 Methodology article
 Open Access
 Published:
A new association test based on disease allele selection for case–control genomewide association studies
BMC Genomics volume 15, Article number: 358 (2014)
Abstract
Background
Current robust association tests for case–control genomewide association study (GWAS) data are mainly based on the assumption of some specific genetic models. Due to the richness of the genetic models, this assumption may not be appropriate. Therefore, robust but powerful association approaches are desirable.
Results
In this paper, we propose a new approach to testing for the association between the genotype and phenotype for case–control GWAS. This method assumes a generalized genetic model and is based on the selected disease allele to obtain a pvalue from the more powerful onesided test. Through a comprehensive simulation study we assess the performance of the new test by comparing it with existing methods. Some real data applications are also used to illustrate the use of the proposed test.
Conclusions
Based on the simulation results and real data application, the proposed test is powerful and robust.
Background
In a case–control genomewide association study (GWAS), to detect the associated singlenucleotide polymorphisms (SNPs), we need to conduct a test for each individual SNP data, which are summarized as a 2by3 table. Although Pearson’s chisquare test can be used, it is usually less powerful than the CochranArmitage trend test (CATT) if the genetic model is known [1–3]. However, if the genetic models are unknown or various, the optimal scores in the CATTs are difficult or unable to find. If we use a CATT with fixed scores for all SNPs, we may lose power for some situations [4–13]. To circumvent this disadvantage, in the literature, many robust association tests have been proposed [7, 12, 14–22]. Those tests do not rely solely on one specific genetic model; rather they consider several possible genetic models simultaneously. In addition, many of them are based on the assumption that the underlying genetic model is one of the following three: additive, recessive, and dominant. For example, the maxmin efficiency robust test (MERT) by Gastwirth [23, 24], and the maximum of the three optimal CATTs under recessive, additive, and dominant models (MAX3) have been studied [6]. Zheng and Ng proposed a twophase procedure called genetic model selection (GMS) method [12] which selects a genetic model from the three models in its first stage. On the contrary, Joo et al. proposed a test which eliminates genetic models [25]. Due to the environmental interaction, there are unlimited genetic models besides the three ideal models (recessive, additive, and dominant). Chen and Ng proposed a robust association test based on the generalized genetic model (GGM) [19], which includes the recessive, additive, and dominant models as special cases. Their approach obtains a pvalue from a onesided test for each of the two possible disease alleles. With the uncertainty of the disease allele, the overall pvalue is then approximated from the two dependent tests.
In this paper, we propose a new robust association test which utilizes the GGM and obtains a onesided pvalue based on the selected disease allele. The performance of the new test is compared with existing methods in terms of controlling type I error rate and detecting power. Some real data applications are also used to demonstrate the use of the new test.
Methods
GGM and existing methods
The data of a diallelic SNP with alleles A and a in a case control GWAS are summarized in Table 1, where r_{1}, r_{2}, and r_{3} are the frequencies of genotypes AA, Aa, and aa, respectively, for the r cases; and s_{1}, s_{2}, and s_{3} are the frequencies of genotypes AA, Aa, and aa, respectively, for the s controls.
Among the three genotypes AA, Aa, and aa, the relative risks of genotypes Aa and aa to AA are defined as:
