Calculation of exact p-values when SNPs are tested using multiple genetic models
- Rajesh Talluri^{1},
- Jian Wang^{1} and
- Sanjay Shete^{1, 2}Email author
https://doi.org/10.1186/1471-2156-15-75
© Talluri et al.; licensee BioMed Central Ltd. 2014
Received: 5 March 2014
Accepted: 27 May 2014
Published: 20 June 2014
Abstract
Background
Several methods have been proposed to account for multiple comparisons in genetic association studies. However, investigators typically test each of the SNPs using multiple genetic models. Association testing using the Cochran-Armitage test for trend assuming an additive, dominant, or recessive genetic model, is commonly performed. Thus, each SNP is tested three times. Some investigators report the smallest p-value obtained from the three tests corresponding to the three genetic models, but such an approach inherently leads to inflated type 1 errors. Because of the small number of tests (three) and high correlation (functional dependence) among these tests, the procedures available for accounting for multiple tests are either too conservative or fail to meet the underlying assumptions (e.g., asymptotic multivariate normality or independence among the tests).
Results
We propose a method to calculate the exact p-value for each SNP using different genetic models. We performed simulations, which demonstrated the control of type 1 error and power gains using the proposed approach. We applied the proposed method to compute p-value for a polymorphism eNOS -786T>C which was shown to be associated with breast cancer risk.
Conclusions
Our findings indicate that the proposed method should be used to maximize power and control type 1 errors when analyzing genetic data using additive, dominant, and recessive models.
Keywords
Background
Genome-wide association studies (GWAS) and candidate gene association studies are commonly performed to test the association of genetic variants with a particular phenotype. Typically, hundreds of thousands of single-nucleotide polymorphisms (SNPs) are tested for association in these studies. Associations between the SNPs and the phenotypes are determined on the basis of differences in allele frequencies between cases and controls [1]. Several statistical methods have been proposed to control the family-wise error rate (FWER) for multiple comparison testing.
A simple approximation can be used to obtain a FWER of α by utilizing the Bonferroni adjustment [2] of ${\mathit{\alpha}}^{*}=\frac{\mathit{\alpha}}{\mathit{n}}$ and using α* as the threshold for significance for each test. Bonferroni adjustment tends to be conservative when the tests are correlated. In genetic association studies, the SNPs being tested are typically in linkage disequilibrium (LD), which leads to correlation among the tests. An alternative approximation to the Bonferroni adjustment is Sidak’s correction [3, 4], ${\mathit{\alpha}}^{*}=1-{\left(1-\mathit{\alpha}\right)}^{\frac{1}{\mathit{n}}}$ which assumes independence among tests. Conneely and Boehnke [5] proposed a correction that does not assume independence among tests but assumes joint multivariate normality of all test statistics. Other methods to control the FWER include using the false discovery rate (FDR) [6, 7].
In genetic association studies, three genetic models--additive, dominant, and recessive--are generally used to test each SNP using the Cochran-Armitage (CA) trend test [8–12]. In association studies the true underlying genetic model is unknown. Some investigators report the smallest p-value obtained from the three tests corresponding to the three genetic models. However, such a procedure inherently leads to an inflated type 1 error rate. Also, FDR-based methods to control FWER are not applicable in this situation because the hypotheses are highly correlated, as the same SNP is tested using different genetic models.
Thus, there is a need to correct for multiple comparisons corresponding to the three genetic tests performed for testing the association of a single SNP. These three tests are not only correlated but also functionally dependent. The standard methods for correcting for multiple testing referred to above are either too conservative or fail to meet the assumptions underlying these methods (e.g., asymptotic multivariate normality, independence among tests). Several approaches have been proposed to account specifically for the multiple comparisons of these three genetic models [13–15]. However, these approaches assume asymptotic tri-variate normality for the additive, dominant and recessive test statistics. While this is a reasonable approximation to correct for multiple comparisons, preliminary investigations regarding the joint distribution of the three test statistics revealed the following insights: 1) the joint distribution of the test statistics is discrete and the grids at which the probability mass function is positive is few and far between; 2) The distribution is highly multimodal in most of the situations, particularly, when the number of cases and controls are different and unimodal only in a handful of situations (e.g. when the number of cases and controls are equal). Therefore, we propose a method to compute the exact joint distribution of the three CA trend tests corresponding to the additive, dominant, and recessive genetic models. We used this joint distribution to compute the exact p-value for testing each SNP using the different genetic models. We performed simulations to demonstrate control of type 1 errors and power gains using the proposed approach. Finally, we applied the proposed approach to assess the significance of the association between a promoter polymorphism, eNOS-786T>C and breast cancer risk.
