We applied a new approach based on Mantel statistics to analyze the Genetic Analysis Workshop 14 simulated data with prior knowledge of the answers. The method was developed in order to improve the power of a haplotype sharing analysis for gene mapping in complex disease. The new statistic correlates genetic similarity and phenotypic similarity across pairs of haplotypes from case-control studies. The genetic similarity is measured as the shared length between haplotype pairs around a genetic marker. The phenotypic similarity is measured as the mean corrected cross-product based on the respective phenotypes. Cases with phenotype P1 and unrelated controls were drawn from the population of Danacaa. Power to detect main effects was compared to the X2-test for association based on 3-marker haplotypes and a global permutation test for haplotype association to test for main effects. Power to detect gene × gene interaction was compared to unconditional logistic regression. The results suggest that the Mantel statistics might be more powerful than alternative tests.
Recently we proposed a flexible approach to gene mapping of complex diseases, whereby we combine Mantel statistics for space-time clustering with genetic information obtained from haplotypes . It has been shown that haplotype sharing methods are well suited for mapping such genes [2-5]. Mantel statistics were introduced in 1967 to correlate temporal and spatial distributions of cancer, notably childhood leukemia, in a generalized regression approach . The Mantel statistic M is the sum of the cross product of the spatial similarity Xij multiplied by the temporal similarity Yij across all pairs of cases i and j:
The idea behind this approach is that in the presence of space-time clustering the values of spatial similarity Xij correspond to the values of temporal similarity Yij for correlated cases i and j.
Mantel statistics using haplotypes
Here we apply the general approach of Mantel's statistics for space-time clustering (Equation 1) to correlate genetic and phenotypic similarity, and to test for gene × gene interaction. The first statistic has the form:
where x denotes a genetic marker, and i and j are haplotypes. Lij(x) denotes the genetic similarity between the haplotypes i and j at x, and is defined as the number of intervals surrounding x that are flanked by markers with the same alleles, i.e., that are identical by state (IBS). The phenotypic similarity for two haplotype copies i and j derived from individuals si and sj is defined as the mean corrected product Ysisj = (ysi - μ)(ysi - μ), where ysi and ysj are the phenotypes of si and sj, and μ denotes the expectation of the phenotype. Here, we chose μ as the sample mean, i.e., μ = 0.5. Concordant pairs of affected and concordant pairs of unaffected individuals have the weights Ysisj = 0.25, while discordant pairs have the weights Ysisj = -0.25. Alternative measures of phenotypic similarity were discussed in the framework of sib-pair analysis, e.g., the Haseman-Elston method  and the weighted pair-wise correlation statistics , as well as in family-based association analysis . The summation is over all pairwise comparisons of haplotypes for i ≠ j, where the haplotypes are derived from case-control studies.
The second statistic is constructed to test for the combined effect of two loci:
The information of the first locus x is incorporated as the shared length Lij(x). At the second locus only genotype information is used. The variable zsi is coded in a dominant way, i.e., zsi is 1, if the individual si carries at least one mutant allele, and 0 otherwise. The measure of genotypic similarity Zsisj is then 1, if zsi = zsj, and 0 otherwise.
The summands of the Mantel statistic are highly correlated, and any statistical procedure to test for significance has to take into account the interrelationship of the data. Here, we use a Monte Carlo permutation approach to test for significance, as proposed by Mantel . For M0(x) the phenotype ysi is permuted over the individuals. The definition of Z is such that M1(x) is the sum over all comparisons of haplotypes from individuals who have the same genotype coding z at the second locus. To derive the null hypothesis of no statistical interaction, the phenotype ysi and the genotype coding zsi at the second locus for individual si are permuted jointly over the individuals, and thus the comparisons of haplotypes derived from discordant individuals are incorporated under the null hypothesis.
Statistical tests for comparison
We used two alternative tests for power comparison.
1. We applied the X2-test for association to 3-marker haplotypes. The region of interest was covered by overlapping sliding windows. The haplotypes consisted of 3 consecutive genetic markers. The test was based on a 2xk X2-table, with k-1 degrees of freedom, where k denotes the number of haplotypes that occurred in either the case or the control sample. A p-value was assigned to the marker in the center of the window. Note that no tests were performed for the marginal markers.
2. The haplotype assignment software PHASE [10,11] performs a global permutation test for significant differences in haplotype frequencies in case and control groups. PHASE tests the null hypothesis that the case and control haplotypes are a random sample from a single set of haplotype frequencies, versus the alternative that cases are more similar to other cases than to controls. Here, this test was based on 100 permutations due to computational burden.
Gene × gene interaction
We compared the test statistic M1(x) using haplotypes to unconditional logistic regression based on the genotypes at 2 genetic markers . The respective genotypes were coded for both the recessive and the dominant model.
Datasets and genetic data
The case-control study samples for two different samples sizes were drawn from the population Danacaa to limit the analysis to individuals defined by phenotype P1.
In this dataset, two major genes, D1 and D2, interacted in an epistatic model. Mode of inheritance is dominant for both D1 and D2.
