—to the extent that these widespread scientific practices are effective—the claim that the probabilities involved in genetic sampling are causal probabilities receives additional support.

Moreover, techniques using and other statistical methods (including simulation-based methods just described) are explicitly designed to distinguish between natural selection, drift, migration, and so forth using such implicit assumptions about probabilities. It therefore appears that natural selection, drift, and so forth are different kinds of causes of evolution because they depend on differences in causal probabilities.

8. Conclusion

I have tried to illustrate an implicit philosophical commitment of a significant class of research practices that use . For debates about whether natural selection, drift, and so forth are causes of evolution, it is important to see what widespread biological practices are committed to and how deeply those commitments go. My focus on the use of in practice should allow discussion of the role of causation in evolution to engage with evolutionary theory and practice in new ways. More specifically, I have argued that the way in which many biologists estimate (in diverse research, often successful, presumably) makes this practice implicitly committed to the assumption that there is some kind of causal probability that governs much about the evolution of populations, according to which different probability distributions correspond to interactions between different evolutionary forces, including natural selection and drift. I have further motivated the suggestion that these probabilities are causal by examining the way in which computer simulations that involve causal probabilities are used to model processes in the world.