Supporting Information
For: Robinson & Beckerman - Quantifying multivariate plasticity: genetic variation in
resource acquisition drives plasticity in resource allocation to components of life
history
The following Supporting Information contains an outline of the simulated data used, and the
results of an analysis of this data to assess bias and accuracy of both of our proposed
approaches in describing plasticity and genotype-environment interactions (GEI) for multiple
traits.
SI Methods
Data Simulation for the Matrix Comparison Approach.
Individual phenotypes were simulated by a Gaussian random process:
[S1]
𝑦(𝑥) = 𝑔(𝑥) + 𝑒(𝑥)
where g and e are independent Gaussian processes. Genetic values g, were simulated for 200
lines. Each genetic (breeding) value is then replicated four times to produce four replicates
per genetic level, and then a further j times to produce multiple measures per replicate across
j environments. To these values, environmental effects, e, were then added to each
measurement by sampling from an independent Gaussian distribution, with a mean of zero
and a variance of 2. This gave phenotypic measurements, for each trait y, on individuals from
four replicates of 200 genetic groups, each measured across j environments.
We simulated genetic effects for multiple traits across environments by sampling g from
multivariate Gaussian distribution. We specified a full genetic variance covariance matrix for
multiple traits, n, containing variance and covariance of traits within each environment Gj =
n(n+1)/2, specified in a series of diagonal blocks, and the covariance between traits across
environments Gi,Gj = n(n+1), specified in a series of off-diagonal blocks. For example, genetic
values for two traits across two environments can be simulated from a genetic variance
covariance matrix G:
𝐺=
2
𝜎𝑡1,𝑗1
𝜎𝑡1𝑗1,𝑡2𝑗2
𝜎𝑡1𝑗1,𝑡1𝑗2
𝜎𝑡1𝑗1,𝑡2𝑗2
𝜎𝑡1𝑗1,𝑡2𝑗2
𝜎𝑡1𝑗1,𝑡1𝑗2
[𝜎𝑡1𝑗1,𝑡2𝑗2
2
𝜎𝑡2,𝑗1
𝜎𝑡2𝑗1,𝑡1,𝑗2
𝜎𝑡2𝑗1,𝑡2𝑗2
𝜎𝑡1𝑡2,𝑗1
2
𝜎𝑡2,𝑗2
]
𝜎𝑡2𝑗1,𝑡1,𝑗2
𝜎𝑡2𝑗1,𝑡2𝑗2
2
𝜎𝑡1,𝑗2
𝜎𝑡1𝑗2,𝑡2,𝑗2
[S2]
2
where𝜎𝑡,𝑗
gives the genetic variance of each trait t in each environment j, and 𝜎𝑡𝑗,𝑡𝑗 gives the
covariance between traits both within and across environments. In this example there are two
environments, with the gray shaded blocks giving the variance of the two traits and their
covariance within each environment, and the off-diagonal blocks giving the covariance
between traits across environments.
Below, we outline the simulation of five scenarios for the genetic variance and covariance of
five traits across two environments (a 10x10 variance covariance matrix). As described above,
for each of these scenarios genetic values for the five traits were simulated for 200 genetic
groups, each with four replicates measured across the two environments, and environmental
effects, e, were added to each measurement by random sampling from an independent
Gaussian distribution, with a mean of zero and a variance of 2.
To these five sets of data, the matrix comparison analysis framework proposed above was
then applied to test the accuracy of this approach to describe patterns of multivariate GEI. We
used Eq. 1 of the Material and Methods to estimate the genetic variance and covariance
matrices within each environment (e.g. estimate both of the shaded blocks above), and we
then used the joint posterior estimates to compare these two matrices using Eq. 2 through 9
outlined in the Material and Methods. We also include some additional metrics within the
simulations that are not described: we estimate the number of dimensions and the
evolvability as outlined in a recent paper (Kirkpatrick 2009) and compare these approaches
to the ones we suggest within this article.
Data Simulation for the Tensor-based Approach.
