Operating Characteristics of Alternative Statistical Methods for Detecting Gene-by-Measured Environment Interaction in the Presence of Gene–Environment Correlation in Twin and Sibling Studies

Abstract

It is likely that all complex behaviors and diseases result from interactions between genetic vulnerabilities and environmental factors. Accurately identifying such gene–environment interactions is of critical importance for genetic research on health and behavior. In a previous article we proposed a set of models for testing alternative relationships between a phenotype (P) and a putative moderator (M) in twin studies. These include the traditional bivariate Cholesky model, an extension of that model that allows for interactions between M and the underling influences on P, and a model in which M has a non-linear main effect on P. Here we use simulations to evaluate the type I error rates, power, and performance of the Bayesian Information Criterion under a variety of data generating mechanisms and samples sizes (n = 2,000 and n = 500 twin pairs). In testing the extension of the Cholesky model, false positive rates consistently fell short of the nominal Type I error rates ($\alpha = 10,.05,.01$). With adequate sample size (n = 2,000 pairs), the correct model had the lowest BIC value in nearly all simulated datasets. With lower sample sizes, models specifying non-linear main effects were more difficult to distinguish from models containing interaction effects. In addition, we provide an illustration of our approach by examining possible interactions between birthweight and the genetic and environmental influences on child and adolescent anxiety using previously collected data. We found a significant interaction between birthweight and the genetic and environmental influences on anxiety. However, the interaction was accounted for by non-linear main effects of birthweight on anxiety, verifying that interaction effects need to be tested against alternative models.

Appendices

Appendix 1

Estimated sample size analysis

We followed the method recommended by Saunders et al. (2003). Assume $\widehat{E}(\chi^{2} ) = Q$for N simulations, where $\widehat{E}(\chi^{2} )$ is the mean LRT between H_A and H₀ across simulations. Then, let $\frac{Q - P}{N} = \hat{k}$ where p is the degrees of freedom for the LRT. From Q and $\hat{k}$, we can compute the χ² non-centrality parameter for a given sample size $\bar{N}$ and thereby generate power. Reversing this process, given p and a target power $(1 - \beta )$, we can, via the non-centrality parameter $\lambda_{p,1 - \beta }$, compute the sample size $\bar{N}$ needed to achieve power $(1 - \beta )$. If we set $\lambda_{p,1 - \beta } = \hat{k}\bar{N}$ then

$$ \bar{N} = \frac{{\lambda_{p,1 - \beta } }}{{\hat{k}}} = \lambda_{p,1 - \beta } \frac{N}{Q - P} $$

The non-centrality parameter was calculated using Stata’s npnchi2 function (Stata Corp. 2009 Stata Statistical Sofware: Release 11. College Station, TX: StataCorp, LP) ${{\uplambda}}_{{{\text{p,1}} - \beta }} {\text{ = npnchi2(p,invchi2(p,(1}} - \alpha ) ) , ( 1- (\beta / 1 0 0 ) / {\text{Q)}}$.

Appendix 2

See (Table 6)

Table 6 Percent of LRT statistics under the null hypothesis exceeding critical value for pairs of nested models (Cholesky GxM and Cholesy) based on 2,000 replicates of n = 4,000 twin pairs

Appendix 3

See (Table 7)

Table 7 Percent of LRT statistics exceeding critical value ($\alpha = 0.05$) for nested models based on 2,000 replicates of either n = 2,000 twin pairs or n = 500 twin pairs