# The causal meaning of Fisher’s average effect

James J. Lee1∗
Carson C. Chow1

1Laboratory of Biological Modeling
National Institute of Diabetes and Digestive and Kidney Diseases
National Institutes of Health
Bethesda, MD 20892, USA

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
RESEARCH PAPER

RUNNING HEAD: Causal meaning of average effect

Summary

In order to formulate the Fundamental Theorem of Natural Selection, Fisher
defined the _average excess_ and _average effect_ of a gene substitution.
Finding these notions to be somewhat opaque, some authors have recommended
reformulating Fisher’s ideas in terms of covariance and regression, which are
classical concepts of statistics. We argue that Fisher intended his two
averages to express a distinction between correlation and causation. On this
view the average effect is a specific weighted average of the actual
phenotypic changes that result from physically changing the allelic states of
homologous genes. We show that the statistical and causal conceptions of the
average effect, perceived as inconsistent by Falconer, can be reconciled if
certain relationships between the genotype frequencies and non-additive
residuals are conserved. There are certain theory-internal considerations
favoring Fisher’s original formulation in terms of causality; for example, the
frequency-weighted mean of the average effects equaling zero at each locus
becomes a derivable consequence rather than an arbitrary constraint. More
broadly, Fisher’s distinction between correlation and causation is of critical
importance to gene-trait mapping studies and the foundations of evolutionary
biology.

Keywords: quantitative genetics, causality, confounding, selection bias,
natural selection

## 1\. Introduction

Darwin perceived that hereditary variation in fitness leads to an increase in
adaptive complexity. In an attempt to provide a Mendelian and mathematical
formulation of this profound insight, Fisher expounded the Fundamental Theorem
of Natural Selection (FTNS), which in a modern paraphrase states that the
partial increase in population mean fitness ascribable solely to changes in
allele frequencies by natural selection is equal to the additive genetic
variance in fitness [1, 33, 51, 13, 15, 16, 30, 9, 10, 36, 45]. In the
discrete-time formulation of the FTNS, the additive genetic variance is
proportional to this partial increase, as it must be divided by the mean
fitness.

In his exposition of the FTNS, Fisher took some pains to define the concepts
of _average excess_ and _average effect_. In his own words,

> Let us now consider the manner in which any quantitative individual
> measurement, such as human stature, may depend upon the individual genetic
> constitution. We may imagine, in respect of any pair of alternative
> [alleles], the population divided into two portions, each comprising one
> homozygous type together with half of the heterozygotes, which must be
> divided equally between the two portions. The difference in average stature
> between these two groups may then be termed the _average excess_ (in
> stature) associated with the gene substitution in question…. [21, p. 30,
> emphasis added]

In contrast,

> [b]y whatever rules …the frequency of different gene combinations, may be
> governed, the substitution of a small proportion of the genes of one
> [allelic] kind by the genes of another will produce a definite proportional
> effect upon the average stature. The amount of the difference produced, on
> the average, in the total stature of the population, for each such gene
> substitution, may be termed the _average effect_ of such substitution, in
> contra-distinction to the average excess as defined above…. [21, p. 31,
> emphasis added]
>
> It is natural to conceive [of the average effect] as the actual increase in
> the total of the measurements of a population, when without change in the
> environment, or the mating system, the gene substitution is _experimentally_
> brought about, as it might be by mutation. [24, p. 373, emphasis added]

This paper addresses a puzzle raised by [18] in his brilliant explication of
Fisher’s two genetic averages. Falconer assumed that what Fisher meant by the
quoted definition of the average effect was as follows. We randomly sample a
zygote immediately after fertilization but before the onset of any
developmental events. If the zygote’s genotype contains a gene of a certain
allelic type, say $\mathcal{A}_{1}$, we change it to $\mathcal{A}_{2}$. This
experimental intervention may lead to a value of the focal phenotype at the
time of measurement that differs from what it would have been if the
intervention had not been performed. Falconer reasoned that the expected
magnitude of this difference corresponds to Fisher’s verbal definition of the
average effect.

Falconer then showed that [24]’s (fisher:1941) now widely accepted
mathematical definition of the average effect—the partial regression
coefficient of gene count in the linear regression of the phenotype on all
loci in the genome—does not generally coincide with the definition in terms of
experimental gene substitutions performed at random. Falconer expressed
surprise at the apparent invalidity of the latter definition, given that
“Fisher uses the imaginary replacement of one allele by another as a verbal
description to introduce the idea of average effect, and it seems to have been
seen by him as the basis for the concept” (p. 334).

Falconer correctly perceived the importance of experimental intervention to
Fisher’s conception of the average effect. Indeed, Fisher did not even bother
to spell out his regression definition in the first edition of _The Genetical
Theory of Natural Selection_. Furthermore, to any reader familiar with
Fisher’s work on experimental design and his controversial stance on the
tobacco-cancer connection, the quotations given above must bring into mind his
repeated admonition that an observed _excess_ in the average measurement of
one group over another can always be interpreted as the causal _effect_ of the
factor distinguishing the groups under the following circumstance: the
allotment of members to groups has been randomized in a controlled experiment
[23, 27]. This preoccupation with causation is one of the stark contrasts
between Fisher and his nemesis Karl Pearson; contrary to the intellectual
fashion of the Edwardian era, Fisher did not regard causality as a meaningless
concept. In the inaugural issue of the journal _Philosophy of Science_ , the
word _cause_ and its derivatives appear in [22] no fewer than seventy times.
Over much resistance by seasoned experimenters [3], Fisher advocated
randomization in experimental design for the precise purpose of distinguishing
causation from spurious correlations brought about by confounding variables.
There is thus compelling reason to believe that the notion of experimental
control revealing causation is critical to the proper interpretation of the
average effect.

We argue that a more nuanced reading of Fisher’s writings can bring his
experimental and regression definitions of the average effect into full
agreement in certain special cases. We then provide reasons to favor the
experimental definition in more general situations. A striking disadvantage of
the regression definition is that its use invalidates the FTNS if some of the
variance in fitness has environmental causes.

For simplicity our main text mostly follows [18] in treating the case of a
single locus with two alleles. We provide the generalization to multiple
alleles and loci in two of the later sections. Some interesting new concepts
do arise in this generalization, but the central ideas can be conveyed without
multilocus notation, which seems inevitably to be either cumbersome or opaque.

## 2\. A Notation for Causal Notions

A formal symbolic language to distinguish causal relations from merely
correlational ones, such as the counterfactual notation of [42] and [52], 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 [46]. We are to interpret an
expression such as $\mathbb{E}[Y\,|\,do(x)]$ 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

$\mathbb{P}(mud\,|\,rain)\geq\mathbb{P}(mud)\quad\textrm{and}\quad\mathbb{P}(rain\,|\,mud)\geq\mathbb{P}(rain)$
(1)

and

$\mathbb{P}[mud\,|\,do(rain)]\geq\mathbb{P}(mud)\quad\textrm{and}\quad\mathbb{P}[rain\,|\,do(mud)]=\mathbb{P}(rain).$
(2)

(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
[57]. Letting $Y$ denote the phenotype of interest, we can say that a genetic
variant is a causal variant if the equality

$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]=\mathbb{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2})]$
(3)

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 [47] 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
[58]’s (wright:1968) 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
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$ 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
$\mathcal{A}_{1}\mathcal{A}_{1}$ 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” [20, 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 $\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})$ are
precisely equal to $\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$ at
the single causal locus. Until we say otherwise, we assume the stochastic
independence of genotypes and environments.

Following the notation of [19], we let $P$, $2Q$, and $R$ denote the
respective frequencies of the genotypes $\mathcal{A}_{1}\mathcal{A}_{1}$,
$\mathcal{A}_{1}\mathcal{A}_{2}$, and $\mathcal{A}_{2}\mathcal{A}_{2}$. Given
that a zygote’s genotype is $\mathcal{A}_{1}\mathcal{A}_{1}$, we write the
expected phenotypic effect of changing a gene’s allelic type from
$\mathcal{A}_{1}$ to $\mathcal{A}_{2}$ as

$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2}),\mathcal{A}_{1}\mathcal{A}_{1}]-\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}).$
(4)

There is no contradiction in conditioning on both the observation of
$\mathcal{A}_{1}\mathcal{A}_{1}$ and the experimental setting of the genotype
to $\mathcal{A}_{1}\mathcal{A}_{2}$. 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 $\mathcal{A}_{1}\mathcal{A}_{1}$. Similarly, we define

$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2}=\mathbb{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2}),\mathcal{A}_{1}\mathcal{A}_{2}]-\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}).$
(5)

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 $\mathcal{A}_{2}$ for $\mathcal{A}_{1}$ 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
$\mathcal{A}_{2}$, we leave it alone. If the chosen gene is of type
$\mathcal{A}_{1}$, we change it to $\mathcal{A}_{2}$. The expected phenotypic
effect of the gene substitutions performed under this scheme is thus

$\frac{P(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2})+Q(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2})}{P+Q}.$
(6)

Falconer pointed out that (6) does not agree with the regression definition of
the average effect that [24] gave in an article criticizing Sewall Wright for
conflating the average excess and average effect. This article 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

$P[\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})-\nu+\alpha]^{2}+2Q[\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2})-\nu]^{2}+R[\mathbb{E}(Y\,|\,\mathcal{A}_{2}\mathcal{A}_{2})-\nu-\alpha]^{2},$
(7)

where $\nu$ 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

$P[\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})-\mu-2\alpha_{1}]^{2}+2Q[\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2})-\mu-\alpha_{1}-\alpha_{2}]^{2}\\\
+R[\mathbb{E}(Y\,|\,\mathcal{A}_{2}\mathcal{A}_{2})-\mu-2\alpha_{2}]^{2}\quad\textrm{where
$\mu=\mathbb{E}(Y)$}.$ (8)

In the definition (7), then, the average effect $\alpha$ is the slope in the
regression of the phenotype on gene count. $\alpha_{1}$ and $\alpha_{2}$ in
(8) are the average effects of the two alleles individually—a notion to which
we will return. For now we simply note that $\alpha$ will turn out to equal
$\alpha_{2}-\alpha_{1}$ in magnitude. There is some ambiguity in the
literature over whether the outcome variable in the regression should be
defined, as in (8), with the subtraction of the unconditional phenotypic mean
[28, 51, 16]. 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 $\alpha=\alpha_{2}-\alpha_{1}$. In our later discussion of
individual average effects, we will give a compelling reason to favor the mean
subtraction.