Methods
Genotypic counts, parameterizations, and notations for various parameters used in the model formulation
Genotype | ||||
---|---|---|---|---|
AA | Aa | aa | Sum | |
Cases (X) | X _{1} | X _{2} | X _{3} | R _{ X } |
Controls (Y) | Y _{1} | Y _{2} | Y _{3} | R _{ Y } |
Sum | C _{1} | C _{2} | C _{3} | N |
where the weight, t_{ i }, is chosen on the basis of the genetic model considered: additive, dominant, or recessive. The additive model assumes the deleterious effect is linearly related to the number of deleterious alleles. The dominant model assumes the deleterious effect is related to the presence of the deleterious allele. And the recessive model assumes the deleterious effect is related to the presence of both the deleterious alleles. The weights t = [t_{1}, t_{2}, t_{3}] corresponding to each of the models are as follows: additive model: t = [0, 1, 2], dominant model: t = [0, 1, 1], and recessive model: t = [0, 0, 1] for genotypes AA, Aa, and aa, respectively. Let the three test statistics corresponding to the additive, dominant, and recessive models be T_{1}, T_{2}, and T_{3}, respectively.
The joint distribution
Simulations
We performed simulations to evaluate the performance of the proposed method and compared our approach with standard approaches used in the literature. All the simulation results were based on 1000 replicate data sets. Each replicate dataset comprised 1000 cases and 1000 controls. The disease status for each data set was obtained using the logistic regression model logit(P(Z = 1)) = β_{0} + β_{1}X, where X is the indicator for genotype, Z is the disease status, β_{0} is the intercept, and β_{1} is the log odds ratio for the SNP. The genotype data for a SNP were simulated using a minor allele frequency (MAF) of 40% for the null hypothesis and two MAFs of 40% and 20% for the power comparisons. For the type 1 error comparisons, we simulated 1000 replicate datasets from the null hypothesis (i.e., the SNP was not associated with disease status), with β_{0} = − 2.5 and β_{1} = log (1). For the power comparisons, we simulated 1000 replicate datasets for 40% and 20% MAFs from the alternate hypothesis (i.e., the SNP was associated with disease status) for each of the three scenarios: (1) additive model with odds ratio of 1.2, (2) dominant model with odds ratio of 1.3, and (3) recessive model with odds ratio of 1.3. The methods we compared were as follows: performing only additive analyses (additive-only), performing only dominant analyses (dominant-only), performing only recessive analyses (recessive-only), using the p-value based on reporting the smallest p-value of the three genetic models (min-p), using the Bonferroni correction approach, and using the proposed exact p-value method.