Second, for the tensor-approach we specified genetic variance and covariance of multiple
traits across environments using a linear function, giving an intercept term for the average
genetic variance across environments, and a slope term for the genetic variance of the linear
change in genetic values across an environmental gradient. For example, genetic values for
two traits can be simulated across an environmental covariate from a genetic variance
covariance matrix G:
𝐺=
2
𝜎𝑡1
𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
𝜎𝑡1 𝑖𝑛𝑡,𝑠𝑙𝑜𝑝𝑒
𝜎𝑡1 𝑖𝑛𝑡,𝑠𝑙𝑜𝑝𝑒
𝜎𝑡1 𝑖𝑛𝑡,𝑡2 𝑖𝑛𝑡
𝜎
[ 𝑡1 𝑖𝑛𝑡,𝑡2 𝑠𝑙𝑜𝑝𝑒
2
𝜎𝑡1
𝑠𝑙𝑜𝑝𝑒
𝜎𝑡1𝑠𝑙𝑜𝑝𝑒,𝑡2 𝑖𝑛𝑡
𝜎𝑡1 𝑠𝑙𝑜𝑝𝑒,𝑡2 𝑠𝑙𝑜𝑝𝑒
𝜎𝑡1 𝑖𝑛𝑡,𝑡2 𝑖𝑛𝑡
𝜎𝑡1𝑠𝑙𝑜𝑝𝑒,𝑡2 𝑖𝑛𝑡
2
𝜎𝑡2𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
𝜎𝑡2 𝑖𝑛𝑡,𝑠𝑙𝑜𝑝𝑒
𝜎𝑡1 𝑖𝑛𝑡,𝑡2 𝑠𝑙𝑜𝑝𝑒
𝜎𝑡1 𝑠𝑙𝑜𝑝𝑒,𝑡2 𝑠𝑙𝑜𝑝𝑒
𝜎𝑡2 𝑖𝑛𝑡,𝑠𝑙𝑜𝑝𝑒
2
𝜎𝑡2𝑠𝑙𝑜𝑝𝑒
]
[S3]
where 𝜎𝑡2𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 gives the intercept of trait t and 𝜎𝑡2𝑠𝑙𝑜𝑝𝑒 gives the genetic variance for the
change in genentic value across an environmental covariate (genetic variance of the slope) of
trait t. The off-diagonal blocks describe the average relationship between the two traits, and a
change in that relationship across an environmental covariate.
Below, we outline the simulation of six scenarios for the genetic variance and covariance of
two traits across five environments. For each of these scenarios genetic values for the
intercept and slope of the two traits were simulated for 200 genetic groups, each with four
replicates. An environmental covariate j was created with five points (-2, -1, 0, 1, 2). The
phenotype y for each trait n, for individuals of genetic group g measured at each point i across
the environmental covariate j was then calculated as:
𝑦𝑛,𝑔,𝑖 = 𝑖𝑛𝑡𝑔 + (𝑠𝑙𝑜𝑝𝑒𝑔 × 𝑗𝑖 ) + 𝑒
[S4]
where 𝑖𝑛𝑡𝑔 is the intercept value for line g, 𝑠𝑙𝑜𝑝𝑒𝑔 is the slope value for line g, 𝑗𝑖 is the value
for the environmental covariate at point i, and e is an environmental (residual deviate) effect
added to each measurement by random sampling from an independent Gaussian distribution,
with a mean of zero and a variance of 2.
To these data, the tensor-based analysis framework proposed above was then applied to test
the accuracy of this approach to describe patterns of multivariate GEI across a series of
environments. We used Eq. 10 of the Material and Methods to estimate the genetic variance
and covariance between the two traits within each of the five environments, and we then used
the joint posterior estimates within Eq. 11 and 12 of the Material and Methods to describe the
patterns of multivariate GEI with a geometric approach.
This approach simulates data that fits a ‘reaction norm’ where GEI is simulated as a slope,
with genetic variance responding linearly to a change in an environmental covariate. This
type of data is typically analysed using a ‘reaction norm’ model, such as a random regression.
Note that the tensor-based approach is not restricted to the analysis of data where GEI can be
described as a function, but for simplicity and as GEI is often considered as a reaction norm,
we restrict our analyses presented here to this scenario.
SI Results and Discussion
Results of Simulations for the Matrix Comparison Approach.
First simulation scenario. We simulated a variance of 1 for each trait within each of the two
environments, and a covariance of 0.8 between traits both within and across environments.
