Improved branch and bound algorithm for detecting SNP-SNP interactions in breast cancer
- Li-Yeh Chuang^{1},
- Hsueh-Wei Chang^{2, 3, 4},
- Ming-Cheng Lin^{5} and
- Cheng-Hong Yang^{5}Email author
https://doi.org/10.1186/2043-9113-3-4
© Chuang et al.; licensee BioMed Central Ltd. 2013
Received: 18 August 2012
Accepted: 6 February 2013
Published: 14 February 2013
Abstract
Background
Single nucleotide polymorphisms (SNPs) in genes derived from distinct pathways are associated with a breast cancer risk. Identifying possible SNP-SNP interactions in genome-wide case–control studies is an important task when investigating genetic factors that influence common complex traits; the effects of SNP-SNP interaction need to be characterized. Furthermore, observations of the complex interplay (interactions) between SNPs for high-dimensional combinations are still computationally and methodologically challenging. An improved branch and bound algorithm with feature selection (IBBFS) is introduced to identify SNP combinations with a maximal difference of allele frequencies between the case and control groups in breast cancer, i.e., the high/low risk combinations of SNPs.
Results
A total of 220 real case and 334 real control breast cancer data are used to test IBBFS and identify significant SNP combinations. We used the odds ratio (OR) as a quantitative measure to estimate the associated cancer risk of multiple SNP combinations to identify the complex biological relationships underlying the progression of breast cancer, i.e., the most likely SNP combinations. Experimental results show the estimated odds ratio of the best SNP combination with genotypes is significantly smaller than 1 (between 0.165 and 0.657) for specific SNP combinations of the tested SNPs in the low risk groups. In the high risk groups, predicted SNP combinations with genotypes are significantly greater than 1 (between 2.384 and 6.167) for specific SNP combinations of the tested SNPs.
Conclusions
This study proposes an effective high-speed method to analyze SNP-SNP interactions in breast cancer association studies. A number of important SNPs are found to be significant for the high/low risk group. They can thus be considered a potential predictor for breast cancer association.
Background
At present, identifying SNP-SNP interactions in genome-wide case–control studies is computationally and methodologically challenging [1]. To better understand the complex disease characteristics in case–control studies, we extended previous research of a breast cancer study and simultaneously explored single nucleotide polymorphism (SNP) combinations in low and high risk groups [2]. In complex diseases and cancers, joint genetic effects (epistasis) across the whole genome need to be considered. In a recent study, Phillips identifies three types of epistasis: compositional epistasis, statistical epistasis and functional epistasis [3]. Compositional epistasis blocks the effect of one allele by another at a different locus, statistical epistasis constitutes a statistical deviation from the additive effects of two loci on the phenotype, and functional epistasis addresses molecular interactions [3, 4].
Many methods have been developed to detect epistasis on the basis of a statistical definition to explore gene-gene interactions or SNP-SNP interactions (epistasis) in complex diseases; these include logic regression [5, 6], Multifactor-Dimensionality Reduction (MDR) [7], Polymorphism Interaction Analysis (PIA) [8], Bayesian model selection [9], SNPruler [10], random jungle [11], genetic algorithms [12] and other methods [13–16]. The challenges posed by traditional parametric statistical methods (e.g., logistic regression models) have been detailed in Hahn [6]. The MDR method is inspired by the combinatorial partitioning method, in which a data-reduction method effectively reduces the genotype predictors from n dimensions to one dimension. However, the computational load can be excessive when dealing with more than 10 polymorphisms [17]. PIA uses a case-based exclusion for missing SNP data, i.e., only those subjects for which all SNPs are identified (in a particular combination) are used in the analysis. SNPruler is a statistical method for identifying SNP combinations; it uses the Chi-square test to design the bound in the original Branch and Bound algorithm. Unlike our study, which focuses on the difference between cases and controls, SNPruler focuses on the ratio between cases and controls. Although these methods are widely used, they can still be improved upon. As a test data set increases in size, the run time increases exponentially with the order of interaction. However, few studies address SNP-SNP interactions for multiple SNPs. Hence, when a data set is sufficiently large, selecting an appropriate method becomes important.