Perhaps frustrated by Fisher’s concise style, Falconer concluded his article
by approvingly quoting [51]’s (price:1972) 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” [24, 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 $F$ [7]. 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 $\lambda=Q^{2}/(PR)$, the ratio of the squared (ordered)
heterozygote frequency to the product of the homozygote frequencies. $\lambda$
can be written in the symmetrical form

$\frac{\mathbb{P}(\mathcal{A}_{2}\,|\,\mathcal{A}_{1})}{\mathbb{P}(\mathcal{A}_{1}\,|\,\mathcal{A}_{1})}\cdot\frac{\mathbb{P}(\mathcal{A}_{1}\,|\,\mathcal{A}_{2})}{\mathbb{P}(\mathcal{A}_{2}\,|\,\mathcal{A}_{2})},$

and it attains the constant value of unity if the population mates randomly, a
fact first noted by [31].

Let $p=Q+R$ denote the frequency of $\mathcal{A}_{2}$, and write the
population mean of $Y$ as a function of allele frequency and the rules of
combination, $\mu(p,\lambda)$. We now show that the expression
$\mu(p+dp,\lambda)-\mu(p,\lambda)$ is proportional to the average effect,
$\alpha$, obtained from regression equation (7). In other words the ratio
$\lambda$ 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

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

The average effect is then proportional to the change of $\mu$ with respect to
$p$ while holding $\lambda$ constant. We can increase $p$ by carrying out
either the intervention $\mathcal{A}_{1}\mathcal{A}_{1}$ $\rightarrow$
$\mathcal{A}_{1}\mathcal{A}_{2}$ or $\mathcal{A}_{1}\mathcal{A}_{2}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{2}$. As detailed in the Appendix,
upon noting that the differential of $Q^{2}=\lambda PR$ for constant $\lambda$
yields the differential equation

$\frac{dP}{P}+\frac{dR}{R}=\frac{2dQ}{Q},$ (10)

we find that Fisher’s average effect is

$\alpha=\frac{c_{1}(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2})+c_{2}(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2})}{c_{1}+c_{2}},$
(11)

where the weights are

$\displaystyle c_{1}$ $\displaystyle=P(Q+R),$ $\displaystyle c_{2}$
$\displaystyle=R(P+Q).$

Let us recapitulate the meaning of (11). Immediately after fertilization we
take a random sample of the zygotes bearing the genotype
$\mathcal{A}_{1}\mathcal{A}_{1}$. We then randomly assign some of these
zygotes to the “treatment,” which consists of changing the allelic type of a
gene from $\mathcal{A}_{1}$ to $\mathcal{A}_{2}$. 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 $\mathcal{A}_{1}\mathcal{A}_{2}$ to
$\mathcal{A}_{2}\mathcal{A}_{2}$. The weighted average of the two causal
effects—where the weights $c_{1}$ and $c_{2}$ are chosen so as to preserve
$\lambda$ 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 [10]. Suppose that the effect
of changing an $\mathcal{A}_{1}\mathcal{A}_{1}$ individual to
$\mathcal{A}_{1}\mathcal{A}_{2}$ is 3 phenotypic units, whereas the effect of
changing $\mathcal{A}_{1}\mathcal{A}_{2}$ to $\mathcal{A}_{2}\mathcal{A}_{2}$
is $-2$. Suppose also that the numbers of the genotypes
$\mathcal{A}_{1}\mathcal{A}_{1}$, $\mathcal{A}_{1}\mathcal{A}_{2}$, and
$\mathcal{A}_{2}\mathcal{A}_{2}$ 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
$\lambda$ are affected by each step in a sequence of gene substitutions
leading to an increase in $p$ but tending to keep $\lambda$ 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 $\lambda$ after the
gene substitution.

Table 1: _Sequence of experimental gene substitutions yielding the average effect._ experimental change | genotype numbers | $\frac{\Delta(\mu N)}{\textrm{number of changes}}$ | $\lambda$
---|---|---|---
— | 40, 40, 20 | — | 1/2
$\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$ | 39, 41, 20 | 3 | .5387821
$\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2}$ | 39, 40, 21 | 1/2 | .4884005
$\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$ | 38, 41, 21 | 4/3 | .5266291
$\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2}$ | 38, 40, 22 | 1/2 | .4784689
$\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$ | 37, 41, 22 | 1 | .5162776
$\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2}$ | 37, 40, 23 | 1/2 | .4700353
$\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$ | 36, 41, 23 | 6/7 | .5075483

It is readily confirmed that the final value of $\lambda$ is the closest to
the starting value of $1/2$ that can be achieved with 7 gene substitutions. If
we take population size to infinity, we can make the discrepancy between the
original and new values of $\lambda$ as small as we please.

In the special case of genotype-environment independence considered so far,
where equalities such as
$\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$
always hold, Fisher’s experimental and regression definitions of the average
effect coincide for constant $\lambda$. In the example above, 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 $\mathcal{A}_{2}$ 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. But 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_ $\rightarrow$ _lactose tolerance_ $\rightarrow$ _dairy consumption_
$\rightarrow$ _stature_ ultimately begins with a genetic variable. This
subtlety may have been among the reasons why Fisher favored “speaking of the
residue as non-genetic, rather than environmental …” [2, 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 [19, 24, 26, 29], and it has been asserted that Fisher’s
biometrical theory is meaningless if these conditions are not met [56, _e.g._
,].

As [51] 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” [21, 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. [2, 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,

$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{i}=\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{j}$
(12)

for any choice of environments $\mathcal{E}_{i}$ and $\mathcal{E}_{j}$. 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 favorable 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
$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$
is not necessarily equal to $\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{1}\mathcal{A}_{1}$,
since the genotypes $\mathcal{A}_{1}\mathcal{A}_{1}$ and
$\mathcal{A}_{1}\mathcal{A}_{2}$ may tend to be found in different
environments. This difficulty can be overcome by redefining expressions such
as $\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$
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

$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]-\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})].$

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 $\lambda$ constant as a means to
address gene-gene interaction. In order to formulate the FTNS, Fisher wished
to quantify the causal effect of changing allele frequency while holding the
environment constant. In his view the way in which alleles combine to form
genotypes, as parameterized by $\lambda$, 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
$\lambda$, it becomes reasonable when we realize that $\lambda$ 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 [29, pp. 283–284]. The
experimenter may wish to know whether the fertilizer affects sugar purity
through a direct causal path, _fertilizer_ $\rightarrow$ _sugar purity_ ,
distinct from the indirect path _fertilizer_ $\rightarrow$ _root weight_
$\rightarrow$ _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 $\lambda$,
we can still mathematically define an average effect stipulating that
$\lambda$ remains clamped to a constant value. This point of view is similar
to one expressed by [45].

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
$\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$
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

$\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})=\sum_{i}\mathbb{P}(\mathcal{E}_{i}\,|\,\mathcal{A}_{1}\mathcal{A}_{1})\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1},\mathcal{E}_{i})$
(13)

for each genotype. Because genotypes and environments exhaust all possible
causes of phenotypic variation,
$\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1},\mathcal{E}_{i})$ is
equivalent to
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1}),do(\mathcal{E}_{i})]$. 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
$\mathbb{P}(\mathcal{E}_{i}\,|\,\mathcal{A}_{1}\mathcal{A}_{1})$. A candidate
definition for the average effect is then

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

The problem with this candidate definition, however, is that it can lead to a
nonzero 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

$\displaystyle\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$
$\displaystyle=\sum_{i}\mathbb{P}[\mathcal{E}_{i}\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]\,\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1}),\mathcal{E}_{i}]$
$\displaystyle=\sum_{i}\mathbb{P}(\mathcal{E}_{i})\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1},\mathcal{E}_{i}).$
(14)

The left-hand side is the expected phenotypic value upon sampling a zygote at
random and, if its genotype is not $\mathcal{A}_{1}\mathcal{A}_{1}$, making it
so. Since changing the genotype of a zygote cannot affect its environment, we
have
$\mathbb{P}[\mathcal{E}_{i}\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]=\mathbb{P}(\mathcal{E}_{i})$
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 nonzero average effect is indeed an average of genetic effects, at
least one of which would turn out to be nonzero under experimental control.

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

$\alpha=\frac{\sum_{i}c_{1,i}(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{i})+c_{2,i}(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2},\mathcal{E}_{i})}{\sum_{i}c_{1,i}+c_{2,i}},$
(15)

where

$\displaystyle c_{1,i}=c_{1}\mathbb{P}(\mathcal{E}_{i}),$ $\displaystyle
c_{2,i}=c_{2}\mathbb{P}(\mathcal{E}_{i}).$

## 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
${n\choose 2}$ 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” [28, p. 35]. This definition can be explicated with respect to a
given allele, say $\mathcal{A}_{1}$, 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 $\mathcal{A}_{1}$ in such a way that
the probabilities of selection are equal to the allele frequencies. That is,
if the vector of allele frequencies is $(p_{1},\ldots,p_{n})$, then the gene
to be changed is $\mathcal{A}_{1}$ with probability $p_{1}$, $\mathcal{A}_{2}$
with probability $p_{2}$, and so on. If the gene to be changed happens to be
$\mathcal{A}_{1}$ itself, then the $\mathcal{A}_{1}$ $\rightarrow$
$\mathcal{A}_{1}$ 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 $\alpha_{1}$, the average effect of $\mathcal{A}_{1}$.

