### Introduction

### Methods

*T*denote the survival time for the

_{i}*i*individual and

^{th}*x*and

_{i}*z*denote the predictor variable vector coding a gene-gene and gene-environment interaction of interest and the vector coding for the covariates, respectively. Let β and γ be the corresponding parameter vectors to

_{i}*x*and

_{i}*z*, respectively. Then, we call β the target effects and γ the covariate effects. The parametric regression model represents the linear relationship between the log survival time and covariates as follows:

_{i}*X*=

*Z*= 0, σ is a scale parameter, and

*W*is the error distribution. When

*T*has a Weibull distribution,

*W*has a standard extreme value distribution. For a log-logistic distribution,

*W*has a standard logistic distribution.

*i*individual,

^{th}*S*, as 1 for the high-risk group and 0 otherwise. In the second step, the variable

*S*is considered with the other adjusting covariates in the accelerated failure time regression model. The testing for the significance of

*S*implies that there is a significant gene-gene interaction associated with survival time. For testing the significance of

*S*, a Wald-type test statistic is used, and its asymptotic distribution is a chi-square distribution under the null hypothesis of no gene-gene interaction. However, as described in Yu et al. [9], the asymptotic distribution of the Wald-type test statistic is not a central chi-square distribution, because the expectation of the test statistic is not 0 under the null hypothesis. To adjust for the bias of the test statistic, non-centrality is estimated by a small number of permutations—say, 5 or 10 times. Based on the non-central chi-square test statistic, the significance of a gene-gene interaction can be tested for all possible pairs of SNPs without any intensive permutations. The proposed method easily tests the significance of a gene-gene interaction for all possible higher-order pairs of SNPs in the framework of a regression model. It allows for the adjustment of covariates and the main effect of SNPs, while the original MDR method cannot.

### Results

*f*=

_{ij}*P*(

*high risk*|

*SNP*

_{1}=

*i*,

*SNP*

_{2}=

*j*),

*Z*~

*N*(0,1),

*W*~N(0,1). Here,

*f*is an element from the

_{ij}*i*row and the

^{th}*j*column of a penetrance function, which defines a probabilistic relationship between a status of high-risk or low-risk and SNPs. We consider 14 different combinations of two different minor allele frequencies of (0.2 and 0.4) and seven different heritabilities of (0.01, 0.025, 0.05, 0.1, 0.2, 0.3, and 0.4) and 70 epistatic models with 70 various penetrance functions, as described by Velez et al. [10]. We also consider four censoring fractions (0.0, 0.1, 0.3, and 0.5).

^{th}*PBonf*and

*PRank*.

*PBonf*is the proportion of p-values less than the nominal sizes after adjusting for multiple testing among 100 cases. However, the power of AFT-MDR is defined as the percentage of times that it correctly chooses the disease-causal model out of each set of 100 datasets. Thus, we comparably define the power of the proposed method as

*PRank*, which is estimated as the percentage of times that the causal model has the smallest p-value out of all possible multi-locus models. We compared these three powers in the simulation study. In addition, we simulated two different scenarios according to the main effect.

*PBonf*,

*PRank*, and AFT-MDR methods over various combinations of two different minor allele frequencies, seven different heritabilities, and four different censoring fractions. As indicated in Fig. 2,

*PRank*is greater than

*PBonf*and the power of AFT-MDR for all cases, whereas the power of AFT-MDR is less than

*PBonf*, although the difference is smaller, as the censoring fraction is larger than 0.5. The trend of these three powers is similar, in the sense that they increase as the heritability increases, whereas they decrease as the censoring fraction increases. In addition, it is shown that the power is relatively larger when the minor allele frequency (MAF) is 0.2 than when the MAF is 0.4. When the heritability is smaller than 0.1, the power is not larger than 0.3, but

*PRank*rapidly increases as the heritability is greater than 0.1, but both

*PBonf*and the power of AFT-MDR slowly increase.

_{3}as follows:

*S*, is defined, and the significance of the gene-gene interactions between SNP

_{1}and SNP

_{2}is tested under the following model:

*PBonf*,

*PRank*, and AFT-MDR is similarly obtained and displayed in Fig. 3 and Table 3.

*PRank*shows a similar trend, as shown in Fig. 2, when the main effect is not considered. However, the size of

*PBonf*is also smaller, as shown in Fig. 2, and has almost no power when the MAF is 0.4 and censoring fraction is larger than 0.1. The effect of the censoring fraction is much larger on these three powers when the main effect is considered. Comparing

*PRank*with the power of AFT-MDR, AFT-MDR hardly detects any interaction effect when the main effect of SNPs is considered in the model. However,

*PRank*has slightly moderate power when MAF is 0.2 and the censoring fraction is smaller than 0.5.