Table 1 Regression models included in the paper. The 'Model' column refers to the number that identifies each model in the paper.

Main genetic effect

(2)

E(Y _i |X, Z) = β₀ + β₁X _i + β₂Z _i

Ordinary linear regression model

(3)

E(Y _i |X, Z) = β₀ + β₁(X _i - E(X _i |g _im , g _if )) + β₂Z _i

Adjusted version of (2). The model is adjusted by the expected value of the offspring's genotype conditional to the parental genotypes

(4)

E(Y _i |X, Z) = β_0M+ β₁(X _i - E(X _i |g _im , g _if )) + β₂Z _i

Gauderman's model (QTDT_M) adjusted for the covariate Z

(5)

E(Y _i |X, Z) = β_0M+ β₁X _i + β₂Z _i

(4) equivalent to (5)

(10)

${\chi }_{FBAT}^2=\frac{U^2}{Var\left(U\right)}$

FBAT statistic

Gene-environment interaction

(1)

E(Y _i |X, Z) = β₀ + β₁X _i + β₂Z _i + β₃X _i Z _i

Ordinary linear regression model

(6)

E(Y _i |X, Z) = β_0M+ β₁X _i + β₂Z _i + β₃X _i Z _i

Gauderman's model (QTDT_M)

(7)

E(Y _i |X, Z) = β_0M+ β₁[X _i - E(X _i |g _im , g _if )] + β₂Z _i + β₃Z _i [X _i - E(X _i |g _im , g _if )]

(6) is not equivalent to (7) when the environment covariate (Z _i ) is not constant within mating type.

(8)

E(Y _i |X, Z) = β_0M+ β₁[X _i - E(X _i |g _im , g _if )] + β_2MZ _i + β₃Z _i [X _i - E(X _i |g _im , g _if )]

Adjusted QTDT_M

(9)

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

(8) equivalent to (9)

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

Ordinary linear mixed model (OLMM)

(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

Adjusted linear mixed model (ALMM)

Gene-environment interaction

(11)

Y _ij = 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 + b_1i+b_2it _ij + e _ij

Ordinary linear mixed model

(12)

FEF_2575ij= α_0M+ α_1Mt _ij + α₂Z _i + α₃[X _i - E(X _i |g _im , g _if )] + α₄[X _i - E(X _i |g _im , g _if )]Z _i + α₅[X _i - E(X _i |g _im , g _if )]t _ij + α_6MZ _i t _ij + α₇[X _i - E(X _i |g _im , g _if )]Z _i t _ij + b_1i+b_2it _ij + e _ij

Adjusted linear mixed model (ALMM)

(13)

FEF_2575ij= 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 + b_1i+b_2it _ij + e _ij

(13) is equivalent to (12)