The following G matrix was simulated:
1.0
0.8
0.8
0.8
0.8
𝑮𝟏 =
0.8
0.8
0.8
0.8
[ 0.8
0.8 0.8
1.0 0.8
0.8 1.0
0.8 0.8
0.8 0.8
0.8 0.8
0.80.8
0.80.8
0.80.8
0.80.8
0.8 0.8
0.8 0.8
0.8 0.8
1.0 0.8
0.8 1.0
0.8 0.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
1.00.8
0.81.0
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
0.80.8
1.00.8
0.81.0
0.80.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
1.0 ]
where the gray boxes show the within-environment diagonal blocks. Table S1 gives the
results of the analysis of this simulated data, which reveals that none of the derived statistics
provide any support for GEI.
Second simulation scenario. We simulated a variance of 1 for each trait within each of the
two environments, and a covariance of 0.8 between traits within environments, but a
covariance of -0.8 across environments. The following G matrix was simulated:
1.0
0.8
0.8
0.8
0.8
𝑮𝟐 =
−0.8
−0.8
−0.8
−0.8
[−0.8
0.8 0.8
1.0 0.8
0.8 1.0
0.8 0.8
0.8 0.8
−0.8−0.8
−0.8−0.8
−0.8−0.8
−0.8−0.8
−0.8−0.8
0.8 0.8 −0.8−0.8 −0.8−0.8 −0.8
0.8 0.8 −0.8−0.8 −0.8−0.8 −0.8
0.8 0.8 −0.8−0.8 −0.8−0.8 −0.8
1.0 0.8 −0.8−0.8 −0.8−0.8 −0.8
0.8 1.0 −0.8 −0.8 −0.8 −0.8 −0.8
−0.8−0.8
1.0 0.8
0.8 0.8
0.8
−0.8−0.8
0.8 1.0
0.8 0.8
0.8
−0.8−0.8
0.8 0.8
1.0 0.8
0.8
−0.8−0.8
0.8 0.8
0.8 1.0
0.8
−0.8−0.8
0.8 0.8
0.8 0.8
1.0]
where the gray boxes show the within-environment diagonal blocks.
Table S1 gives the results of the analysis of this simulated data, which reveals that none of the
derived statistics provide any support for multivariate GEI. This is expected as the estimates
of the two block diagonals, should be identical. The cross-environment correlations represent
perfect crossing over of the “reaction norms” of each trait, however the negative covariance is
close to -1 and thus the traits have the same genetic basis across environments, but the effects
of the underlying causal variants are in an opposing direction. Even though there is plasticity
for each individual trait, all traits respond in the same way, resulting in constant multivariate
trait variance and covariance within environments.
Third simulation scenario. We simulated a variance of 1 for each trait within the first
environment, and then increasing variance across traits within the second environment. We
simulated the covariance both within and across environments to give a constant correlation
of 0.8. This provides an example where genetic variance increases across environments, but
genetic relationships among traits remain constant. The following G matrix was simulated:
1.000 0.800 0.800 0.800 0.800 0.800 0.980 1.131 1.265 1.386
0.800 1.000 0.800 0.800 0.800 0.800 0.980 1.131 1.265 1.386
0.800 0.800 1.000 0.800 0.800 0.800 0.980 1.131 1.265 1.386
0.800 0.800 0.800 1.000 0.800 0.800 0.980 1.131 1.265 1.386
0.800 0.800 0.800 0.800 1.000 0.800 0.980 1.131 1.265 1.386
𝑮𝟑 =
0.800 0.800 0.800 0.800 0.800 1.000 0.980 1.131 1.265 1.386
0.980 0.980 0.980 0.980 0.980 0.980 1.500 1.386 1.549 1.697
1.131 1.131 1.131 1.131 1.131 1.131 1.386 2.000 1.789 1.960
1.265 1.265 1.265 1.265 1.265 1.265 1.549 1.789 2.500 2.191
[1.386 1.386 1.386 1.386 1.386 1.386 1.697 1.960 2.191 3.000]
where the gray boxes show the within-environment diagonal blocks.
Table S1 gives the results of the analysis of this simulated data, which reveals a substantial
increase in the volume of the G in the second environment reflecting the simulated pattern.
The underlying genetic basis of the multivariate phenotype has changed from environment 1
to two (Ovaskainen’s D), suggesting that additional allelic effects in environment 2 that are
not present in environment 1.
