2. A notation for causal notions

A formal symbolic language to distinguish causal relations from merely correlational ones, such as the counterfactual notation of Neyman (Reference Neyman1923) and Rubin (Reference Rubin2005), was not to our knowledge ever adopted by Fisher. This is despite the fact that he frequently wrote about this distinction. Although such formalisms lack the elegance of Fisher's prose, adopting the appropriate formalism is an aid to understanding.

For this purpose, we adopt the do operator of Pearl (Reference Pearl1995). We are to interpret an expression such as ${\mathbb E}[ Y\vert do(x) ] $ to mean the expectation of Y given that the random variable X has been experimentally fixed to the value x. The contrast between conditional quantities containing the do symbol and traditional conditional quantities is evident in the expressions

(1)$$\eqalign{{\mathbb P}( {\rm mud}\vert {\rm rain}) \geqslant {\mathbb P}( {\rm mud}) \quad {\rm and}\quad {\mathbb P}( {\rm rain}\vert {\rm mud}) \geqslant {\mathbb P}( {\rm rain})} $$

and

(2)$$\eqalign{{\mathbb P}[ {\rm mud}\vert do( {\rm rain}) ] \geqslant {\mathbb P}( {\rm mud}) \quad {\rm and}\cr \quad {\mathbb P}[ {\rm rain}\vert do( {\rm mud}) ] = {\mathbb P}( {\rm rain})} .$$

Equation (1) indicates that we are more likely to find mud if we have already observed rain. Because co-occurrence is symmetric, it also becomes more likely that it has rained if we have already observed mud. On the other hand, (2) symbolizes the much stronger and asymmetrical assertion that rain causes mud and not vice versa; muddying up the backyard with a garden hose will not make it rain.

This notation and its associated machinery may be of some benefit in the burgeoning field of genome-wide association studies (GWAS), where it is important to single out genetic variants with a causal effect on a given phenotype from markers that are merely associated with the phenotype for other reasons, including linkage disequilibrium (LD) with a nearby causal variant (Visscher et al., Reference Visscher, Brown, McCarthy and Yang2012). Letting Y denote the phenotype of interest, we can say that a genetic variant is a causal variant if the equality

(3)$${\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) ] = {\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{2}} ) ] = {\mathbb E}[ Y\vert do( {\cal A}_{{2}} {\cal A}_{{2}} ) ] $$

does not hold. The expectation is taken over the space of all possible multilocus genotypes and environments. Note that the equality does in fact hold for a non-causal marker locus in LD with a causal locus. If we could experimentally mutate a randomly chosen zygote's genotype at a biologically inert marker locus immediately before the onset of development, we would not expect any ensuing change in the phenotype.

The do notation is more than a convenient means of fixing ideas. The treatise of Pearl (Reference Pearl2009) grounds this symbol in a rich syntax and semantics. From one point of view, the work of Pearl can be regarded as a vast generalization of Wright's (Reference Wright1968) path analysis.

For simplicity, we will speak of events in the life cycle such as fertilization, development and phenotypic measurement as if all individuals experienced each such event at the same time – a convention that is appropriate for an organism with a life cycle consisting of discrete and non-overlapping generations. We can then speak of selecting one zygote for an experimental treatment from all those zygotes making up the current generation. Our discussion also applies, however, to organisms with a life cycle consisting of continuous and overlapping generations. In this case, a quantity such as ${\mathbb E}[ Y\vert {do}( {\cal A}_{{1}} {\cal A}_{{1}} ) ]$ is to be interpreted as the present phenotypic value that a randomly selected organism would have been expected to obtain if its genotype could have been converted to ${\cal A}_{{1}} {\cal A}_{{1}} $ immediately after its own fertilization. Fisher's own writings suggest the importance of counterfactual thinking. In a summary of his work on the correlations between relatives, he wrote: ‘[I]t should be clearly understood what we mean by a cause of variability. If we say, “This boy has grown tall because he has been well fed,” we are not merely tracing out cause and effect in an individual instance; we are suggesting that he might quite probably have been worse fed, and that in this case he would have been shorter’ (Fisher, Reference Fisher1919, p. 214, emphasis in original). The do operator bears both interventional and counterfactual interpretations. If necessary, each organism can be weighted by reproductive value.

3. Falconer's interpretation of the experimental average effect

We can use the do operator to symbolize the gene substitutions in Fisher's thought experiment. Here, we use it to review Falconer's understanding of this experiment for a single biallelic locus. We first note that if genotypic and environmental causes of phenotypic variation act additively and independently, then quantities such as ${\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} ) $ are precisely equal to ${\mathbb E}[ Y\vert {do}( {\cal A}_{{1}} {\cal A}_{{1}} ) ]$ at the single causal locus. Until we say otherwise, we assume the stochastic independence of genotypes and environments.

Following the notation of Fisher (Reference Fisher1918), we let P, 2Q, and R denote the respective frequencies of the genotypes ${\cal A}_{{1}} {\cal A}_{{1}} $, ${\cal A}_{{1}} {\cal A}_{{2}} $, and ${\cal A}_{{2}} {\cal A}_{{2}} $. Given that a zygote's genotype is ${\cal A}_{{1}} {\cal A}_{{1}} $, we write the expected phenotypic effect of changing a gene's allelic type from ${\cal A}_{{1}} $ to ${\cal A}_{{2}} $ as

(4)$${\rm \Delta }Y\vert {\cal A}_{{1}} {\cal A}_{{1}}\! \to\! {\cal A}_{{1}} {\cal A}_{{2}}\! = {\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{2}} ) , {\cal A}_{{1}} {\cal A}_{{1}} ] - {\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} ) .$$

There is no contradiction in conditioning on both the observation of ${\cal A}_{{1}} {\cal A}_{{1}} $ and the experimental setting of the genotype to ${\cal A}_{{1}} {\cal A}_{{2}} $. This simply means that instead of performing the experiment on a zygote sampled at random from the entire population, we perform it specifically on a zygote that would otherwise have borne the genotype ${\cal A}_{{1}} {\cal A}_{{1}} $. Similarly, we define

(5)$${\rm \Delta }Y\vert {\cal A}_{{1}} {\cal A}_{{2}}\! \to\! {\cal A}_{{2}} {\cal A}_{{2}}\! = {\mathbb E}[ Y\vert do( {\cal A}_{{2}} {\cal A}_{{2}} ) , {\cal A}_{{1}} {\cal A}_{{2}} ] - {\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{2}} ) .$$

