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REGULAR A RTI CLE
The Impact of Gene–Environment Interactionand Correlation on the Interpretation of Heritability
Omri Tal
Received: 16 November 2010 / Accepted: 10 August 2011
� Springer Science+Business Media B.V. 2011
Abstract The presence of gene–environment statistical interaction (GxE) and
correlation (rGE) in biological development has led both practitioners and philos-
ophers of science to question the legitimacy of heritability estimates. The paper
offers a novel approach to assess the impact of GxE and rGE on the way genetic and
environmental causation can be partitioned. A probabilistic framework is developed,
based on a quantitative genetic model that incorporates GxE and rGE, offering a
rigorous way of interpreting heritability estimates. Specifically, given an estimate of
heritability and the variance components associated with estimates of GxE and rGE,
I arrive at a probabilistic account of the relative effect of genes and environment.
Keywords Heritability � Quantitative genetics � Probability � GxE interaction �G–E correlation
1 Introduction
We may want to get an intuitive feeling for the ‘importance’ of genetic
variation in the population and a reasonable measure of this relative
importance is the broad heritability.
—Lewontin (1975)
Can the broad heritability provide reliable intuition on the importance of genetic
causes in producing phenotypic variation? Heritability is variably expressed as the
correspondence between a latent genetic variable and a measurable phenotypic one,
the slope of the parent-offspring regression, or the proportion of phenotypic
variance due to genetic differences. This latter formulation is common in the
O. Tal (&)
School of Philosophy and The Cohn Institute for the History and Philosophy of Science and Ideas,
Tel Aviv University, Tel Aviv 69978, Israel
e-mail: talomri@post.tau.ac.il
123
Acta Biotheor
DOI 10.1007/s10441-011-9139-8
literature, and perhaps the least confusing to conceptualize. However, the varianceis only one measure of population dispersion in a certain quantity. The standarddeviation (SD), specified in the underlying units of the target variables, is arguably a
more natural description, and mathematically, a particular variance ratio VG/VP
necessarily implies a different (higher) ratio of SD. But even the SD is not naturally
congruent with how individual differences are commonly conceptualized, since it is
essentially based on squared distances from the population mean, rather than on
pairwise differences. Moreover, heritability estimates are population measures, and
consequently say little about the relative impact of genetic and environmental causal
factors on individual phenotypic development. When nonlinear and interactive
elements are introduced into the developmental framework, the interpretation and
usage of heritability estimates become even more contentious. In a discussion on the
role of genes in development Rose (1999)criticizes the use of heritability estimates,
arguing that such estimates are meaningful only in the absence of gene–environment
interaction and under a random distribution of genotypes across environments,
If genotypes are distributed randomly across environments, it is possible to
estimate heritability, which defines the proportion of the variance which is
genetically determined. However, the mathematics only works if all the
relevant simplifying assumptions are made. If there is a great deal of
interaction between genes and environment, that is if genes behave according
to Dobzhansky’s (1973) vision of norms of reaction, if genes interact with
each other, and if the relationships are not linear and additive but interactive,
the entire mathematical apparatus of heritability estimates falls apart. Thus themeaningful application of heritability estimates is only possible in very specialcases, from which the majority of traits of interest outside the special world of
artificial selection are likely to escape [added emphasis].
The following observation may shed some light on Rose’s assertion. A partition of
the phenotypic variance into genetic and nongenetic components is related to the
correspondence of a latent variable to a measured one. The quantification of the
correspondence between phenotypic and genotypic values is central to the analysis
of response to selection, familial resemblance and phenotypic development in
quantitative genetics (Lynch and Walsh 1998). A useful measure of the linearcorrespondence of P and G under an additive linear model (P = G ? E) is the
squared correlation coefficient, or the coefficient of determination. Formally, this
measures the proportion of the variance in P that is explained by assuming that the
true regression E[P|G] is linear (ibid., p. 47). From basic principles,
q2ðP; GÞ ¼ COVðGþ E; GÞrP � rG
� �2
¼ VG þ COVðG; EÞð Þ2
VP � VG¼ VG
VP:
Crucially, the coefficient of determination is equivalent to the common formulation
for broad heritability—the proportion of the phenotypic variance due to genetic
variation—only when the phenotypic model is additive and COV(G, E) is zero.