Under the null hypothesis that there is no association between the genotype and phenotype, we have Λ_{1} = Λ_{2} = 1. Regarding the alternative hypothesis, we assume the underlying genetic model is a GGM, which is also called orderrestricted relative risks model [19]. For the case where a is the disease allele, GGM assumes Λ_{1} ≥ 1 and Λ_{2} ≥ Λ_{1} with at least one of the inequalities is strictly greater than. It is easy to see that the aforementioned ideal models, recessive (Λ_{1} = 1, Λ_{2} > Λ_{1}), additive (Λ_{1} = (1 + Λ_{2})/2), and dominant (Λ_{1} = Λ_{2} > 1), are all special cases of the generalized model. If A, rather a, is the disease allele, GGM assumes 1 ≥ Λ_{1} and Λ_{1} ≥ Λ_{2} with at least one of the inequalities is strict.
Suppose the frequencies for AA, Aa, and aa are p_{ 1 }, p_{ 2 }, p_{ 3 } for cases and q_{ 1 }, q_{ 2 }, and q_{ 3 } for controls, respectively. Under the null hypothesis that there is no association between the disease and the genotype, it is easy to show p_{ 1 } = q_{ 1 }, p_{ 2 } = q_{ 2 }, and p_{ 3 } = q_{ 3 }. In this paper, we assume the cases and controls follow trinomial distributions TN(r, p_{ 1 }, p_{ 2 }, p_{ 3 }) and TN (s, q_{ 1 }, q_{ 2 }, q_{ 3 }), respectively.
The test statistics for some wellknown existing methods are summarized as follows:
The CATT test statistic is [8]:
They are the scores assigned to the three columns.
The statistic for MAX3 is [6]:
The statistic for GMS is [12, 13]:
where I is the indicator function, and the HardyWeinberg disequilibrium trend test (HWDTT) statistic is given by [15]:
${\mathit{Z}}_{\mathit{\text{HWDTT}}}=\frac{{\left(\mathit{rs}/\mathit{n}\right)}^{1/2}\left({\widehat{\mathrm{\Delta}}}_{\mathit{P}}{\widehat{\mathrm{\Delta}}}_{\mathit{Q}}\right)}{\left\{1{\mathit{n}}_{2}/\mathit{n}{\mathit{n}}_{1}/\left(2\mathit{n}\right)\left\}\right\{{\mathit{n}}_{2}/\mathit{n}+{\mathit{n}}_{1}/\left(2\mathit{n}\right)\right\}},$${\widehat{\mathrm{\Delta}}}_{\mathit{P}}={\mathit{r}}_{2}/\mathit{r}{\left({\mathit{r}}_{2}/\mathit{r}+{\mathit{r}}_{1}/\left(2\mathit{r}\right)\right)}^{2},.$$\phantom{\rule{0.25em}{0ex}}{\widehat{\mathrm{\Delta}}}_{\mathit{Q}}={\mathit{s}}_{2}/\mathit{s}{\left({\mathit{s}}_{2}/\mathit{s}+{\mathit{s}}_{1}/\left(2\mathit{s}\right)\right)}^{2},$ and c is a constant and usually chosen as 1.645. Here n_{i} = s_{i} + r_{i} (i = 1,2,3), and n = n_{1} + n_{2} + n_{3}.
The statistic for MERT is [24]:
The proposed test
Let $\mathit{T}=\left[\begin{array}{c}\hfill {\mathit{T}}_{1}\hfill \\ \hfill {\mathit{T}}_{2}\hfill \\ \hfill {\mathit{T}}_{3}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\mathit{r}}_{2}{\mathit{s}}_{1}{\mathit{r}}_{1}{\mathit{s}}_{2}\hfill \\ \hfill {\mathit{r}}_{3}\mathit{s}\mathit{r}{\mathit{s}}_{3}\hfill \\ \hfill {\mathit{r}}_{3}{\mathit{s}}_{1}{\mathit{r}}_{1}{\mathit{s}}_{3}\hfill \end{array}\right]$, it can be shown that under the null hypothesis of no association, the mean of the vector E[T] = 0, and variancecovariance matrix is.
We define the following “standardized” statistics:
It is easy to prove the following result.
Theorem 1. Asymptotically, under the null hypothesis the statistics in (2) follow a multivariate normal distribution, Z ~ MVN (0, Σ_{ Z }), where the variancecovariance matrix is.
From theorem 1, Z_{1}, Z_{2} and Z_{3} are linear dependent; it is not difficult to show that asymptotically Z_{3} = aZ_{1} + bZ_{2}, where $\mathit{a}=\sqrt{\frac{{\mathit{p}}_{2}{\mathit{p}}_{3}}{\left(1{\mathit{p}}_{2}\right)\left(1{\mathit{p}}_{3}\right)}}$, and $\mathit{b}=\sqrt{\frac{{\mathit{p}}_{1}}{\left(1{\mathit{p}}_{2}\right)\left(1{\mathit{p}}_{3}\right)}}$. It can also be shown that under the GGM, if a (or A) is the disease allele, the expectations of Z_{i} (i = 1,2,3) in (2) are all greater (or less) than 0. In addition, under the GGM, if z_{1} is close to 0, the genetic model is close to the recessive model. On the other hand, the genetic model is unlikely to be the recessive model if z_{1} is far from 0.
