The role of linkage disequilibrium in case-only studies of gene–environment interactions

Abstract

Gene–environment (G × E) interactions have been invoked to account, at least in part, for the gap between the known heritability of common human diseases and the phenotypic variation hitherto explained by genetic variants. Noteworthy in this context, a case-only (CO) design has been proposed in the past as a means to detect G × E interactions possibly more efficiently than by using classical case–control and cohort designs. So far, however, most CO studies have followed a candidate (or single) gene approach, and the genome-wide utility of the CO design is still more or less unknown. In particular, the way in which linkage disequilibrium (LD) impacts upon the chance to detect G × E interaction through the analysis of proxy markers has not been studied in much detail before. Therefore, we systematically assessed the power to indirectly detect a given G × E interaction through exploiting LD in a CO design. Our simulations revealed a strong relationship between LD and detection power that was subsequently validated in a real colorectal cancer data set.

Appendix: Estimating the level of G × E interaction in case-only studies

ORs for the joint effect of G and E and for the marginal effects of G and E are calculated as follows (adapted from Gatto et al. 2004; see “Materials and methods” of the main text for definitions):

Joint effect of G and E			Marginal effect of G (given E = 0)			Marginal effect of E (given G = 0)
	D = 1	D = 0		D = 1	D = 0		D = 1	D = 0
G = 1, E = 1	a 1	b 1	G = 1	c1	d 1	E = 1	a 2	b 2
G = 0, E = 0	c 2	d 2	G = 0	c2	d 2	E = 0	c 2	d 2
${\text{OR}}_{\text{GE}} = \frac{{a_{ 1} /b_{1} }}{{c_{2} /d_{2} }}$			${\text{OR}}_{G} = \frac{{c_{1} /d_{1} }}{{c_{2} /d_{2} }}$			${\text{OR}}_{E} = \frac{{a_{2} /b_{2} }}{{c_{2} /d_{2} }}$

1. Note that OR _G and OR _E entail the genotype and exposure odds given E = 0 and G = 0, respectively
2. Rearranging the above tables and stratifying the G and E data by disease status allows calculation of the gene–environment OR in both cases and controls (denoted as Term I and Term II below)

Cases (D = 1)			Controls (D = 0)
	G = 1	G = 0		G = 1	G = 0
E = 1	a 1	a 2	E = 1	b 1	b 2
E = 0	c 1	c 2	E = 0	d 1	d 2
$G - E {\text{ OR}}_{\text{cases}} = \frac{{a_{1} /a_{2} }}{{c_{1} /c_{2} }}$			$G - E {\text{ OR}}_{\text{controls}} = \frac{{b_{1} /b_{2} }}{{d_{1} /d_{2} }}$
Term 1			Term 2

Conceptually, multiplicative interaction between G and E (G × E _OR) refers to any deviation of the product of OR _G and OR _E from OR_GE. Therefore, substituting OR_GE, OR _G and OR _G by the respective terms yields

$$G \times E_{\text{OR}} = \frac{{{\text{OR}}_{\text{GE}} }}{{{\text{OR}}_{G}\cdot {\text{OR}}_{E} }} = \frac{{\frac{{a_{1} /b_{1} }}{{c_{2} /d_{2} }}}}{{\frac{{c_{1} /d_{1} }}{{c_{2} /d_{2} }} \cdot \frac{{a_{2} /b_{2} }}{{c_{2} /d_{2} }}}} = \frac{{\left. { \frac{{a_{1} c_{2} }}{{a_{2} c_{1} }} } \right\} {\text{Term I}} }}{{\left. {\frac{{b_{1} d_{2} }}{{b_{2} d_{1} }} } \right\} {\text{Term II}}}}$$

Under the assumptions that G and E are independent in the general population and that the disease is rare (i.e. that controls are almost representative of the population as a whole), Term II will be equal to 1. Then, G × E _OR can be estimated by the gene environment OR in cases (see Piegorsch et al. 1994 for further details).