From this definition we can derive some important consequences. Let $N_{k}$
stand for the number of $\mathcal{A}_{k}$ genes in the population. The total
number of genes is $\sum_{k=1}^{n}N_{k}=N$. 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

$\sum_{k=1}^{n}\frac{N_{k}}{N}\left\\{\sum_{\ell=1}^{n}\frac{N_{\ell}}{N}\left[\left(\frac{N_{1}}{N},\ldots,\frac{N_{n}}{N}\right)+\frac{1}{N}\mathbf{e}_{k}-\frac{1}{N}\mathbf{e}_{\ell}\right]\right\\},$
(16)

where $\mathbf{e}_{k}$ 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 $N_{1}\left(\sum
N_{k}\right)^{2}/N^{3}=p_{1}$, the second is $N_{2}\left(\sum
N_{k}\right)^{2}/N^{3}=p_{2}$, 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

$\sum_{k=1}^{n}p_{k}\alpha_{k}=0.$ (17)

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)
[16], 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 $\mathcal{A}_{2}$ genes in this
population of 200 total genes. If we perform the experiment defining
$\alpha_{1}$, then with probability .40 the population gene numbers remain at
$(80,120)$ and with probability .60 the numbers become $(81,119)$. In the
event of a non-null substitution, with probability $4/7$ (given by
$\frac{c_{1}}{c_{1}+c_{2}}$) the change is $\mathcal{A}_{1}\mathcal{A}_{2}$
$\rightarrow$ $\mathcal{A}_{1}\mathcal{A}_{1}$ and with probability $3/7$
(given by $\frac{c_{2}}{c_{1}+c_{2}}$) it is $\mathcal{A}_{2}\mathcal{A}_{2}$
$\rightarrow$ $\mathcal{A}_{1}\mathcal{A}_{2}$. 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 $\lambda$. Using simple
probability calculus, we can calculate that the numerical value of
$\alpha_{1}$ is $-18/35$.

In summary, the experiment defining $\alpha_{1}$ will lead to the null
substitution $\mathcal{A}_{1}$ $\rightarrow$ $\mathcal{A}_{1}$ with
probability $p_{1}$ (in which case the causal effect is zero) and to the
substitution $\mathcal{A}_{2}$ $\rightarrow$ $\mathcal{A}_{1}$ 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 $\alpha_{1}$ must be equal to $(p_{1})(0)+(p_{2})(-\alpha)$, and
from this we can use $p_{1}\alpha_{1}+p_{2}\alpha_{2}=0$ to derive
$\alpha=\alpha_{2}-\alpha_{1}$ algebraically. The meaning of this relation
among the three average effects is as follows. The expected outcome of the
experiment defining $\alpha_{2}$ is a population with gene numbers
$(79.6,120.4)$ and nearly the same value of $\lambda$. Now suppose that we
perform the “opposite” of the experiment defining $\alpha_{1}$, on average
reducing the number of $\mathcal{A}_{1}$ genes rather than increasing them. We
compose this experiment with the one defining $\mathcal{A}_{2}$, which in our
example has a numerical value of $12/35$. The population is thus expected to
proceed through the sequence $(80,120)$ $\rightarrow$ $(79.4,120.6)$
$\rightarrow$ $(79,121)$, preserving $\lambda$ at each step. The final state
is precisely the one expected upon performing the experiment defining
$\alpha$, 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

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

where $\mu$ 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. (18) fills the lacuna in Wright’s
casual use of the expression

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

to which [24] strongly objected. The explicit dependence of $\mu$ on
$\lambda$, 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 $\lambda$, 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 $k$, $\mathcal{A}^{(k)}_{i_{k}}$, as

$\alpha^{(k)}_{i_{k}}=\frac{1}{2}\frac{\partial\mu(\mathbf{p},\bm{\lambda})}{\partial
p^{(k)}_{i_{k}}},$ (19)

where $\mathbf{p}$ is now a vector of allele frequencies at several loci,
$p^{(k)}_{i_{k}}$ being the element corresponding to
$\mathcal{A}^{(k)}_{i_{k}}$, and $\bm{\lambda}$ 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 $\mathcal{A}^{(\ell)}_{i_{\ell}}$
$(i_{\ell}=1,\ldots,n_{\ell})$ at locus $\ell$. We have already stipulated
that $p^{(\ell)}_{i_{\ell}}$ is the frequency of allele
$\mathcal{A}^{(\ell)}_{i_{\ell}}$. Put $i=(i_{1},\ldots,i_{L})$ and denote the
gamete $\mathcal{A}^{(1)}_{i_{1}}\cdots\mathcal{A}^{(L)}_{i_{L}}$ 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_ ,

$\theta_{ij}=\frac{P_{ij}}{\prod_{k}p^{(k)}_{i_{k}}p^{(k)}_{j_{k}}},$ (20)

as the ratio of the (ordered) whole-genome genotype $ij$ to the products of
its constituent allele frequencies. The $\theta_{ij}$ 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 [33], although
[41] 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

$\mathring{\theta}_{ij}=\frac{\Delta
P_{ij}}{P_{ij}}-\sum_{k}\left(\frac{\Delta
p^{(k)}_{i_{k}}}{p^{(k)}_{i_{k}}}+\frac{\Delta
p^{(k)}_{j_{k}}}{p^{(k)}_{j_{k}}}\right)$ (21)

denote the relative change in $\theta_{ij}$. In the limit of infinitesimal
changes, this is equivalent to the logarithmic derivative of $\theta_{ij}$.

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

$\mathbb{E}[Y\,|\,do(ij)]=\mu+\alpha_{ij}+\varepsilon_{ij},\quad\textrm{where
$\alpha_{ij}=\alpha_{i}+\alpha_{j}$,
$\alpha_{i}=\sum_{k=1}^{L}\alpha^{(k)}_{i_{k}}$},$ (22)

to the treatment means thus obtained. The $\alpha^{(k)}_{i_{k}}$ are the
average effects of the individual alleles. The residuals $\varepsilon_{ij}$
will reflect both dominance and epistasis, and in the general case it does not
seem profitable to separate the two in the manner that [33] attempted. The
fitting is accomplished by seeking the vector of average effects,
$\bm{\alpha}$, that minimizes the sum of squares

$\sum_{i,j}P_{ij}\varepsilon_{ij}^{2}.$ (23)

Whereas the minimization defines the $\varepsilon_{ij}$ uniquely, the
$\alpha^{(k)}_{i_{k}}$ 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 [16]. 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

$\overline{\varepsilon\,\mathring{\theta}}=\sum_{i,j}P_{ij}\varepsilon_{ij}\mathring{\theta}_{ij}$
(24)

is equal to zero. Another way to phrase this key result is that the vanishing
of $\overline{\varepsilon\,\mathring{\theta}}$ is a necessary and sufficient
condition for the regression and experimental average effects to coincide in
the case of genotype-environment independence. [33] showed that constancy of
$\lambda$ suffices for $\overline{\varepsilon\,\mathring{\theta}}$ 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 $\mathring{\theta}_{ij}$
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
$\bm{\alpha}$. 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 $\overline{\varepsilon\,\mathring{\theta}}$ along each possible branch of
the random experiment implies that the expected change in phenotypic mean must
be equal to

$2\sum_{i_{k}}^{n_{k}}\mathbb{E}\left(\Delta
p^{(k)}_{i_{k}}\right)\alpha^{(k)}_{i_{k}},$ (25)

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

The vanishing of $\overline{\varepsilon\,\mathring{\theta}}$ preserves the
population’s rules of genotype formation in the following sense. Although the
number of parameters required to describe departure 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. $\overline{\varepsilon\,\mathring{\theta}}$ 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

$\alpha^{(k)}_{i_{k}}=\frac{1}{2}\left(\frac{\partial}{\partial
p^{(k)}_{i_{k}}}\sum_{i,j}P_{ij}\mathbb{E}[Y\,|\,do(ij)]\right)_{\overline{\varepsilon\,\mathring{\theta}}=0}$
(26)

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 $\mathcal{A}^{(k)}_{i_{k}}$ 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 $\frac{1}{2}$ is owed to diploidy.

It may seem from the form of the constrained derivative that this definition
contains an element of circularity, since the $\varepsilon_{ij}$ 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 [38, 5], 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

$E(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})=E(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2})=E(Y\,|\,\mathcal{A}_{2}\mathcal{A}_{2})$
(27)

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 nonzero “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 modeling are typically employed [49, 59]. 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 [11]. However, one can construct
examples where partialing out other loci fails to deal with confounding [39],
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 $\mathcal{A}_{2}$ is initially 1/2, and the
population mated at random in the previous generation. The genotypic
fitnesses—the values of
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$,
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]$,
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2})]$—are .4, .5, and .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 .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: _Evolutionary change across time intervals in a simulated organism._ | $\beta$ | $\Delta p$ | $\Delta A$
---|---|---|---
fertilization | .100 | $9.33\times 10^{-3}$ | $1.87\times 10^{-3}$
time 1 | .091 | $8.07\times 10^{-3}$ | $1.61\times 10^{-3}$
time 2 | .084 | $5.86\times 10^{-3}$ | $1.17\times 10^{-3}$
time 3 | .079 | $6.11\times 10^{-3}$ | $1.22\times 10^{-3}$
time 4 | .073 | $5.08\times 10^{-3}$ | $1.02\times 10^{-3}$
time 5 | .069 | $4.72\times 10^{-3}$ | $9.45\times 10^{-4}$
time 6 | .065 | $4.10\times 10^{-3}$ | $8.20\times 10^{-4}$
time 7 | .062 | $3.47\times 10^{-3}$ | $6.95\times 10^{-4}$
time 8 | .060 | $3.00\times 10^{-3}$ | $5.91\times 10^{-4}$
time 9 | .059 | $1.28\times 10^{-3}$ | $2.57\times 10^{-4}$
time 10 | .060 | — | —

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
$\mathcal{A}_{2}$ gene count among those individuals alive at the beginning of
the time interval; $\beta$ is the conventional notation for a regression
coefficient. The third column gives the change in $\mathcal{A}_{2}$ 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 $\mathcal{A}_{2}$ for $\mathcal{A}_{1}$ does not
depend on the allelic type of the undisturbed gene, the experimental average
effect is of course .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 $A$.