Results
Type 1 error comparisons for different approaches at the 0.05 level of significance for 1000 replicates, each replicate representing a data set containing 1000 cases and 1000 controls
Method | α =0.05 |
---|---|
Additive Only | 0.044 |
Dominant Only | 0.045 |
Recessive Only | 0.056 |
Min-p | 0.105 |
Bonferroni | 0.030 |
Exact p-value | 0.047 |
Power comparisons for different approaches at the 0.05 level of significance for 3 different simulation scenarios using genotypes coded as additive, dominant, and recessive, respectively, for 40% and 20% MAFs
Genotype model | ||||
---|---|---|---|---|
MAF | Method | Additive model | Dominant model | Recessive model |
Odds ratio = 1.2 | Odds ratio = 1.3 | Odds ratio = 1.3 | ||
Additive Only | 0.816 | 0.660 | 0.410 | |
Dominant Only | 0.676 | 0.803 | 0.116 | |
40% | Recessive Only | 0.588 | 0.158 | 0.589 |
Bonferroni | 0.721 | 0.671 | 0.452 | |
Exact p-value | 0.743 | 0.726 | 0.517 | |
Additive Only | 0.656 | 0.774 | 0.116 | |
Dominant Only | 0.603 | 0.823 | 0.061 | |
20% | Recessive Only | 0.306 | 0.102 | 0.249 |
Bonferroni | 0.556 | 0.715 | 0.168 | |
Exact p-value | 0.584 | 0.782 | 0.197 |
When the data were simulated using the dominant model (column 4, Table 3), the additive-only, dominant-only and recessive-only analyses had powers of 0.660, 0.803, and 0.158, respectively, for 40% MAF and 0.774, 0.823, and 0.102, respectively for 20% MAF, at the 0.05 level of significance. Once again, as expected, the powers of the dominant-only analysis were the highest because the data were generated using the dominant model. The proposed exact p-value method had powers of 0.726 and 0.782 for the 40% and 20% MAFs, respectively, which were higher than the Bonferroni method which had powers of 0.671 and 0.715 for the 40% and 20% MAFs, respectively. When the data were simulated using the recessive model (column 5, Table 3), the additive-only, dominant-only and recessive-only analyses had powers of 0.410, 0.116, and 0.589, respectively, for 40% MAF and 0.116, 0.061, and 0.249, respectively, for 20% MAF. The proposed exact p-value method had powers of 0.517 and 0.197 for the 40% and 20% MAFs, respectively, which were higher than the Bonferroni method (0.452 and 0.168 for 40% and 20% MAFs, respectively).
P-values computed using various approaches for association of eNOS -786T> C with breast cancer
Genotype Data for eNOS -786T> C | Method | p-value | ||
---|---|---|---|---|
Controls | Cases | Additive Only | 0.0045 | |
Total | 423 | 421 | Dominant Only | 0.0148 |
TT | 203 | 167 | Recessive Only | 0.0313 |
CT | 185 | 200 | Bonferroni | 0.0135 |
CC | 35 | 54 | Exact p-value | 0.0021 |
Discussion
In this paper, we proposed a method to calculate the exact p-value for testing a single SNP using multiple genetic models. We recommend using the proposed method to maximize power and control type 1 errors when analyzing genetic data using additive, dominant, and recessive models. The proposed method is robust to model misspecifications and different SNP minor allele frequencies. Furthermore, similar to the computation of Fisher’s exact p-value, the proposed approach does not depend on asymptotic distributions.
In our simulation study, where replicate datasets were simulated using the null hypothesis, we found that the proposed method had well-controlled type 1 error probabilities. In contrast, the method of reporting the smallest p-value of the three genetic models tested had the highest false-positive rate and was found to be invalid. And, as expected, the type 1 error of the Bonferroni correction approach was well controlled but conservative, which typically led to a loss in power for identifying genetic variants.
We also simulated replicate datasets under an alternative hypothesis using the different genetic models: additive, dominant, and recessive. In these simulations, we observed that no single method: additive-only, dominant-only, or recessive-only, had higher power in all three scenarios. Each of these methods had higher power only when the model used to analyze the data was the same as the true model used to generate the data. However, because the true mode of disease inheritance is usually unknown, analyses using all three genetic models are necessary. In general, the Bonferroni correction approach led to higher power than using a model that did not correspond to the true model. The proposed exact p-value method was an improvement over the Bonferroni method. The conservativeness of the Bonferroni method may be due to its inability to account for the functional dependence between the three test statistics. In contrast, our proposed approach accounts for this functional dependence by computing p-values from the joint probability mass function. Finally, we analyzed breast cancer study data in which the polymorphism eNOS -786T>C, was found to be significant [17].
The computation time needed to obtain the exact p-value is substantial. The problem is very closely related to Fisher’s exact test, and there are many patterns inherent in the structure of the problem that could be exploited to calculate the p-values more efficiently. In the Appendix, we present several novel optimization techniques to efficiently compute the test statistics in a reasonable time (e.g., approximately 15 min for a 1000 cases and 1000 controls dataset). The software to compute exact p-values is available at http://odin.mdacc.tmc.edu/~rtalluri/index.html.