This study proposes a method based on statistical epistasis and an improved branch and bound algorithm combined with feature selection (IBBFS) to explore combinations of SNP-SNP interactions in a breast cancer association study. The proposed method can reduce the search time and accurately determine the difference between cases and controls in low and high risk groups. Finally, we use the odds ratio (OR) as a quantitative measure to assess combinations of SNPs in the case–control studies. The odds ratio is a commonly-used statistic that expresses the strength of association between exposure and disease [18–20]. Experimental results show that the IBBFS method can determine risk factors in breast cancers.
Results
Identification of best SNP-SNP interaction combinations with maximal difference between cases and controls
Estimated best combinations of two SNPs on the occurrence of breast cancer
Combined SNP number (specific SNPs) | SNP Genotypes | Control number / Case number | CC | SN | SP | Average | Odds Ratio (95%CI) | p-value | |
---|---|---|---|---|---|---|---|---|---|
High-risk | Two SNPs | Other | 330/209 | ||||||
SNPs (4, 7) | 2-3 | 4/11 | 0.616 | 0.050 | 0.988 | 0.551 | 4.342 (1.259-16.394) | 0.013 | |
(Diff. = 7) | |||||||||
Two SNPs | Other | 327/209 | |||||||
SNPs (4, 6) | 3-2 | 7/11 | 0.610 | 0.050 | 0.979 | 0.546 | 2.459 (0.867-7.143) | 0.084 | |
(Diff. = 4) | |||||||||
Two SNPs | Other | 289/172 | |||||||
SNPs (3, 5) | 2-1 | 45/48 | 0.608 | 0.218 | 0.865 | 0.564 | 1.792 (1.118-2.875) | 0.014 | |
(Diff. = 3) | |||||||||
Low-risk | Two SNPs | Other | 197/151 | ||||||
SNPs (3, 4) | 1-1 | 137/69 | 0.480 | 0.314 | 0.589 | 0.461 | 0.657 (0.452-0.955) | 0.025 | |
(Diff. = 68) | |||||||||
Two SNPs | Other | 226/174 | |||||||
SNPs (3, 7) | 1-2 | 108/46 | 0.491 | 0.209 | 0.676 | 0.459 | 0.553 (0.364-0.839) | 0.004 | |
(Diff. = 62) | |||||||||
Two SNPs | Other | 223/168 | |||||||
SNPs (1, 3) | 2-1 | 111/52 | 0.496 | 0.236 | 0.668 | 0.467 | 0.622 (0.415-0.931) | 0.017 | |
(Diff. = 59) |
Estimated best combinations of SNPs on the occurrence of breast cancer in the high risk group
Combined SNP number (specific SNPs) | SNP Genotypes | Control number / Case number | CC | SN | SP | Average | Odds Ratio (CI) | p-value |
---|---|---|---|---|---|---|---|---|
Two SNPs | Other | 330/209 | ||||||
SNPs (4, 7) | 2-3 | 4/11 | 0.615 | 0.050 | 0.988 | 0.551 | 4.342 (1.259-16.934) | 0.013 |
(Diff. = 7) | ||||||||
Three SNPs | Other | 314/191 | ||||||
SNPs (3, 5, 6) | 2-1-1 | 20/29 | 0.619 | 0.132 | 0.940 | 0.564 | 2.384 (1.263-4.518) | 0.005 |
(Diff. = 9) | ||||||||
Four SNPs | Other | 325/203 | ||||||
SNPs (3, 4, 5, 6) | 2-1-1-1 | 9/17 | 0.617 | 0.077 | 0.973 | 0.556 | 3.024 (1.246-7.491) | 0.008 |
(Diff. = 8) | ||||||||
Five SNPs | Other | 329/210 | ||||||
SNPs (1, 3, 4, 5, 6) | 1-2-1-1-1 | 5/10 | 0.612 | 0.045 | 0.985 | 0.547 | 3.133 (0.969-10.680) | 0.031 |
(Diff. = 5) | ||||||||
Six SNPs | Other | 332/214 | ||||||
SNPs (1, 2, 3, 5, 6, 7) | 1-2-2-1-1-2 | 0/4 | 0.610 | 0.018 | 1.000 | N.E | ||
(Diff. = 4) | ||||||||
Seven SNPs | Other | 333/216 | ||||||