Fourth simulation scenario. We simulated a variance of 1 for each trait and a constant
covariance of 0.8 among traits within the first environment. For the second environment, we
simulated increasing variance across traits, and changing covariance among traits, which also
created changing covariance among traits across environments. This provides an example
where genetic variance increases across environments, genetic relationships among traits
also vary. The following G matrix was simulated:
1.0 0.8 0.8 0.8 0.8 0.8 0.6 0.4 0.2 0.0
0.8 1.0 0.8 0.8 0.8 0.8 0.6 0.4 0.2 0.0
0.8 0.8 1.0 0.8 0.8 0.8 0.6 0.4 0.2 0.0
0.8 0.8 0.8 1.0 0.8 0.8 0.6 0.4 0.2 0.0
0.8 0.8 0.8 0.8 1.0 0.8 0.6 0.4 0.2 0.0
𝑮𝟒 =
0.8 0.8 0.8 0.8 0.8 1.0 0.6 0.4 0.2 0.0
0.6 0.6 0.6 0.6 0.6 0.6 1.5 0.8 0.4 0.2
0.4 0.4 0.4 0.4 0.4 0.4 0.8 2.0 1.0 0.4
0.2 0.2 0.2 0.2 0.2 0.2 0.4 1.0 2.5 1.2
[0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.4 1.2 3.0]
where the gray boxes show the within-environment diagonal blocks.
Table S1 gives the results of the analysis of this simulated data, which reveals changes in
variance and covariance across the two environments. As expected, the number of vectors
differs between the environments. This is associated with a significant Ovaskainen’s D, a
rotation of the major axis of variation, a decrease in variance explained by Gmax in
environment 2, higher genetic variation overall, and a shift in the trait membership linked to
the major axis in each environment. The methods thus suggest allelic sensitivity to the
environment and environment specific loci which alter the covariance between traits across
environments.
Fifth simulation scenario. We simulated a variance of 1 for each trait, but different
relationships between traits within the first environment. For the second environment, there
was increasing variance across traits, and changing covariance among traits, which also
created changing covariance among traits across environments. This provides an example
where genetic variance increases across environments, genetic relationships among traits
also vary. The following G matrix was simulated:
1.0 0.6 0.4 0.2 0.0 − 0.8 − 0.6 − 0.4 − 0.2
0.6 1.0 0.6 0.4 0.2 − 0.6 − 0.8 − 0.6 − 0.4
0.4 0.6 1.0 0.6 0.4 − 0.4 − 0.6 − 0.8 − 0.6
0.2 0.4 0.6 1.0 0.6 − 0.2 − 0.4 − 0.6 − 0.8
0.0 0.2 0.4 0.6 1.0 0.0 − 0.2 − 0.4 − 0.6
𝑮𝟓 =
−0.8 − 0.6 − 0.4 − 0.2 0.0 1.0 0.6 0.4 0.2
−0.6 − 0.8 − 0.6 − 0.4 − 0.2 0.6 1.5 1.0 0.4
−0.4 − 0.6 − 0.8 − 0.6 − 0.4 0.4 1.0 2.0 1.2
−0.2 − 0.4 − 0.6 − 0.8 − 0.6 0.2 0.4 1.2 2.5
[ 0.0 − 0.2 − 0.4 − 0.6 − 0.8 0.0 0.2 0.4 1.4
0.0
− 0.2
− 0.4
− 0.6
− 0.8
0.0
0.2
0.4
1.4
3.0 ]
where the gray boxes show the within-environment diagonal blocks.
Table S1 gives the results of the analysis of this simulated data, which reveals volume
differences, a significant Ovaskainen’s D, a significant difference in angle, accompanied by a
change in the traits (TCI) and estimation of two vectors in each environment. As in Scenario 4,
the methods detect both differences in variance and changes in covariance across
environments, but in this scenario there is the detection of two significant vectors of genetic
variation in each environment, whose variance and orientation differ across environments.
Summary of the matrix comparison results. We found that the matrix comparison approach
outlined in the Methods generally described changes in the within- and across-environment
variance and covariance accurately. Therefore, this approach can be applied to statistically
test for changes in both multivariate variance and covariance across discrete treatments or
environments. As the approach only requires estimation of the block-diagonal, it may thus be
advantageous for examining plasticity of multiple traits across discrete treatment groups, or
environments, whilst avoiding having to estimate the full across-environment G matrix.
Results of Simulations for the Tensor-based Approach.
First simulation scenario. We simulated data where there was no GEI for either of two traits,
and a genetic covariance of zero between traits. The following G matrix was used:
1.00
𝐺1 = [0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
0.00
0.00
0.00]
0.00
0.00
where an intercept variance of 1 is simulated for each trait, and a slope variance of zero. This
gives a constant genetic variance of 1 across the five environments, with constant differences
between genetic groups for each trait, and a consistent relationship of zero between traits.