We will select the disease allele based on the sign of Z_{3}, and combine the two onesided pvalues obtained based on the selected disease allele. Following the idea of combining pvalues from independent studies using robust test [26], we define the test statistic as follows:
where Φ(∙) is the cumulative density function (CDF) of the standard normal distribution (N(0,1)) and F^{ 1}(∙) is the inverse of the CDF of the chisquare distribution with one degree of freedom.
Usually it is difficult to directly calculate the pvalue (or critical value) for statistic C in (3). However, the pvalue can be easily estimated using resampling method. Specifically, we first simulate two independent samples, z_{1}, and z_{2}, both from N(0,1). Then we calculate ${\mathit{z}}_{3}=\hat{\mathit{a}}{\mathit{z}}_{1}+\hat{\mathit{b}}{\mathit{z}}_{2}$, where $\hat{\mathit{a}}=\sqrt{\frac{{\hat{\mathit{p}}}_{2}{\hat{\mathit{p}}}_{3}}{\left(1{\hat{\mathit{p}}}_{2}\right)\left(1{\hat{\mathit{p}}}_{3}\right)}}$, $\hat{\mathit{b}}=\sqrt{\frac{{\hat{\mathit{p}}}_{1}}{\left(1{\hat{\mathit{p}}}_{2}\right)\left(1{\hat{\mathit{p}}}_{3}\right)}}$ are the estimates of a and b, and ${\hat{\mathit{p}}}_{\mathit{i}}=\frac{{\mathit{n}}_{\mathit{i}}}{\mathit{n}}\phantom{\rule{0.25em}{0ex}}$ are the estimates for p_{ i } (i = 1, 2, 3) from the data. Next we calculate.
$\mathit{g}=\left\{\begin{array}{c}\hfill {\mathrm{F}}^{1}\left(\mathrm{\Phi}\left({\mathit{z}}_{1}\right)\right)+{\mathrm{F}}^{1}\left(\mathrm{\Phi}\left({\mathit{z}}_{2}\right)\right)\phantom{\rule{0.25em}{0ex}}\mathit{if}\phantom{\rule{0.25em}{0ex}}{\mathit{z}}_{3}\ge 0\hfill \\ \hfill {\mathrm{F}}^{1}\left(\mathrm{\Phi}\left({\mathit{z}}_{1}\right)\right)+{\mathrm{F}}^{1}\left(\mathrm{\Phi}\left({\mathit{z}}_{2}\right)\right)\phantom{\rule{0.25em}{0ex}}\mathit{if}\phantom{\rule{0.25em}{0ex}}{\mathit{z}}_{3}<0\hfill \end{array}\right.$. We repeat the above steps K times and get K values for g. The pvalue can then be estimated as (number of g ' s that are greater than or equal to c)/K, where c is the observed statistic calculated from data using (3).
Results and discussion
Simulation study
In this section, we will assess the performance of the proposed test by comparing it with existing methods in terms of controlling type I error rate and detecting power through a comprehensive simulation study. In the simulation study we assume that cases and controls in Table 1 follow trinomial distributions with probabilities p = (p_{ 1 },p_{ 2 },p_{ 3 }) and q = (q_{ 1 },q_{ 2 },q_{ 3 }), respectively. It can be shown that, for given q_{ i }’s, and relative risks Λ_{1}, Λ_{2}, the values of the corresponding p_{ i }’s can be obtained as follows [17–21]: ${\mathit{p}}_{1}=\frac{{\mathit{q}}_{1}}{{\mathit{q}}_{1}+{\mathit{\Lambda}}_{1}{\mathit{q}}_{2}+{\mathit{\Lambda}}_{2}{\mathit{q}}_{3}}$, ${\mathit{p}}_{2}=\frac{{\mathit{\Lambda}}_{1}{\mathit{q}}_{2}}{{\mathit{q}}_{1}+{\mathit{\Lambda}}_{1}{\mathit{q}}_{2}+{\mathit{\Lambda}}_{2}{\mathit{q}}_{3}}$, and ${\mathit{p}}_{3}=\frac{{\mathit{\Lambda}}_{2}{\mathit{q}}_{3}}{{\mathit{q}}_{1}+{\mathit{\Lambda}}_{1}{\mathit{q}}_{2}+{\mathit{\Lambda}}_{2}{\mathit{q}}_{3}}$.
We first assume Hardy–Weinberg equilibrium (HWE) holds for controls, and the minor allele frequencies (MAF) are 0.3 and 0.5. The disease allele is either the minor or the major allele. The numbers of cases (r) and controls (s) are both set to be 1000. Different pairs of Λ_{1} and Λ_{2} are used in the simulation to compare the performance of our proposed method with those of GMS, MERT, MAX3, Pearson’s chisquare test, and CATT with x = 0.5, which is the commonly used test under the assumption of additive genetic model. More specifically, we fix Λ_{2} to be 1.4 and let Λ_{1} vary from 1.0 to 1.4 with increment 0.1. Therefore, the three special genetic models are included in the simulation study. To assess the robustness of the proposed test, we also simulate data from genetic models other than the GGM: overdominant model (Λ_{1} = 1.5, Λ_{2} = 1.4 ) and underdominant model (Λ_{1} = 0.9, Λ_{2} = 1.4). The significance level is set to be 0.05, and the type I error rate and power are estimated by the proportions of rejections from 1000 replicates. To estimate the pvalue for the proposed test, we resample 10,000 times (i.e., K = 10,000). The pvalues from MAX3, GMS, and MERT are obtained by using the R package “Rassoc” [13].