SNPs (1, 2, 3, 4, 5, 6, 7) | 2-2-2-1-1-1-1 | 1/4 | 0.608 | 0.014 | 0.997 | 0.540 | 6.167 (0.648-145.871) | 0.084 |
(Diff. = 3) |
Estimated best combinations of SNPs on the occurrence of breast cancer in the low risk group
Combined SNP number (specific SNPs) | SNP Genotypes | Control number / Case number | CC | SN | SP | Average | Odds Ratio (CI) | p-value |
---|---|---|---|---|---|---|---|---|
Two SNPs | Other | 197/151 | ||||||
SNPs (3, 4) | 1-1 | 137/69 | 0.480 | 0.314 | 0.600 | 0.465 | 0.657 (0.452-0.955) | 0.025 |
(Diff. = 68) | ||||||||
Three SNPs | Other | 260/189 | ||||||
SNPs (1, 3, 5) | 2-1-1 | 74/31 | 0.525 | 0.141 | 0.778 | 0.481 | 0.576 (0.355-0.934) | 0.020 |
(Diff. = 43) | ||||||||
Four SNPs | Other | 294/207 | ||||||
SNPs (1, 2, 3, 4) | 2-2-1-1 | 40/13 | 0.554 | 0.059 | 0.880 | 0.498 | 0.462 (0.228-0.919) | 0.018 |
(Diff. = 27) | ||||||||
Five SNPs | Other | 310/215 | ||||||
SNPs (1, 2, 3, 4, 5) | 2-2-1-1-1 | 24/5 | 0.569 | 0.023 | 0.928 | 0.507 | 0.300 (0.099-0.846) | 0.011 |
(Diff. = 19) | ||||||||
Six SNPs | Other | 323/218 | ||||||
SNPs (1, 2, 3, 4, 5, 6) | 2-2-1-1-1-2 | 11/2 | 0.587 | 0.009 | 0.967 | 0.521 | 0.269 (0.041-1.301) | 0.070 |
(Diff. = 9) | ||||||||
Seven SNPs | Other | 325/219 | ||||||
SNPs (1, 2, 3, 4, 5, 6, 7) | 2-2-1-1-1-2-1 | 9/1 | 0.588 | 0.005 | 0.973 | 0.522 | 0.165 (0.008-1.277) | 0.098 |
(Diff. = 8) |
Analysis of combinations of SNP (4, 7) and combinations of SNP (3, 4) in breast cancer
Odds ratio ( OR ) (95% CI) ( p -value) for SNP interactions in SNP (4, 7) combinations
Statistics | CXCL12^{*} | ||||
---|---|---|---|---|---|
Genotypes (X) | |||||
GG | AG | AA | |||
KITLG ^{ * } | TT | OR ^{ a } | 1.047 | 1.163 | 1.685 |
Genotypes (Y) | CI | 0.709-1.546 | 0.763-1.773 | 0.706-4.030 | |
p-value | 0.849 | 0.471 | 0.214 | ||
(Ca./Co.) | (66/97) | (54/73) | (13/12) | ||
CT | OR ^{ a } | 0.460 | 0.823 | 0.499 | |
CI | 0.265-0.793 | 0.503-1.341 | 0.106-2.034 | ||
p-value | 0.003 | 0.484 | 0.379 | ||
(Ca./Co.) | (22/65) | (33/59) | (3/9) | ||
CC | OR ^{ a } | 0.811 | 4.342 | N.E | |
CI | 0.288-2.221 | 1.259-16.394 | |||
p-value | 0.817 | 0.013 | |||
(Ca./Co.) | (7/13) | (11/4) |
Odds ratio ( OR ) (95% CI) ( p -value) for SNP interactions in SNP (3, 4) combinations
Statistics | CXCR4^{*} | ||||
---|---|---|---|---|---|
Genotypes (X) | |||||
CC | CT | TT | |||
KITLG ^{ * } | GG | OR ^{ a } | 0.657 | 1.557 | 1.528 |
Genotypes (Y) | CI | 0.459-0.940 | 0.936-2.590 | 0.414-5.636 | |
p-value | 0.025 | 0.110 | 0.719 | ||
(Ca./Co.) | (69/137) | (33/34) | (4/4) | ||
AG | OR ^{ a } | 1.092 | 1.222 | 1.012 | |
CI | 0.757-1.576 | 0.712-2.098 | 0.201-5.111 | ||
p-value | 0.639 | 0.484 | 1.000 | ||
(Ca./Co.) | (70/100) | (26/33) | (2/3) | ||
AA | OR ^{ a } | 1.076 | 1.012 | N.E | |
CI | 0.511-2.266 | 0.303-3.382 | |||
p-value | 0.848 | 1.000 | |||
(Ca./Co.) | (12/17) | (4/6) |
Rank analysis of odds ratios for breast cancer