The problem with identifying the effect of a gene substitution – as in identifying the effect of an alteration to any nonlinear causal system – is that the expected change depends on the context. In other words (4) and (5) are not equal in general. Falconer supposed that Fisher arrived at the ‘average effect’ of substituting ${\cal A}_{{2}} $ for ${\cal A}_{{1}} $ by averaging (4) and (5) in the following way. We sample a zygote at random and then select one of its genes at random. If the chosen gene is of allelic type ${\cal A}_{{2}} $, we leave it alone. If the chosen gene is of type ${\cal A}_{{1}} $, we change it to ${\cal A}_{{2}} $. The expected phenotypic effect of the gene substitutions performed under this scheme is thus

(6)$${{P(\rm \Delta Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{1}} \to {\cal A}_{{1}} {\cal A}_{{2}} ) + Q(\rm \Delta Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{2}} \to {\cal A}_{{2}} {\cal A}_{{2}} ) } \over {P + Q}}.$$

Falconer pointed out that (6) does not agree with the regression definition of the average effect that Fisher (Reference Fisher1941) gave in an article criticizing Sewall Wright for conflating the average excess and average effect. This paper required explicit expressions for the two genetic averages in traditional notation, and Fisher obtained an expression for the average effect adequate for demonstrating its distinctness from the average excess by minimizing the sum of squares

(7)$$\eqalign{P[ {\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} ) - \nu + \alpha ] ^{{2}} + 2Q[ {\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{2}} ) - \nu ] ^{{2}} \cr \quad + R[ {\mathbb E}( Y\vert {\cal A}_{{2}} {\cal A}_{{2}} ) - \nu - \alpha ] ^{{2}} , \cr} $$

where ν is the regression constant. Using a notation that generalizes to a locus with more than two alleles, we can express this sum of squares equivalently as

(8)$$\eqalign{ P[ {\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} )\! - \mu \!- 2\alpha _{{1}} ] ^{{2}}\! + 2Q[ {\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{2}} ) - \mu - \alpha _{{1}} - \alpha _{{2}} ] ^{{2}} \cr + R[ {\mathbb E}( Y\vert {\cal A}_{{2}} {\cal A}_{{2}} ) - \mu - 2\alpha _{{2}} ] ^{{2}} , \quad {\rm where}\quad \mu = {\mathbb E}( Y) .\cr} $$

In definition (7), then, the average effect α is the slope in the regression of the phenotype on gene count. α₁ and α₂ in (8) are the average effects of the two alleles individually – a notion to which we will return. For now we simply note that α will turn out to equal α₂–α₁ in magnitude. There is some ambiguity in the literature over whether the outcome variable in the regression should be defined, as in eqn (8), with the subtraction of the unconditional phenotypic mean (Fisher, Reference Fisher1958b; Price, Reference Price1972; Ewens, Reference Ewens2011). However, this choice simply adds a constant term to the average effects of the individual alleles, and this term disappears in the biallelic average effect α=α₂–α₁. In our later discussion of individual average effects, we will give a compelling reason to favour the mean subtraction.

Perhaps frustrated by Fisher's concise style, Falconer concluded his article by approvingly quoting Price's (Reference Price1972) remark that Fisher's ideas can be translated into well-understood concepts such as covariance and regression without dealing with his ‘special’ notions of average excess and average effect.

In the following, we show that the two definitions of the average effect can be reconciled, in the case of genotype–environment independence, for a specific weighting of the two possible substitutions. However, if such independence fails to hold, it is not possible to dispense with Fisher's ‘special’ definition in terms of experimental gene substitutions.

4. Fisher's experimental average effect

Fisher conditioned the gene substitutions in his hypothetical experiment on the ‘rules’ by which ‘the frequency of different gene combinations may be governed.’ It is this difficult subtlety that Falconer did not take into account. In The Genetical Theory Fisher's wording seems to imply that it is only the mating scheme that determines how different alleles combine to form whole-genome genotypes. Later he acknowledged that other factors also influence the departure of genotype frequencies from random combination of genes, explicitly mentioning ‘the partial isolation of sections of the population’ (Fisher, Reference Fisher1941, p. 54). The implication for the experimental gene substitutions is that they must be carried out in a manner that does not disturb the arrangement of alleles into genotypes called for by the population's rules of formation.

The three genotype frequencies sum to unity, as do the frequencies of the two alleles. Thus, given the frequency of one allele, one more parameter is required to specify the genotype frequencies. There appears to be complete freedom in the choice of this parameter. For example, one possibility is Wright's inbreeding coefficient F (Crow & Kimura, Reference Crow and Kimura1956). As we later show, if we require the experimental average effect to coincide with the regression average effect in the case of genotype–environment independence, then we must choose the parameter to be λ=Q ²/(PR), the ratio of the squared (ordered) heterozygote frequency to the product of the homozygote frequencies. λ can be written in the symmetrical form

$${{{\mathbb P}( {\cal A}_{{2}} \vert {\cal A}_{{1}} ) } \over {{\mathbb P}( {\cal A}_{{1}} \vert {\cal A}_{{1}} ) }} \cdot {{{\mathbb P}( {\cal A}_{{1}} \vert {\cal A}_{{2}} ) } \over {{\mathbb P}( {\cal A}_{{2}} \vert {\cal A}_{{2}} ) }}, $$

and it attains the constant value of unity if the population mates randomly, a fact first noted by Hardy (Reference Hardy1908).

Let p=Q+R denote the frequency of ${\cal A}_{{2}} $, and write the population mean of Y as a function of allele frequency and the rules of combination, μ(p, λ). We now show that the expression μ(p+dp, λ)− μ(p, λ) is proportional to the average effect, α, obtained from regression eqn (7). In other words, the ratio λ must be kept constant under this manipulation, whatever the population's rules of formation have determined this ratio to be, in order for the experimental gene substitutions to yield what Fisher intended by the average effect.

