Improved statistics for genome-wide interaction analysis

Abstract

Recently, Wu and colleagues [1] proposed two novel statistics for genome-wide interaction analysis using case/control or case-only data. In computer simulations, their proposed case/control statistic outperformed competing approaches, including the fast-epistasis option in PLINK and logistic regression analysis under the correct model; however, reasons for its superior performance were not fully explored. Here we investigate the theoretical properties and performance of Wu et al.'s proposed statistics and explain why, in some circumstances, they outperform competing approaches. Unfortunately, we find minor errors in the formulae for their statistics, resulting in tests that have higher than nominal type 1 error. We also find minor errors in PLINK's fast-epistasis and case-only statistics, although theory and simulations suggest that these errors have only negligible effect on type 1 error. We propose adjusted versions of all four statistics that, both theoretically and in computer simulations, maintain correct type 1 error rates under the null hypothesis. We also investigate statistics based on correlation coefficients that maintain similar control of type 1 error. Although designed to test specifically for interaction, we show that some of these previously-proposed statistics can, in fact, be sensitive to main effects at one or both loci, particularly in the presence of linkage disequilibrium. We propose two new "joint effects" statistics that, provided the disease is rare, are sensitive only to genuine interaction effects. In computer simulations we find, in most situations considered, that highest power is achieved by analysis under the correct genetic model. Such an analysis is unachievable in practice, as we do not know this model. However, generally high power over a wide range of scenarios is exhibited by our joint effects and adjusted Wu statistics. We recommend use of these alternative or adjusted statistics and urge caution when using Wu et al.'s originally-proposed statistics, on account of the inflated error rate that can result.

Figure 1. Chi-squared (1 df) Q-Q plot for Scenario 1 (Global Null).

Top panels ((a), (b) and (c)): Case/Control not in LD; Middle panels ((d), (e) and (f)): Case/Control in LD; Bottom panels ((g), (h) and (i)): Case-Only not in LD; FE: Fast-Epistasis; AFE: Adjusted FE; Wu: Wu et al. statistic; AWu: Adjusted Wu statistic; IWu: Ideal Wu statistic; WZ: Wellek and Ziegler statistic; JE: Joint Effects statistic.

Figure 2. Chi-squared (1 df) Q-Q plot for Scenario 2 (Recessive effect at locus G).

Top panels ((a), (b) and (c)): Case/Control not in LD; Middle panels ((d), (e) and (f)): Case/Control in LD; Bottom panels ((g), (h) and (i)): Case-Only not in LD; FE: Fast-Epistasis; AFE: Adjusted FE; Wu: Wu et al. statistic; AWu: Adjusted Wu statistic; IWu: Ideal Wu statistic; WZ: Wellek and Ziegler statistic; JE: Joint Effects statistic.

Figure 3. Chi-squared (1 df) Q-Q plot for Scenario 5c (Rare disease, Additive effects at both loci).

Top panels ((a), (b) and (c)): Case/Control not in LD; Middle panels ((d), (e) and (f)): Case/Control in LD; Bottom panels ((g), (h) and (i)): Case-Only not in LD; FE: Fast-Epistasis; AFE: Adjusted FE; Wu: Wu et al. statistic; AWu: Adjusted Wu statistic; IWu: Ideal Wu statistic; WZ: Wellek and Ziegler statistic; JE: Joint Effects statistic.

Figure 4. Chi-squared (1 df) Q-Q plot for Scenario 5d (Rare disease, Recessive effects at both loci).

Top panels ((a), (b) and (c)): Case/Control not in LD; Middle panels ((d), (e) and (f)): Case/Control in LD; Bottom panels ((g), (h) and (i)): Case-Only not in LD; FE: Fast-Epistasis; AFE: Adjusted FE; Wu: Wu et al. statistic; AWu: Adjusted Wu statistic; IWu: Ideal Wu statistic; WZ: Wellek and Ziegler statistic; JE: Joint Effects statistic.

Figure 5. Power curves for Scenario 6 (Recessive Recessive).

Power to achieve significance level . Top panels ((a) and (b)): Case/Control not in LD; Middle panels ((c) and (d)): Case/Control in LD; Bottom panels ((e) and (f)): Case-Only not in LD; Left hand panels ((a), (c) and (e)): No main effect; Right hand panels ((b), (d) and (f)): Locus G has main effect; FE: Fast-Epistasis; AFE: Adjusted FE; Wu: Wu et al. statistic; AWu: Adjusted Wu statistic; WZ: Wellek and Ziegler statistic; JE: Joint Effects statistic; IWu: Ideal Wu statistic; C: Logistic regression using correct coding; IC: Logistic regression using incorrect ( = RecessiveDominant) coding; WZC: Wellek and Ziegler case-only statistic using correct coding; WZIC: Wellek and Ziegler case-only statistic using incorrect ( = RecessiveDominant) coding.

Figure 6. Power curves for Scenario 7 (Dominant Dominant).

Power to achieve significance level . Top panels ((a) and (b)): Case/Control not in LD; Middle panels ((c) and (d)): Case/Control in LD; Bottom panels ((e) and (f)): Case-Only not in LD; Left hand panels ((a), (c) and (e)): No main effect; Right hand panels ((b), (d) and (f)): Locus G has main effect; FE: Fast-Epistasis; AFE: Adjusted FE; Wu: Wu et al. statistic; AWu: Adjusted Wu statistic; WZ: Wellek and Ziegler statistic; JE: Joint Effects statistic; IWu: Ideal Wu statistic; C: Logistic regression using correct coding; IC: Logistic regression using incorrect ( = DominantRecessive) coding; WZC: Wellek and Ziegler case-only statistic using correct coding; WZIC: Wellek and Ziegler case-only statistic using incorrect ( = DominantRecessive) coding.