A conditional synergy index to assess biological interaction

Abstract

In genetic studies of complex diseases, a crucial task is to identify and quantify gene–gene interactions which are often defined as deviance from genetic additive effects. This statistical definition, however, does not need to reflect the biological interactions of genes. We propose a new method to detect gene–gene interactions. This new approach exploits the concept of synergy and antagonism that is appropriate to capture biological relationships. The conditional synergy index (CSI) describes the extent of interaction on the penetrance scale. We develop the CSI for two-locus disease models and cohort data. The index assumes genotypes to be dichotomized into risk-genotypes (exposed) and non-risk-genotypes (unexposed) but it does not assume the loci to be in linkage equilibrium. We investigate the performance of the CSI and compare it to classical epidemiological interaction measures like Rothman’s synergy index (S) and the attributable proportion due to interaction (AP). In addition, the performance of an estimator of this new parameter is illustrated in a practical example.

Technical appendix

Let us assume an independent identically distributed (iid) sequence of multinomially distributed random vectors ${\bf X}=(X_{000},\ldots, X_{JKL})$ with parameters $P(D=j,G_A=k,G_B=l) = p_{jkl}, p_{jkl} \in (0,1),\,j=0,\ldots,J,$ k = 0,…, K, l = 0,…, L, and sample sizes N _jkl for each combination j, k, l, $\sum_{j=0}^J \sum_{k=0}^K \sum_{l=0}^L N_{jkl} = N.$ The unbiased maximum-likelihood (ML) estimator of p _jkl is given as $\hat{p}_{jkl}= \frac{X_{jkl}}{N}$ with ${var}(\hat{p}_{jkl})=p_{jkl}(1-p_{jkl})/N.$

Consistency of $\widehat{CSI}$

Theorem 1 Assuming the multinomial model described above and K = L = 1, $\widehat{CSI}$ converges almost surely to CSI for N → ∞ and $\lim_{N \rightarrow \infty} \hat{\delta}_{dab} \neq 0.$

Proof It holds for the ML estimators $\hat{p}$: $\lim_{N \rightarrow \infty} \hat{{\bf p}} = \lim_{N \rightarrow \infty} \left(\frac{X_{000}}{N}, \ldots, \frac{X_{111}}{N} \right) = {\bf p}$ , j, k, l = 0,1. Since $\hat{\delta}_{dab}$ is a continuous function of $\hat{{\bf p}}_{jkl}$ Slutsky’s Theorem can be applied which yields

$$ \begin{aligned} \lim_{N \rightarrow \infty} \hat{\delta}_{dab} = &\lim_{N \rightarrow \infty} \frac{1}{N} \left[ \left(\sum_j X_{jab} - \sum_j X_{ja'b'}\right) \frac{X_{dab}} {\sum_j X_{jab}} \right.\\ &+ \left. \left(\sum_j \sum_l X_{ja'l}\right) \frac{X_{da'b}} {\sum_j X_{ja'b}} + \left(\sum_j \sum_k X_{jkb'}\right) \frac{X_{dab'}} {\sum_j X_{jab'}} \right] \\ =&\lim_{N \rightarrow \infty} \frac{1}{N} \left[ \left(\sum_j \hat{p}_{jab}N - \sum_j \hat{p}_{ja'b'}N \right) \frac{\hat{p}_{dab}N}{\sum_j \hat{p}_{jab}N}\right.\\ &+ \left.\left(\sum_j \sum_l \hat{p}_{ja'l}N \right) \frac{\hat{p}_{da'b}N} {\sum_j \hat{p}_{ja'b}N} + \left(\sum_j \sum_k \hat{p}_{jkb'}N \right) \frac{\hat{p}_{dab'}N} {\sum_j \hat{p}_{jab'}N} \right] \\ =& \left(\sum_j p_{jab} - \sum_j p_{ja'b'}\right) \frac{p_{dab}} {\sum_j p_{jab}} \\ &+ \left(\sum_j \sum_l p_{ja'l}\right) \frac{p_{da'b}} {\sum_j p_{ja'b}} + \left(\sum_j \sum_k p_{jkb'}\right) \frac{p_{dab'}}{\sum_j p_{jab'}}\\ =& \delta_{dab}. \end{aligned} $$

It follows directly that

$$ \lim_{N \rightarrow \infty} \widehat{CSI} = \lim_{N \rightarrow \infty} \left( \frac{\frac{\hat{\delta}_{111}} {\hat{\delta}_{011}} + \frac{\hat{\delta}_{100}} {\hat{\delta}_{000}}} {\frac{\hat{\delta}_{101}} {\hat{\delta}_{001}} + \frac{\hat{\delta}_{110}} {\hat{\delta}_{010}}} \right) = \frac{\frac{\lim_{N \rightarrow \infty}\hat{\delta}_{111}} {\lim_{N \rightarrow \infty}\hat{\delta}_{011}} + \frac{\lim_{N \rightarrow \infty}\hat{\delta}_{100}}{\lim_{N \rightarrow \infty}\hat{\delta}_{000}}} {\frac{\lim_{N \rightarrow \infty}\hat{\delta}_{101}} {\lim_{N \rightarrow \infty}\hat{\delta}_{001}} + \frac{\lim_{N \rightarrow \infty}\hat{\delta}_{110}} {\lim_{N \rightarrow \infty}\hat{\delta}_{010}}} = \frac{\frac{\delta_{111}} {\delta_{011}} + \frac{\delta_{100}} {\delta_{000}}} {\frac{\delta_{101}} {\delta_{001}} + \frac{\delta_{110}} {\delta_{010}}}= CSI. $$