The population mean is given by the expression

(9)$$\eqalign{ {\mu = P\,{\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) ] + 2Q\,{\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{2}} ) ]} \cr + R\,{\mathbb E}[ Y\vert do( {\cal A}_{{2}} {\cal A}_{{2}} ) ] . \cr} $$

The average effect is then proportional to the change of μ with respect to p while holding λ constant. We can increase p by carrying out either the intervention ${\rm {\cal A}}_1 {\rm {\cal A}}_1 \to {\rm {\cal A}}_1 {\rm {\cal A}}_2 $ or ${\cal A}_{{1}} {\cal A}_{{2}} \to {\cal A}_{{2}} {\cal A}_{{2}} $. As detailed in the Appendix, upon noting that the differential of Q ²=λPR for constant λ yields the differential equation

(10)$${{{\rm d}}P \over P} + {{{\rm d}}R \over R} = {{2{\rm d}}Q \over Q}, $$

we find that Fisher's average effect is

(11)$$\alpha = {{c_{{1}} (\rm \Delta Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{1}} \to {\cal A}_{{1}} {\cal A}_{{2}} ) + c_{{2}} (\rm \Delta Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{2}} \to {\cal A}_{{2}} {\cal A}_{{2}} ) } \over {c_{{1}} + c_{{2}} }}, $$

where the weights are

$$c_{{1}} = P( Q + R) , $$

$$c_{{2}} = R( P + Q) .$$

Let us recapitulate the meaning of (11). Immediately after fertilization, we take a random sample of the zygotes bearing the genotype ${\cal A}_{{1}} {\cal A}_{{1}} $. We then randomly assign some of these zygotes to the ‘treatment,’ which consistsof changing the allelic type of a gene from ${\cal A}_{{1}} $ to ${\cal A}_{{2}} $. The expected difference in phenotype between treatments and controls at the time of measurement is the causal effect of the gene substitution. We perform the analogous experiment to determine the causal effect of changing ${\cal A}_{{1}} {\cal A}_{{2}} $ to ${\cal A}_{{2}} {\cal A}_{{2}} $. The weighted average of the two causal effects – where the weights c ₁ and c ₂ are chosen so as to preserve λ if the two types of gene substitutions are applied to the population in the ratio c ₁/c ₂ – is the average effect of gene substitution holding constant the rules governing the frequencies of the different genotypes.

Now that the average effect has been defined in (11), we can apply it to an example of a population changing in mean phenotypic value under a sequence of gene substitutions (Table 1). This example may be seen as a numerical counterpart to the diagrammatic illustration by Edwards (Reference Edwards2002). Suppose that the effect of changing an ${\cal A}_{{1}} {\cal A}_{{1}} $ individual to ${\cal A}_{{1}} {\cal A}_{{2}} $ is 3 phenotypic units, whereas the effect of changing ${\cal A}_{{1}} {\cal A}_{{2}} $ to${\cal A}_{{2}} {\cal A}_{{2}} $ is −2. Suppose also that the numbers of the genotypes ${\cal A}_{{1}} {\cal A}_{{1}} $, ${\cal A}_{{1}} {\cal A}_{{2}} $, and ${\cal A}_{{2}} {\cal A}_{{2}} $ in this population are 40, 40, and 20, respectively. These genotype frequencies imply that (c ₁, c ₂) is proportional to (4, 3). Table 1 shows how the average phenotypic change and λ are affected by each step in a sequence of gene substitutions leading to an increase in p but tending to keep λ constant. The first column gives the gene substitution. In this sequence, the two types of substitution alternate, but this is not an essential feature. The second column gives the numbers of the genotypes after the gene substitution. The third column gives the cumulative change in the total phenotypic measurements (the mean phenotype times the population size) divided by the number of gene substitutions. The fourth column gives the new value of λ after the gene substitution.

Table 1. Sequence of experimental gene substitutions yielding the average effect.

It is readily confirmed that the final value of λ is the closest to the starting value of 1/2 that can be achieved with seven gene substitutions. If we take population size to infinity, we can make the discrepancy between the original and new values of λ as small as we please.

In the special case of genotype–environment independence considered so far, where equalities such as ${\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} ) = {\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) ] $ always hold, Fisher's experimental and regression definitions of the average effect coincide for constant λ. In the above example, after assigning each genotype an expected phenotypic value consistent with the magnitudes of the experimental effects, it is easily verified that the slope in the least-squares regression of phenotypic value on ${\cal A}_{{2}} $ gene count is 6/7.

5. Gene–environment correlation and interaction

As a preliminary matter, we note that any variable along a causal path (in the sense of Wright and Pearl) from genotype to phenotype must not be counted as environmental. For example, if dairy consumption affects stature, it is tempting to regard dairy consumption as an environmental (non-genotypic) variable with respect to stature. However, if genetic variation affects lactose tolerance and thus the amount of milk consumed, assigning the effect of dairy consumption on stature to the environment ignores the fact that the path genotype → lactose tolerance → dairy consumption → stature ultimately begins with a genetic variable. This subtlety may have been among the reasons why Fisher favoured ‘speaking of the residue as non-genetic, rather than environmental …’ (Bennet, Reference Bennett1983, p. 260).

It is worth asking whether Fisher intended the average effect to be defined in the event that genotypic and environmental causes are either dependent or non-additive. In many places, he certainly assumed or argued for independence and additivity (Fisher, Reference Fisher1918, Reference Fisher1941, Reference Fisher1953, Reference Fisher1970), and it has been asserted that Fisher's biometrical theory is meaningless if these conditions are not met (e.g. Vetta, Reference Vetta1980).

As Price (Reference Price1972) has pointed out, Fisher's exposition in The Genetical Theory leaves much to be desired. A close reading of this text and Fisher's other writings, however, turns up many reasons to suspect that Fisher regarded independence and additivity as reasonable specifications for certain demonstrations and not as strictly necessary conditions for the average effect to be defined.

1. 1. In the discussion of the average effect in The Genetical Theory, Fisher did not explicitly refer to his other work where he made special assumptions regarding the environment.

2. 2. The average effect is a key concept in the FTNS, which Fisher regarded as an exact and rigorous statement. One would like to believe that Fisher, having been trained in mathematical physics, would not have compared the FTNS to the second law of thermodynamics if the FTNS depended on assumptions regarding the environment that must always be approximations at best.

3. 3. We can read that ‘[t]he genetic variance as here defined is only a portion of the variance determined genotypically, and this will differ from, and usually be somewhat less than, the total variance to be observed’ (Fisher, Reference Fisher1930, p. 34). The genotypic variance is greater than the total variance only if ‘good’ genotypes tend to be found in ‘bad’ environments, and thus Fisher was clearly allowing for the possibility of dependence.

4. 4. In a letter to J. A. Fraser Roberts, Fisher wrote that

   [t]here is one point in which Hogben and his associates are riding for a fall, and that is in making a great song about the possible, but unproved, importance of non-linear interactions between hereditary and environmental factors. … What they do not see is that we ordinarily count as genetic only such part of the genetic effect as may be included in a linear formula and that we make a present to the environmentalists of such variation due to the combined action of genetic and environmental factors as is not expressible in such a formula (Bennett, Reference Bennett1983, p. 260).

These remarks clearly show that Fisher did not regard genotype–environment interaction as an obstacle to defining the average effect. Emboldened by this evidence regarding the intended generality of the average effect, we extend our treatment to encompass gene–environment correlation and interaction.