Tables 2 and 3 show the estimated effects (CC, SN, SP, OR and 95% CI) of certain specific SNP combinations on the occurrence of breast cancer. These specific SNP combinations (two to seven SNPs) had a 0.657 to 0.165 risk of breast cancer (Table 3). In addition, these specific SNP combinations (two to seven SNPs) also show a higher risk (OR > 1) of breast cancer in Table 2. When the OR value is larger than 1, the proportion of subjects with breast cancer is higher. On the other hand, when the proportion of subjects with breast cancer is smaller than 1, the OR values are lower than 1.
Discussions
The representative difference of the [Control-Breast Cancer] occurrence value by PSO and GA
Combination of SNPs | SNP Genotype | Control (n)/ Breast Case (n) | Difference of Control -Breast Case (n) | Occurrence | |
---|---|---|---|---|---|
PSO | GA | ||||
SNPs (3,4) | (1–1) | 137/69 | 68 | 7 | 7 |
SNPs (3,7) | (1–2) | 108/46 | 62 | 2 | 1 |
SNPs (3,5) | (1–1) | 162/103 | 59 | 1 | 2 |
We focus on understanding the breast cancer risk of functionally-relevant joint effects of combinatorial SNPs within and between different cancer pathways. We calculated the same data set by exhaustive search (ES) using two to seven SNP combinations to find SNP interactions which determine an optimal solution. These calculations were rather time-consuming and the ES method of calculating combinations of SNPs is thus impracticable for large data sets. From a practical standpoint, the main difference between the aforementioned methods is the computational time required to reach an improvement. The IBBFS method found optimal solutions faster for a high order of interaction combinations by cutting off unnecessary paths. IBBFS guarantees that each result contains an optimal solution through the use the integrated feature selection method. The selected number of features is r = n-m + 1, where r is the number of features used, and n and m are the total number of SNPs and the number of selected SNPs, respectively. Examples of the ES and IBBFS calculations are respectively shown in Additional file 1: Figure S1 and Additional file 1: Figure S2. The number of possible solutions calculated by ES was 30229, whereas IBBFS reduced this number to 348. IBBFS is thus better suited to deal with large data sets. IBBFS allows for the investigation of an almost unlimited number of SNP combinations, whereas traditional algorithms are rather limited. The experiments show that IBBFS has great potential for the identification of complex biological relationships among cancer processes during the development of breast cancer.
Conclusion
This study focused on the selection of SNP combinations that give a maximal difference between case and control groups. Evaluating a large number of SNPs associated with a disease requires a strategy for focusing on only a select number of complex interactions. IBBFS was used on complex SNP-SNP interactions and was demonstrated to provide the best SNP-SNP interactions for predicting breast cancer susceptibility. The odds ratio (OR) was used as a quantitative measure of the breast cancer risk. Experimental results indicate that the proposed IBBFS method can identify the complex interactions of the tested SNPs both in the low and high risk groups. In the future, the IBBFS method can potentially be applied to SNP-SNP interactions (epistasis) in other association studies.