Conclusions
In genetic association studies, three genetic models--additive, dominant, and recessive--are generally used to test each SNP using the Cochran-Armitage trend test. Reporting the minimum p-value of the three genetic models leads to inflated type 1 errors. We proposed an approach to compute the exact p-value when genomic data is analyzed using the three genetic models. The proposed approach leads to higher power while controlling the type 1 error.
Appendix
Optimization techniques for computing the exact p-value
Recall that X_{1}, X_{2}, X_{3} and Y_{1}, Y_{2}, Y_{3} are the number of individuals with genotypes AA, Aa, and aa in cases and controls, respectively, with X_{1} + X_{2} + X_{3} = R_{ X } and Y_{1} + Y_{2} + Y_{3} = R_{ Y }. The three genotype counts in cases (X_{1}, X_{2}, X_{3}) and the three genotype counts in controls (Y_{1}, Y_{2}, Y_{3}) follow a multinomial distribution with probabilities (p_{1}, p_{2}, p_{3}) and (q_{1}, q_{2}, q_{3}), respectively. The probability mass function (pmf) of (X_{ 1 }, X_{ 2 }, X_{ 3 }) is ${\mathit{f}}_{\mathit{X}}\left(\mathit{X}\right)=\frac{{\mathit{R}}_{\mathit{X}}!}{{\mathit{X}}_{2}!{\mathit{X}}_{3}!\left({\mathit{R}}_{\mathit{X}}-{\mathit{X}}_{2}-{\mathit{X}}_{3}\right)!}{\mathit{p}}_{1}^{{\mathit{R}}_{\mathit{X}}-{\mathit{X}}_{2}-{\mathit{X}}_{3}}{\mathit{p}}_{2}^{{\mathit{X}}_{2}}{\mathit{p}}_{3}^{{\mathit{X}}_{3}}$ and the pmf of (Y_{1}, Y_{2}, Y_{3}) is ${\mathit{f}}_{\mathit{Y}}\left(\mathit{Y}\right)=\frac{{\mathit{R}}_{\mathit{Y}}!}{{\mathit{Y}}_{2}!{\mathit{Y}}_{3}!\left({\mathit{R}}_{\mathit{Y}}-{\mathit{Y}}_{2}-{\mathit{Y}}_{3}\right)!}{\mathit{q}}_{1}^{{\mathit{R}}_{\mathit{Y}}-{\mathit{Y}}_{2}-{\mathit{Y}}_{3}}{\mathit{q}}_{2}^{{\mathit{Y}}_{2}}{\mathit{q}}_{3}^{{\mathit{Y}}_{3}}$. The three test statistics corresponding to the additive, dominant, and recessive models are, T_{1} = (R_{ Y }X_{2} − R_{ X }Y_{2}) + 2(R_{ Y }X_{3} − R_{ X }Y_{3}) , T_{2} = (R_{ Y }X_{2} − R_{ X }Y_{2}) + (R_{ Y }X_{3} − R_{ X }Y_{3}), and T_{3} = (R_{ Y }X_{3} − R_{ X }Y_{3}) respectively. As T_{3} = T_{1} − T_{2}, we only need to derive the joint distribution of T_{1} and T_{2}. Let $\mathit{T}=\left(\begin{array}{l}{\mathit{T}}_{1}\\ {\mathit{T}}_{2}\end{array}\right)$, $\mathit{X}=\left(\begin{array}{l}{\mathit{X}}_{2}\\ {\mathit{X}}_{3}\end{array}\right)$, and $\mathit{Y}=\left(\begin{array}{l}{\mathit{Y}}_{2}\\ {\mathit{Y}}_{3}\end{array}\right)$. The test statistics can be written as T = AX + BY, where $\mathit{A}=\left[\begin{array}{l}{\mathit{R}}_{\mathit{Y}}\phantom{\rule{0.25em}{0ex}}2{\mathit{R}}_{\mathit{Y}}\\ {\mathit{R}}_{\mathit{Y}}\phantom{\rule{0.25em}{0ex}}{\mathit{R}}_{\mathit{Y}}\end{array}\right]$ and $\mathit{B}=\left[\begin{array}{l}-{\mathit{R}}_{\mathit{X}}\phantom{\rule{0.25em}{0ex}}-2{\mathit{R}}_{\mathit{X}}\\ -{\mathit{R}}_{\mathit{X}}\phantom{\rule{0.25em}{0ex}}-{\mathit{R}}_{\mathit{X}}\end{array}\right]$ . We proceed to derive the joint probability mass function of $\mathit{T}=\left(\begin{array}{l}{\mathit{T}}_{1}\\ {\mathit{T}}_{2}\end{array}\right)$.