We first suppose that genotypic and environmental causes act additively but are not independent. Additivity means that the experimental effect of a gene substitution remains the same regardless of the environment in which the experiment is carried out; varying the environment simply raises or lowers the expected phenotypic values of all three genotypes by the same amount. For instance,

(12)$${\rm \Delta }Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{1}} \to {\cal A}_{{1}} {\cal A}_{{2}} , {\cal E}_{i} = {\rm \Delta }Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{1}} \to {\cal A}_{{1}} {\cal A}_{{2}} , {\cal E}_{j} $$

for any choice of environments ${\cal E}_{{i}} $ and ${\cal E}_{{j}} $. In this case, all of the discussion in previous sections continues to apply except for the equivalence of the experimental and regression average effects. If some genotypes are more frequently found in favourable environments for phenotypic development, then the regression of phenotypic value on gene count does not have a simple genetic interpretation.

Non-additivity means that at least one equality of the kind in (12) does not hold. The precise magnitude of the expected change upon an experimental gene substitution now depends on some aspect of the environment that the manipulated zygote will experience between the onset of development and the time of measurement. This case is problematic because now a quantity such as ${\rm \Delta }Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{1}} \to {\cal A}_{{1}} {\cal A}_{{2}} $ is not necessarily equal to ${\rm \Delta }Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{2}} \to {\cal A}_{{1}} {\cal A}_{{1}} $, since the genotypes ${\cal A}_{{1}} {\cal A}_{{1}} $ and ${\cal A}_{{1}} {\cal A}_{{2}} $ may tend to be found in different environments. This difficulty can be overcome by redefining expressions such as ${\rm \Delta }Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{1}} \to {\cal A}_{{1}} {\cal A}_{{2}} $ so that each symbolizes a difference between experimental treatments rather than a difference between a treatment and an unperturbed control group. For example, (4) would become

$${\rm \Delta }Y\vert {\cal A}_{{1}} {\cal A}_{{1}} \to {\cal A}_{{1}} {\cal A}_{{2}} = {\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{2}} ) ] - {\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) ] .$$

Seeking an equivalent generalization that retains the interventional form of (4) and (5), however, sheds substantially greater light on the problem.

Before taking up the issue of gene–environment interaction, it is helpful to review Fisher's motivation for holding λ constant as a means to address gene–gene interaction. In order to formulate the FTNS, Fisher wished to quantify the causal effectof changing allele frequency while holding the environment constant. In his view the way in which alleles combine to form genotypes, as parameterized by λ, should be regarded as part of the environment. Although this choice may initially seem eccentric, because fitness differences among genotypes will typically change both p and λ, it becomes reasonable when we realize that λ may also change as a result of extrinsic events such as the formation or dissolution of geographical hindrances to random mating.

There is an analogy here to Fisher's analysis of covariance to separate the direct and indirect effects of a given experimental manipulation on a focal outcome. For instance, in an experiment to determine whether a given fertilizer affects the purity of sugar extracted from sugar-beets, the experimenter may already know that the fertilizer affects the weight of the beet roots, which in turn affects sugar purity (Fisher, Reference Fisher1970, pp. 283–284). The experimenter may wish to know whether the fertilizer affects sugar purity through a direct causal path, fertilizer→ sugar purity, distinct from the indirect path fertilizer→ root weight → sugar purity. In certain cases adjustment for root weight by analysis of covariance yields the target quantity: the amount by which sugar purity would change upon application of the fertilizer, if root weight could be experimentally clamped to the value that it would have obtained in the control condition. Similarly, while gene substitutions that are not deliberately balanced as in (11) will typically change both p and λ, we can still mathematically define an average effect stipulating that λ remains clamped to a constant value. This point of view is similar to one expressed by Okasha (Reference Okasha2008).

Once we regard any change in how alleles are arranged into genotypes as environmentally caused, it perhaps becomes obvious that we should regard certain changes in the allotment of genotypes to environments as such. After all, a redistribution among environments might lead to changes in the phenotypic means of the genotypes. Such changes in the genotype–phenotype mapping, when caused by extrinsic events such as climate change, are readily classified as environmental in nature. This consideration suggests that the gene substitutions defining the average effect in the presence of genotype–environment interaction should be balanced in such a way that the phenotypic means of the genotypes remain constant.

Since equalities such as ${\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} ) = {\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) ] $ do not hold when genotypes and environments are also dependent, there is ambiguity in what is meant by holding constant the phenotypic means. We first consider holding constant the observed means. If the environments interacting with genotypes can be classified discretely, then we can write an equation like

(13)$${\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} ) = \mathop \sum \limits_{i} {\mathbb P}( {\cal E}_{i} \vert {\cal A}_{{1}} {\cal A}_{{1}} ) {\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} , {\cal E}_{i} ) $$

for each genotype. Because genotypes and environments exhaust all possible causes of phenotypic variation, ${\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} , {\cal E}_{i} ) $ is equivalent to ${\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) , do( {\cal E}_{i} ) ] $. In a sense even the expectation operator is unnecessary because Y is a deterministic function when both genotype and environment are specified.

Constancy of observed means requires constancy of the conditional probabilities taking the form ${\mathbb P}( {\cal E}_{i} \vert {\cal A}_{{1}} {\cal A}_{{1}} ) $. A candidate definition for the average effect is then

$$\eqalign{ 2\alpha dp = \mu [ p + dp, \lambda , {\mathbb P}( {\cal E}_{{1}} \vert {\cal A}_{{1}} {\cal A}_{{1}} ) , \ldots , {\mathbb P}( {\cal E}_{n} \vert {\cal A}_{{2}} {\cal A}_{{2}} ) ] \cr - \mu [ p, \lambda , {\mathbb P}( {\cal E}_{{1}} \vert {\cal A}_{{1}} {\cal A}_{{1}} ) , \ldots , {\mathbb P}( {\cal E}_{n} \vert {\cal A}_{{2}} {\cal A}_{{2}} ) ] . \cr} $$

The problem with this candidate definition, however, is that it can lead to a non-zero average effect even if in each environment neither gene substitution has a causal effect. This is because preserving a genotype's conditional probabilities of being found in the various environments may require that some gene substitutions be accompanied by the placement of the manipulated organism in a different environment; the resulting change in phenotype may then be entirely the result of the environmental change.