Methods
Data sets
Baseline characteristics of breast cancer cases and controls
SNP (Genes) | Chr. | SNP Genotype | Control no. /Case no. | Scoring function | p-value | ||||
---|---|---|---|---|---|---|---|---|---|
CC | SN | SP | Average | Odds Ratio | |||||
1. rs12812942 | 12 | 1-AA | 174/128 | ||||||
(CD4) | 2-AT | 141/76 | 0.482 | 0.372 | 0.552 | 0.469 | 0.733 (0.503-1.068) | 0.10 | |
3-TT | 19/16 | 0.564 | 0.111 | 0.902 | 0.526 | 1.145 (0.536-2.438) | 0.72 | ||
2. rs3136685 | 17 | 1-GG | 107/77 | ||||||
(CCR7) | 2-AG | 180/114 | 0.462 | 0.587 | 0.373 | 0.474 | 0.880 (0.594-1.304) | 0.57 | |
3-AA | 47/29 | 0.523 | 0.274 | 0.695 | 0.357 | 0.857 (0.478-1.536) | 0.68 | ||
3. rs2228014 | 2 | 1-CC | 254/151 | ||||||
(CXCR4) | 2-CT | 73/63 | 0.586 | 0.294 | 0.777 | 0.552 | 1.452 (0.962-2.191) | 0.07 | |
3-TT | 7/6 | 0.622 | 0.382 | 0.973 | 0.659 | 1.442 (0.421-4.880) | 0.57 | ||
4. rs1801157 | 10 | 1-GG | 175/106 | ||||||
(CXCL12) | 2-AG | 136/98 | 0.530 | 0.480 | 0.562 | 0.524 | 1.189 (0.822-1.723) | 0.37 | |
3-AA | 23/16 | 0.597 | 0.131 | 0.884 | 0.537 | 1.149 (0.550-2.387) | 0.73 | ||
5. rs3025039 | 6 | 1-CC | 211/155 | ||||||
(VEGF) | 2-CT | 117/59 | 0.498 | 0.276 | 0.643 | 0.472 | 0.687 (0.463-1.016) | 0.05 | |
3-TT | 6/6 | 0.574 | 0.037 | 0.972 | 0.528 | 1.361 (0.381-4.870) | 0.77 | ||
6. rs2287074 | 16 | 1-GG | 164/113 | ||||||
(MMP2) | 2-AG | 139/93 | 0.505 | 0.451 | 0.541 | 0.499 | 0.971 (0.670-1.408) | 0.93 | |
3-AA | 31/14 | 0.553 | 0.110 | 0.841 | 0.510 | 0.655 (0.316-1.347) | 0.25 | ||
7. rs10506957 | 12 | 1-TT | 182/133 | ||||||
(KITLG) | 2-CT | 133/69 | 0.486 | 0.342 | 0.578 | 0.469 | 0.709 (0.484-1.042) | 0.08 | |
3-CC | 19/18 | 0.568 | 0.119 | 0.905 | 0.531 | 1.296 (0.622-2.700) | 0.08 |
Branch and bound algorithm
The branch and bound algorithm (BB) is a divide-and-conquer approach used to solve global optimization issues [28]. The concept of a BB is based on constructing a search tree. Only feasible solutions are used and explicitly evaluated to detect optimal solutions. A BB algorithm requires two steps. First, a branching procedure is used to define the tree structure (the search tree). Then a bounding procedure that computes upper and lower bounds for the evaluation value (evaluation nodes) is implemented. If the next node (lower bound) in the series does not conform to the evaluation value (set bound value), the node is cut off. Compared to exhaustive search (ES), traditional BB algorithms do not guarantee that enough subtrees are cut off to keep the total number of criteria computations lower than in the ES method [29]. Under most circumstances, a traditional BB algorithm is faster than an exhaustive search. However, many redundant searches are still conducted in a BB algorithm [30]. To overcome this problem, Somol proposed the fast branch and bound algorithm [28] and Chen proposed an improved branch and bound algorithm for optimal feature subset selection [30]. Branch and bound algorithms have been successfully applied in many fields, such as predicting drug-like compounds [31], analysis of protein–protein interactions [32], feature selection problems [29] and data mining problems [33, 34]. In addition, branch and bound performance may be weaker under the following conditions: (1) Nearer to the root, the criterion value computation is usually slower (evaluated feature subsets are larger) and (2) nearer to the root, subtree cut-offs are less frequent (higher criterion values of larger subsets are compared to the bound, which is updated in the leaves) [28]. A possible solution tree is introduced in Additional file 1: Figure S3 and a BB algorithm flowchart is shown in Additional file 1: Figure S4.
