Author: Alexia Jolicoeur-Martineau
Date: 2024-01-23

# GxE Testing

#### Alexia Jolicoeur-Martineau

#### 2024-01-23

Note that you can cite this work as:

Jolicoeur-Martineau, A., Wazana, A., Szekely, E., Steiner, M., Fleming, A. S., Kennedy, J. L., Meaney M. J. & Greenwood, C.M. (2017). Alternating optimization for GxE modelling with weighted genetic and environmental scores: examples from the MAVAN study. arXiv preprint arXiv:1703.08111.

Jolicoeur-Martineau, A., Belsky, J., Szekely, E., Widaman, K. F., Pluess, M., Greenwood, C., & Wazana, A. (2017). Distinguishing differential susceptibility, diathesis-stress and vantage sensitivity: beyond the single gene and environment model. arXiv preprint arXiv:1712.04058.

## 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://osf.io/preprints/psyarxiv/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:

\[\mathbf{g}_j \sim Binomial(n=1,p=.30) \\ j = 1, 2, 3, 4\] \[\mathbf{e}_l \sim 10 Beta(\alpha=2,\beta=2) \\ l = 1, 2, 3\] \[\mathbf{g} = .30\mathbf{g}_1 + .10\mathbf{g}_2 + .20\mathbf{g}_3 + .40\mathbf{g}_4 \] \[ \mathbf{e} = .45\mathbf{e}_1 + .35\mathbf{e}_2 + .20\mathbf{e}_3\] \[\mathbf{\mu} = 3 + \beta_E(\mathbf{e}-c) + 2\mathbf{g}(\mathbf{e}-c) \] 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)
    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")

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    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"    ""
    ## G + E only                         "1237.56" NA        ""
    ## G only                             "1285.07" NA        ""
    ## E only                             "1308.15" NA        ""
    ## Intercept only                     "1341.59" NA        ""
    ##                                    Within observable range?
    ## Vantage sensitivity WEAK           ""
    ## Differential susceptibility WEAK   "No"
    ## Differential susceptibility STRONG "No"
    ## Vantage sensitivity STRONG         ""
    ## Diathesis-stress WEAK              ""
    ## Diathesis-stress STRONG            ""
    ## G + E only                         ""
    ## G only                             ""
    ## E only                             ""
    ## Intercept only                     ""
    ##                                    % of observations below crossover
    ## Vantage sensitivity WEAK           "0"
    ## Differential susceptibility WEAK   "0"
    ## Differential susceptibility STRONG "0.376"
    ## Vantage sensitivity STRONG         "0"
    ## Diathesis-stress WEAK              "1"
    ## Diathesis-stress STRONG            "1"
    ## G + E only                         NA
    ## G only                             NA
    ## E only                             NA
    ## Intercept only                     NA

Here we see the results. Notice that vantage sensitivity WEAK is the model with the lowest BIC. Differential susceptibility WEAK has a slightly worse 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")

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    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"      ""
    ## G + E only                         "1237.56" NA        ""
    ## G only                             "1285.07" NA        ""
    ## E only                             "1308.15" NA        ""
    ## Intercept only                     "1341.59" NA        ""
    ##                                    Within observable range?
    ## Vantage sensitivity WEAK           ""
    ## Differential susceptibility WEAK   "No"
    ## Differential susceptibility STRONG "No"
    ## Vantage sensitivity STRONG         ""
    ## Diathesis-stress WEAK              ""
    ## Diathesis-stress STRONG            ""
    ## G + E only                         ""
    ## G only                             ""
    ## E only                             ""
    ## Intercept only                     ""
    ##                                    % of observations below crossover
    ## Vantage sensitivity WEAK           "0"
    ## Differential susceptibility WEAK   "0"
    ## Differential susceptibility STRONG "0.376"
    ## Vantage sensitivity STRONG         "0"
    ## Diathesis-stress WEAK              "1"
    ## Diathesis-stress STRONG            "1"
    ## G + E only                         NA
    ## G only                             NA
    ## E only                             NA
    ## Intercept only                     NA

We obtain similar results.

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")

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    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"    ""
    ## G + E only                         "1170.07" NA        ""
    ## G only                             "1171.24" NA        ""
    ## Intercept only                     "1247.53" NA        ""
    ## E only                             "1248.73" NA        ""
    ##                                    Within observable range?
    ## Differential susceptibility STRONG "No"
    ## Vantage sensitivity STRONG         ""
    ## Vantage sensitivity WEAK           ""
    ## Differential susceptibility WEAK   "No"
    ## Diathesis-stress WEAK              ""
    ## Diathesis-stress STRONG            ""
    ## G + E only                         ""
    ## G only                             ""
    ## Intercept only                     ""
    ## E only                             ""
    ##                                    % 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"
    ## G + E only                         NA
    ## G only                             NA
    ## Intercept only                     NA
    ## E only                             NA

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")

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    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 )"
    ## E only                             "1011.14" NA        ""
    ## G + E only                         "1011.79" NA        ""
    ## Intercept only                     "1149.96" NA        ""
    ## G only                             "1153.2"  NA        ""
    ## Diathesis-stress STRONG            "1157.61" "6.95"    ""
    ## Vantage sensitivity STRONG         "1159.69" "-2.28"   ""
    ##                                    Within observable range?
    ## Differential susceptibility WEAK   "No"
    ## Diathesis-stress WEAK              ""
    ## Vantage sensitivity WEAK           ""
    ## Differential susceptibility STRONG "No"
    ## E only                             ""
    ## G + E only                         ""
    ## Intercept only                     ""
    ## G only                             ""
    ## Diathesis-stress STRONG            ""
    ## Vantage sensitivity STRONG         ""
    ##                                    % of observations below crossover
    ## Differential susceptibility WEAK   "1"
    ## Diathesis-stress WEAK              "1"
    ## Vantage sensitivity WEAK           "0"
    ## Differential susceptibility STRONG "1"
    ## E only                             NA
    ## G + E only                         NA
    ## Intercept only                     NA
    ## G only                             NA
    ## Diathesis-stress STRONG            "1"
    ## Vantage sensitivity STRONG         "0"

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")

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    ## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
    ##   Consider formula(paste(x, collapse = " ")) instead.

    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 WEAK              "834.4"  "9.74"    ""
    ## Vantage sensitivity WEAK           "834.9"  "0.44"    ""
    ## E only                             "848.02" NA        ""
    ## G + E only                         "851.11" NA        ""
    ## Intercept only                     "895.99" NA        ""
    ## G only                             "899.81" NA        ""
    ## Diathesis-stress STRONG            "909.24" "6.77"    ""
    ## Vantage sensitivity STRONG         "913.1"  "-2.45"   ""
    ##                                    Within observable range?
    ## Differential susceptibility STRONG "No"
    ## Differential susceptibility WEAK   "No"
    ## Diathesis-stress WEAK              ""
    ## Vantage sensitivity WEAK           ""
    ## E only                             ""
    ## G + E only                         ""
    ## Intercept only                     ""
    ## G only                             ""
    ## Diathesis-stress STRONG            ""
    ## Vantage sensitivity STRONG         ""
    ##                                    % of observations below crossover
    ## Differential susceptibility STRONG "1"
    ## Differential susceptibility WEAK   "1"
    ## Diathesis-stress WEAK              "1"
    ## Vantage sensitivity WEAK           "0"
    ## E only                             NA
    ## G + E only                         NA
    ## Intercept only                     NA
    ## G only                             NA
    ## Diathesis-stress STRONG            "1"
    ## Vantage sensitivity STRONG         "0"

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)