# The Iterative Exclusion of Compatible Samples Workflow for Multi-SNP Analysis in Complex Diseases
## 1 Introduction  
Single-nucleotide polymorphism (SNP), the most prevalent form of genetic variation in the human genome, represents a third-generation genetic marker [1; 2; 3; 4]. SNPs are connected to the occurrence of inherited diseases in humans [5], while there is still limited understanding regarding the mechanism underlying this phenomenon [6; 7]. Some associations between SNPs and diseases have been discovered, including the primary effect of single SNPs, interactions between SNPs, and interactions between SNPs and the environment [8; 9; 10]. The main effects of single SNPs can be detected by single-point association analysis [11; 12; 13]. However, this approach can only explain a small portion of complex diseases. To explain more complex diseases, epistasis analysis is required to detect SNP-SNP interactions [14; 15]. Studies of epistasis analysis methods start with small datasets. With the development of genome sequencing technologies, extensive volumes of data have been produced, resulting in the widespread implementation of genome-wide association studies (GWAS) [16]. GWAS have been carried out to identify sequence variations in the whole human genome and screen out the SNPs associated with diseases through single-point association analysis and epistasis analysis [17; 18]. With the advancement of bioinformatics, numerous epistasis analysis methods have emerged, but epistasis analysis is faced with the challenge of combinatorial explosion since GWAS data are characterized by high dimensions [19]. For epistasis analysis, the methods can be mainly classified into searching, screening, and machine learning methods. The searching method transforms the mining of SNP-SNP interactions into a problem of searching for SNP combinations in an N-dimensional space. For instance, multifactor dimension reduction (MDR) is a searching method, proposed in 2001 [20], which can transform a structure of high dimensions into a structure of one dimension which consists of two levels (high risk or low risk). Following dimensionality reduction, evaluation of the capability to identify and predict diseases using the one-dimensional multifactor combination can be performed through cross-validation and permutation tests [21]. In the following years, the MDR method has been continuously improved. For instance, an enhanced method named OR-MDR (odds ratio-based MDR) was introduced by incorporating the odds ratio as a risk indicator [22], which greatly improves the recognition ability but increases the amount of calculation. GMDR (generalized multifactor dimensionality reduction) (GMDR) broadens the data range of MDR to continuous variables [23]. The method of MB-MDR (multifactor dimensionality reduction based on models) can be applied to investigate datasets with limited initial sample sizes [24]. MDRGPU is a GPU-based multifactor dimension reduction method with great improvement of computing speed [25]. QMDR is an algorithm that can identify models for quantitative traits [26]. MDR-ER, proposed in 2013, introduces a classifier function, which improves the probability of the correct classification of genotypes but also increases the amount of calculation [27]. Fuzzy MDR combines the fuzzy set theory, in which the conditional variables can be fuzzy data between 0 and 1 [28]. UM-MDR is a unified model-based MDR method that reduces the error rate by using a regression framework with a semi-parametric correction procedure [29]. The combination of classification-based multifactor dimensionality reduction (CMDR) with the differential evolution algorithm has led to the development of an innovative algorithm known as DECMDR, which shows improvement of recognition ability but an increase in the calculation amount [30]. Multi-objective MDR (MOMDR) regards the contingency table of MDR as the target equation and employs the classification accuracy and likelihood ratio to measure SNP-SNP interactions, which improves the recognition ability of MDR [31].