Discussions of gene–environment interaction have been mired with conceptual
confusion. The term ‘GxE interaction’ is often used loosely to mean that both genes
O. Tal
123
and environment contribute to the response variable, but quantitative geneticists
employ the term in the stricter statistical sense. Simply stated, GxE designates a
contribution that some non-additive function of the hidden variables G and E makes to
the phenotypic value, independently of the main effects of these variables. It can also
be conceptualized as a relationship between the environment and a phenotype that
depends on the genotype, or alternatively a genotype-phenotype relationship that
depends on the environment (Carey 2002). Figure 1 is a depiction of GxE in terms of
norms of reaction, reflecting possible relations between the underlying variables.
The term G–E correlation refers to the phenomenon where exposure to
environment may have a genetic basis. Such correlation mainly occurs in
observational studies, whenever the environment cannot be randomly assigned
between genotypes in a controlled setting. More formally, G–E correlation is a
feature of the distribution of genotypes within environments and exists whenever a
genetic disposition leads individuals to develop under certain environments. The
present paper proposes a method of incorporating estimates of G–E interaction and
covariance within a probabilistic framework of genetic effects on individuals.
2 The Model
Quantitative genetic analysis generally proceeds by using measured phenotypic
variances and covariances to estimate latent variance components within the
framework of a postulated quantitative phenotypic model. The foundational discretemodel of GxE interactions is zijk = l ? Gi ? Ej ? Iij ? eijk where zijk denotes the
value of the k’th replicate of genotype i (Gi) in environment j (Ej), Iij denotes the
interaction between genotype i and environment j, and eijk the specific environment,
all terms with a mean of zero; To stress, eijk is the residual deviation of an
individual’s phenotype from the expectation of Gi ? Ej ? Iij, where the residuals
are uncorrelated (Lynch and Walsh 1998, p. 108). A standard abstraction of the
discrete model is P = f(G, E), with a simplified model P = l ? G ? E ? I serving
as a workable approximation. In this model the E variable represents both general
systematic and specific nonsystematic environmental effects. In this respect, the
simplified linear model makes no distinction between epigenetic, systematic,
nonsystematic or stochastic effects.
Fig. 1 The norms-of-reaction for three separate developmental scenarios
Heritability, G 9 E and rGE
123
The partition of phenotypic variance follows directly from the standardquantitative genetic model (Falconer and Mackay 1996),
P ¼ Gþ E þ I ! VP ¼ VG þ VE þ VI þ 2COVðG; EÞ ð1Þ
where the population mean l is either assumed zero or subsumed in the genetic
component. Modeling the measured and latent variables requires several standard
assumptions. It is standard practice to assume the normality of the phenotypic
distribution for many quantitative traits (some traits, such as litter size or lifespan,
are clearly not normally distributed, but adequate transformations can be invoked).
The marginal normality of the two components I and G ? E then follows from the
decomposition of a normal random variable (P = G ? E ? I) and its stability
features under e-independence and deviations from normality (Tal 2009). Finally,
we adopt the standard assumption of the joint-normality of G and E. This is justified
by observations of the linearity of statistical regressions, such as parent-offspring
regression (Lynch and Walsh 1998, p. 552; Tal 2009; Hill 2010), and a marginal
normality of G, due to high proportion of additive genetic variance for complex
traits (Hill et al. 2008). Formally, the quantitative model requires,
[a] P is a standardized normally distributed trait
[b] The joint distribution of G and E is bivariate normal with a possible covariance
term
[c] I is normally distributed
[d] I is statistically independent of G ? E[e] Availability of estimates: heritability h2 = VG/VP, the fraction of VP due to
GxE interaction c2 = VI/VP, and a G–E correlation coefficient, q.