Table S2 gives the results of the tensor-based analysis of this simulated data, which reveals
that the traits contribute in independent directions toward the multivariate variance, but that
there was no significant change in genetic variance across environments for either trait
relative to the other, and no change in the genetic variance of the first or second eigentensor
across environments. These results indicate no multivariate GEI across environments, as
expected from the simulated data.
Second simulation scenario. We simulated data where there was GEI for the first trait but no
GEI for the second trait, and a constant genetic covariance of zero between traits across
environments. The following G matrix was used:
1.00
𝐺2 = [0.00
0.00
0.00
0.00
0.50
0.00
0.00
0.00
0.00
1.00
0.00
0.00
0.00]
0.00
0.00
where an intercept variance of 1 is simulated for each trait, and a slope variance of 0.5 for the
first trait and zero for the second trait. For the first trait, this gives greater genetic variance at
either end of the distribution and different slope among genetic groups across environments,
with no covariance between intercept and slope. The second trait had a constant genetic
variance of 1 across the five environments, with constant differences between genetic groups.
Table S2 gives the results of the tensor-based analysis of this simulated data, which reveals a
significant change in genetic variance for the first trait relative to the second. There was
evidence that the genetic variance of the first vector of multivariate genetic variance changed
across environments, and that this was driven entirely by the first trait. This was evidenced
by non-overlapping proportions of variance in the first eigentensor explained by each trait,
with the first trait explaining most of the variance. In contrast, there was no significant change
in genetic variance for the second eigentensor across environments. Therefore this approach
can accurately partition GEI to the appropriate trait involved.
Third simulation scenario. We simulated data where there was identical patterns of GEI for
both traits, and a consistent genetic covariance of zero between traits across environments.
The following G matrix was used:
1.00
𝐺3 = [0.00
0.00
0.00
0.00
0.50
0.00
0.00
0.00
0.00
1.00
0.00
0.00
0.00]
0.00
0.50
where an intercept variance of 1 is simulated for each trait, and a slope variance of 0.5 both
traits. This gives greater genetic variance at either end of the distribution and variable
differences between genetic groups across environments, with no covariance between
intercept and slope for both traits. This gives GEI for both traits, where although the patterns
of variance change are the same for both traits, there is no relationship between them.
Table S2 gives the results of the tensor-based analysis of this simulated data, which reveals
that there is no change in the variance of the traits relative to each other, and that both traits
contribute independently but equally to the first eigentensor. This is evidenced by
overlapping confidence intervals in both the change in genetic variance and the proportion of
variance explained by each trait for the first eigentensor. There was a significant change in
variance of the first eigentensor across environments, revealing one axis of genetic change
across environments to which two traits contribute equally across environments, with the
covariance between them also remaining constant.
Fourth simulation scenario. We simulated data where there was GEI for both traits, and a
relationship between traits that was environmentally dependent. The following G matrix was
used:
1.00
𝐺4 = [0.00
0.70
0.44
0.00
0.50
0.44
0.20
0.70
0.44
1.00
0.00
0.44
0.20]
0.00
0.50
where an intercept variance of 1 and a slope variance of 0.5 is simulated for each trait. This
gives genetic variance covariance matrix for each trait across five environments as:
3.00
2.00
𝐺𝑡 =
1.00
0.00
[ −1.00
…
2.00
1.50
1.00
0.00
…
…
1.00
1.50
1.00
…
…
…
2.00
2.00
…
…
…
…
3.00 ]
where the genetic variance of each trait t at each environment is given on the diagonal and the
genetic covariance between measures across environments is given on the off-diagonal. In
this scenario, both traits followed the same pattern shown in Gt, where there is increasing
genetic variance at the ends of the distribution, and decreasing genetic covariance between
measures at either end of the distribution. G4 gave the following pattern of genetic
correlations between the traits across environments:
𝐺𝑡1,𝑡2
−0.020
−0.057
= −0.087
−0.113
[ −0.100
…
…
0.047 …
0.212 0.700
0.300 0.931
0.302 0.912
…
…
…
…
…
…
0.995
…
0.989 0.998 ]
where the genetic correlation between trait 1 and trait 2 across the five environments is given
on the diagonal, and the off-diagonal give the correlations between traits where measures
were made in different environments. This gave changing genetic covariance between traits
across environments from around zero to around unity. For example, the correlation between
traits in environment 5 is -0.020 where as the correlation between traits in environment 1 is
0.998.