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 $\Delta A=2\alpha\Delta p$ in fact holds for
each transition from one time interval to the next. The relation $\Delta
A=2\beta\Delta p$, however, does not hold for any transition besides the
first. Note the decline in $\beta$, far greater and more systematic than can
be explained by sampling fluctuations, with the passage of time.

What explains the increasing discrepancy between $\alpha$ and $\beta$? This is
an example of what some methodologists call _selection bias_ [47, _e.g._ ,].
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 favorable environment. This same principle explains why
selection tends to induce deviations from Hardy-Weinberg and linkage
equilibrium [4, 41, 5, 15]; 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 environment 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 [29, 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.

## 9\. Discussion

[18] had the good sense to intuit that sensitivity to physical change was
important to Fisher’s conception of the average effect. Indeed, among all
twentieth-century scientists, Fisher might have been the one most likely to
incorporate the distinction between an observed excess and a causal effect
into a formal theory. The discrepancy that Falconer thought he had uncovered
between Fisher’s regression and experimental definitions of the average effect
can be reconciled, in the case of genotype-environment independence, by using
a specific weighted average of the two possible gene substitutions rather than
a naive average. If the phenotype is affected by one biallelic locus, the
weights are chosen so that a population subject to gene substitutions in
numbers proportional to the weights retains the same value of
$\lambda=Q^{2}/(PR)$, a parameter describing the way in which alleles are
combined into genotypes. If genotypes and environments interact non-
additively, then the gene substitutions must also be balanced with respect to
the marginal distribution of environmental states. This balancing has the
desirable property of preserving the experimentally ascertained phenotypic
means of the genotypes. In the case of multiple loci, there is no longer a
fixed parameterization of genotype formation to which the weightings of the
gene substitutions must conform, but in a loose sense the changes in the
departures from random combination must average out to zero. These
restrictions are requirements for a change in allele frequency “without change
in the environment, or in the mating system [rules of genotype combination].”
When genotypes and environments are dependent—which must always be the case,
even if only slightly, as a result of natural selection—the experimental
definition is to be preferred.

[24] gave one reason why a definition based on experimental gene substitutions
may be inferior to one based on passive observations of a static population
(although later in this paper he reverted to the language of gene
substitutions). He pointed out that changes in the frequencies of the
different genotypes may feed back to change the phenotypic means themselves.
He gave the example of experimental gene substitutions increasing milk yield,
which lead to females in the next generation who can leverage their superior
nourishment to provide even more milk to their own offspring. Fisher wished to
discount such knock-on effects—presumably because they are too complex to form
general rules about them. These knock-on effects can be positive or negative.
When fitnesses are frequency-dependent, the knock-on effects of naturally
selected changes in allele frequencies can steadily decrease the mean fitness
of the population [44]. The approach of a female-skewed sex ratio to a stable
fifty-fifty equilibrium in a polygynous species can be an example of precisely
this phenomenon ([pp. 141-143]fisher:1930:gtns[p. 232]bennett:1983). Therefore
Fisher consigned changes in the genotype-phenotype mapping—the
$\mathbb{E}[Y\,|\,do(ij)]$—brought about by gene substitutions with all other
possible such changes, including those brought about by unpredictable changes
in climate, predators, parasites, and so on. Our preferred resolution of the
dilemma raised by the cascade of additional phenotypic changes that may be
initiated by a physical gene substitution is to stipulate the constancy of (4)
and (5), for instance, in the experimental definition of the average effect.
That is, the average effect is calculated on the assumption that the
prevailing genotype-phenotype mapping will not itself change as a result of
the gene substitutions. This is equivalent to the _stable unit treatment value
assumption_ (SUTVA) in the Neyman-Rubin counterfactual framework.

SUTVA may often have a reasonable interpretation. For example, in the cases of
fecundity selection and frequency-dependent fitnesses of game-theoretic
strategies, we may interpret each causal effect as the expected phenotypic
change upon placing a manipulated organism in a virtual environment containing
the same mixture of types constituting the undisturbed population. In any
event finding an interpretation of SUTVA may not be important in most
biological situations, so long as any frequency-dependent changes ensuing from
the experimental manipulation of a few individuals can be neglected in a
theoretically infinite population.

It is the constancy of the $\mathbb{E}[Y\,|\,do(ij)]$ rather than the
constancy of the corresponding observed phenotypic means that is satisfied by
the gene substitutions defining the average effect in the case of genotype-
environment dependence and interaction. This striking fact further affirms the
priority of causal quantities over observables that may have no causal
interpretation.

A renewed understanding of the average effect is especially timely given the
enablement of GWAS by modern technology and the upsurge of research into the
inheritance of fitness in human populations [54]. The findings of the [12]
indicate that the fine-mapping of the variants with nonzero experimental
average effects responsible for a given association signal may turn out to be
less onerous than was once supposed. However, care is needed as researchers
isolate variants with ever smaller average effects, which will be difficult to
distinguish from spurious signals generated by subtle confounding or selection
bias.

An appealing feature of GWAS is the availability of a complementary study
design, pioneered by [53], that offers nearly the entirety of the benefits
inhering in experimental control. According to Mendel’s laws, a parent passes
on a randomly chosen gene from each of its homologous pairs to a given
offspring. Given the applicability of Mendel’s laws, we can then treat the
genotype of an offspring given the parental genotypes much like a treatment in
a randomized experiment. It follows that a significant association between
transmission of a particular allele and the focal phenotype cannot be the
result of confounding; in the absence of selection bias, the only feasible
explanation is linkage with a locus where the average effect is nonzero.
Fisher himself noted this feature of family-based studies:

> Genetics is indeed in a peculiarly favoured condition in that Providence has
> shielded the geneticist from many of the difficulties of a reliably
> controlled comparison. The different genotypes possible from the same mating
> have been beautifully randomized by the meiotic process. A more perfect
> control of conditions is scarcely possible, than that of different genotypes
> appearing in the same litter. [25, p. 7]

Family-based studies have successfully been used to replicate findings from
studies of nominally unrelated individuals [35, 55], and this is another way
in which the thought experiments defining the average effect are becoming less
like _Gedanken_ and more like routine empirical operations. We note that when
[53] introduced their family-based test, their null hypothesis was no linkage
with a causal locus despite the presence of population association. This test
and its variants have since often been used to test the null hypothesis that
there is neither linkage nor association. We anticipate that there will be a
trend back toward the original form of the test. Because parent-offspring
trios and sets of siblings can be difficult to recruit and require more
genotyping, investigators find it convenient to test for population
association in large samples of unrelated individuals. Those markers showing
evidence of association can then be interrogated, however, for linkage with
loci where there are nonzero average effects. The follow-up cohorts of
families will typically be much smaller and less likely to yield genome-wide
significant $p$-values, but it will be reasonable to require less stringent
evidence or merely overall sign agreement greatly exceeding 50 percent. This
procedure can provide a check on whether the association stage is producing an
acceptably low rate of false positives with respect to the causal hypothesis
of a nonzero average effect—which, of course, is not strictly the same as the
statistical hypothesis of a nonzero partial regression coefficient.

We note that family-based studies are not immune to selection bias intervening
between fertilization and the time of measurement, which may rise to an
appreciable level in studies of phenotypes strongly affecting fitness. This
may be a challenge for gene-trait mapping studies conducted in the near
future.

It may be tempting to define the average effect in terms of a hypothetical
family-based study. However, whereas rejecting the null hypothesis of a zero
average effect requires only the assumptions of Mendel’s laws, effect
estimation requires additional assumptions and thus does not seem particularly
suited for a theoretical definition after all [17].

Finally, we comment on the role of the average effect in the FTNS. We write
the breeding (additive genetic) value of a given individual as

$A=\sum_{\ell=1}^{L}\sum_{i_{\ell}=1}^{n_{\ell}}\chi\left(\mathcal{A}^{(\ell)}_{i_{\ell}}\right)\alpha^{(\ell)}_{i_{\ell}},$
(28)

where $\chi(\cdot)$ is a function giving the number of
$\mathcal{A}^{(\ell)}_{i_{\ell}}$ genes (0, 1, or 2) present in the
individual’s genotype. The variance in breeding values, $\textrm{Var}(A)$, is
now called the _additive genetic variance_ , and the ratio
$\textrm{Var}(A)/\textrm{Var}(Y)$ the _heritability in the narrow sense_. It
is important to keep in mind that these breeding values are linear functions
of _experimental_ average effects; we are building up a predicted value for a
given individual from the causal effects of the genes present in the genotype.

The FTNS states that the partial change in mean fitness attributable to
changes in allele frequencies caused by natural selection is proportional to
the additive genetic variance in fitness, which can be shown to equal

$2\sum_{\ell=1}^{L}\sum_{i_{\ell}=1}^{n_{\ell}}p^{(\ell)}_{i_{\ell}}a^{(\ell)}_{i_{\ell}}\alpha^{(\ell)}_{i_{\ell}},$
(29)

where the meaning of $a^{(\ell)}_{i_{\ell}}$ is as follows. If genotypes and
environments are independent, then this quantity is the average excess of
$\mathcal{A}^{(\ell)}_{i_{\ell}}$, which is usually defined as the difference
in mean fitness between the bearers of the given allele and the entire
population. (29) is invariably derived under the assumption that genotypes and
environments are independent. Because under our definitions the values of the
experimental average effects do not depend on the extent of genotype-
environment dependence, it follows that the breeding values and hence the
additive genetic variance are also insensitive to genotype-environment
dependence. The equality of (29) with $\textrm{Var}(A)$ is thus fully valid in
our account—given the following modification regarding
$a^{(\ell)}_{i_{\ell}}$.