The goal is to arrive at an expression for prob(|G| [ |E| | P). Since we are
modeling only the deviations from the means we can standardize P and assume zero
means for all variables without losing information. Thus VP = 1 and the variance of
G becomes h2. We then get from Eq. 1 and the above model assumptions the
following marginal distributions and a covariance term,
P�N 0; 1ð Þ; G�N 0; h2� �
; E�N 0; VEð Þ; I�N 0; VIð Þ; COV G; Eð Þ¼ q � h �
ffiffiffiffiffiffiVE
p: ð2Þ
Naturally, all the variances and SD should be positive, i.e., h, c and rE [ 0,
where rE2 = VE. In contrast, the correlation coefficient is allowed full range,
-1 \ q\ 1. We would like to express rE in terms of the given estimates, h2, c2 and
q. From Eqs. 1 and 2 we have,
1 ¼ h2 þ r2E þ c2 þ 2qhrE: ð3Þ
Solving this as a quadratic equation for rE we get,
rE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� h2 1� q2ð Þ � c2
p� q � h: ð4Þ
A corollary of Eq. 4 is that h2 B (1 - c2)/(1 - q2), with a stricter upper bound
h2 \ 1 - c2 for q[ 0, from rE [ 0. Since (from assumptions) the probability
O. Tal
123
density function (pdf) for G and E is bivariate normal with correlation of q, the
general expression is,
fh2;rE ;qðg; eÞ ¼ 1
2phrE
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2ð Þ
p � e� 1
2� 1�q2ð Þg2
h2�2qgh� e
rEþe2
r2E
h i: ð5Þ
To formalize the probability that |G| [ |E| we first need to arrive at the joint
density function of G and E conditional on P, denoted F. From first principles of
conditional probability we have,
Fp;h2;c2;qðg; eÞ ¼ pG;EjPðg; ejpÞ ¼ pG;E;Pðg; e; pÞpPðpÞ
¼pPjG;Eðpjg; eÞ � pG;Eðg; eÞ
pPðpÞ¼ pIðp� g� eÞ � pG;Eðg; eÞ
pPðpÞ: ð6Þ
Note that pIðp� g� eÞ ¼ pPjG;Eðpjg; eÞ since P = G ? E ? I; hence pPjG;Eðpjg; eÞhas same variance as I (c2) but a mean of g ? e. The pdf of a normal random
variable X with zero mean and non-zero variance v (we assume G, E, and IGxE in
Eq. 1 are non-degenerate random variables) is given by
uvðxÞ ¼1ffiffiffiffiffiffiffiffi2pvp e�x2=2v: ð7Þ
In terms of uvðxÞ and f we have,
Fp;h2;c2;qðg; eÞ ¼uc2ðp� g� eÞ � fh2;rE ;qðg; eÞ
u1ðpÞð8Þ
which results in a bivariate normal pdf after explicit substitution and arranging of
terms (see ‘‘Appendix’’),
Fðg; eÞ ¼ 1
2pr1r2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
12
� �q e
�1
2� 1�q212ð Þ
g�l1r1
� �2
�2q12g�l1r1
� �� e�l2
r2
� �þ e�l2
r2
� �2
: ð9Þ
Figure 2 depicts F within the domain defined by |G| [ |E|.
Finally, we employ F to express the conditional probability, prob(|G| [ |E| | P).
This probability is in fact the integral of F in the domain that satisfies |G| [ |E|,
expressed as a function of h2, and denoted Mp;q;c2 h2ð Þ,
Mp;q;c2 h2� �
¼ prob Gj j[ Ej jjP ¼ pð Þ ¼Z1
�1
Zjgj
�jgj
Fp;h2;c2;qðg; eÞde dg
¼Z0
�1
Z�g
g
Fp;h2;c2;qðg; eÞdedgþZ1
0
Zg
�g
Fp;h2;c2;qðg; eÞde dg: ð10Þ
It can easily be shown that the probability is independent of the sign of P, i.e.,
prob(|G| [ |E| | P = p) = prob(|G| [ |E| | P = -p) for any p; the probability is the
same, whether the phenotype is below or above the population mean. Figure 3a–d
depict several instances of M(h2) for combinations of P, GxE and rGE, as a function
of heritability.
Heritability, G 9 E and rGE
123
It is instructive to observe some features that are immediately discernable from
the probability graphs. Figure 3a–c show a consistent pattern: as genes and
environment become more positively correlated, or as GxE effects increase, the
probability of |G| [ |E| is higher across the whole range of heritability values.