When the null hypothesis is true (Λ_{1} = Λ_{2} = 1), all methods have rejection proportions close to the preset significance level 0.05 (For power values, see Additional file 1: Power plots of Figures S1S5 for Tables 26); this indicates that they can control type I error rate. Table 2 reports the empirical powers of each method when HWE holds for controls with MAF equals to 0.3, and the minor allele is the risk allele. The chisquare test is usually less powerful than the proposed test except for the condition where the genetic model is overdominant, under which the proposed test is more powerful than other methods. The CATT and MERT perform relatively poorly when the genetic models are far from the additive model. MAX3 and GMS perform similarly and have comparable power values with the proposed test; while the new test performs better when the genetic model is dominant or overdominant. Table 3 lists the power values when HWE holds for controls with MAF equals to 0.3, and the major allele is the risk allele. Once again, the CATT and MERT lose power dramatically compared to other methods when the genetic model is recessive or underdominant. The proposed test is more powerful than the MAX3 and GMS when the genetic models are close to recessive model. The chisquare test is less powerful than the proposed test when the genetic models are between recessive and dominant.
Table 4 gives the empirical powers when HWE holds for controls with MAF equals to 0.5. As seen in Tables 2 and 3, the CATT and MERT are less powerful when the genetic model is dominant or overdominant. On the contrary, chisquare test is reasonably powerful under those situations. The proposed test has the largest powers when the genetic model is close to the dominant one (i.e., Λ_{1} = 1.3, 1.4, 1.5). For other situations, the new test has comparable powers with those from MAX3 and GMS.
Tables 5 and 6 report the empirical powers when HWE does not hold with the genotypic frequencies for controls are (0.1, 0.36, 0.54), and the disease risks are minor allele and major allele, respectively. Once again we can see that CATT and MERT are not robust; they may lose power dramatically under some conditions (e.g., overdominant in Table 5 and underdominant in Table 6). The proposed test has comparable power values with those from the MAX3 and GMS; under some situations, it is more powerful (e.g., overdominant model in Table 5).
The method based on GGM by Chen and Ng (GGM in Tables 2–6) has similar performance as the proposed test. However, if the disease allele is the minor allele, the new test is more powerful than the GGM method when the genetic model is dominant or overdominant (see Tables 2, 4 and 5). In addition, unlike the proposed test, the GGM method doesn’t report the disease allele.
Real data application
To illustrate the use of the proposed test, we apply it to some real data. The SNP rs181489 has been shown to be associated with Parkinson disease [27]. Table 7 lists the genotypic counts of cases and controls of this SNP from fifteen sites [27]. We apply the proposed test and others to the data of each site to obtain pvalues. Table 8 reports the results along with the three statistics, z_{1}, z_{2} and z_{3}. Out of the 15 populations, 14 have positive values for z_{3}, indicating the disease allele is A rather than G for this SNP. Recall the statistic z_{1} compares genotype GG to GA between controls and cases. A small z_{1} indicates the relative risk Λ_{1} is close to 1, and therefore the genetic model is