If genotypes and environments are not independent, $a^{(\ell)}_{i_{\ell}}$ in
(29) is not exactly the same as the average excess defined by [28, p. 35 ]. It
is rather the average excess that _would_ be observed if genotypes were
distributed randomly among environments. In other words each
$a^{(\ell)}_{i_{\ell}}$ only reflects confounding with other genetic loci and
not with environmental causes. To repeat, this is a consequence of the fact
that our experimental average effects—and hence all quantities derived from
them, including the additive genetic variance—are sensitive only to the
marginal distribution of environmental states. Every factor in (29), including
the $a^{(\ell)}_{i_{\ell}}$, must therefore be equal to whatever they would be
under genotype-environment independence, the standard setting in which (29) is
calculated. If the “full” average excesses were substituted into (29), then
the expression would no longer be interpretable as a variance; it could then
possibly be negative.

It is well known that the change in the frequency of
$\mathcal{A}^{(\ell)}_{i_{\ell}}$ is proportional to the product of
$p^{(\ell)}_{i_{\ell}}$ and the actual difference in mean fitness between the
bearers of the given allele and the entire population [50, _e.g._ ,]. From the
fact that the difference is not necessarily equal to our
$a^{(\ell)}_{i_{\ell}}$, we learn that there is partition of the total change
in allele frequency between the change caused by natural selection and the
change attributable to how genotypes are distributed across environments
varying in severity. This partition is in the same spirit of Fisher’s
conditions discussed previously. Like changes in the rules of genotype
formation or the $\mathbb{E}[Y\,|\,do(ij)]$, deviations from genotype-
environment independence cannot generally lead to an increase in fitness, and
indeed the example set out in Table 2 demonstrates that the dependence induced
by natural selection itself tends to retard the frequency increase of the
superior allele.

Each increment of naturally selected change in allele frequency is a direct
cause of a change in the mean fitness equal to $2\alpha^{(\ell)}_{i_{\ell}}$.
Any discrepancy between the total change and this partial change, summed over
all loci and alleles, is owed to indirect effects acting through changes in
the rules of genotype formation, the distribution of environmental states, or
some other determinant of fitness. This completes the FTNS: the _increase_ in
the mean fitness of a population caused exclusively by the effect of natural
selection on allele frequencies—setting aside those _changes_ in fitness
(which can be positive or negative) ascribable to other causes—is equal to the
additive genetic variance in fitness.

Fisher’s contributions to biology and applied mathematics were of course
numerous and profound. Judging from his writing in _The Genetical Theory_ ,
however, we surmise that he considered the FTNS to be the most important of
his achievements. The FTNS quantifies Darwin’s notion of hereditary variation
in fitness leading to adaptation and provides a deeper understanding of it. It
is interesting that (29), Fisher’s “supreme law of the biological sciences,”
explicitly encodes a distinction between an observed excess and a causal
effect, the same distinction that animated his work on experimental design,
which [43] praised as the greatest of Fisher’s contributions to statistics.
The FTNS was thus another blow struck by Fisher against his scientific
adversary Karl Pearson, who believed it was possible both to study evolution
mathematically and to discard the notion of causality. If causality appears
inevitably in the formulation of a phenomenon as fundamental as evolution by
natural selection, then it surely cannot be a dispensable “fetish amidst the
inscrutable arcana of modern science” [48, p. xii].

## Acknowledgements

We thank A. W. F. Edwards for sharing his unpublished work and his
correspondence with the late Douglas Falconer, Sabin Lessard for helpfully
answering one of our queries, and the reviewers for comments and suggestions
that have greatly improved this paper. This work was supported by the
Intramural Research Program of the NIDDK, NIH.

## Appendix

Here we explicitly derive the conditions under which the regression and
experimental definitions of average effect are equivalent. We assume that the
equivalence can always be secured in a meaningful way, either because
genotypes and environments are independent or because the regression has been
performed on the experimental genotypic means rather than the observed
genotypic means. We will often refer to an experimental average effect in the
sense of an arbitrary linear combination of relevant causal effects
(differences between genotypic means) and narrow down our reference to
particular linear combinations as the given argument proceeds. We first treat
the case of a single biallelic locus, which is of special interest because it
is possible here to find explicit expressions for the weights $c_{1}$ and
$c_{2}$ in (11).

Let $i$ stand for $\textrm{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$, $j$
for $\textrm{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]$, and $k$ for
$\textrm{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2})]$. This notation is
similar to that of [19, 24]. By using the $do$ symbol, however, our argument
below is meaningful even if genotypes and environments are dependent and non-
additive.

To minimize the sum of squares

$P(i-\nu+\alpha)^{2}+2Q(j-\nu)^{2}+R[k-\nu-\alpha]^{2},$

we take partial derivatives with respect to $\nu$ and $\alpha$ and set them
equal to zero. Solving the two resulting equations gives

$\alpha=\frac{P(Q+R)(j-i)+R(P+Q)(k-j)}{PQ+QR+2PR},$ (A1)

which can easily be recognized as equivalent to (11) in the case that
genotypes and environments act additively. Using (5\. Gene-Environment
Correlation and Interaction) to expand each experimental mean, we find that
the numerator of (A1) becomes

$c_{1}\left[\sum_{i}\textrm{Pr}(\mathcal{E}_{i})\textrm{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{i})-\textrm{Pr}(\mathcal{E}_{i})\textrm{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1},\mathcal{E}_{i})\right]\\\
+c_{2}\left[\sum_{i}\textrm{Pr}(\mathcal{E}_{i})\textrm{E}(Y\,|\,\mathcal{A}_{2}\mathcal{A}_{2},\mathcal{E}_{i})-\textrm{Pr}(\mathcal{E}_{i})\textrm{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{i})\right],$
(A2)

which means that (A1) is also equivalent to (15).

Now consider the change in the mean phenotype caused by experimental gene
substitutions. The contribution to the population mean phenotype by the
experimental means of the genotypes is given by

$\mu=iP+2jQ+kR,$ (A3)

and the change in the population mean upon effecting the gene substitutions is

$d\mu=idP+2jdQ+kdR.$ (A4)

The changes $dP,dQ,dR$ have two degrees of freedom. To express the changes in
terms of a single change $dp$, we must obtain another condition, which can be
expressed without loss of generality as $f(P,Q,R)=0$. [24] gave the condition
that $\lambda=Q^{2}/(PR)$ remains constant, but his concise argument has
puzzled many commentators.

It turns out that Fisher set $d\mu=idP+2jdQ+kdR$ equal to $2\alpha dp$ and
equated the coefficients of $i,j,k$ [8], which yields

$\displaystyle dP$ $\displaystyle=-2P(Q+R)dp/S,$ $\displaystyle dQ$
$\displaystyle=Q(P-R)dp/S,$ $\displaystyle dR$ $\displaystyle=2R(P+Q)dp/S,$
(A5)

where $S=P(Q+R)+R(P+Q)$. The function $f$ satisfies the differential equation

$\frac{\partial f}{\partial P}dP+\frac{\partial f}{\partial
Q}dQ+\frac{\partial f}{\partial R}dR=0.$ (A6)

Inserting (A5) into (A6) gives

$-2P(Q+R)\frac{\partial f}{\partial P}+Q(P-R)\frac{\partial f}{\partial
Q}+2R(P+Q)\frac{\partial f}{\partial R}=0.$ (A7)

Now $(-2P(Q+R),Q(P-R),2R(P+Q))$ and $(\frac{\partial f}{\partial
p},\frac{\partial f}{\partial Q},\frac{\partial f}{\partial R})$ can be
regarded as two orthogonal vectors in three-space. We want the second
condition to be independent of the conservation of probability condition and
not to be the trivial zero vector. By inspection, we see that a solution is
given by

$\displaystyle\frac{\partial f}{\partial P}$ $\displaystyle=\frac{\phi}{P},$
$\displaystyle\frac{\partial f}{\partial Q}$ $\displaystyle=\frac{-2\phi}{Q},$
$\displaystyle\frac{\partial f}{\partial R}$ $\displaystyle=\frac{\phi}{R},$
(A8)

where $\phi$ is an arbitrary function of $P,Q,R$. A simple solution is given
by setting $\phi$ equal to the constant $a$, whereupon (A8) can be integrated
to obtain

$f=-2a\ln Q+a\ln P+a\ln R+a\ln\lambda,$ (A9)

which gives the condition $Q^{2a}=(\lambda PR)^{a}$. $a=1$ gives the condition
expressed in terms of the classic Fisher parameter. Conversely, if we let
$\phi=PRQ^{-2}$ then we get $f=PRQ^{-2}-(1/\lambda)$, which also gives the
Fisher parameter.

Taking the partial second derivatives gives the compatibility conditions that
$\phi$ must satisfy:

$\displaystyle\frac{1}{P}\frac{\partial\phi}{\partial R}$
$\displaystyle=\frac{1}{R}\frac{\partial\phi}{\partial P},$
$\displaystyle\frac{1}{P}\frac{\partial\phi}{\partial Q}$
$\displaystyle=\frac{-2}{Q}\frac{\partial\phi}{\partial P},$
$\displaystyle\frac{1}{R}\frac{\partial\phi}{\partial Q}$
$\displaystyle=\frac{-2}{Q}\frac{\partial\phi}{\partial R}.$ (A10)

Hence, any differentiable function of $PRQ^{-2}$ is a solution. This then
implies that $f$ can be any differentiable function of $PRQ^{-2}$ as well.
This shows that the average phenotypic increment caused by a number of
experimental gene substitutions is the same as the slope in the regression of
the phenotype on the experimental genotypic means if the substitutions are
performed in a background where any function of $PRQ^{-2}$ is held constant,
with $\lambda$ being the simplest one.

We now treat a phenotype affected by an arbitrary number of multiallelic loci.
As shown in Section 7, the experimentally determined phenotypic means of the
whole-genome genotypes can be expressed as

$\mathbb{E}[Y\,|\,do(ij)]=\mu+\alpha_{ij}+\varepsilon_{ij}.$

In the remainder we abbreviate $\mathbb{E}[Y\,|\,do(ij)]$ as $G_{ij}$ and set
$a_{ij}=G_{ij}-\mu$, which obeys the condition $\sum_{i,j}P_{ij}a_{ij}=0$.

