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SUPPLEMENTARY METHODS
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AMBROSIA: The pseudocode for the AMBROSIA algorithm is shown in Supplementary
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Figure 1.
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Case Study 1, Interactions in Data with Simulated LD Patterns: The gene-gene
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interaction model used for this case study is shown in Supplementary Figure 2A.
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We simulated 60 SNPs in four groups, G1 through G4, with linkage disequilibrium within
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each group (Supplementary Figure 2B). The disease-causing SNPs corresponding to
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SNPs i and j in Supplementary Figure 2A were SNP 7 (red in G1) and 22 (red in G2),
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respectively. The minor allele frequencies (MAF) of the central SNP in each group were
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0.5; the MAF of the flanking SNPs and LD pattern were the same within each group
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and are shown in Supplementary Figure 2B. The penetrance matrix corresponding to
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the disease associated gene-gene interaction between SNP 7 and SNP 22 was:
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BB
Bb
bb
AA
p
p 1 p
Aa
p
p 1 p
aa 1 p 1 p 1 p
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The major and minor alleles for SNP 7 are A and a, respectively, whereas the major
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and minor alleles for SNP 22 are B and b, respectively. The probability of disease in
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individuals with the risk-associated genotype combinations, {AA, BB}, {AA, Bb}, {Aa,
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BB}, {Aa, Bb}, is p. The probability of disease in individuals without the risk-associated
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genotypes was set to (1 – p). In this context, the relative risk was the ratio of p to (1 –
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p). The case-control status phenotype variable was denoted by Y.
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The relative risk values used in simulation were 1.2, 1.5, 1.8, 2.0 and 2.5. The case-
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control design with 2500 cases and 2500 controls was used.
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Power was assessed from 100 independent repetitions of the simulation procedure.
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Power was defined as the proportion of models that contained the interacting
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combinations. The average number of false combinations per model (FCM) was
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defined as the number of false combinations in each model averaged over the 100
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independent repetitions. For Case Study 1, the AMBIENCE input parameter values
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were  = 10 and = 2. All power calculations in AMBROSIA were done with  = 0.001.
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Case Study 2, Gene-Gene Interactions with Genetic Heterogeneity: To challenge
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the capabilities of AMBROSIA, we created a simulation with two pairs of gene-gene
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interactions.
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The model for Case Study 2, summarized in Supplementary Figure 2C, contained 120
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SNPs in eight groups. The allele frequencies of the central SNPs were all 0.5; the MAF
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of the flanking SNPs and the LD patterns within each group were similar to Case Study
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1.
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The model contained genetic heterogeneity (GH) with two pairs of interacting loci, SNP
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7 with SNP 22 and SNP 67 with SNP 82 that each increased risk in half of the cases.
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Both pairs of interactions followed the penetrance matrix of Case Study 1. The relative
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values used in simulation were 1.2, 1.5, 1.8, 2.0 and 2.5 and a case-control design with
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2500 cases and 2500 controls was used. For Case Study 2, the AMBIENCE input
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parameter values were  = 10 and = 2.
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Comparisons with Multi-factor Dimensionality Reduction (MDR)
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The AMBROSIA method was compared head to head with MDR
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implementation was downloaded from http://sourceforge.net/projects/mdr/.
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Comparisons to MDR using Case Study 2. The MDR method was first run to conduct
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an exhaustive search of all possible 1 to 4-locus interactions with each data repetition
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of Case Study 2. However, each of these MDR runs failed to complete successfully
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The MDR
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after 48 hours of run time. To enable comparison of AMBROSIA to MDR, we created a
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smaller data set by simplifying Case Study 2 to contain only 24 SNPs. The central SNP
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and one SNP with R2 = 0.9 and one SNP with R2 = 0.8 were selected from each of the
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eight Groups. The LD patterns of this reduced data set are summarized in
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Supplementary Figure 2D.
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The model contained genetic heterogeneity (GH) with two pairs of interacting loci:
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Interaction 1 consisted of SNP 1 and SNP 4 and Interaction 2 consisted SNP 13 and
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SNP 16 that each increased risk in half of the cases. Both pairs of interactions followed
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the penetrance matrix of Case Study 1. The simulation strategy was similar to Case
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Study 1.