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Variance of $\widehat{CSI}$

To find the variance of $CSI(\hat{{\bf p}}),$ we use the standard delta method based on a stochastic expansion of CSI at ${\bf p},$ which yields:

$$ \begin{aligned} {var}_A(\widehat{CSI})= &\frac{1}{N} \left( {\bf\phi}' {\bf diag(p)} {\bf\phi} - ({\bf\phi}' {\bf p})^2 \right)\\ =& \frac{1}{N} \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 p_{jkl} \left(\frac{\partial g({\bf p})}{\partial p_{jkl}}\right)^2 - \frac{1}{N} \left(\sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 p_{jkl} \frac{\partial g({\bf p})}{\partial p_{jkl}}\right)^2\\ =& \frac{1}{N \left( Q_{10} + Q_{01} \right) ^2} \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 p_{jkl}(1-p_{jkl}) \left(\pi^c_{jkl} + \pi^d_{jkl} \right)^2, \end{aligned} $$

where

$$ \begin{aligned} \pi^c_{jkl} =& \frac{\frac{-p_{1k'l'} +Q_{k'l'}p_{0k'l'}} {\sum_{j^*}p_{j^*k'l'}} + \frac{p_{1kl'} -Q_{k'l'}p_{0kl'}} {\sum_{j^*}p_{j^*kl'}} +\frac{p_{1k'l}-Q_{k'l'}p_{0k'l}} {\sum_{j^*}p_{j^*k'l}}}{\delta_{0k'l'}}\\ & + \frac{(-1)^{1-{\mathcal{I}}_j} \cdot \left({\mathcal{I}}_j + (1-{\mathcal{I}}_j) Q_{kl} - p_{j'kl}\frac{\sum_{j^*} p_{j^*k'l'}} {(\sum_{j^*}p_{j^*kl})^2}\left[1 +Q_{kl} \right] \right)} {\delta_{0kl}} \\ & - CSI \left((-1)^{1-{\mathcal{I}}_j} \cdot \frac{{\mathcal{I}}_j +(1-{\mathcal{I}}_j) Q_{kl'} + p_{j'kl} \frac{\sum_{j^*}p_{j^*k'l}}{(\sum_{j^*}p_{j^*kl})^2} \left[1 + Q_{kl'} \right]} {\delta_{0kl'}}\right.\\ & +\left.(-1)^{1-{\mathcal{I}}_j} \cdot\frac{{\mathcal{I}}_j + (1-{\mathcal{I}}_j) Q_{k'l} + p_{j'kl}\frac{\sum_{j^*}p_{j^*kl'}} {(\sum_{j^*}p_{j^*kl})^2} \left[1 +Q_{k'l} \right]} {\delta_{0k'l}} \right) \\ \pi^d_{jkl} =&(-1)^{1-{\mathcal{I}}_j} \cdot \frac{{\mathcal{I}}_j +(1-{\mathcal{I}}_j) Q_{kl'} + p_{j'kl} \frac{\sum_{j^*}p_{j^*k'l}}{(\sum_{j^*}p_{j^*kl})^2} \left[1 + Q_{kl'} \right]} {\delta_{0kl'}}\\ & +(-1)^{1-{\mathcal{I}}_j} \cdot \frac{{\mathcal{I}}_j +(1-{\mathcal{I}}_j) Q_{k'l} + p_{j'kl} \frac{\sum_{j^*}p_{j^*kl'}}{(\sum_{j^*}p_{j^*kl})^2} \left[1 + Q_{k'l} \right]} {\delta_{0k'l}}\\ &- CSI \left( {\mathcal{I}}_{k> l} \cdot \frac{\left[\frac{p_{1k'l'} - Q_{k'l'}p_{0k'l'}} {\sum_{j^*}p_{j^*k'l'}}\right]} {\delta_{0k'l'}} + (1- {\mathcal{I}}_{k> l})\frac{\left[\frac{-p_{1k'l'} + Q_{k'l'}p_{0k'l'}}{\sum_{j^*}p_{j^*k'l'}}\right]} {\delta_{0k'l'}} \right.\\ & +\frac{ \left[\frac{p_{1kl'} - Q_{k'l'}p_{0kl'}}{\sum_{j^*}p_{j^*kl'}} + \frac{p_{1k'l}-Q_{k'l'}p_{0k'l}}{\sum_{j^*}p_{j^*k'l}}\right]} {\delta_{0k'l'}} \\ & +\left.\frac{(-1)^{1-{\mathcal{I}}_j} \cdot \left(\right.{\mathcal{I}}_j +(1-{\mathcal{I}}_j) Q_{kl} - p_{j'kl} \frac{\sum_{j^*} p_{j^*k'l'}}{(\sum_{j^*}p_{j^*kl})^2} \left[1 + Q_{kl} \right]} {\delta_{0kl}}\right ), \end{aligned} $$

and ${\mathcal{I}}_j, {\mathcal{I}}_{k> l}$ are indicator functions that assign the values 0 and 1 as follows:

$$ {\mathcal{I}}_j = \left\{\begin{array}{ll} 1 & j=1\\ 0 & j=0 \end{array}\right. \quad \quad {\mathcal{I}}_{k> l} = \left\{\begin{array}{ll}1 & k> l \\ 0 & k< l\end{array}\right. $$