# The Iterative Exclusion of Compatible Samples Workflow for Multi-SNP Analysis in Complex Diseases
## 2 Materials and Methods
### Iterative Exclusion of Compatible Samples Workflow
The Alzheimer's disease data were also analyzed by MDR and BOOST, and the results and comparisons with those of IECS are shown in \[\text{Coverage}(X_{i}\leq Y_{i})=\sum[\min(X_{i},Y_{i})]/\sum(Y_{i}) \tag{2}\]0 IECS obtained three significant items with a runtime of 4.496 s. MDR obtained one significant item, namely, the simultaneous mutation of IV S22 + 36 C > A and 3'UT R159 C > T, with a runtime of 42.591 s. BOOST obtained an insignificant item, namely, the simultaneous mutation of IV S10 \(-\) 5 C > T and IV S22 + 36 C > A, with a runtime of 0.368 s. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Noise (\%)**} & \multicolumn{2}{c}{**Runtime (Seconds)**} & \multirow{2}{*}{**Samples**} & \multicolumn{3}{c}{**Runtime (Seconds)**} \\  & & **BOOST** & & **IECS** & & **MDR** & **BOOST** & **IECS** \\ \hline
10 & 14.258 & 0.056 & 0.962 & 200 & 13.783 & 0.056 & 0.953 \\
20 & 14.241 & 0.056 & 1.034 & 400 & 17.243 & 0.059 & 1.013 \\
30 & 14.238 & 0.056 & 1.069 & 600 & 20.658 & 0.063 & 1.070 \\
40 & 14.221 & 0.056 & 1.040 & 800 & 24.159 & 0.074 & 1.151 \\
50 & 14.104 & 0.056 & 0.969 & 1000 & 26.810 & 0.068 & 1.162 \\
60 & 12.393 & 0.056 & 0.783 & 1200 & 30.203 & 0.071 & 1.229 \\
70 & 11.990 & 0.058 & 0.651 & 1400 & 33.574 & 0.075 & 1.288 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Runtimes of IECS, MDR, and BOOST for simulated data with different noise percentages and numbers of samples. Previous research has demonstrated the interactive association of Alzheimer's disease with two SNPs, namely, IV S22 + 36 C \(>\) A and 3'UT R159 C \(>\) T, located within introns [22]. This interaction was determined using the multifactor dimensionality reduction method based on a log-linear model and the multifactor dimensionality reduction algorithm. IV S17 \(-\) 294 C \(>\) T in introns was associated with an increase in the risk for Alzheimer's disease, as indicated by the statistical analysis and the haplotype analysis; in addition, IV S22 + 36 C \(>\) A in introns was also related to a higher risk of Alzheimer's disease [48]. Among IECS, MDR, and BOOST for the Alzheimer's disease dataset, IECS could obtain more results than MDR and BOOST, its runtime is relatively short, and the results are all supported by the literature, demonstrating that IECS can detect multiple SNPs related to complex diseases. According to the significance of necessary conditions, we took [PERSON]'s disease as the outcome and eight SNPs as the conditional variables to perform the chi-square test. The relation of necessary conditions was expressed as \"No A, then no B\" logically. A conditional variable is deemed necessary if its consistency exceeds 0.9 and its coverage surpasses 0.5 [49]. There are four conditional variables necessary for [PERSON]'s disease: \(\sim\)\(-\)204 G \(>\) C, \(\sim\)C\(\cdot\)401 A \(>\) G, \(\sim\)IV S10 \(-\) 5 C \(>\) T, and \(\sim\)IV S15 + 144 T \(>\) A (\"\(\sim\)\" denotes no mutation of the SNP). According to the significance of necessary conditional variables, a chi-square test was conducted by taking the disease as the outcome. The results are shown in Table 4. \"No mutation of \(-\)204 G \(>\) C\" under Alzheimer's disease is significant with a \(p\)-value of 0.008, suggesting that if the SNP of \(-\)204 G \(>\) C is mutated, the individual will not get Alzheimer's disease. The other three conditional variables are not significant. In a previous study, analysis of the transcription factor binding site performed by Consite showed that the mutation at the position of \(-\)204 G \(>\) C enables it to enhance the expression of neprilysin and reduce the accumulation of A \(\beta\) (amyloid beta) in the brain, which possibly hinders Alzheimer's disease [48].