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The relative risk value of 2.0 was studied and a case-control design with 2500 cases
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and 2500 controls was used.
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The MDR method was first run to conduct an exhaustive search of all possible 1 to 4-
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locus interactions with each data repetition. Statistical significance in MDR was
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obtained by comparing the observed prediction error for each MDR model to the null
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distribution obtained from 10,000 permutations.
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The AMBIENCE input parameter values were  = 50 and = 2. The power and FCM of
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MDR and AMBROSIA were computed from 100 independent simulations.
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Comparisons with MDR using Models Based on the MDR Power Paper. Four two-
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locus interaction models employed in the MDR power evaluation paper by Ritchie et al.
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1
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models used is shown in Table 1 of the main paper.
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A case-control study design with 2500 cases and 2500 controls was assumed. 24
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diallelic SNPs were simulated in eight groups (Supplementary Figure 1D). For each
were used for comparison against AMBROSIA. The penetrance matrices for the
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group with 3 SNPs, the central SNP is in LD with R2 = 0.9 and R2 = 0.8 with the other
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two SNPs of that group respectively.
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Four interaction models were simulated with genetic heterogeneity. Models 1-GH, 2-
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GH, 3-GH and 4-GH contained genetic heterogeneity (GH) with two pairs of interacting
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loci, SNP(1) with SNP(4), defined as Interaction 1 and SNP(13) with SNP(16), defined
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as Interaction 2. The corresponding penetrance matrices in Table 1 were used for
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simulations for both pairs of interacting loci for each model such that each Interaction
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increased disease risk in half of the cases. The remaining SNPs were not associated
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with the phenotype. For models 1-GH and 2-GH, within each 3-SNP group, the allele
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frequency for the central SNP was 0.5; the minor allele frequencies (MAF) for the other
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two SNPs were 0.445 (R2 of 0.9 with the central SNP) and 0.474 (R2 of 0.8 with the
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central SNP) respectively. For model 3-GH, within each 3-SNP group, the MAF for the
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central SNP was 0.25; MAF for the other two SNPs were 0.232 (R2 of 0.9 with the
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central SNP) and 0.211 (R2 of 0.8 with the central SNP) respectively. For model 4-GH,
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within each 3-SNP group, the MAF for the central SNP was 0.1; MAF for the other two
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SNPs were 0.091 (R2 of 0.9 with the central SNP) and 0.082 (R2 of 0.8 with the central
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SNP) respectively. For each model, we simulated 100 data sets. Genotypes were
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assumed to be in Hardy-Weinberg equilibrium proportions.
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SUPPLEMENTARY RESULTS
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Demonstration Run with Case Study 1, Interactions in Data with Simulated LD
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Patterns: We used the top 10 one-SNP and two-SNP combinations with the highest
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KWII values identified by AMBIENCE to build the most parsimonious model using
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AMBROSIA.
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In the first step of AMBROSIA, we confirmed that all the combinations identified by
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AMBIENCE were significant using permutation based approaches with  = 0.001. The
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KWII values for the combinations are summarized in Supplementary Table 1. The value
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of  was set to 0.001.
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AMBROSIA starts with the combination {22, Y}, which has the highest KWII, as the first
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member combination in the model. In the next step, it assesses whether {7, Y} can be
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added to the model. It evaluates KWII(7, 22, Y) = 0.012 > 0 and because MICC = 254,
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which makes e–MICC < . As a result {7, Y} gets added to the model, so that M = {{22, Y},
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{7, Y}}. In the same step, the significant combination {7, 22, Y} is also added to the
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model because it has positive KWII and since its sub-combinations are present in the
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model, the overall model complexity is not increased. Next the combination {23, Y} is
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evaluated; however, KWII(22, 23, Y) = -0.036 < 0, so {23, Y} is rejected. Repeating
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these steps for the 1-way and 2-way combinations listed in Table 2 does not add any
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more combinations to M. Finally M = {{22, Y}, {7, Y}, {7,22, Y}} emerged as the most
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informative and parsimonious model explaining the disease phenotype. The
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significance of the overall model was p < 0.0001. These results from AMBROSIA are
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consistent with model used for simulation.