close to the recessive one. On the other hand, if z_{1} is large, the underlying genetic model is unlikely to be a recessive model. For population D, the estimated z_{1} is 0.090, the genetic model is close to a recessive one, under which the CATT test is less powerful than other robust methods. In fact, for this population CATT obtains the largest pvalue (0.024). Similar observation can be found for population G. In contrast, z_{1} and z_{2} from populations H and O are both large, indicating the underlying genetic models are close to the additive model, under which the CATT is more powerful than others. Therefore, there is no surprise that CATT obtains the smallest pvalues from those two populations.
Conclusions
Although CATT has been widely used in case–control GWAS with the assumption that the underlying genetic model is additive, its performance may be very poor if the true genetic model is not additive. Therefore, robust but powerful association tests are more appropriate when detecting the associated SNPs. Many existing association tests make the assumption that the genetic model is one of the three special genetic models (recessive, additive, and dominant), which may be a too strong assumption in practice. In this paper, we propose a robust association test without making strong assumption about the genetic model. Our simulation results show that even the assumption of GGM is violated (e.g., over and under dominant models), the proposed test still has reasonable power; indicating it is a robust test. In terms of computational cost, the proposed test is reasonably fast. For instance, it took my desktop about 70 seconds to get the results in Table 8 for the real data application. Our simulation study also confirmed that the proposed test can control type I error rate with smaller cutoff pvalue, e.g., 10^{4} and 10^{5} (see Additional file 2: Table S1S2).
The test statistic in (3) is defined based on the idea of combining pvalues from independent studies using chisquare distribution with 1 degree of freedom [26]. Although there are many other approaches available in the literature [26, 28–31], it remains a research topic to choose the best one if there is any. However, it should be noticed that for case control GWAS, a robust method, such as the proposed test, is desirable due to the various underlying genetic models. In addition, when we combine the pvalues from the 15 independent studies using the chisquare distribution with 1 degree of freedom [26], the overall pvalue is 2.6 × 10^{11}.
Through simulation studies and real data applications, we have shown that the proposed test is robust and powerful. In addition, the three statistics, z_{1}, z_{2}, and z_{3}, may also provide useful information about the disease allele and the genetic model.
Abbreviations
 GWAS:

Genomewide association study
 SNP:

Singlenucleotide polymorphism
 CATT:

CochranArmitage trend test
 MERT:

Maxmin efficiency robust test
 MAX3:

The maximum of the three optimal
 CATTs:

Under recessive, additive, and dominant models
 GMS:

Genetic model selection
 GGM:

Generalized genetic model
 CDF:

Cumulative density function
 HWDTT:

HardyWeinberg disequilibrium trend test
 HWE:

Hardy–Weinberg equilibrium.
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Aknowledgement
The author would like to thank the support from several faculty research funds awarded to the author by the Indiana University School of Public HealthBloomington. The author is also grateful to the three anonymous referees for their constructive comments which highly improved the presentation of the manuscript.
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Chen, Z. A new association test based on disease allele selection for case–control genomewide association studies. BMC Genomics 15, 358 (2014) doi:10.1186/1471216415358
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Keywords
 Generalized genetic model
 Robust test
 Singlenucleotide polymorphism