. Author manuscript; available in PMC: 2015 Aug 1.

Published in final edited form as: Twin Res Hum Genet. 2015 Jan 13;18(1):19–27. doi: 10.1017/thg.2014.81

Table 1.

Bivariate variance components models for latent component-by-measured environment (G×M) interaction.

Model

Equation

Classical ACE model^a

M = μ_{M} + a_{M} A_{M} + c_{M} C_{M} + e_{M} E_{M}

(1)

Bivariate cholesky

P = μ_{P} + a_{C} A_{M} + c_{C} C_{M} + e_{C} E_{M} + a_{U} A_{U} + c_{U} C_{U} + e_{U} E_{U}

(2)

Bivariate cholesky with G×M

P = μ_{P} + (a_{C} + α_{C} M) A_{M} + (c_{C} + κ_{C} M) C_{M} + (e_{C} + ɛ_{C} M) E_{M} + (a_{U} + α_{U} M) A_{U} + (c_{U} + κ_{U} M) C_{U} + (e_{U} + ɛ_{U} M) E_{U} .

(3)

Nonlinear main effects model with G×M

P = μ_{P} + β_{1} M + β_{2} M^{2} + (a_{U} + α_{U} M) A_{U} + (c_{U} + κ_{U} M) C_{U} + (e_{U} + ɛ_{U} M) E_{U} .

(4)

Nonlinear main effects

P = μ_{P} + β_{1} M + β_{2} M^{2} + a_{U} A_{U} + c_{U} C_{U} + e_{U} E_{U} .

(4*)

Linear main effects only

P = μ_{P} + β_{1} M + a_{U} A_{U} + c_{U} C_{U} + e_{U} E_{U} .

(4†)

shown here for the putative moderator.

Note: P refers to a phenotype of interest, M refers to a putative moderator, a, c, and e refer to additive genetic, shared environment, and non-shared environment influences respectively, and μ is the mean. A, C, and E are standard normal latent variables. The subscript “C” refers to factors that affect both M and P, and the subscript “U” refers to factors that are unique to P.