Figure 3a and b depict a divergence of the curves as we traverse the heritability
axis: at low heritability estimates and for individuals close to the population mean,
the probability that |G| [ |E| is insensitive to the presence of either interaction or
correlation effects. For instance, Fig. 3a shows that for phenotypic values close to
the mean, at a heritability range up to 0.4 there is less than a 10% change in the
resulting probability across a wide range of G–E correlation. A different pattern is
discernable for high phenotypic deviations: the probability curves re-converge at the
high heritability range, as Fig. 3c depicts, for P = 3. This means that for high
heritability values the probability that |G| [ |E| is less sensitive to the presence of
interaction for individuals that deviate largely from the mean. Figure 3d shows how
the probability curve depends on the phenotypic value. Crucially, all the probability
curves intersect at a single heritability threshold corresponding to the neutral
probability of 50%, where prob(|G| [ |E| | P) = 0.5. It is also discernable that up to
this heritability threshold, for any combination of GxE and rGE, lower values of |P|
correspond to a higher probability that |G| [ |E|; above that threshold the pattern is
reversed—lower values of |P| correspond to lower probabilities.
The probabilities discussed so far are conditional on a phenotypic value. The
unconditional probability of |G| [ |E| is simply the average over the phenotypic
distribution. We denote by M the expected value of M across the population,
Mq;c2 h2� �
¼ prob Gj j[ Ej jð Þ ¼ EðMðpÞÞ ¼Z1
�1
u1ðpÞMðpÞdp
¼ 1� 1ffiffiffiffiffiffi2pp
Z1
�1
e�p2
2 �Z0
�1
Z�g
g
Fðg; eÞdedgþZ1
0
Zg
�g
Fðg; eÞde dg
0B@
1CAdp:
ð11ÞFigure 4 depicts M for a particular instance of GxE and rGE.
1.00.5
0.0
0.5
1.0
G
1.0
0.5
0.0
0.5
E
0.0
0.5
1.0
1.5
FG
,E
Fig. 2 An instance of thebivariate normal density F,given h2 = 0.6, c2 = 0.1,q = 0.3 and p = 0.2, where F isrestricted to the domainsatisfying |G| [ |E|
O. Tal
123
We now turn to extracting similar probabilities from a purely additive model—
where GxE interaction is absent but G–E correlation is potentially present. Note we
cannot merely plug c2 = 0 in expressions (10) and (11), since the general model in
Eq. 1 assumes that a GxE factor exists and has non-zero variance. However, the
probability M converges to M0 in Eq. 17 as c2 approaches zero. This will cover those
study designs that do not model GxE but still factor a variance component from
rGE. We therefore proceed in a similar fashion to Eq. 1, briefly,
P ¼ Gþ E! VP ¼ VG þ VE þ 2COVðG; EÞ: ð12ÞFormally, the quantitative framework is,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.10.20.30.40.50.60.70.80.9
1
Heritability
Pro
b (|
G|>
|E| |
P)
P = 3, G−E correlation = 0, GxE
0.5
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.10.20.30.40.50.60.70.80.9
1
Heritability
Pro
b (|
G|>
|E| |
P)
P= 0.5, G−E correlation = 0.2, GxE
0.5
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Heritability
Pro
b (|
G|>
|E| |
P)
P = 0.25, GxE variance component = 0.2,
0.3
−0.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.10.20.30.40.50.60.70.80.9
1
Heritability
Pro
b (|
G|>
|E| |
P)
G−E correlation = 0.3, GxE variance
p=0
p=2
component = 0.1, P: 0 ... 2
variance component: 0.1 ... 0.5 G−E correlation −0.3 ... 0.3
variance component: 0.1 ... 0.5
A B
C D