In this scenario there were two non-zero eigentensors which explained 84.634% (71.226,
91.086) and 15.619% (8.587, 28.534) of the variance respectively. Table S2 gives the results
of the tensor-based analysis of this simulated data, which reveals that both traits contribute
towards the variance of the first tensor, which has greater variance in environments four and
five. This suggests that the first tensor represents the identical patterns of variance in both
traits across environments. Both traits also contribute towards the second eigentensor, which
has the greatest variance in environment 1, suggesting that this axis represents trait variance
which results in changing genetic covariance between traits across environments. Therefore
there are two dimensions of multivariate genetic variation, the first representing an identical
change in both traits, and the second representing changing covariance between traits.
Fifth simulation scenario. We simulated data where there was GEI for both traits, where the
following G matrix was used:
1.00 −0.30
−0.30
0.50
𝐺5 = [
−0.70 −0.44
−0.44 0.35
−0.70
−0.44
1.00
0.30
−0.44
0.35 ]
0.30
0.50
where an intercept variance of 1 and a slope variance of 0.5 is simulated for each trait, which
gives genetic variance covariance matrix for trait one across five environments as:
𝐺𝑡1
and for trait 2:
4.20 2.90
2.90 2.10
=
1.60 1.30
0.30 0.50
[ −1.00 − 0.30
1.60
1.30
1.00
0.70
0.40
0.30 − 1.00
0.50 − 0.30
0.70 0.40
0.90 1.10
1.10 1.80 ]
𝐺𝑡2
1.80 1.10
1.10 0.90
=
0.40 0.70
0.30 0.50
[ −1.00 − 0.30
0.40 − 0.30 − 1.00
0.70 0.50 − 0.30
1.00 1.30 1.60
1.30 2.10 2.90
1.60 2.90 4.20 ]
where the genetic variance of the trait at each environment is given on the diagonal and the
genetic covariance between measures across environments is given on the off-diagonal. Both
traits followed an opposing pattern of (co)variance change across environments, where there
is increasing genetic variance at opposing ends of the distribution, and decreasing genetic
covariance between measures at either end of the distribution. G5 gave the following pattern
of genetic correlations between the two traits:
𝐺𝑡1,𝑡2
0.895
…
…
0.679
0.386
…
=
0.087 − 0.179 − 0.700
−0.323 − 0.500 − 0.987
[ −0.500 − 0.620 − 0.771
…
…
…
− 0.895
− 0.879
…
…
…
…
− 0.668 ]
where the genetic correlation between trait 1 and trait 2 across the five environments is given
on the diagonal, and the off-diagonal give the correlations between traits where measures
were made in different environments. This gave changing genetic covariance between traits
across environments from positive to negative. For example, the correlation between traits in
environment 5 is -0.468 where as the correlation between traits in environment 1 is 0.895.
In this scenario there were two non-zero eigentensors which explained 70.072% (61.589,
77.746) and 30.295% (21.865, 38.182) of the variance respectively. Table S2 gives the results
of the tensor-based analysis of this simulated data, which reveals that the first eigentensor
shows consistent variance across environments, with the second eigentensor representing
the changes in (co)variance that occur across environments. Both traits contribute to each
eigentensor in opposing directions. This suggests, that multivariate genetic variation can be
described by one dimension of consistent variance, representing the strongly negative and
positive associations at either ends of the environmental distribution, and a second dimension
of changing variance representing the weaker correlations between the first trait in
environment 1 and trait 2 measured in environments 2 through 5.
Sixth simulation scenario. We simulated data where there was GEI for only the first trait,
which created changing covariance between traits. The following G matrix was used:
1.00 −0.30
𝐺6 = [−0.30 0.50
−0.70 −0.44
−0.44 0.35
−0.70
−0.44
1.00
0.00
−0.44
0.35 ]
0.00
0.00
where an intercept variance of 1 and a slope variance of 0.5 is simulated for the first trait, and
just an intercept variance of 1 simulated for the second trait. This gives GEI for the first trait,
but no GEI for the second trait whose variance will be constant 1 across environments with a
constant covariance of 1. G4 gave the following pattern of genetic correlations between the
two traits:
𝐺𝑡1,𝑡2
0.995
…
…
0.911
0.366
…
=
0.180 − 0.179 − 0.700
−0.468 − 0.500 − 0.987
[ −0.995 − 0.996 − 0.999
…
…
…
− 0.984
− 0.979
…
…
…
…
− 0.790 ]
where the genetic correlation between trait 1 and trait 2 across the five environments is given
on the diagonal, and the off-diagonal give the correlations between traits where measures
were made in different environments. This gave changing genetic covariance between traits
across environments from positive to negative, driven only by GEI for the first trait.