The average effects can be written as
$\alpha_{ij}=\alpha_{i}+\alpha_{j}=\sum_{\ell}(\alpha_{i_{\ell}}^{(\ell)}+\alpha_{j_{\ell}}^{(\ell)})$
and are obtained by minimizing

$\sum_{i,j}P_{ij}(a_{ij}-\alpha_{ij})^{2}.$ (A11)

The minimum obeys the condition

$p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}=\sum_{i\nmid
i_{k}}\sum_{j}P_{ij}\alpha_{ij}=\sum_{i\nmid
i_{k}}\sum_{j}P_{ij}\sum_{\ell}\left(\alpha_{i_{\ell}}^{(\ell)}+\alpha_{j_{\ell}}^{(\ell)}\right),$
(A12)

where

$p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}=\sum_{i\nmid i_{k}}\sum_{j}a_{ij}P_{ij}$ (A13)

defines the average excesses. A sum running over $i\nmid i_{k}$ should be
understood as a sum over all multi-indices $i$ where the $k$th element is
fixed to $i_{k}$. These relations imply that $\sum_{ij}P_{ij}\alpha_{ij}=0$,
which also implies that $\sum_{ij}P_{ij}\varepsilon_{ij}=0$.

Equation (A12) can be rewritten as

$p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}=\left(p_{i_{k}}^{(k)}+q_{i_{k}i_{k}}^{(kk)}\right)\alpha_{i_{k}}^{(k)}+\sum_{\ell\neq
k}\sum_{i_{\ell}}p_{i_{k}i_{\ell}}^{(k\ell)}\alpha_{i_{\ell}}^{(\ell)}+\sum_{\ell}\sum_{j_{\ell}\neq
i_{k}}q_{i_{k}j_{\ell}}^{(k\ell)}\alpha_{j_{\ell}}^{(\ell)}\\\
\equiv\sum_{j_{\ell}}H^{(k\ell)}_{i_{k}j_{\ell}}\alpha_{j_{\ell}}^{(\ell)},$
(A14)

where

$p^{(k\ell)}_{i_{k}i_{\ell}}=\sum_{i\nmid i_{k},i_{\ell}}\sum_{j}P_{ij}$

denotes the frequency of gametes that carry $\mathcal{A}^{(k)}_{i_{k}}$ and
$\mathcal{A}^{(\ell)}_{i_{\ell}}$ and

$q^{(k\ell)}_{i_{k}j_{\ell}}=\sum_{i\nmid i_{k}}\sum_{j\nmid j_{\ell}}P_{ij}$

denotes the frequency of all multilocus genotypes that carry
$\mathcal{A}^{(k)}_{i_{k}}$ and $\mathcal{A}^{(\ell)}_{j_{\ell}}$ on different
chromosomes. The matrix $\mathsf{H}$ in (A14) is constructed as follows. Let
$\mathbf{p}$ denote the vector of allele frequencies, $\mathbf{a}$ the vector
of average excesses, and $\bm{\alpha}$ the vector of average effects. These
vectors have length $\sum_{k}^{L}n_{k}$, and their elements are ordered by
locus. We can then define

$\mathsf{H}=\mathsf{D}+\mathsf{P}+\mathsf{Q},$ (A15)

where $\mathsf{D}$ is the diagonal matrix with the components of $\mathbf{p}$
on the diagonal, $\mathsf{P}$ is the matrix with entries
$p^{(k\ell)}_{i_{k}i_{\ell}}$ if $k\neq\ell$ and 0 otherwise, and $\mathsf{Q}$
is the matrix with entries $q^{(k\ell)}_{i_{k}j_{\ell}}$ [14, 6]. We will use
the notation $\mathbf{p}\cdot\mathbf{a}$ to designate the component-wise
product of the vectors $\mathbf{p}$ and $\mathbf{a}$, i.e.,
$(\mathbf{p}\cdot\mathbf{a})_{i}=\mathbf{p}_{i}\mathbf{a}_{i}$. (A14) can thus
be rewritten again as

$\bm{\alpha}=\mathsf{H}^{-1}(\mathbf{p}\cdot\mathbf{a})$ (A16)

subject to suitable constraints on $\bm{\alpha}$. We will shortly see that
these constraints turn out to be (17) for each locus. Given our ordering
convention, the element $H^{(k\ell)}_{i_{k}j_{\ell}}$ lies in the row of
$\mathsf{H}$ corresponding to allele $\mathcal{A}^{(k)}_{i_{k}}$ and the
column corresponding to $\mathcal{A}^{(\ell)}_{i_{\ell}}$.

The total change in $\mu$ is

$d\mu=\sum_{i_{k}}\sum_{i,j}G_{ij}\frac{\partial P_{ij}}{\partial
p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)}=\sum_{i_{k}}\sum_{i,j}(\mu+\alpha_{ij}+\varepsilon_{ij})\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)}\\\
=\sum_{i_{k}}\sum_{i,j}(\alpha_{ij}+\varepsilon_{ij})\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)}$ (A17)

upon performing a number of experimental gene substitutions at locus $k$.
Agreement of the experimental and regression average effects implies that this
change must equal the change predictable from the breeding values,

$d\mu=\sum_{i_{k}}\sum_{i,j}\alpha_{ij}\frac{\partial P_{ij}}{\partial
p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)},$ (A18)

which implies in turn that

$\sum_{i,j}\sum_{i_{k}}\varepsilon_{ij}\frac{\partial P_{ij}}{\partial
p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)}=0$ (A19)

is a necessary and sufficient condition for the experimental and regression
average effects to coincide. The bald statement that the changes in genotype
frequencies must somehow nullify the non-additive residuals, however, is not
very revealing. We can render (A19) into a more insightful form by noting that

$\sum_{i,j}P_{ij}\sum_{\ell=1}^{L}\left(\frac{\partial
p_{i_{\ell}}}{p_{i_{\ell}}}+\frac{\partial
p_{j_{\ell}}}{p_{j_{\ell}}}\right)\varepsilon_{ij}=0$ (A20)

because the sum over $\ell$ is a constant determined by the experimenter.
Using this, from (A19) we obtain

$\sum_{ij}P_{ij}\left[\frac{1}{P_{ij}}\frac{\partial P_{ij}}{\partial
p^{(k)}_{i_{k}}}-\sum_{\ell}\left(\frac{\partial
p_{i_{\ell}}}{p_{i_{\ell}}}+\frac{\partial
p_{j_{\ell}}}{p_{j_{\ell}}}\right)\right]\varepsilon_{ij}=0,$ (A21)

which leads to (24). This argument, which simplifies one given by [36], can be
used to construct a variety of quantities measuring departures from random
combination. The $\theta_{ij}$ appear to be the simplest such quantities.

The criterion (A21) does not pick out a unique weighting of the possible gene
substitutions for a given genetic architecture. It would be of great
significance if a subset of the possible weights could be characterized in a
manner that does not depend on the non-additive residuals. We have done this
for a single biallelic locus, where the subset contains the singleton
weighting of the two possible gene substitutions that conserves $\lambda$. If
a general procedure for constructing such a residual-free characterization for
any number of loci exists, then the following argument should be able to find
it.

The contribution of the experimental genotypic means to the population mean is

$\mu=\sum_{i,j}G_{ij}P_{ij}.$ (A22)

The definition of the experimental average effect can be written as

$\alpha_{i_{k}}^{(k)}=\frac{1}{2}\frac{\partial\mu}{\partial
p_{i_{k}}^{(k)}}.$ (A23)

Imposing constancy of the experimental means, we can write the change in the
population mean due to a change in frequency of allele
$\mathcal{A}^{(k)}_{i_{k}}$ as

$\frac{\partial}{\partial p^{(k)}_{i_{k}}}\mu=\sum_{i,j}G_{ij}\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}}=\sum_{i,j}(G_{ij}-\mu)\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}}=\sum_{i,j}a_{ij}\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}},$ (A24)

using the fact that $\sum_{i,j}\frac{\partial}{\partial
p^{(k)}_{i_{k}}}P_{ij}=0$. The indeterminacy in the partial derivatives with
respect to allele frequency will be resolved by the properties of
$\bm{\lambda}$ in (19) that emerge from the subsequent analysis.

Substituting (A24) and (A23) into (A14) using (A13) gives the condition

$\sum_{i\nmid
i_{k}}\sum_{j}a_{ij}P_{ij}=\frac{1}{2}\sum_{j}\sum_{j_{\ell}}H^{(k\ell)}_{i_{k}j_{\ell}}\sum_{m,n}a_{mn}\frac{\partial
P_{mn}}{\partial p^{(\ell)}_{j_{\ell}}}$ (A25)

for each $i_{k}$ and $k$. Closed-form solutions of these partial differential
equations will not exist in general. However, using symmetry conditions and
properties of $\mathsf{H}$, we may infer some necessary conditions on the
genotype frequencies that must be satisfied.

We first note that the image space of $\mathsf{H}$ contains all permissible
vectors of allele-frequency changes [37]. Since $\mathsf{H}$ is invertible on
its image space, we may operate on (A25) by the inverse of $\mathsf{H}$ (which
we call $\mathsf{J}$) and thereby separate the PDE system into a set of $\sum
n_{\ell}$ ordinary differential equations, which we denote by

$2\sum_{i_{k}}J_{j_{k}i_{k}}p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}=\sum_{m,n}a_{mn}\frac{\partial
P_{mn}}{\partial p^{(k)}_{j_{k}}}.$ (A26)

We may now select any row of (A26), expand the
$p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}$ in terms of the $a_{mn}$, and equate the LHS
and RHS coefficients of $a_{mn}$. This will result in a set of $\prod
n_{\ell}\times\prod n_{\ell}$ ordinary differential equations of the form

$\frac{1}{P_{mn}}\frac{\partial P_{mn}}{\partial
p^{(k)}_{j_{k}}}=\phi_{mn}(j_{k}),$ (A27)

where $\phi_{mn}(j_{k})$ is some linear combination of the elements of the
vector $J_{j_{\ell}=j_{k},i_{\ell}}$. From this point the
$a_{mn}=\alpha_{mn}+\varepsilon_{mn}$ no longer appear in the argument, and it
follows that we must be finding properties of a solution that depends on
neither the breeding values nor the non-additive residuals.