the resulting pmf of (Z_{1}, Z_{2}) is a one-to-one function of the pmf of (T_{1}, T_{2}). Hence, the p-value obtained will be the same when using (Z_{1}, Z_{2}) instead of (T_{1}, T_{2}). The resulting pmf of (Z_{1}, Z_{2}) can be derived using the same method as with (T_{1}, T_{2}).
- 1.
X _{3}, Y _{3}, X _{2} and Y _{2} are integers
- 2.
X _{3}, Y _{3}, X _{2} and Y _{2} ≥ 0
- 3.
X _{3} + X _{2} ≤ R _{ X }
- 4.
Y _{3} + Y _{2} ≤ R _{ Y }
On the basis of these four constraints the solution space can be calculated. While the exact solution space could not be found, it follows a pattern that can be enumerated.
Figure 1 depicts the pmf of the scenario with R_{ X } = 19 and R_{ Y } = 2 where a pattern of six triangles can be visualized from the figure. Similarly, Figure 2 depicts the pmf of the scenario with R_{ X } = 20 and R_{ Y } = 3, where a pattern of ten triangles can be visualized from the picture. This trend can be generalized for all values of R_{ X } and R_{ Y }.
Generalizing the above scenario, there are $\left[1+2+\cdot \cdot \cdot +\left({\mathit{R}}_{\mathit{Y}}+1\right)=\frac{\left({\mathit{R}}_{\mathit{Y}}+1\right)\left({\mathit{R}}_{\mathit{Y}}+2\right)}{2}\right]$ triangles for the solution space. In each triangle, there are $\left[1+2+\cdot \cdot \cdot +\left({\mathit{R}}_{\mathit{X}}+1\right)=\frac{\left({\mathit{R}}_{\mathit{X}}+1\right)\left({\mathit{R}}_{\mathit{X}}+2\right)}{2}\right]$ elements that correspond to all possible combinations of X_{3} + X_{2} ≤ R_{ X }. In each triangle, the values of Y_{3} and Y_{2} are constant and the $\frac{\left({\mathit{R}}_{\mathit{Y}}+1\right)\left({\mathit{R}}_{\mathit{Y}}+2\right)}{2}$ triangles correspond to all possible combinations of Y_{3} + Y_{2} ≤ R_{ Y }, which make up the whole solution space.
Another important fact is that these triangles may overlap, reducing the solution space, which is depicted in Figures 3 and 4. Figure 3 depicts the pmf of the scenario with R_{ X } = 10 and R_{ Y } = 2 where a pattern of six triangles can be visualized from the figure. The overlap of the triangles can be observed when compared to Figure 1. Figure 4 depicts the pmf of the scenario with R_{ X } = 5 and R_{ Y } = 5 where a pattern of 21 triangles can be visualized from the figure, where most of the triangles are overlapping one another. The additional computational burden is to determine where the solution space triangles overlap and how many triangles are overlapping at a particular location. This is a function of the greatest common divisor (GCD) of R_{ X } and R_{ Y }. If R_{ X } and R_{ Y } are co-prime (GCD=1), only three triangles overlap at a single point (Z_{1} = 0, Z_{2} = 0) which requires no additional computation. When R_{ X } and R_{ Y } are not co-prime, the triangles overlap at multiples of the GCD of R_{ X } and R_{ Y }. In this scenario, multiple values of X_{3}, Y_{3}, X_{2,} and Y_{2} contribute to the same (Z_{1}, Z_{2}).