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These promising results provide proof of concept that AMBROSIA can be used for
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model synthesis in the presence of the confounding effects of LD.
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TABLES
Supplementary Table 1. Top 10 one and two-way combinations for Case Study 1.
1-way Combinations
KWII
2-way Combinations
KWII
{22, Y}
0.026
{7, 22, Y}
0.450
{7, Y}
0.026
{7, 21, Y}
0.380
{6, Y}
0.023
{8, 21, Y}
0.379
{8, Y}
0.023
{7, 23, Y}
0.379
{21, Y}
0.023
{6, 22, Y}
0.378
{23, Y}
0.023
{8, 22, Y}
0.323
{5, Y}
0.020
{8, 23, Y}
0.322
{24, Y}
0.020
{6, 21, Y}
0.322
{20, Y}
0.020
{6, 23, Y}
0.321
{9, Y}
0.020
{7, 24, Y}
0.319
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Supplementary Table 2. Interactions in the GAW15 data set.
Locus
Chr
SNP #
Phenotype
Effects
DR
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152-155
RA
Affects risk of RA
A
16
30-31
RA
Controls effect of DR on RA risk
B
8
442
RA
Controls effect of smoking on RA risk
C
6
152-155
RA
Increases RA risk only in women
D
6
161-162
RA
Rare allele increases RA risk 5-fold
E
18
268-269
RA, Anti-CCP
F
11
387-389
IgM
G
9
185-186
Severity
25% QTL for severity
H
9
192-193
Severity
25% QTL for severity
Age
-
RA
Affects RA risk through smoking and sex ratio
Sex
-
RA
Affects RA risk with Locus C
Smoking
-
RA, IgM
Affects of DR on anti-CCP and increases RA risk
QTL for IgM
Affects RA risk with Locus B and through IgM
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FIGURE LEGENDS
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Supplementary Figure 1. Pseudocode for the AMBROSIA algorithm.
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Supplementary Figure 2. Supplementary Figure 1 shows the gene-gene interaction
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model used to generate the data for Case Study 1. The red bars correspond to the
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disease-associated interacting central SNPs whereas the blue bars represent the non-
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interacting central SNPs in each group. The height of the bars corresponds to the
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extent of linkage disequilibrium from the central SNP as measured by R2. The R2 values
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are indicated and range from 0.2-0.9 in intervals of 0.1 for Figure 1B and 1C and are
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0.9 and 0.8.
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SUPPLEMENTARY FIGURE 1
Algorithm : Ambrosia
Input :
 : Set of combinations sorted by KWII.
 : Threshold for complexity filter.
Output :
 : Model
1. Compute combination p - values with permutations.
2.   0; # MIC; N  no. of samples in the data;   1;
3. 1  2N    2   ;
4.   ;
5. while  is not empty do
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C  Combination with sub-combinations already in M or the one with highest KWII from
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 temp    {C};
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  Degrees of freedom of M
temp
;
9.  2  2N  (  KWII(C))  2   ;
10.    2  1; # MICC
11. if exp(- )   # Complexity Filter#
12.
13.
14.
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if redundancy(M, C) is false # Redundancy Filter#
M  M temp ;
    KWII(C);
endif
16. endif
17.    \ {C};
18. endwhile
19. return M;
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;
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SUPPLEMENTARY FIGURE 2
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REFERENCES
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Ritchie MD, Hahn LW, Moore JH (2003). Power of multifactor dimensionality reduction
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for detecting gene-gene interactions in the presence of genotyping error, missing data,
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phenocopy, and genetic heterogeneity. Genet Epidemiol 24(2): 150-157.
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Ritchie MD, Hahn LW, Roodi N, Bailey LR, Dupont WD, Parl FF et al (2001).
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Multifactor-dimensionality reduction reveals high-order interactions among estrogen-
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metabolism genes in sporadic breast cancer. Am J Hum Genet 69(1): 138-147.
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