Improved branch and bound algorithm with feature selection
As previously stated, traditional BB algorithms that search for all possible combinations are impractical since the number of combinations increases exponentially as the dimensionality increases [30]. Hence, we propose the use of a BB algorithm combined with a feature selection technique to reduce the necessary calculation time. Feature selection algorithms are special from a theoretical perspective. It can be shown that optimal feature selection for supervised learning problems requires an exhaustive search of all possible subsets of features of the chosen cardinality [30]. A large number of features is thus impractical. By using a subset of features, the processing time required by the classification process can be reduced. This improved branch and bound algorithm has several advantages when combined with feature selection (IBBFS). It not only reduces the search time but, more importantly, also sorts the results into low and high risk groups (discussed in the bound evaluation section). The IBBFS algorithm is very efficient because it avoids exhaustive searches (ES) by rejecting suboptimal subsets. It also guarantees that a selected subset yields the best global value. A flowchart of this process is shown in Additional file 1: Figure S5. The IBBFS pseudo-code is given in below.
IBBFS pseudo-code
B: Defined as 0.; R: Number of features used.; N: Total number of SNPs.; M: Total number of selected SNPs.; AVAIL: List of available feature values that LIST(m) can assume.; LIST(m): List of the features that can be assumed at level m.; Φ: Empty set.
Step 1: Initialize
Level m=1, AVAIL ={node_{ m } 1, node_{ m } 2, node_{ m }j, _{…}, node_{ m }r | node_{ m-1 } ≠Φ,
r=(n-m+1) × (n-(m-1)), j is the jth node}
Step 2: Generate branch
LIST(m)={AVAIL | select top r node based on their bound value}
If LIST(m) =Φ, go to step 5.
Step 3: Select node
Select the rightmost node in LIST(m), i.e., if node_{ m }j=max(LIST(m)) remove the
rightmost node in LIST(m)
Step 4: Calculate bound value
If bound(node_{ m }j) > B, return node_{ m }j to AVAIL and go to Step 5.
If last node in level m
If level m= higher level, go to Step 6, otherwise, m=m+1 and go to Step 2.
Step 5: Backtrack
If LIST(m) is empty, set m=m-1. If m=0, terminate the algorithm,
otherwise, go to Step 3.
Step 6: Higher level,
Sort nodes_{ m } based on bound value
Return best node_{ m }.
IBBFS uses top-down and right-left search strategies together with backtracking. We define the update bound value as 0, which means that, if the number of cases and controls is 0, the node should be not explored. If the bound value at a node j at level m is larger than the current bound value B, then the paths originating from that node to the bottom of the tree should still be explored. We select the top r node based on the bound value for exploration to the next level. Omitting the evaluation of bound values for a set of successor nodes (i.e., j < r at some parent nodes) is key to an efficient IBBFS. Backtracking is used until all successors or nodes and paths with bounds larger than the current bound value B have been searched. The computational savings in the IBBFS occur when the bound value at a node j at a higher level in the tree is the best value.
