Specification, testing, and interpretation of gene-by-measured-environment interaction models in the presence of gene-environment correlation

Abstract

Purcell (Twin Res 5:554-571, 2002) proposed a bivariate biometric model for testing and quantifying the interaction between latent genetic influences and measured environments in the presence of gene-environment correlation. Purcell's model extends the Cholesky model to include gene-environment interaction. We examine a number of closely related alternative models that do not involve gene-environment interaction but which may fit the data as well as Purcell's model. Because failure to consider these alternatives could lead to spurious detection of gene-environment interaction, we propose alternative models for testing gene-environment interaction in the presence of gene-environment correlation, including one based on the correlated factors model. In addition, we note mathematical errors in the calculation of effect size via variance components in Purcell's model. We propose a statistical method for deriving and interpreting variance decompositions that are true to the fitted model.

Figure 1

Path diagrams for several models which may or may not contain GxM. (a) Purcell’s (2002) model for testing and quantifying GxM in the presence of rGM. (b) Main effects model containing no A_M-by-M or E_M-by-M terms. (c) Alternative model containing main effects of M and M² instead of A_M-by-M terms. (d) Alternative model containing no multiplicative terms in M. (e) Correlated factors model for testing and quantifying GxM in the presence of rGM. Note: All latent variables have mean 0 and variance 1. Dashed arrows pointing from a variable to a path indicate moderation of the indicated path by that variable.

Figure 2

Variance decompositions for first hypothetical gene-environment interaction model for P moderated by M, presented in § 4.3. Left: Incorrect variance decomposition according to formula (12). Center: Correct variance decomposition for unique effects g₂ and g₃ controlling M. Right: Correct variance decomposition for environmental effects h₂ controlling genetic effects. Note: For center panel, effect variance due jointly to A_M and E_M combined is 0.18 and is not shown in the plot because it is not a function of M. For right panel, effect variance due to all genetic effects A_M and A_U is 1.03 and is not shown in the plot because it is not a function of A_M or A_U.

Figure 3

Variance decompositions for second hypothetical gene-environment interaction model for P moderated by M, presented in § 4.3. Left: Incorrect variance decomposition according to formula (12). Center: Correct variance decomposition for unique effects g₂ and g₃ controlling M. Right: Correct variance decomposition for environmental effects h₂ controlling genetic effects. Note: For center panel, effect variance due jointly to A_M and E_M combined is 0.92 and is not shown in the plot because it is not a function of M. For right panel, effect variance due to all genetic effects A_M and A_U is 1.05 and is not shown in the plot because it is not a function of A_M or A_U.