If we instead consider holding constant the experimental means, then we obtain

(14)$$\eqalign{ {\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) ] = \mathop \sum \limits_{i} {\mathbb P}[ {\cal E}_{i} \vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) ] \,{\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) , {\cal E}_{i} ]\hskip-32pt \cr = \mathop \sum \limits_{i} {\mathbb P}( {\cal E}_{i} ) {\mathbb E}( Y\vert {\cal A}_{{1}} {\cal A}_{{1}} , {\cal E}_{i} ) . \cr} $$

The left-hand side (LHS) is the expected phenotypic value upon sampling a zygote at random and, if its genotype is not ${\cal A}_{{1}} {\cal A}_{{1}} $, making it so. Since changing the genotype of a zygote cannot affect its environment, we have ${\mathbb P}( {\cal E}_{i} ) \vert do( {\cal A}_{{1}} {\cal A}_{{1}} ) ] = {\mathbb P}( {\cal E}_{i} ) $ for each i and thus a justification of the second line. Therefore preserving the experimental means only requires a constant marginal distribution of environmental states. Of course, we can always abide by this constraint if we never foster any manipulated organism in a different environment. This ensures that a non-zero average effect is indeed an average of genetic effects, at least one of which would turn out to be non-zero under experimental control.

Hence, a natural definition of the average effect in the presence of genotype–environment interaction is

(15)$$\eqalign{\alpha = {{\mathop \sum \nolimits_{i} c_{{1}, i} (\rm \Delta Y\vert {\cal A}_{{1}} {\cal A}_{{1}} \to {\cal A}_{{1}} {\cal A}_{{2}} , {\cal E}_{i} ) + c_{{2}, i} (\rm \Delta Y\vert {\cal A}_{{1}} {\cal A}_{{2}} \to {\cal A}_{{2}} {\cal A}_{{2}} , {\cal E}_{i} ) } \over {\mathop \sum \nolimits_{i} c_{{1}, i} + c_{{2}, i} }},} $$

where

$$c_{{1}, i} = c_{{1}} {\mathbb P}( {\cal E}_{i} ) , $$

$$c_{{2}, i} = c_{{2}} {\mathbb P}( {\cal E}_{i} ) .$$

6. Average effects of individual alleles

We will now explain how the experimental average effect of an individual allele may be defined for a locus with any number of alleles. Since there are $\left({\matrix{ n \cr 2 \cr} } \right)$ possible gene substitutions at a locus with n alleles, we can no longer speak of a single average effect in the case of n>2, and thus an extension of this kind is plainly necessary. In the second edition of The Genetical Theory, we can read that ‘[w]ith multiple allelomorphism it is convenient to define [the average effect of an allele] by the effect of substituting any chosen gene for a random selection of the genes homologous with it’ (Fisher, Reference Fisher1958b, p. 35). This definition can be explicated with respect to a given allele, say ${\cal A}_{{1}} $, as follows. Immediately after fertilization but before the onset of any developmental events, we select the allelic type of a gene to be changed into ${\cal A}_{{1}} $ in such a way that the probabilities of selection are equal to the allele frequencies. That is, if the vector of allele frequencies is (p ₁, …, p_n), then the gene to be changed is ${\cal A}_{{1}} $ with probability p ₁, ${\cal A}_{{2}} $ with probability p ₂, and so on. If the gene to be changed happens to be ${\cal A}_{{1}} $ itself, then the ${\cal A}_{{1}} \to {\cal A}_{{1}} $ change will have no phenotypic consequence. For all changes other than the null change, the choice of the undisturbed gene in the genotype is made in such a way that the population's rules of genotype formation are preserved. If genotypes and environments are both dependent and interacting, then the marginal distribution of environmental states must be considered as in (15). The expected change in the phenotype of the manipulated organism is then α₁, the average effect of ${\cal A}_{{1}} $.

From this definition we can derive some important consequences. Let N_k stand for the number of ${\cal A}_{k} $ genes in the population. The total number of genes is $\sum _{k = {1}}^{n} N_{k} $. Among the n experiments defining the individual average effects, choose one to perform with a probability equal to its corresponding allele frequency. The expected vector of allele frequencies following the randomly chosen experiment is then

(16)$$\eqalign{\mathop \sum \limits_{k = {1}}^{n} {{N_{k} } \over N}\left\{ {\mathop \sum \limits_{\ell = {1}}^{n} {{N_{\ell } } \over N}\left[ {\left( {{{N_{{1}} } \over N}, \ldots , {{N_{n} } \over N}} \right) + {1 \over N}{\bf e}_{k} - {1 \over N}{\bf e}_{\ell } } \right]} \right\}}, $$

where e_k is the vector of length n with element unity at position k and zeroes elsewhere. After some algebra we find that the first element of the expected vector is N ₁(ΣN_k)²/N ³=p ₁, the second is N ₂(ΣN_k)²/N ³=p ₂, and so on. The expected outcome of the randomly chosen experiment is a population with exactly the same allele frequencies, rules of genotype formation, and phenotypic mean. We have thus proved that the experimental average effects satisfy

(17)$$\mathop \sum \limits_{k = {1}}^{n} p_{k} \alpha _{k} = 0.$$

With the generalization of the experimental average effect given in the next section, (17) holds at any one of arbitrarily many multiallelic loci. In the case of a single locus, (17) holds for the regression average effects in (8) (Ewens, Reference Ewens2011), and agreement of the regression and experimental average effects thus requires the mean subtraction in that expression.

Let us apply the definition of the individual average effect to the biallelic example in Table 1. There are initially 120 ${\cal A}_{{2}} $ genes in this population of 200 total genes. If we perform the experiment defining α₁, then with probability 0·40 the population gene numbers remain at (80, 120) and with probability 0·60 the numbers become (81, 119). In the event of a non-null substitution, with probability 4/7 (given by ${\textstyle{{c_{{1}} } \over {c_{{1}} + c_{{2}} }}}$) the change is ${\cal A}_{{1}} {\cal A}_{{2}} \to {\cal A}_{{1}} {\cal A}_{{1}} $ and with probability 3/7 (given by ${\textstyle{{c_{{2}} } \over {c_{{1}} + c_{{2}} }}}$) it is ${\cal A}_{{2}} {\cal A}_{{2}} \to {\cal A}_{{1}} {\cal A}_{{2}} $. The expected outcome of the experiment is thus a population with gene numbers (80·6, 119·4) and, up to the limits of finite size, the same value of λ. Using simple probability calculus, we can calculate that the numerical value of α₁ is −18/35.