Fig. 3 The graphs of prob(|G| [ |E| | P) as a function of the phenotypic value, G–E correlation and theGxE variance component. a The graphs of M(h2) for various values of G–E correlation. These are theprobabilities that the genetic deviation of an individual |G| had a greater effect than its environmentaldeviation |E| on its phenotypic deviation |P|, for p = 0.25 and GxE variance component of 0.2. At the lowheritability range the probability is insensitive to the presence of G–E correlation. b The graphs of M(h2)for various values of the GxE effect for p = 0.5 and a G–E correlation of 0.2. At the low heritabilityrange the probability is insensitive to GxE. c The graphs of M(h2) for various values of the GxE effect forp = 3 and a G–E correlation of 0. The probability curves converge at high heritability—the effect ofGxE is reduced. d The graphs of M(h2) for various values of P for a G–E correlation of 0.3 and aGxE variance component of 0.1. The probability curves intersect at a certain heritability value where theprobability is insensitive to the phenotypic value
Heritability, G 9 E and rGE
123
[a’] P = G ? E is a normally distributed trait, standardized to unit variance and
zero mean
[b’] G and E are bivariate normal, with possible non-zero covariance
[c’] We obtain estimates of h2 and the rGE correlation coefficient q
This implies,
P�N 0; 1ð Þ; G�N 0; h2� �
; E�N 0; r2E
� �; COV G; Eð Þ ¼ qhrE: ð13Þ
From Eqs. 12 and 13 we have 1 = h2 ? rE2 ? 2qhrE and solving this as a
quadratic equation for rE we get,
rE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� h2 1� q2ð Þ
p� q � h: ð14Þ
Note that q does not restrict the range of h2. The bivariate normal pdf for G and Ewith correlation of q, denoted f0, is,
f 0h2;qðg; eÞ ¼ 1
2phrE
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2ð Þ
p � e�1
2� 1�q2ð Þg2
h2�2qgh� e
rEþe2
r2E
h i: ð15Þ
To formalize the probability F0 that |G| [ |E| we need to arrive at the joint density
function of G conditional on P. From first principles of conditional probability and
Eq. 12 we have,
F0p;h2;qðgÞ ¼ pGjPðgjpÞ ¼pP;Gðp; gÞ
pPðpÞ¼
pEjGðp� gjgÞ � pGðgÞpPðpÞ
¼ pG;Eðg; p� gÞpPðpÞ
¼f 0h2;qðg; p� gÞ
u1ðpÞ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2p � h2r2E 1� q2ð Þ
p � e� g�ph
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�r2
E1�q2ð Þ
pð Þ2
2�h2r2E
1�q2ð Þ : ð16Þ
0.0 0.2 0.4 0.6 0.80.0
0.2
0.4
0.6
0.8
1.0
Heritability
G−E Corr =0.3 , GxE Var component = 0.1 , averaged over P
Pro
b (|
G|>
|E|)
Fig. 4 The graph of M for a G–E correlation of 0.3 and a GxE variance component of 0.1: the probabilitythat |G| [ |E| averaged over the phenotypic distribution
O. Tal
123
F0 is expressed in Eq. 16 in a form easily recognizable as a normal pdf, where the
variance is h2r2E 1� q2ð Þand the mean is ph
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2
E 1� q2ð Þp
. The expression for
M0 ¼ prob Gj j[ Ej jjPð Þ is finally,
M0p;h2 ¼ prob Gj j[ Ej jjPð Þ ¼
R1p=2
F0p;h2;qðgÞdg if p� 0
Rp=2
�1F0p;h2;qðgÞdg if p\0
8>>><>>>:
: ð17Þ
The expected value of M0 across the population is denoted M0,
M0 h2� �
¼ prob Gj j[ Ej jð Þ ¼ E M0h2ðpÞ� �
¼Z1
�1
u1ðpÞM0h2ðpÞdp
¼Z0
�1
u1ðpÞZp=2
�1
F0p;h2;qðgÞdgdpþZ1
0
u1ðpÞZ1
p=2
F0p;h2;qðgÞdgdp
¼ 2
Z1
0
1ffiffiffiffiffiffi2pp e
�p2
2
Z1
p=2
F0p;h2;qðgÞdgdp: ð18Þ
Figure 5a and b depict the probabilities generated within this additive framework
for the most basic scenario where G–E correlation is zero.