- 1.
GCD(R _{ X }, R _{ Y }) = 1
- 2.
GCD(R _{ X }, R _{ Y }) = R _{ X } = R _{ Y }
- 3.
GCD(R _{ X }, R _{ Y }) < min(R _{ X }, R _{ Y })
Scenario 1
When R_{ X } and R_{ Y } are co-prime, the triangles only overlap at a single point (Z_{1} = 0, Z_{2} = 0); therefore, we can independently evaluate each of the possible values of the solution space. The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, so we evaluate the probabilities of each of the possible values of the test statistics one at a time. Hence, the p-value is the sum of all the probabilities of test statistics that are lower than the probability of the observed test statistic. Using this procedure there is no need to store any data, which leads to faster computation of the p-value from the joint distribution.
Scenario 2
- 1.
Let R _{ X } = R _{ Y } = R. The solution space can then be constrained to a matrix with 2R + 1 rows and 2R + 1 columns. Let the center of the matrix correspond to the test statistic (Z _{1} = 0, Z _{2} = 0).2. Now, as we can see from Figure 5, we need to compute the colored cells in quadrants 3 and 4. In quadrant 3, the cells with the same number of overlapping triangles are placed diagonally, and in quadrant 4, they are placed horizontally and then vertically. We exploit the pattern that follows from the same number of triangles overlapping at a particular cell.
- 3.
For i = 1: R start at (Z _{1} = − (R − i), Z _{2} = − 1). Find the possible combinations of X _{3}, Y _{3}, X _{2} and Y _{2} that contribute to the cell corresponding to (Z _{1} = − (R − i), Z _{2} = − 1). Compute the probabilities for the cells along the diagonal path in quadrant 3, until Z _{1} = 0. Here X _{3} and X _{2} remain the same; hence, it is trivial to compute the probabilities for each cell.
- 4.
Then in quadrant 4, compute the probabilities for the cells along the horizontal path until Z _{1} = R − (i − 1); here X _{3} remains the same and X _{2new} = X _{2} + Z _{2}.
- 5.
Then continue vertically until Z _{2} = 0; here X _{3} and X _{2} remain the same.
This algorithm reduces the computational burden by computing the possible combinations of X_{3}, Y_{3}, X_{2} and Y_{2} that contribute to all the cells only R times, as opposed to computing once for each cell (approximately 4R^{2} times).
Scenario 3
- 1.
For each possible (Z _{1}, Z _{2}) compute the triangles that contribute to this particular point.
- 2.
Add up the probabilities of each of the elements of these triangles to compute the p-value of that particular (Z _{1}, Z _{2}).
Declarations
Acknowledgements
This work was supported by National Institutes of Health grants R01CA131324 (SS), NIH R25 DA026120 (SS), and R01DE022891 (SS). This research was supported in part by Barnhart Family Distinguished Professorship in Targeted Therapy (SS). This research was supported in part by a cancer prevention fellowship for Rajesh Talluri supported by a grant from the National Institute of Drug Abuse (NIH R25 DA026120).