Bound evaluation
In Eq. (1), N represents the number of samples in the control data, and C represents the number of samples in the case data. Check_control_{ n } (i) and Check_case_{ c } (i) are respectively checked as to whether or not the node i (i.e., SNP combination) matches the n sample in the control data and the c sample in the case data. If a match occurs, Check_control_{ n } (i)/ Check_case_{ c } (i) is set to 1, otherwise, it is set to 0. $\sum _{n=1}^{N}\left(\mathit{Check}\_\mathit{contro}{l}_{n}\right)$ represents the sum of the Check_control_{ n } (i) from 1 to N, and $\sum _{c=1}^{C}\left(\mathit{Check}\_\mathit{cas}{e}_{c}\right)$ represents the sum of the Check_case_{ c } (i) from 1 to C. If the positive maximum bound value is selected as a feature in the next combination, then the respective OR value indicates a low cancer risk. On the other hand, if the negative maximum bound is selected as a feature in the next combination, then the respective OR value is associated with a high cancer risk. The supplementary example illustrates how the bound values are calculated.
For example, assume that SNPs (3, 4) with genotype (1–1) are the best SNP combination. SNP_{3} (rs2228014) has the three genotypes CC, CT, and TT, which can be respectively represented as 1, 2, and 3, and SNP_{4} (rs1801157) has the three genotypes GG, AG, and AA, which can also be respectively represented as 1, 2, and 3. We compute the number that matches the condition of the SNPs and genotypes for the case and control data. First, we calculate the control number for SNP_{ 3 } with genotype 1 and SNP_{ 4 } with genotype 1. The number of controls that independently match SNP_{ 3 } with genotype 1 and SNP_{ 4 } with genotype 1 are 254 and 175, respectively. The number of controls that match SNP (3, 4) with genotype (1–1) is thus 137. Secondly, we calculate the number of cases independently matching SNP_{ 3 } with genotype 1 and SNP_{ 4 } with genotype 1 as 151 and 106, respectively. The number of cases that match SNP (3, 4) with genotype (1–1) is thus 69. According to Eq. (1), the bound value is determined by subtracting 69 from 137, thus giving 68.
Performance measurement
TP represents the number of true positives, TN represents the number of true negatives, FN represents the number of false negatives, and FP represents the number of false positives.
Illustrative example
The proposed IBBFS algorithm with incorporated feature selection selects the most promising solution and then evaluates only the features of the next SNP combinations of this branch. Furthermore, the algorithm is based on the expansion of two-SNP combinations, which means that the two-SNP combination results are used and expanded until the maximum combination (number of SNPs) is reached. For example, if the SNP (1, 2) with genotype (2–2) combinations constitutes the best result (feature), then combinations of three SNPs that contain SNP (1, 2) with genotype (2–2) are found in the next step. The expanded results are SNP (1, 2, 3) with genotype (2-2-1), SNP (1, 2, 3) with genotype (2-2-2), and SNP (1, 2, 3) with genotype (2-2-3). A detailed example is shown in Additional file 1: Figure S7. These expanded results reduce the search time by cutting off unnecessary paths. The update bound value in this study was set to 0, which means that, if the numbers of cases and controls are 0, the node is cut off. In contrast to the BB algorithm, IBBFS only uses selected features (after sorting the results), which allows it to find an optimal solution by cutting off unnecessary pathways. Although the IBBFS algorithm is of a high time complexity for combinations of two SNPs, it performs better for interaction combinations of a high order. After the best SNP combinations are found, the OR is used in the next step to evaluate each best SNP combination with regard to the susceptibility risk. A simple IBBFS calculation process is shown in the Additional file 1 section.
where n is the total number of SNPs, and m is the number of selected SNP combinations.
When two SNPs are selected and each genotype has three possible state combinations, ES calculates the number of possible solutions as C(4,2)*3^{ 2 }=54. Based on the aforementioned calculation process, the use of traditional BB algorithms or ES to explore combinations of three, four or more SNPs is impractical since the increased number of combinations exponentially increases the time complexity Simple ES, BB and IBBFS calculation processes are shown in the Additional file 1 section.
Declarations
Acknowledgements
This work was partly supported by the National Science Council in Taiwan under grants 101-2622-E-151-027-CC3, 100-2221-E-151-049-MY3, 100-2221-E-151-051-MY2, by the National Sun Yat-Sen University-KMU Joint Research Project (#NSYSU-KMU 102-034) and DOH102-TD-C-111-002.
Authors’ Affiliations
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