Conserved quantities imposed by (A25), which can be used to form elements of
$\bm{\lambda}$, can be constructed by taking linear combinations of the ODEs
such that

$\sum_{m,n}\sigma_{mn}\frac{1}{P_{mn}}\frac{\partial P_{mn}}{\partial
p^{(k)}_{j_{k}}}=0,$ (A28)

from which we obtain conserved measures of departure from random combination
assuming the form

$\frac{\prod_{\\{\sigma>0\\}}P_{\alpha\beta}}{\prod_{\\{\sigma<0\\}}P_{\gamma\delta}}=\lambda_{\sigma},$
(A29)

where $\sigma_{mn}$ is some set of coefficients that are positive, zero, or
negative. These conserved quantities will form a set of necessary conditions
for the equivalence of the experimental and regression definitions of the
average effects.

Note that the coefficients of $a_{mn}$ on the LHS of (A26) are grouped
according to the $a_{i_{k}}^{(k)}$. Thus all of the $a_{mn}$ expressed in a
given $a_{i_{k}}^{(k)}$ will have the same coefficient (one of the elements of
$\mathsf{J}$). We can thus construct conserved measures of Hardy-Weinberg and
linkage disequilibrium without an explicit calculation of $\mathsf{J}$ because
we know which sets of coefficients are equal.

Our first numerical example is of a single locus with three alleles (Table
A1). The case of a single locus with any number of alleles was analytically
treated by [32]. The equating of coefficients along the $i$th row of (A26)
leads to the matrix of equations

$\begin{pmatrix}J_{i1}=\dfrac{1}{P_{11}}\dfrac{\partial P_{11}}{\partial
p_{i}}&J_{i1}=\dfrac{1}{P_{12}}\dfrac{\partial P_{12}}{\partial
p_{i}}&J_{i2}=\dfrac{1}{P_{21}}\dfrac{\partial P_{21}}{\partial p_{i}}\\\
&&\\\ J_{i2}=\dfrac{1}{P_{22}}\dfrac{\partial P_{22}}{\partial
p_{i}}&J_{i1}=\dfrac{1}{P_{13}}\dfrac{\partial P_{13}}{\partial
p_{i}}&J_{i3}=\dfrac{1}{P_{31}}\dfrac{\partial P_{31}}{\partial p_{i}}\\\
&&\\\ J_{i3}=\dfrac{1}{P_{33}}\dfrac{\partial P_{33}}{\partial
p_{i}}&J_{i2}=\dfrac{1}{P_{23}}\dfrac{\partial P_{23}}{\partial
p_{i}}&J_{i3}=\dfrac{1}{P_{32}}\dfrac{\partial P_{32}}{\partial p_{i}}\\\
\end{pmatrix}$ (A30)

for allele $i$. The notation $P_{ij}$ now means the ordered genotype with
alleles $i$ and $j$. This matrix gives a set of nine conditions plus
conservation of probability that must be satisfied to ensure the equality of
(A25). However, given that there are only six unique genotypes, these
conditions are overdetermined and will not necessarily be solvable. We can
attempt to formulate a solvable set by combining these conditions. We can see
that the second and third elements in a given row of this matrix must equal
the sum of the elements in the first column corresponding to the homozygous
bearers of the relevant alleles. For example,

$\frac{1}{P_{12}}\frac{\partial P_{12}}{\partial
p_{i}}+\frac{1}{P_{21}}\frac{\partial P_{21}}{\partial
p_{i}}=\frac{1}{P_{11}}\frac{\partial P_{11}}{\partial
p_{i}}+\frac{1}{P_{22}}\frac{\partial P_{22}}{\partial p_{i}}=J_{i1}+J_{i2},$
(A31)

and these equations lead collectively to the three conserved measures of
Hardy-Weinberg disequilibrium

$\lambda_{12}=\frac{P_{12}^{2}}{P_{11}P_{22}},\quad\lambda_{13}=\frac{P_{13}^{2}}{P_{11}P_{33}},\quad\lambda_{23}=\frac{P_{23}^{2}}{P_{22}P_{33}}.$
(A32)

Two of the allele frequencies and these three conserved quantities appear to
be a complete specification of the six genotype frequencies. By the implicit
function theorem, invertibility of the Jacobian at any solution ($p_{1}$,
$p_{2}$, $\lambda_{12}$, $\lambda_{13}$, $\lambda_{23}$) specifying a valid
vector of genotype frequencies ensures that there are unique solutions for
small perturbations of the allele frequencies. Numerical testing suggests that
invertibility of the Jacobian is a generic property of this five-dimensional
system.

Table A1: A trait affected by a single triallelic locus. genotype | $\mathbb{E}[Y\,|\,do(\cdot)]$ | frequency | $\varepsilon$
---|---|---|---
$\mathcal{A}_{1}\mathcal{A}_{1}$ | 10 | .2 | $-.3402778$
$\mathcal{A}_{2}\mathcal{A}_{2}$ | 13 | .2 | .2152778
$\mathcal{A}_{3}\mathcal{A}_{3}$ | 12 | .2 | $-.6875$
$\mathcal{A}_{1}\mathcal{A}_{2}$ | 11 | .2 | $-.5625$
$\mathcal{A}_{1}\mathcal{A}_{3}$ | 14 | .1 | 2.4861111
$\mathcal{A}_{2}\mathcal{A}_{3}$ | 13 | .1 | .2638889

Given the numerical values in Table A1, what is the experimental average
effect of substituting $\mathcal{A}_{2}$ for $\mathcal{A}_{1}$? There are
three ways in which this gene substitution can be brought about:
$\mathcal{A}_{1}\mathcal{A}_{1}$ $\rightarrow$
$\mathcal{A}_{1}\mathcal{A}_{2}$, $\mathcal{A}_{1}\mathcal{A}_{2}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{2}$, and
$\mathcal{A}_{1}\mathcal{A}_{3}$ $\rightarrow$
$\mathcal{A}_{2}\mathcal{A}_{3}$. The causal effects of these three
substitutions are 1, 2, and $-1$ respectively.

We first attempt to satisfy the weaker criterion that (24) is equal to zero by
determining which weighted average of the first two substitutions yields the
smallest absolute value of $\overline{\varepsilon\,\mathring{\theta}}$. To
calculate a discrete approximation of the $\mathring{\theta}_{ij}$, we use a
population size of 10,000. We examine all integer weights such that the
weights sum to 90. There are 91 such weighted averages, and it turns out that
the weights $(70,20)$ yield the minimum. In fact, the absolute value of
$\overline{\varepsilon\,\mathring{\theta}}$ yielded by these weights is
roughly $1.5\times 10^{-16}$, which is nearly within machine error of zero.
The 90 other weighted averages lead to absolute values of
$\overline{\varepsilon\,\mathring{\theta}}$ exceeding $1\times 10^{-4}$.

These weights lead to an experimental average effect, $\alpha_{2}-\alpha_{1}$,
equaling 11/9. In the case of a single locus, the regression average effects
(which we now denote by $\beta$) do not require the imposition of (17) to be
identified, and the calculations yielding the values of the $\varepsilon_{ij}$
in Table A1 also give us ($-0.7798611$, 0.4423611, 0.39375) as the numerical
value of ($\beta_{1}$, $\beta_{2}$, $\beta_{3}$). It appears that
$\beta_{2}-\beta_{1}$ is exactly equal to 11/9.

We can use a different pair of substitutions, say
$\mathcal{A}_{1}\mathcal{A}_{2}$ $\rightarrow$
$\mathcal{A}_{2}\mathcal{A}_{2}$ and $\mathcal{A}_{1}\mathcal{A}_{3}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{3}$, to yield the experimental
average effect $\alpha_{2}-\alpha_{1}$. We examine all integer weightings of
these two substitutions such that the weights sum to 270. It turns out that
the weighting $(200,70)$ yields the minimum. The absolute value of
$\overline{\varepsilon\,\mathring{\theta}}$ yielded by these weights is
roughly $4\times 10^{-16}$, again nearly within machine error of zero, whereas
the 270 other weighted averages all lead to absolute values of
$\overline{\varepsilon\,\mathring{\theta}}$ exceeding $3\times 10^{-4}$. These
minimizing weights again lead to an experimental average effect of 11/9. It is
rather interesting that the neighboring weights $(199,71)$ and $(201,69)$ lead
to such higher values of $\overline{\varepsilon\,\mathring{\theta}}$ despite
the numerical closeness of these weighted averages and the fineness of our
discretization. In fact, we have chosen to present this example because of
this phenomenon, which we conjecture to be related to the fact that the
$\alpha_{2}-\alpha_{1}$ happens to be rational and thus exactly equal to some
integer-weighted average of the causal effects.

Evidently it should not be possible to obtain a valid average effect by using
only the substitutions $\mathcal{A}_{1}\mathcal{A}_{1}$ $\rightarrow$
$\mathcal{A}_{1}\mathcal{A}_{2}$ and $\mathcal{A}_{1}\mathcal{A}_{3}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{3}$. Examining all integer weights
summing to 1000, we find that $\overline{\varepsilon\,\mathring{\theta}}$
declines linearly from $(0,1000)$ to $(1000,0)$; the absolute minimum of
$\overline{\varepsilon\,\mathring{\theta}}$ is thus attained at a boundary,
and it is not especially small ($\sim 2\times 10^{-2}$).

We examine whether our conception of individual average effects is valid.
Using the method of minimizing $\overline{\varepsilon\,\mathring{\theta}}$, we
find that $\alpha_{2}-\alpha_{3}$ is approximately .049. According to our
notion of substituting $\mathcal{A}_{2}$ for a random homologous gene,
$\alpha_{2}$ must be equal to
$p_{1}(\alpha_{2}-\alpha_{1})+p_{3}(\alpha_{2}-\alpha_{3})$. In our example
($p_{1}$, $p_{2}$, $p_{3}$) happens to be (.35, .35, .30), which leads to
$.4425$ as the approximate numerical value of $\alpha_{2}$. This is in good
agreement with $\beta_{2}$. Continuing this exercise, we can satisfy ourselves
that ($\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$) and ($\beta_{1}$, $\beta_{2}$,
$\beta_{3}$) are equal.

