GxE Testing
Alexia Jolicoeur-Martineau
2017-12-08
GxE testing
From version 1.2.0 of the LEGIT package, we introduce a function to test for different types of gene-by-environment interactions (GxE) as per Belsky et al. (2013). See our preprint: https://psyarxiv.com/27uw8. We want to test if the GxE reflects diathesis stress, differential susceptibility or vantage sensitivity.
The figure below is a graphical representation of the different types of interactions assuming a STRONG model:
knitr::include_graphics("https://raw.githubusercontent.com/AlexiaJM/LEGIT/master/images/GxE_testing_strong.png")
The figure below is a graphical representation of the different types of interactions assuming a WEAK model:
knitr::include_graphics("https://raw.githubusercontent.com/AlexiaJM/LEGIT/master/images/GxE_testing_weak.png")
To test for differential susceptibility, Widaman et al. (2013) estimate the cross-over point of the model. This can be done easily with LEGIT by simply adding a negative intercept to the environment E. This can be seen by the following: \[y = 2 + 3(E-c) + 5G(E-c) \\ E = q_1e_1 + q_2e_2 + ... + q_le_l\] is equivalent to \[y = 2 + 3E' + 5GE' \\ E' = -c + q_1e_1 + q_2e_2 + ... + q_le_l\]
To test for vantage sensitivity and diathesis-stress, we can use the model above but fix the cross-over point to the expected minimum and maximum of \(E\) respectively.
Simulation of diathesis-stress, vantage sensitivity and differential susceptibility
Let’s look at a two-way interaction model with a cross-over point and a continuous outcome:
\[\mathbf{g}_j \sim Binomial(n=1,p=.30) \\ j = 1, 2, 3, 4\] \[\mathbf{e}_l \sim max(Poisson(\mu=0,\sigma=4), 10) \\ l = 1, 2, 3\] \[\mathbf{g} = .30\mathbf{g}_1 + .10\mathbf{g}_2 + .20\mathbf{g}_3 + .40\mathbf{g}_4 \] \[ \mathbf{e} = .45\mathbf{e}_1 + .35\mathbf{e}_2 + .20\mathbf{e}_3\] \[\mathbf{\mu} = 3 + \beta_E(\mathbf{e}-c) + 2\mathbf{g}(\mathbf{e}-c) \] where \(c\) and \(\beta_E\) depend on the model assumed. \(c = 0\) represents vantage sensitivity, \(c = 10\) represents diathesis-stress. We set \(c=5\) to represents differential susceptibility. \(\beta_E = 0\) assumes a STRONG model while \(\beta_E \ne 0\) assumes a WEAK model.
\[ y \sim Normal(\mu,\sigma=1)\]
Note that the environmental score E is in the range [0,10].
Example 1: Vantage sensitivity WEAK
When assuming vantage sensitivity WEAK, we let \(c=0\) so that: \[\mathbf{\mu} = 3 + (\mathbf{e}-0) + 2\mathbf{g}(\mathbf{e}-0) \]
We generate N=250 observations and test all 4 possible models based on the BIC (other choices are also available: AIC, AICc, cross-validation and cross-validated AUC). It is important to not include G or E in the model formula because they will be added automatically. We start by assuming that we don’t know the true minimum of the environmental score, thus we use option crossover = c(“min”,“max”) to find the minimum/maximum from the observed data (this is the default).
set.seed(777)
library(LEGIT)
## Loading required package: formula.tools
## Loading required package: operator.tools
ex_dia = example_with_crossover(N=250, c=0, coef_main = c(3,1,2), sigma=1, seed=7)
GxE_test_BIC = GxE_interaction_test(data=ex_dia$data, genes=ex_dia$G, env=ex_dia$E, formula_noGxE = y ~ 1, crossover = c("min","max"), criterion="BIC")