Authors’ Affiliations
References
- Clarke GM, Anderson CA, Pettersson FH, Cardon LR, Morris AP, Zondervan KT: Basic statistical analysis in genetic case–control studies. Nat Protoc. 2011, 6 (2): 121-133. 10.1038/nprot.2010.182.PubMedPubMed CentralView ArticleGoogle Scholar
- Dunn OJ: Multiple comparisons among means. J Am Stat Assoc. 1961, 56 (293): 52-64. 10.1080/01621459.1961.10482090.View ArticleGoogle Scholar
- Sidak Z: On multivariate normal probabilities of rectangles - their dependence on correlations. Ann Math Stat. 1968, 39 (5): 1425-1434.Google Scholar
- Sidak Z: Probabilities of rectangles in multivariate student distributions - their dependence on correlations. Ann Math Stat. 1971, 42 (1): 169-175. 10.1214/aoms/1177693504.View ArticleGoogle Scholar
- Conneely KN, Boehnke M: So many correlated tests, so little time! Rapid adjustment of P values for multiple correlated tests. Am J Hum Genet. 2007, 81 (6): 1158-1168. 10.1086/522036.PubMedPubMed CentralView ArticleGoogle Scholar
- Benjamini Y, Hochberg Y: Controlling the false discovery rate - a practical and powerful approach to multiple testing. J Roy Stat Soc B Methods. 1995, 57 (1): 289-300.Google Scholar
- Benjamini Y, Yekutieli D: The control of the false discovery rate in multiple testing under dependency. Ann Math Stat. 2001, 29 (4): 1165-1188. 10.1214/aos/1013699998.View ArticleGoogle Scholar
- Agresti A: Categorical Data Analysis. 2002, New York: John Wiley & SonsView ArticleGoogle Scholar
- Armitage P: Tests for linear trends in proportions and frequencies. Biometrics. 1955, 11 (3): 375-386. 10.2307/3001775.View ArticleGoogle Scholar
- Barrett JH, Iles MM, Harland M, Taylor JC, Aitken JF, Andresen PA, Akslen LA, Armstrong BK, Avril MF, Azizi E, Bakker B, Bergman W, Bianchi-Scarra G, Bressac-de Paillerets B, Calista D, Cannon-Albright LA, Corda E, Cust AE, Debniak T, Duffy D, Dunning AM, Easton DF, Friedman E, Galan P, Ghiorzo P, Giles GG, Hansson J, Hocevar M, Hoiom V, Hopper JL, et al: Genome-wide association study identifies three new melanoma susceptibility loci. Nat Genet. 2011, 43 (11): 1108-1113. 10.1038/ng.959.PubMedPubMed CentralView ArticleGoogle Scholar
- Cochran WG: Some methods for strengthening the common X2 tests. Biometrics. 1954, 10 (4): 417-451. 10.2307/3001616.View ArticleGoogle Scholar
- Lewis CM, Knight J: Introduction to genetic association studies. Cold Spring Harb Protoc. 2012, 2012 (3): 297-306.PubMedView ArticleGoogle Scholar
- Freidlin B, Zheng G, Li ZH, Gastwirth JL: Trend tests for case–control studies of genetic markers: power, sample size and robustness. Hum Hered. 2002, 53 (3): 146-152. 10.1159/000064976.PubMedView ArticleGoogle Scholar
- Gonzalez JR, Carrasco JL, Dudbridge F, Armengol L, Estivill X, Moreno V: Maximizing association statistics over genetic models. Genet Epidemiol. 2008, 32 (3): 246-254. 10.1002/gepi.20299.PubMedView ArticleGoogle Scholar
- Hothorn LA, Hothorn T: Order-restricted scores test for the evaluation of population-based case–control studies when the genetic model is unknown. Biometrical J. 2009, 51 (4): 659-669. 10.1002/bimj.200800203.View ArticleGoogle Scholar
- Purcell S, Neale B, Todd-Brown K, Thomas L, Ferreira MAR, Bender D, Maller J, Sklar P, de Bakker PIW, Daly MJ, Sham PC: PLINK: a tool set for whole-genome association and population-based linkage analyses. Am J Hum Genet. 2007, 81 (3): 559-575. 10.1086/519795.PubMedPubMed CentralView ArticleGoogle Scholar
- Lu JC, Wei QY, Bondy ML, Yu TK, Li DH, Brewster A, Shete S, Sahin A, Meric-Bernstam F, Wang LE: Promoter polymorphism (−786T > C) in the endothelial nitric oxide synthase gene is associated with risk of sporadic breast cancer in non-Hispanic white women age younger than 55 years. Cancer. 2006, 107 (9): 2245-2253. 10.1002/cncr.22269.PubMedView ArticleGoogle Scholar
- Casella G, Berger RL: Statistical Inference. 2002, Australia ; Pacific Grove, CA: Thomson Learning, 46-54. 2Google Scholar
- Mordell LJ: Diophantine Equations. 1969, Academic P: London, New YorkGoogle Scholar
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