Table S1 List of models used in the simulation process. The column model refers to the number that identifies each model in the paper. X _i is a fixed variable that translates an offspring genotype to a numeric value; Z _i is an observed environmental covariate, either continuous or dichotomous; g_im, g_if are the parental genotypes (mother and father, respectively); E(X _i |g_im, g_if) is calculated under segregation and independent assortment assumptions using Mendel's law; M = 1, 2, ..., 6 are the six possible mating types; i = 1, 2, 3, ..., n subjects; j = 1, 2, 3, ..., m measurement occasions into the subject; t _ij is the repeated ozone exposure variable.

(15)

E(Y _ij |X, Z, t) = α₀ + α₁t _ij + α₂Z _i + α₃X _i + α₄X _i Z _i + α₅X _i t _ij + α₆Z _i t _ij + α₇X _i Z _i t _ij

Homogeneous population

α₁ = -0.8; α₂ = -0.2; α₃ = -0.05; α₄ = 0.2;

α₆ = 0.6; α₇ = 1.0

P(A) = 0.6

P(a) = 0.4

Admixed population

α₁ = -0.8; α₂ = -0.2; α₃ = -0.05; α₄ = 0.2;

α₆ = 0; α₇ = 1.0

Population 1

P(A) = 0.4

P(a) = 0.6

α₁ = -0.8; α₂ = -0.2; α₃ = -0.05; α₄ = 0.2;

α₆ = 2.0; α₇ = 1.0

Population 2

P(A) = 0.8

P(a) = 0.2

Assessing models

Model

Independent subjects design

Case-parent design

Comments

Two-step modelling approach

Main genetic effect

(16)

E(slope _i |X, Z) = β₀ + β₁X _i + β₂Z _i

Taken from model (2)

(17)

E(slope _i |X, Z) = β_0M+ β₁X _i + β₂Z _i

Taken from model (5)

(18)

${\chi }_{FBAT}^2=\frac{\sum \left[\left(slop{e}_i-E\left(slop{e}_i\right)\right)*\left({\mathsf{\text{X}}}_{\mathsf{\text{i}}}-\mathsf{\text{E}}\left({\mathsf{\text{X}}}_{\mathsf{\text{i}}}|{\mathsf{\text{g}}}_{\mathsf{\text{im}}},{\mathsf{\text{g}}}_{\mathsf{\text{if}}}\right)\right)\right]}{{{\sum }_i\left(slop{e}_i-E\left(slop{e}_i\right)\right)}^2*\mathsf{\text{Var}}\left({\mathsf{\text{X}}}_{\mathsf{\text{i}}}|{\mathsf{\text{g}}}_{\mathsf{\text{im}}},{\mathsf{\text{g}}}_{\mathsf{\text{if}}}\right)}$

Taken from model (10)

Gene-environment interaction

(21)

E(slope _i |X, Z) = β₀ + β₁X _i + β₂Z _i + β₃Z _i X _i

Taken from model (1)

(22)

E(slope _i |X, Z) = β_0M+ β₁X _i + β_2MZ _i + β₃Z _i X _i

Taken from model (9)

Models for longitudinal data

Main genetic effect

(19)

E(Y _ij |X, Y, t) = α₀ + α₁t _ij + α₂Z _i + α₃X _i + α₄X _i Z _i + α₅X _i t _ij + α₆Z _i t _ij

Taken from model (15) with α₇ = 0

(20)

E(Y _ij |X, Y, t) = α_0M+ α_1Mt _ij + α₂Z _i + α₃X _i + α₄X _i Z _i + α₅X _i t _ij + α_6MZ _i t _ij

Taken from model (13) with α₇ = 0

Gene-environment interaction

(23)

E(Y _ij |X, Z, t) = α₀ + α₁t _ij + α₂Z _i + α₃X _i + α₄X _i Z _i + α₅X _i t _ij + α₆Z _i t _ij + α₇X _i Z _i t _ij

Taken from model (15)

(24)

E(Y _ij |X, Y, t) = α_0M+ α_1Mt _ij + α₂Z _i + α₃X _i + α₄X _i Z _i + α₅X _i t _ij + α_6MZ _i t _ij + α₇X _i Z _i t _ij

Taken from model (13)