In this scenario there were two non-zero eigentensors which explained 97.561% (90.423,
99.486) and 2.295% (0.324, 8.503) of the variance respectively, suggesting one predominant
axis of multivariate genetic variance. Table S2 gives the results of the tensor-based analysis of
this simulated data, which reveals that greater than 99% of the variance of the first
eigentensor can be attributed to the first trait, and that the genetic variance of this
eigentensor changes across environments. This suggests therefore, that all of the changes in
the multivariate genetic variance across environments can be attributed to the first trait,
accurately reflecting the dimensionality of the simulated scenario.
Summary of the tensor results. We found that the tensor-based approach outlined in the
Methods generally described changes in the within- and across-environment variance and
covariance accurately. As we demonstrate above, this approach can be applied to statistically
test for changes in both multivariate variance and covariance across a continuous gradient,
whilst accurately describing the dimensionality of multivariate GEI, and partitioning the
variance of the axes into components attributed by each trait. As the approach only requires
estimation of the block-diagonal at each point of measurement, it may be advantageous for
examining plasticity of multiple traits across gradients, whilst avoiding having to estimate the
full within- and across-environment G matrix, or avoiding the use of functions to describe
(co)variance changes.
Table S1. Matrix comparison statistics for the five simulated data scenarios.
Table S2.Eigentensor statistics for the five simulated data scenarios.
Table S3. Multivariate GEI in crickets, Gryllus firmus, across resource environments described by estimates of the genetic correlations
across treatments.
Traits
Low food
Medium food
High food
Reproductive allocation – Flight allocation
-0.618 (-0.829, -0.215)
-0.429 (-0.778, -0.231)
-0.680 (-0.846, -0.378)
Trade-off acquisition – Flight allocation
0.666 (0.348, 0.869)
0.314 (-0.251, 0.769)
-0.514 (-0.776, -0.114)
Trade-off allocation – Flight allocation
0.915 (0.734, 0.973)
0.798 (0.206, 0.927)
0.836 (0.628, 0.911)
Total acquisition – Flight allocation
-0.089 (-0.472, 0.247)
-0.205 (-0.650, 0.174)
-0.449 (-0.755, -0.122)
Trade-off acquisition - Reproductive allocation
-0.180 (-0.455, 0.450)
0.566 (-0.405, 0.415)
0.871 (0.674, 0.941)
Trade-off allocation – Reproductive allocation
-0.816 (-0.939, -0.580)
-0.894 (-0.967, 0.367)
-0.963 (-0.984, -0.898)
Total acquisition – Reproductive allocation
0.239 (-0.173, 0.589)
-0.199 (-0.805, 0.394)
0.511 (0.184, 0.729)
Trade-off allocation – Trade-off acquisition
0.426 (-0.048, 0.765)
-0.106 (-0.448, 0.471)
-0.814 (-0.941, -0.591)
Total acquisition – Trade-off acquisition
0.579 (0.301, 0.799)
0.561 (0.266, 0.816)
0.765 (0.634, 0.908)
Total acquisition – Trade-off allocation
-0.071 (-0.511, 0.247)
-0.138 (-0.596, 0.531)
-0.489 (-0.745, -0.223)
Posterior mode for the pairwise genetic correlations between multiple traits across food treatments, along with their 95% credible intervals
shown in brackets. Values in bold indicate correlations whose 95% credible intervals do not overlap across treatments.
Figure S1. Subspace representations of the differences across environments between the first (red) and second (blue)
within-environment genetic variance covariance matrices simulated for each of five scenarios. The phenotypic
covariance matrices are shown for scenario 4, which is used as an example in Box1.
Scenario 1
Scenario 2
Scenario 3
Scenario 4 – Genetic (co)variance
Scenario 4 – Phenotypic (co)variance
Scenario 5