3 Discussion
The framework outlined in this paper allows incorporating estimates of gene–
environment interaction and covariance within a probabilistic interpretation ofheritability (Tal 2009; see Tal et al. 2010, for an extension that includes a putative
epigenetic variable). Specifically, given estimates of heritability and the variance
components associated with GxE and rGE, a method based on the standard quantitativemodel generates the conditional probability that genetic factors had a greater effect than
environmental factors on a deviation from the population mean. Previous approaches of
reformulating heritability arise from the application of more complex mixedquantitative models (see Oakey et al. 2007, for a ‘‘generalized heritability’’ that
incorporates pedigree information to form an extended pedigree model).
Two underlying assumptions of the present framework should be reemphasized.
First, the model takes as a point of departure the standard linear quantitative modelwith interaction, P = G ? E ? I, where the three latent terms are modeled as
continuous variables. Second, the probabilistic framework relies on the availability of
estimates of variance components describing the non-additive effects. In this respect,
it is a methodological issue whether a particular sample size or genetic model has
sufficient power to detect and quantify GxE for a target experimental design or
observational study. Indeed, it is well known that the power to detect interaction is
considerably less than that of the main effects (Wahlsten 1990; Sternberg and
Heritability, G 9 E and rGE
123
Grigorenko 1997; Plomin et al. 2008). There are a few experiential strategies for
increasing the power of GxE detection, primarily, using a greater sample size.
However, in controlled experimental designs the detection of GxE may be an artifact
of the maximization of the phenotypic variance, due to the relatively higher additive
genotypic variance compared to natural randomly breeding populations. Consider-
ations of strain replication in the sample are also pertinent. Using high replication of
fewer genotypes in less environments yields greater sensitivity to GxE detection than
low replication of more genotypes in more environments. This illustrates a basic
tension in defining research targets: whether to increase replication of a few
genotypes in a few environments or to better represent the range of genetic or
environmental variation possible at a cost to the sensitivity to detect small differences
(Hodgins-Davis and Townsend 2009). A related methodological issue with quanti-
tative phenotypes is the mathematical ability to induce or remove interaction effects
simply by transformation of scale (Wahlsten 1990). Ultimately, it is an empiricalissue whether GxE interaction exists with respect to a particular target trait, the given
distribution of genotypes and the range of environments.
An important theoretical issue whether the presence of any amount of
interaction, or even the possibility of undetected interaction, renders the partition
of the phenotypic variance meaningless in terms of its causal explanatory content
(see for e.g., Lewontin 1974; Sarkar 1998, for a critical perspective; Sesardic 2005
for an opposing view; Oftedal 2005 for an attempt at conciliation; Tabery 2009 for
the relation to difference mechanisms). Arguably, there is no clear criterion that
allows distinguishing ‘‘strong’’ from ‘‘weak’’ interactions. Is it only when norms of
reaction cross, as in the rightmost graph of Fig. 1? Is it when the variance due to
GxE reaches a certain proportion of the total phenotypic variance? What is clear is
that attempting to separate the effects of genes and environment under substantialGxE is futile, since the interaction component is ultimately some unknowncombination of G and E. Given the standard model with interaction,
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Heritability
Zero correlation and interaction,
0 0.2 0.4 0.6 0.8 10
0.10.20.30.40.50.60.70.80.9
1
Heritability
Pro
b( |G
| > |E
| | P
)
Pro
b (|
G| >
|E|)
Zero correlation and interaction,
p=0
p=0.25
p=2
for varions values of P: 0...2 Averaged over P
Fig. 5 a The graphs of M0 from Eq. 17 for various values of P (positive or negative deviations) when themodel used is P = G ? E and VP = VG ? VE, i.e., no GxE interaction or G–E correlation. Theprobability curves intersect at h2 = 0.5 and the probability for p = 0 is 0.5 irrespective of h2 (compare to
Fig. 3d). b The graph of M0 from Eq. 18, the probability that |G| [ |E| averaged over the distribution of P
O. Tal
123
P = G ? E ? I, if I is large compared to G and E, our probabilistic framework
cannot capture the full relation between the latent genetic and environmental values.
On the other hand, the presence of strong G–E correlation does not pose such
difficulty. This follows from the fact that the correlation is not a separate component
of the phenotypic value, P = G ? E ? I, but only a characteristic of the joint
distribution of G and E, which is fully captured by the conditional probability
prob(|G| [ |E| | P).