We now attempt to satisfy the stronger criterion that the quantities in (A32)
remain constant. The numerical value of ($p_{1}$, $p_{2}$, $\lambda_{12}$,
$\lambda_{13}$, $\lambda_{23}$) is (35/100, 35/100, 1/4, 1/16, 1/16), and a
perturbation of ($-1/1000$, $1/1000$, 0, 0, 0) leads to a numerical solution
that specifies another valid vector of genotype frequencies. The weighting of
the possible gene substitutions satisfying the changes in genotype frequencies
is typically not unique. In a population of size $10^{8}$, one permissible
vector of weights for our example can be reasonably well approximated by

${\mathcal{A}_{1}\mathcal{A}_{1}}$${\mathcal{A}_{1}\mathcal{A}_{2}}$${\mathcal{A}_{2}\mathcal{A}_{2}}$${\mathcal{A}_{3}\mathcal{A}_{3}}$${\mathcal{A}_{1}\mathcal{A}_{3}}$${\mathcal{A}_{2}\mathcal{A}_{3}}$88,82188,951622,2226
(A33)

where the label of each arrow indicates how many gene substitutions of that
kind are to be performed. Notice that there are 12 gene substitutions
involving a genotype containing the allele $\mathcal{A}_{3}$. For each
$\mathcal{A}_{3}$ gene created by $\mathcal{A}_{1}\mathcal{A}_{3}$
$\rightarrow$ $\mathcal{A}_{3}\mathcal{A}_{3}$, another $\mathcal{A}_{3}$ is
destroyed by $\mathcal{A}_{2}\mathcal{A}_{3}$ $\rightarrow$
$\mathcal{A}_{2}\mathcal{A}_{2}$, and the net result is the same frequency of
$\mathcal{A}_{3}$. These 12 substitutions turn out to be a way of decreasing
the number of $\mathcal{A}_{1}$ genes and increasing the number of
$\mathcal{A}_{2}$ without directly converting one to the other. We might as
well pair each $\mathcal{A}_{1}\mathcal{A}_{3}$ $\rightarrow$
$\mathcal{A}_{3}\mathcal{A}_{3}$ with $\mathcal{A}_{2}\mathcal{A}_{3}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{2}$, treating each such pair as a
single substitution. The weighted average of the gene substitutions is then

$\frac{88,821(1)+88,951(2)+22,222(-1)+6(-2+0)}{88,821+88,951+22,222+6},$

which diverges from 11/9 at the fourth decimal place.

We now apply our argument to the case of two biallelic loci. Here we will
encounter a contradiction.

The equating of coefficients along the row of (A26) corresponding to allele
$\mathcal{A}^{(k)}_{i_{k}}$ now leads to the matrix of equations

$\begin{pmatrix}J_{i1}+J_{i3}=\frac{1}{P_{11,11}}\frac{\partial
P_{11,11}}{\partial
p^{(k)}_{i_{k}}}&J_{i1}+J_{i4}=\frac{1}{P_{12,11}}\frac{\partial
P_{12,11}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i3}=\frac{1}{P_{21,11}}\frac{\partial
P_{21,11}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i4}=\frac{1}{P_{22,11}}\frac{\partial
P_{22,11}}{\partial p^{(k)}_{i_{k}}}\\\
J_{i1}+J_{i3}=\frac{1}{P_{11,12}}\frac{\partial P_{11,12}}{\partial
p^{(k)}_{i_{k}}}&J_{i1}+J_{i4}=\frac{1}{P_{12,12}}\frac{\partial
P_{12,12}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i3}=\frac{1}{P_{21,12}}\frac{\partial
P_{21,12}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i4}=\frac{1}{P_{22,12}}\frac{\partial
P_{22,12}}{\partial p^{(k)}_{i_{k}}}\\\
J_{i1}+J_{i3}=\frac{1}{P_{11,21}}\frac{\partial P_{11,21}}{\partial
p^{(k)}_{i_{k}}}&J_{i1}+J_{i4}=\frac{1}{P_{12,21}}\frac{\partial
P_{12,21}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i3}=\frac{1}{P_{21,21}}\frac{\partial
P_{21,21}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i4}=\frac{1}{P_{22,21}}\frac{\partial
P_{22,21}}{\partial p^{(k)}_{i_{k}}}\\\
J_{i1}+J_{i3}=\frac{1}{P_{11,22}}\frac{\partial P_{11,22}}{\partial
p^{(k)}_{i_{k}}}&J_{i1}+J_{i4}=\frac{1}{P_{12,22}}\frac{\partial
P_{12,22}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i3}=\frac{1}{P_{21,22}}\frac{\partial
P_{21,22}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i4}=\frac{1}{P_{22,22}}\frac{\partial
P_{22,22}}{\partial p^{(k)}_{i_{k}}}\\\ \end{pmatrix}$ (A34)

plus conservation of probability that must be satisfied to ensure the equality
of (A25). An argument analogous to the one below (A30) shows that six
quantities of the form

$\lambda_{ij}=\frac{P_{ij}^{2}}{P_{ii}P_{jj}}$ (A35)

must be conserved. If we do not assume that the double heterozygotes are
phenotypically equivalent, then these six measures of Hardy-Weinberg
disequilibrium, the allele frequencies at the two loci, and conservation of
probability leave one more condition to specify ten genotype frequencies.

Rearrange each element of (A34) to put the genotype frequency on one side and
form the four column sums. Each such sum is the marginal frequency of a
gamete. For example, we have

$P_{11}=(J_{i1}+J_{i3})^{-1}\sum_{11,j}\frac{\partial P_{11,j}}{\partial
p^{(k)}_{i_{k}}},$ (A36)

which implies that

$J_{i1}+J_{i3}=\frac{1}{P_{11}}\frac{\partial P_{11}}{\partial
p^{(k)}_{i_{k}}}.$ (A37)

Combining all columns, we get

$\frac{\partial P_{11}}{\partial p^{(k)}_{i_{k}}}+\frac{\partial
P_{22}}{\partial p^{(k)}_{i_{k}}}-\frac{\partial P_{12}}{\partial
p^{(k)}_{i_{k}}}-\frac{\partial P_{21}}{\partial p^{(k)}_{i_{k}}}=0,$ (A38)

which yields the condition that

$\zeta=\frac{P_{11}P_{22}}{P_{12}P_{21}}$ (A39)

remains constant. $\zeta$ is the measure introduced by [34], and the multi-
index notation immediately reveals that it is equal to unity in linkage
equilibrium.

The equality of the regression and experimental average effects for constant
$\bm{\lambda}=(\lambda_{11,21},\ldots,\lambda_{12,22},\zeta)$ appears to
conflict with the result of [40] that the stipulation of $\Delta\zeta=0$ and
random mating to reset the $\lambda_{ij}$ to unities among zygotes does not
lead to the change in the mean phenotype equaling the summed products of
average effects and changes in allele frequencies (in the case that the
phenotype is fitness). Our next numerical example shows that we have indeed
reached a contradiction (Table A2).

Table A2: A trait affected by two biallelic loci. genotype | $\mathbb{E}[Y\,|\,do(\cdot)]$ | frequency | $\varepsilon$
---|---|---|---
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}$ | 17 | .054 | 5.0100265
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}$ | 12 | .036 | $-1.438691$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}$ | 13 | .257 | $-.8874187$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}$ | 14 | .140 | $-.3345667$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}$ | 18 | .080 | $-.7832893$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}$ | 10 | .039 | $-4.7832893$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}$ | 16 | .066 | 4.7679882
$\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}$ | 15 | .041 | $-.6791599$
$\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}$ | 11 | .029 | $-3.2178824$
$\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}$ | 20 | .258 | .4233950

Numerical testing suggests that invertibility of the Jacobian is also a
generic property of the nine-dimensional system ($p^{(1)}$, $p^{(2)}$,
$\lambda_{11,21}$, …, $\lambda_{12,22}$, $\zeta$). We numerically update the
vector of genotype frequencies in Table A2 by increasing the frequency of
allele $\mathcal{A}^{(1)}_{2}$ by $10^{-6}$. The regression average effect at
locus 1, as determined by the Levenberg-Marquardt algorithm, is approximately
2.4934. However, when we multiply this by two times $10^{-6}$, the result does
not closely agree with $G_{ij}\Delta P_{ij}$. The discrepancy is close to 12
percent and does not diminish as $\Delta p^{(1)}$ is made smaller. We conclude
that we have falsified our initial assumption that a residual-free description
of the average effects always exists.

Sampling vectors of initial genotype frequencies from the Dirichlet
distribution, we find that the changes implied by constancy of $\bm{\lambda}$
in the case of two biallelic loci do not typically produce such a large
discrepancy. The error is usually less than 7 percent. This suggests to us
that there may exist a subset of weights, distinguished by the changes in the
departures from random combination all being “small” in some sense, that can
be mathematically described. We leave this issue to future research.

The vanishing of $\overline{\varepsilon\,\mathring{\theta}}$ is still an
applicable criterion. For example, the genotype
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}$/
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}$ can be transformed into either
double heterozygote, depending on whether the left or right gene at locus 1 is
the target of the substitution. In one case the causal effect is 6, and in the
other it is $-2$. Among all integer weightings of these two substitutions
summing to 1000, the weights (562, 438) yield the minimum. The corresponding
weighted average of the causal effects, $\alpha^{(1)}_{2}-\alpha^{(1)}_{1}$,
equals 2.496 and is also the closest to
$\beta^{(1)}_{2}-\beta^{(1)}_{1}\approx 2.493$ that can be obtained given our
discretization. The replacement of randomly chosen homologous genes can now be
used to determine ($\alpha^{(1)}_{1}$, $\alpha^{(1)}_{2}$).

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