In summary, the experiment defining α₁ will lead to the null substitution ${\cal A}_{{1}} \to {\cal A}_{{1}} $ with probability p ₁ (in which case the causal effect is zero) and to the substitution ${\cal A}_{{2}} \to A_{{1}} $ with probability p ₂ (in which case the effect is equal in magnitude to the average effect of gene substitution with respect to the entire locus). Therefore α₁ must be equal to (p ₁)(0)+(p ₂)(−α), and from this we can use p ₁α₁+ p ₂α₂=0 to derive α=α₂–α₁ algebraically. The meaning of this relation among the three average effects is as follows. The expected outcome of the experiment defining α₂ is a population with gene numbers (79·6, 120·4) and nearly the same value of λ. Now suppose that we perform the ‘opposite’ of the experiment defining α₁, on average reducing the number of ${\cal A}_{{1}} $ genes rather than increasing them. We compose this experiment with the one defining ${\cal A}_{{2}} $, which in our example has a numerical value of 12/35. The population is thus expected to proceed through the sequence (80, 120) → (79·4, 120·6) → (79, 121), preserving λ at each step. The final state is precisely the one expected upon performing the experiment defining α, the average effect of gene substitution for the entire locus. We can see in what sense the average effect of gene substitution (6/7) is equal to the effect of removing one gene (18/35) and then replacing it with another (12/35).

7. Average effects in the case of multiple loci

In the case of a single locus with two alleles, we can just as well define the average effect of gene substitution as

(18)$$\alpha = {1 \over 2}{{\partial \mu ( p, \lambda ) } \over {\partial p}}, $$

where μ is defined as in (9). From this starting point, we can derive the equivalence of the regression (7) and experimental (11) definitions in the case of genotype–environment independence. Equation (18) fills the lacuna in Wright's casual use of the expression

$${{d\overline{W}} \over {dp}}, $$

to which Fisher (Reference Fisher1941) strongly objected. The explicit dependence of μ on λ, a measure of departure from random combination of genes, meets the criticism that ‘the numerator involves the average of [the phenotype] for a number of different genotypes … exceeding the number of gene frequencies p on which their frequencies are taken to depend’ (p. 57).

It is interesting that the only genetic condition governing the gene substitutions defining the average effect for a single biallelic locus is the constancy of λ, a parameter that depends on the genotype frequencies but not the genotypic means. One might have thought that these means, appearing as they do in (7), must play some role in the weighting of the two possible gene substitutions. It is then natural to ask whether the generalization to multiple loci retains the appealing feature that constancy of appropriately quantified departures from Hardy-Weinberg and linkage disequilibrium is sufficient – without any additional information regarding the genotypic means – for an experimental average effect to agree with its corresponding partial regression coefficient. According to our analysis in the Appendix, the multilocus average effects do not in fact retain this feature. That is, we would like to define the multilocus average effect of allele i_k at locus k, ${\cal A}_{i_{k} }^{( k) } $, as

(19)$$\alpha _{i_{k} }^{( k) } = {1 \over 2}{{\partial \mu ( {\bf p}, {\lambda} ) } \over {\partial p_{i_{k} }^{( k) } }}, $$

where p is now a vector of allele frequencies at several loci, $p_{i_{k} }^{( k) } $ being the element corresponding to ${\cal A}_{i_{k} }^{( k) } $, and λ is a vector of whatever measures of departure from random combination are preserved under the appropriately balanced gene substitutions. However, as will be demonstrated, such a mean-invariant description of the average effects does not seem to exist.

To set up a weaker definition of the multilocus average effects, we require some additional definitions and notational conventions. Suppose that there are L causal loci, in the sense of (3), affecting the focal phenotype. Suppose also that there are $n_{\ell } $ alleles ${\cal A}_{i_{\ell } }^{( \ell ) } $$( i_{\ell } = 1, \ldots , n_{\ell } ) $ at locus $\ell $. We have already stipulated that $p_{i_{\ell } }^{( \ell ) } $ is the frequency of allele ${\cal A}_{i_{\ell } }^{( \ell ) } $. Put i=(i ₁, …, i_L) and denote the gamete ${\cal A}_{i_{{1}} }^{( {1}) } \cdots {\cal A}_{i_{L} }^{( L) } $ by the multi-index i. In addition, denote the frequency of the ordered multilocus genotype containing gametes i and j as P_ij.

Define the coefficient of departure from random combination,

(20)$$\theta _{ij} = {{P_{ij} } \over {\mathop \prod \nolimits_{k} p_{i_{k} }^{( k) } p_{j_{k} }^{( k) } }}, $$

as the ratio of the (ordered) whole-genome genotype ij to the products of its constituent allele frequencies. The θ_ij are thus measures of both Hardy–Weinberg and linkage disequilibrium; they are all equal to unity if and only if the rules of genotype formation call for the random combination of all genes. Special cases of this coefficient were introduced by Kimura (Reference Kimura1958), although Nagylaki (Reference Nagylaki1992) has pointed out that some of Kimura's expressions employing these coefficients are incorrect. To capture how the experimental gene substitutions defining the average effects change the departures from random combination, let

(21)$$\mathop {\theta _{ij} }\limits^{ \circ } = {{{\rm\Delta} P_{ij} } \over {P_{ij} }} - \mathop \sum \limits_{k} \left( {{{{\rm\Delta} p_{i_{k} }^{( k) } } \over {p_{i_{k} }^{( k) } }} + {{{\rm\Delta} p_{j_{k} }^{( k) } } \over {p_{j_{k} }^{( k) } }}} \right)$$

denote the relative change in θ_ij. In the limit of infinitesimal changes, this is equivalent to the logarithmic derivative of θ_ij.

Now the experimenter must ascertain the mean of each whole-genome genotype by experimental control and then fit the equation

(22)$$\eqalign{ {\mathbb E}[ Y\,\vert \,do( ij) ] = \mu + \alpha _{ij} + \varepsilon _{ij} , \quad {\rm where} \cr \alpha _{ij} = \alpha _{i} + \alpha _{j}, \;\alpha _{i} = \mathop \sum \limits_{k = {1}}^{L} \alpha _{i_{k} }^{( k) } , \cr} $$

to the treatment means thus obtained. The $\alpha _{i_{k} }^{( k) } $ are the average effects of the individual alleles. The residuals ε_ij will reflect both dominance and epistasis, and in the general case it does not seem profitable to separate the two in the manner that Kimura (Reference Kimura1958) attempted. The fitting is accomplished by seeking the vector of average effects, α, that minimizes the sum of squares

(23)$$\mathop \sum \limits_{i, j} P_{ij} \varepsilon _{ij}^{{2}} .$$

Whereas the minimization defines the ε_ij uniquely, the $\alpha _{i_{k} }^{( k) } $ are so far defined only up to a constant term in the sense that one constant may be added to the average effects at one locus and the same constant subtracted from the average effects at another locus without changing the minimum sum of squares (Ewens, Reference Ewens2011). The experimental average effect of a given allele, however, is obviously not defined only up to a constant term but rather must be equal to the precise number determined by the experiment of replacing a random homologous gene with a gene of the given allelic kind. In the Appendix, we show that performing a non-null substitution in this experiment, in a manner preserving the rules of genotype formation, amounts to weighting the possible gene substitutions such that the scalar quantity