GxE_test_BIC$results
## BIC crossover crossover 95%
## Vantage sensitivity WEAK "772.27" "0.45" ""
## Differential susceptibility WEAK "775.08" "-0.4" "( -0.73 / -0.07 )"
## Differential susceptibility STRONG "870.68" "2.22" "( 1.79 / 2.64 )"
## Vantage sensitivity STRONG "930.4" "0.44" ""
## Diathesis-stress WEAK "971.12" "9.75" ""
## Diathesis-stress STRONG "1123.61" "9.74" ""
## Within observable range?
## Vantage sensitivity WEAK ""
## Differential susceptibility WEAK "No"
## Differential susceptibility STRONG "No"
## Vantage sensitivity STRONG ""
## Diathesis-stress WEAK ""
## Diathesis-stress STRONG ""
## % of observations below crossover
## Vantage sensitivity WEAK "0"
## Differential susceptibility WEAK "0"
## Differential susceptibility STRONG "0.004"
## Vantage sensitivity STRONG "0"
## Diathesis-stress WEAK "1"
## Diathesis-stress STRONG "1"
Here we see the results. Notice that vantage sensitivity WEAK is the model with the lowest BIC. Differential susceptibility WEAK has a slightly lower fit and the crossover point is not within observable range, thus we reject it. Note that we only show the element results of the output, but the 6 models (weak/strong vantage sensitivity, diathesis-stress and differential susceptibility) are also returned.
Considering that we know that the minimum and maximum of E are 0 and 10 respectively, we can also refit the models with crossover = c(0, 10).
GxE_test_BIC_ = GxE_interaction_test(data=ex_dia$data, genes=ex_dia$G, env=ex_dia$E, formula_noGxE = y ~ 1, crossover = c(0, 10), criterion="BIC")
GxE_test_BIC_$results
## BIC crossover crossover 95%
## Vantage sensitivity WEAK "770.07" "0" ""
## Differential susceptibility WEAK "775.08" "-0.4" "( -0.73 / -0.07 )"
## Differential susceptibility STRONG "870.68" "2.22" "( 1.79 / 2.64 )"
## Vantage sensitivity STRONG "948.33" "0" ""
## Diathesis-stress WEAK "965.08" "10" ""
## Diathesis-stress STRONG "1125.2" "10" ""
## Within observable range?
## Vantage sensitivity WEAK ""
## Differential susceptibility WEAK "No"
## Differential susceptibility STRONG "No"
## Vantage sensitivity STRONG ""
## Diathesis-stress WEAK ""
## Diathesis-stress STRONG ""
## % of observations below crossover
## Vantage sensitivity WEAK "0"
## Differential susceptibility WEAK "0"
## Differential susceptibility STRONG "0.004"
## Vantage sensitivity STRONG "0"
## Diathesis-stress WEAK "1"
## Diathesis-stress STRONG "1"
This gave the same results because it already estimated properly the minimum and maximum environmental score, but it might not always.
We can then plot the best model:
plot(GxE_test_BIC$fits$vantage_sensitivity_WEAK, xlim=c(0,10), ylim=c(3,13),cex.leg=1.4, cex.axis=1.5, cex.lab=1.5)
Example 2: Vantage sensitivity STRONG
When assuming vantage sensitivity STRONG, we let \(c=0\) and \(\beta_E=0\) so that: \[\mathbf{\mu} = 3 + 2\mathbf{g}(\mathbf{e}-0) \]
We generate N=250 observations and test all 6 possible models based on the BIC assuming we don’t know what is the minimum environmental score.
ex_dia_s = example_with_crossover(N=250, c=0, coef_main = c(3,0,2), sigma=1, seed=7)
GxE_test_BIC = GxE_interaction_test(data=ex_dia_s$data, genes=ex_dia_s$G, env=ex_dia_s$E, formula_noGxE = y ~ 1, crossover = c("min","max"), criterion="BIC")
GxE_test_BIC$results
## BIC crossover crossover 95%
## Differential susceptibility STRONG "770.17" "-0.5" "( -1.22 / 0.23 )"
## Vantage sensitivity STRONG "771.77" "0.45" ""
## Vantage sensitivity WEAK "772.93" "0.45" ""
## Differential susceptibility WEAK "775.68" "-0.45" "( -1.18 / 0.27 )"
## Diathesis-stress WEAK "967.38" "9.74" ""
## Diathesis-stress STRONG "1104.84" "9.75" ""
## Within observable range?
## Differential susceptibility STRONG "No"
## Vantage sensitivity STRONG ""
## Vantage sensitivity WEAK ""
## Differential susceptibility WEAK "No"
## Diathesis-stress WEAK ""
## Diathesis-stress STRONG ""
## % of observations below crossover
## Differential susceptibility STRONG "0"
## Vantage sensitivity STRONG "0"
## Vantage sensitivity WEAK "0"
## Differential susceptibility WEAK "0"
## Diathesis-stress WEAK "1"
## Diathesis-stress STRONG "1"
We conclude that vantage sensitivity STRONG is the best model.