Table 1 shows the samples that were used to test for main effects. Major gene D1 is located on chromosome 1. We chose flanking single-nucleotide polymorphisms (SNPs) of the disease locus between C01R0045 and C01R0055 from the initial set of markers (sample A), and additional SNPs and microsatellites from packages 28 and 29 (samples B to D). For major gene D2, which is located at the very end of chromosome 3, we analyzed 6 flanking SNPs C03R0276–0281 (samples E and F). To test for gene × gene interaction, information from both disease loci D1 and D2 were used to define the measures L and Z for M1(x). For samples A-D, the markers in Table 1 were used to define the variable L at gene D1, and the SNPs C03R0276–C03R0281 at gene D2 to define the variable Z. For samples E and F, the markers in Table 1 were used to define the variable L at gene D2, and the SNPs C01R0050–C01R0053 at gene D1 to define the variable Z.
Table 1. Study samples used in the analysis
Haplotype pairs assigned to the unrelated individuals were estimated by the use of the PHASE program [10,11]. PHASE lists the most likely pairs of haplotypes for each individual, together with their posterior probability. The most likely (best) estimate of haplotype pairs was chosen for our analysis. SAS 8.02 (SAS Institute Inc., Cary, NC, USA) was used to test for normality and for logistic regression. All other calculations were performed with software developed within our group. Software for the proposed Mantel statistics is available upon request.
Table 2 shows the results for the analysis of main effects of genetic markers close to D1 and D2. For D1, the Mantel statistic M0(x) yielded point-wise significant results at the marker position C01R052 (p = 0.042), which is the marker closest to D1 for the small sample B. For the large sample D, which included additional SNPs, M0(x) yielded the most significant result at SNP C01R0045 (p = 0.014).
Table 2. Results of the Mantel statistic (x) and the haplotype-based X2hap – test for main effects
M0(x) did not yield significant results for the markers flanking D2 with small sample size. The most significant SNP in the large sample was C03R0280 (p = 0.002). The X2hap-test for association, however, did not produce significant results with either the small or the large samples. The permutation test yielded one globally significant p-value of 0.03 in the large sample D.
Gene × gene interaction
Table 3 shows the results for M1(x). The genetic similarity was defined by the same marker sets as in Table 1. M1(x) yielded significant results for all samples except sample A. The most significant results were at the closest markers for D2 (samples E and F), but not for D1. Logistic regression did not reveal significant results for interaction between SNPs surrounding gene D1 and SNPs flanking D2 for the different samples (results not shown).
Table 3. Results of the Mantel statistic M1(x) to test for gene × gene interaction
We successfully employed a new approach to map disease predisposing genes in case-control studies based on Mantel statistics that correlate genetic and phenotypic similarity. Two types of gene effects involved in complex diseases were considered: main effects and joint effects.
1. The Mantel statistic M0(x) identified the major gene D2 on chromosome 3 given adequate sample size, whereas the alternative methods failed. Major gene D1 on chromosome 1 was simulated without linkage disequilibrium (LD). LD is necessary for haplotype association methods, therefore M0(x)-as expected-did not map D1 correctly.
We acknowledge that the comparison against the X2 association test for 3 marker haplotypes is somewhat unfair, but we know of no other standard association test examining longer haplotypes that is not confronted with problems of huge degrees of freedom and sparse data. Additionally, other more sophisticated haplotype-based methods cannot yet be regarded as standard.
2. The Mantel statistic M1(x) accounted for the joint effects of 2 putative disease loci. Taking the combined effects into account, the results were significant for the major genes D1 and D2 and showed lower p-values than the results obtained when considering main effects only.
These results show that main effects might not be detectable if gene × gene interaction is present and not considered in the analysis. Our proposed method M1(x) revealed significant statistical interaction between the genes analyzed in contrast to the results obtained in the logistic regression model.
The proposed Mantel statistics employ haplotypes from case-control data and might not be robust to population stratification. In our analysis, we used samples drawn from the Danacaa population and affection status defined by phenotype P1 to reduce heterogeneity in the data. Population stratification is therefore not a major concern in this analysis. We did not adjust the p-values for multiple comparisons in this candidate analysis.
Multiple testing is a serious problem especially if all possible gene × gene interactions increase the multiplicity. We solved the problem in the mean time by implementing a step-down algorithm to take into account multiple testing [13,14].
Comprehensive power comparisons are currently being carried out to reveal under which conditions our approach is more powerful than alternative methods.
GAW14: Genetic Analysis Workshop 14
IBS: Identical by state
SNP: Single-nucleotide polymorphism
LB participated in planning, interpreting data, carrying out the statistical analysis, and drafting the manuscript. CF participated in planning, interpreting data, writing the manuscript. MO and MR participated in computation and statistical analysis. JC-C participated in planning, interpreting data, and writing the manuscript. All authors read and approved the final manuscript.
This work was supported by a Deutsche Forschungsgemeinschaft grant (CH117/3-1) (LB, MR, MO). We thank Kati Smit for technical assistance.
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