(24)$${{{\lt\varepsilon\mathop \theta \limits^{ \circ } }}\gt} = \mathop \sum \limits_{i, j} P_{ij} \varepsilon _{ij} \mathop {\theta _{ij} }\limits^{ \circ } $$

is equal to zero. Another way to phrase this key result is that the vanishing of $\lt{{\varepsilon \hskip-1pt \mathop \theta \limits^{ \hskip3\circ }\hskip-1pt\gt }} $ is a necessary and sufficient condition for the regression and experimental average effects to coincide in the case of genotype–environment independence. Kimura (Reference Kimura1958) showed that constancy of λ suffices for $\lt{{\varepsilon \hskip-1pt \mathop \theta \limits^{ \hskip3\circ }\hskip-1pt\gt }} $ to vanish in the case of a single biallelic locus; it is worth mentioning that even in this simplest possible case there do not generally exist changes in the genotype frequencies such that each individual $\mathop {\theta _{ij} }\limits^{\vskip2pt \circ } $ vanishes.

Our theoretical experimenter can of course perform all $\sum _{k = {1}}^{L} n_{k} $ experiments to determine the unique values of the elements in the vector α. However, given our demonstration that the mean of the experimental average effects at any given locus is equal to zero, it suffices to impose (17) for each locus as a constraint on the minimization of (23). The proof of (17) is still valid for each of multiple loci because the vanishing of $\lt{{\varepsilon \hskip-1pt \mathop \theta \limits^{ \hskip3\circ }\hskip-1pt\gt }} $ along each possible branch of the random experiment implies that the expected change in phenotypic mean must be equal to

(25)$$2\mathop \sum \limits_{i_{k} }^{n_{k} } {\mathbb E}\left( {{\rm \Delta }p_{i_{k} }^{( k) } } \right)\alpha _{i_{k} }^{( k) } , $$

and since the expected outcome of the experiment is a population with the same allele frequencies, (17) is assured.

The vanishing of $\lt{{\varepsilon \hskip-1pt \mathop \theta \limits^{ \hskip3\circ }\hskip-1pt\gt }} $ preserves the population's rules of genotype formation in the following sense. Although the number of parameters required to describe departures from random combination of genes increases very rapidly with the number of alleles and loci, (24) implies it is not necessary for each and every such parameter to stay constant. It is enough, roughly speaking, for the average change in these parameters to equal zero. $\lt{{\varepsilon \hskip-1pt \mathop \theta \limits^{ \hskip3\circ }\hskip-1pt\gt }} $ is similar in form to the weighted average of the relative changes in the departures from random combination, those genotypes with large non-additive residuals being weighted more heavily.

The expression

(26)$$\alpha _{i_{k} }^{( k) } = {1 \over 2}\left( {{\partial \over {\partial p_{i_{k} }^{( k) } }}\mathop \sum \limits_{i, j} P_{ij} {\mathbb E}[ Y\vert do( ij) ] } \right)_{{{\lt\varepsilon\mathop \theta \limits^{ \circ } }}\gt = {0}} $$

may therefore serve as the definition of the experimental average effect in the case of multiple loci.

Let us recapitulate the meaning of (26). Our variable of interest is the population average of the experimentally determined phenotypic means of the genotypes. If genotypes and environments are dependent, this variable is not the same as the population mean ${\mathbb E}( Y) $. Partial differentiation with respect to the frequency of allele ${\cal A}_{i_{k} }^{( k) } $ indicates that we examine how our variable of interest responds to the replacement of a small number of randomly chosen homologous genes with genes of the given allelic kind. The constraint on the partial derivative indicates that we consider only those counterfactual populations that can be reached from the original population by experimental replacements that result in the vanishing of (24). The factor of ${\textstyle{1 \over 2}}$ is owed to diploidy.

It may seem from the form of the constrained derivative that this definition contains an element of circularity, since the ε_ij are defined relative to the average effects in (22). Any such concern should be dispelled by the fact that (26) fully encodes our argument from (22) to (25), which provides an unambiguous sequence of instructions for the theoretical experimenter to follow. The Appendix provides some numerical examples.

8. Average effects and natural selection

At this point the reader may be questioning the need for defining the average effect in terms of causality, as might be revealed by experimentally controlled gene substitutions. Modern texts give only the regression definition (Lynch & Walsh, Reference Lynch and Walsh1998; Bürger, Reference Bürger2000), and those who are accustomed to these accounts may resist the new notation and new way of thinking.

We have already given one strong motivation to adopt the criterion of sensitivity to experimental manipulation: the need to distinguish a causal variant from the non-causal markers in LD with it. Another motivation is that dependence of genotypes and environments is a frequent occurrence. For instance, a major concern in GWAS is ensuring that discovered associations are not attributable to population stratification, which is essentially a form of confounding. A well-known apocryphal example is the ‘chopstick gene’. A geneticist performing a GWAS of chopstick skill in a large sample containing both Europeans and East Asians will undoubtedly find many marker loci failing to satisfy the equality

(27)$$E( Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{1}} ) = E( Y\,\vert \,{\cal A}_{{1}} {\cal A}_{{2}} ) = E( Y\,\vert \,{\cal A}_{{2}} {\cal A}_{{2}} ) $$

even if, unbeknownst to the geneticist, the corresponding equality (3) is obeyed at all loci linked to the statistically significant markers. This is because the Europeans and East Asians differ both in allele frequencies at these loci and in the prevalence of chopstick use; the latter difference presumably has arisen for reasons having nothing to do with genetics. A regression of the observed phenotypic values on gene count will nevertheless lead to a non-zero ‘average effect’ in violation of both Fisher's verbal definition and common sense.

GWAS investigators attempt to control confounding by including all other genotyped markers in the regression. Since the number of genotyped markers typically exceeds the sample size, techniques such as principal components and mixed linear modelling are typically employed (Price et al., Reference Price, Patterson, Plenge, Weinblatt, Shadick and Reich2006; Zhou & Stephens, Reference Zhou and Stephens2012). The reason for the frequent effectiveness of these techniques is that genomic background become an extremely good proxy for the subpopulation to which a given sample member belongs as the number of loci grows large (Edwards, Reference Edwards2003). However, one can construct examples where partialing out other loci fails to deal with confounding (Mathieson & McVean, Reference Mathieson and McVean2012), and in any case a theoretical definition whose usefulness depends on contingent quantities such as genome size and genetic diversity is inherently unattractive.