We can then plot the best model:
plot(GxE_test_BIC$fits$vantage_sensitivity_STRONG, xlim=c(0,10), ylim=c(3,13),cex.leg=1.4, cex.axis=1.5, cex.lab=1.5)
Example 3: Differential susceptibility WEAK
When assuming differential susceptibility WEAK, we let \(c=5\) (and use 8 instead of 3 as intercept to keep a similar outcome range) so that: \[\mathbf{\mu} = 8 + (\mathbf{e}-5) + 2\mathbf{g}(\mathbf{e}-5) \]
We generate N=250 observations and test all 6 possible models based on the BIC assuming we don’t know what is the minimum environmental score.
ex_ds = example_with_crossover(N=250, c=5, coef_main = c(3+5,1,2), sigma=1, seed=7)
GxE_test_BIC = GxE_interaction_test(data=ex_ds$data, genes=ex_ds$G, env=ex_ds$E, formula_noGxE = y ~ 1, crossover = c("min","max"), criterion="BIC")
GxE_test_BIC$results
## BIC crossover crossover 95%
## Differential susceptibility WEAK "776.06" "4.83" "( 4.5 / 5.16 )"
## Diathesis-stress WEAK "834.53" "9.74" ""
## Vantage sensitivity WEAK "835.05" "0.45" ""
## Differential susceptibility STRONG "864.07" "4.78" "( 4.37 / 5.2 )"
## Diathesis-stress STRONG "869.9" "4.49" ""
## Vantage sensitivity STRONG "1118.98" "0.18" ""
## Within observable range?
## Differential susceptibility WEAK "Yes"
## Diathesis-stress WEAK ""
## Vantage sensitivity WEAK ""
## Differential susceptibility STRONG "Yes"
## Diathesis-stress STRONG ""
## Vantage sensitivity STRONG ""
## % of observations below crossover
## Differential susceptibility WEAK "0.488"
## Diathesis-stress WEAK "1"
## Vantage sensitivity WEAK "0"
## Differential susceptibility STRONG "0.476"
## Diathesis-stress STRONG "0.396"
## Vantage sensitivity STRONG "0.088"
We conclude that differential susceptibility WEAK is the best model.
We can then plot the best model:
plot(GxE_test_BIC$fits$diff_suscept_WEAK, xlim=c(0,10), ylim=c(3,13),cex.leg=1.4, cex.axis=1.5, cex.lab=1.5)
Example 4: Differential susceptibility STRONG
When assuming differential susceptibility STRONG, we let \(c=5\) and \(\beta_E=0\) (and use 8 instead of 3 as intercept to keep a similar outcome range) so that: \[\mathbf{\mu} = 8 + 2\mathbf{g}(\mathbf{e}-5) \]
We generate N=250 observations and test all 6 possible models based on the BIC assuming we don’t know what is the minimum environmental score.
ex_ds_s = example_with_crossover(N=250, c=5, coef_main = c(3+5,0,2), sigma=1, seed=7)
GxE_test_BIC = GxE_interaction_test(data=ex_ds_s$data, genes=ex_ds_s$G, env=ex_ds_s$E, formula_noGxE = y ~ 1, crossover = c("min","max"), criterion="BIC")
GxE_test_BIC$results
## BIC crossover crossover 95%
## Differential susceptibility STRONG "771.2" "4.84" "( 4.13 / 5.55 )"
## Differential susceptibility WEAK "776.64" "4.84" "( 4.12 / 5.55 )"
## Diathesis-stress STRONG "784.55" "4.16" ""
## Diathesis-stress WEAK "834.4" "9.74" ""
## Vantage sensitivity WEAK "834.9" "0.44" ""
## Vantage sensitivity STRONG "886.96" "0.16" ""
## Within observable range?
## Differential susceptibility STRONG "Yes"
## Differential susceptibility WEAK "Yes"
## Diathesis-stress STRONG ""
## Diathesis-stress WEAK ""
## Vantage sensitivity WEAK ""
## Vantage sensitivity STRONG ""
## % of observations below crossover
## Differential susceptibility STRONG "0.484"
## Differential susceptibility WEAK "0.484"
## Diathesis-stress STRONG "0.312"
## Diathesis-stress WEAK "1"
## Vantage sensitivity WEAK "0"
## Vantage sensitivity STRONG "0.104"
We conclude that differential susceptibility STRONG is the best model.
We can then plot the best model:
plot(GxE_test_BIC$fits$diff_suscept_STRONG, xlim=c(0,10), ylim=c(3,13),cex.leg=1.4, cex.axis=1.5, cex.lab=1.5)