Finally, an application of the probabilistic framework would involve plugging the
various estimates of variance components in the probability function M in Eq. 10, or
M in Eq. 11. For instance, if rGE = 0.3 and the variance portion due to GxE is 0.2,
then for a heritability of 0.7 the probability that |G| [ |E| is 0.8 for individuals at �SD from the population mean (see Fig. 3a). It is instructive to compare such results
with the probabilities generated from a model that ignores GxE and rGE, via M0 in
Eq. 17. Using, for comparative illustration, a phenotypic deviation of � SD and
heritability of 0.7, we have M0(0.7) = prob(|G| [ |E| | P = �) = 0.55, quite in
contrast with the higher probability M(0.7) = 0.8, based on a model with some GxEand rGE effects. In a similar fashion, one could compare the averaged probabilities
from M in Eq. 11 with M0 in Eq. 18. Such comparisons are most pertinent in the
context of studies that employ a dual design: a reduced-fit additive model that
ignores GxE and rGE effects, and a better-fit model that incorporates these effects.
The probabilities generated using variance component estimates from the better-fit
model may better describe the relative weight of genes and environment on the
deviation of a target trait from the population mean.
Acknowledgments I would like to thank Jim Tabery, Eva Jablonka, John Loehlin, John C. DeFries,
Neven Sesardic, Samir Okasha, Tamir Tassa and two anonymous reviewers for insightful feedback and
suggestions.
Appendix: Derivation of the Bivariate Normal Form for F
We wish to express F from Eq. 8 such that it complies with a bivariate normal form
Eq. 9, if indeed it can be expressed as such. Towards that end we need to find
expressions for r1, r2, l1, l2 and q12 in terms of the parameters of F. Therefore, we
write F such that terms involving powers of its independent variables g and e are
separately expressed,
Fðg; eÞ ¼ 1
2p � h � c � rE �ffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
p � eX
2�c2 1�q2ð Þh2r2E ð19Þ
where,
X ¼ �g2 1� q2� �
h2hr2E þ c2r2
E
� �� e2 1� q2
� �h2r2
E þ c2h2� �
þ ðgþ eÞ 2p 1� q2� �
h2r2E
� �� 2ge 1� q2
� �h2r2
E þ c2qhrE
� �� p2 1� q2
� �h2r2
E 1� c2� �
ð20Þ
and where rE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� h2 1� q2ð Þ � c2
p� q � h, as derived in Eq. 3.
Heritability, G 9 E and rGE
123
Equating similar terms in Eq. 9 with Eqs. 19 and 20 leads to seven simultaneous
equations for the five unknowns. For instance, equating the g2 term, we get,
�g2 1� q2ð Þh2r2E þ c2r2
E
� �2 � c2 1� q2ð Þh2r2
E
¼ �g2
2 1� q212
� �r2
1
and similarly for the terms related to e2, g, e, ge, the constant within the exponent
and the coefficient of the exponent. The mean, variance and covariance parameters
for F resulting are,
q12 ¼ S �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c2 1� q2ð Þ
1� q2ð Þr2E þ c2ð Þ 1� q2ð Þh2 þ c2ð Þ
s
where,
S ¼�1 if q[ c2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffic4þ4h2r2
E
p2hrE
0 if q ¼ c2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffic4þ4h2r2
E
p2hrE
þ1 if q\ c2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffic4þ4h2r2
E
p2hrE
8>>><>>>:
and,
r1 ¼ hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1� q2
�r2
E þ c2
q; r2 ¼ rE
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1� q2
�h2 þ c2
q;
l1 ¼p
c2
��1� q2
�r2
E þ c2�h2
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi��
1� q2�r2
E þ c2���
1� q2�h2 þ c2
�� c2
�1� q2
q �� rEh
!;
l2 ¼p
c2
��1� q2
�h2 þ c2
�r2
E
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi��
1� q2�r2
E þ c2���
1� q2�h2 þ c2
�� c2
�1� q2
�q� rEh
!;
noting that rE itself is a function of the input parameters, p, h2, c2 and q2, expressed
in Eq. 3.
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