Perhaps the most conspicuous failure of the regression definition occurs in the very situation that motivated Fisher to define the average effect. This is when the phenotype is fitness itself. In this case, the regression average effect will generically fail to be proportional to the partial change in genetic mean per change in allele frequency even if the genotypic and environmental causes of fitness variation are additive and initially independent.

A simple simulation will bear out this perhaps surprising claim. The simulated organism follows a life cycle consisting of non-overlapping generations. The population size is 20 000. Fitness is determined by a single locus and the environment; the frequency of ${\cal A}_{{2}} $ is initially 1/2, and the population mated at random in the previous generation. The genotypic fitnesses – the values of ${\mathbb E}[ Y\vert {do}( {\cal A}_{{1}} {\cal A}_{{1}} ) ]$, ${\mathbb E}[ Y\vert do( {\cal A}_{{1}} {\cal A}_{{2}} ) ] $, ${\mathbb E}[ Y\vert do( {\cal A}_{{2}} {\cal A}_{{2}} ) ] $ – are 0·4, 0·5, and 0·6 respectively. We determine the phenotypic fitness of each individual in the following way. Immediately after fertilization but before the onset of viability selection, an environmental disturbance of 0·3 in absolute value is added to each individual's genotypic fitness. Positive and negative disturbances are equally probable. This scheme ensures that genotypes and environments are independent at this time.

Whether an individual withstands viability selection to mate with a random fellow survivor is determined by a discrete approximation of an exponential process. We stipulate ten discrete time intervals between fertilization and reproduction, each of which an individual survives with a probability chosen so that the probability of surviving all ten intervals is equal to the individual's phenotypic fitness. By dividing the time between fertilization and mating into more intervals, we could more closely approach a true continuous-time model, where the logarithm of phenotypic fitness would be similar to the Malthusian parameter. Ten intervals, however, suffice to make the point at issue.

Table 2 shows the evolution of this population from fertilization to mating. The first column gives the time interval. The second column gives the regression average effect – the slope in the regression of phenotypic values on ${\cal A}_{{2}} $ gene count among those individuals alive at the beginning of the time interval; β is the conventional notation for a regression coefficient. The third column gives the change in ${\cal A}_{{2}} $ frequency from the beginning of the current time interval to the beginning of the next. The fourth column gives the change in the mean genotypic fitness from the beginning of the current time interval to the beginning of the next. Because the effect of substituting ${\cal A}_{{2}} $ for ${\cal A}_{{1}} $ does not depend on the allelic type of the undisturbed gene, the experimental average effect is of course 0·10. In this case of additive gene action, the genotypic value is the same as the ‘breeding’ or ‘additive genetic’ value, which is now often denoted by the symbol A.

Table 2. Evolutionary change across time intervals in a simulated organism.

Immediately after fertilization, the regression and experimental average effects coincide, as expected from the fact that genetic values and environmental disturbances are initially independent. The change in mean genetic value from fertilization to the beginning of the first time interval is equal to two times the experimental average effect times the change in allele frequency. The relation ΔA=2αΔp in fact holds for each transition from one time interval to the next. The relation ΔA=2βΔp, however, does not hold for any transition besides the first. Note the decline in β, far greater and more systematic than can be explained by sampling fluctuations, with the passage of time.

What explains the increasing discrepancy between α and β? This is an example of what some methodologists call selection bias (e.g. Pearl, Reference Pearl2009). Suppose that intelligence and athletic ability are uncorrelated in the population at large. However, if we limit our observations to the students attending a university that uses both of these attributes as admissions criteria, then we will find that intelligence and athleticism are negatively correlated. If we learn that a student at this university is academically undistinguished, then it becomes more probable that the student is a good athlete. Otherwise the student would likely not have been admitted.

Similarly, if there is some relation between fitness at different points of the lifespan, then with the passage of time the genetic and environmental causes of fitness will tend to become correlated even if they were initially independent. If we learn that a particular survivor of a rigorous selection scheme has an unfit genotype, then it becomes more probable that the organism has benefited from a favourable environment. This same principle explains why selection tends to induce deviations from Hardy–Weinberg and linkage equilibrium (Bulmer, Reference Bulmer1980; Nagylaki, Reference Nagylaki1992; Bürger, Reference Bürger2000; Ewens, Reference Ewens2004); if we find that a survivor has an unfit gene at one genomic position, it becomes more probable that the survivor bears fit genes at other positions. As stated previously, the dependence of genotypes and environments leads to a divergence between the experimental and regression average effects, and the latter then has no straightforward genetic interpretation.

It is important to note that our example does not necessarily impugn the validity of the FTNS, under the regression definition of the average effect, with respect to organisms living in discrete time. This is because in this model the FTNS has come to be interpreted as concerning the change in mean breeding value between generations, and the correctness of the FTNS is preserved when the mean is measured upon fertilization and the regression average effect is measured at the beginning of the parental generation. However, because our model places deaths along a temporal dimension between birth and mating, it should properly be classified as a continuous-time model. The FTNS is intended to apply at every point in continuous time, and therefore our argument for the experimental definition of the average effect retains its full force for organisms following such a life cycle.

Fisher knew that selection bias with respect to the outcome variable prevents regression coefficients from being interpretable. In Statistical Methods for Research Workers, he pointed out that the application of a selection process to the outcome variable will change the regression line (Fisher, Reference Fisher1970, p. 130). It is thus rather curious that Fisher never mentioned this principle in connection with natural selection, a form of selection bias that is always and everywhere operating.

The regression definition is made viable by stipulating the use of ‘true’ or ‘intrinsic’ phenotypic measurements as the outcome variable rather than the actual measurements. This approach, which we adopt in the Appendix, may be natural and inevitable in the case of multiple loci. Because of the need to know the residuals in the multilocus case, it does not seem possible to banish the concept of least-squares linear regression from the theory of average effects. The concepts of regression and causality need to work together. Needless to say, the notion of causality remains an essential partner in this collaboration. A definition calling for the regression of ‘true’ phenotypic measurements on gene content really amounts to replacing the observed phenotypic means of the three genotypes in (7) with the experimental means, which requires the same do operator incorporated in (11) and (15). The instance of do in (26) actually covers two points where we must invoke experimental control: once in the determination of the genotypic means, breeding values, and non-additive residuals, and again in the replacement of randomly chosen homologous genes to resolve the non-uniqueness of the individual average effects. To capture what Fisher intended by the average effect in a formal and transparent way, we cannot easily avoid a special notation for singling out causal relations from merely correlational ones.