Bayesian analysis of genetic interactions in case-control studies, with application to adiponectin genes and colorectal cancer risk

Abstract

Complex diseases such as cancers are influenced by interacting networks of genetic and environmental factors. However, a joint analysis of multiple genes and environmental factors is challenging, owing to potentially large numbers of correlated and complex variables. We describe Bayesian generalized linear models for simultaneously analyzing covariates, main effects of numerous loci, gene-gene and gene-environment interactions in population case-control studies. Our Bayesian models use Student-t prior distributions with different shrinkage parameters for different types of effects, allowing reliable estimates of main effects and interactions and hence increasing the power for detection of real signals. We implement a fast and stable algorithm for fitting models by extending available tools for classical generalized linear models to the Bayesian case. We propose a novel method to interpret and visualize models with multiple interactions by computing the average predictive probability. Simulations show that the method has the potential to dissect interacting networks of complex diseases. Application of the method to a large case-control study of adiponectin genes and colorectal cancer risk highlights the previous results and detects new epistatic interactions and sex-specific effects that warrant follow-up in independent studies.

Figure 1

Analysis I: jointly fitting age, sex, all main effects of the ten SNPs and sex-gene interactions with three link functions, logit (left), probit (middle) and cloglog (right). The notation for main effects, a and d, indicate additive and dominance effects, respectively. The term X₁.X₂ represents interaction between X₁ and X₂. Estimated effects of age and sex are not displayed. The points, short lines and numbers at the right side represent estimates of effects, ± 2 standard errors, and p-values, respectively. The deviance (Dev) and Akaike information criterion (AIC) under each model are also shown.

Figure 2

Analysis II: jointly fitting age, sex, all main effects of the ten SNPs, sex-gene and epistatic interactions with three link functions, logit (left), probit (middle) and cloglog (right). The notation for main effects, a and d, indicate additive and dominance effects, respectively. The term X₁.X₂ represents interaction between X₁ and X₂. Estimated effects of age and sex are not displayed. Only epistatic interactions with p-value smaller than 0.05 are displayed. The points, short lines and numbers at the right side represent estimates of effects, ± 2 standard errors, and p-values, respectively. The deviance (Dev) and Akaike information criterion (AIC) under each model are also shown.

Figure 3

Average predictive probability for a) each genotype, b–c) each combination of sex and genotypes of SNPs rs1273385 and rs266729, and d–f) each two-locus genotype at SNPs that show significant interactions. The genotypes c, h, and r represent common homozygote, heterozygote, and rare homozygote, and the notation M and F represent male and female, respectively. The vertical (a) and horizontal (b–f) dotted gray line represents the mean of probabilities.

Figure 4

Simulation I: jointly fitting age, sex, all main effects of the ten SNPs and sex-gene interactions using the proposed priors. The left panel shows the frequency of each effect estimated with p-value smaller than 0.05 over 1000 replicates with three link functions, logit (circle), probit (square) and cloglog (diamond). The right panel shows the assumed values (circle), the estimated means (point) and the 95% intervals (gray line) with the logit link function. Only effects with non-zero simulated value are labeled. The notation for main effects, a and d, indicate additive and dominance effects, respectively. The term X₁.X₂ represents interaction between X₁ and X₂.

Figure 5

Comparison with existing models (I): jointly fitting age, sex, all main effects of the ten SNPs and sex-gene interactions using different priors. Frequency of each effect estimated with p-value smaller than 0.05 over 1000 replicates using 1) uniform prior (ν_j, s_j) = (∞, ∞) (□), 2) normal prior (ν_j, s_j) = (∞, 2.5) (◊), and 3) Jeffreys’ prior (ν_j, s_j) = (0, 0) (∇), for all effects. The points (•) represent the analysis using the proposed priors. Only effects with non-zero simulated value are labeled. The notation for main effects, a and d, indicate additive and dominance effects, respectively. The term X₁.X₂ represents interaction between X₁ and X₂.

Figure 6

Simulation II: jointly fitting age, sex, all main effects of the ten SNPs, sex-gene and epistatic interactions using the proposed priors. The left panel shows the frequency of each effect estimated with p-value smaller than 0.05 over 1000 replicates with three link functions, logit (circle), probit (square) and cloglog (diamond). The right panel shows the assumed values (circle), the estimated means (point) and the 95% intervals (gray line) with the logit link function. Only effects with non-zero simulated value are labeled. The notation for main effects, a and d, indicate additive and dominance effects, respectively. The term X₁.X₂ represents interaction between X₁ and X₂. Only 15 epistatic interactions with the smallest p-values are displayed.

Figure 7

Comparison with existing models (II): jointly fitting age, sex, all main effects of the ten SNPs, sex-gene and epistatic interactions using different priors. Frequency of each effect estimated with p-value smaller than 0.05 over 1000 replicates using 1) normal prior (ν_j, s_j) = (∞, 2.5) (□), 2) t prior (ν_j, s_j) = (1, 2.5) (◊), 3) t prior (ν_j, s_j) = (1, 0.27) (Δ), and 4) Jeffreys’ prior (ν_j, s_j) = (0, 0) (∇), for all effects. The points (•) represent the analysis using the proposed priors. Only effects with non-zero simulated value are labeled. The notation for main effects, a and d, indicate additive and dominance effects, respectively. The term X₁.X₂ represents interaction between X₁ and X₂. Only 15 epistatic interactions with the smallest p-values are displayed.

Figure 8

Receiver operating characteristic (ROC) curves for risk prediction with four models simultaneously fitting: 1) age and sex (gray solid), 2) age, sex, and main effects of SNPs (gray dotted), 3) age, sex, main effects of SNPs, and sex-gene (black solid), and 3) age, sex, main effects of SNPs, sex-gene and epistatic interactions (black dotted). The areas under the ROC curves (AUC) for these four models are 0.79, 0.81, 0.82, and 0.87, respectively.