Abstract
Background
Amyotrophic lateral sclerosis (ALS) is a fatal, degenerative neuromuscular disease characterized by a progressive loss of voluntary motor activity. About 95% of ALS patients are in "sporadic form"meaning their disease is not associated with a family history of the disease. To date, the genetic factors of the sporadic form of ALS are poorly understood.
Methods
We proposed a twostage approach based on seventeen biological plausible models to search for twolocus combinations that have significant joint effects to the disease in a genomewide association study (GWAS). We used a twostage strategy to reduce the computational burden associated with performing an exhaustive twolocus search across the genome. In the first stage, all SNPs were screened using a singlemarker test. In the second stage, all pairs made from the 1000 SNPs with the lowest pvalues from the first stage were evaluated under each of the 17 twolocus models.
Results
we performed the twostage approach on a GWAS data set of sporadic ALS from the SNP Database at the NINDS Human Genetics Resource Center DNA and Cell Line Repository http://ccr.coriell.org/ninds/ webcite. Our twolocus analysis showed that two twolocus combinationsrs4363506 (SNP1) and rs3733242 (SNP2), and rs4363506 and rs16984239 (SNP3)  were significantly associated with sporadic ALS. After adjusting for multiple tests and multiple models, the combination of SNP1 and SNP2 had a pvalue of 0.032 under the Dom∩Dom epistatic model; SNP1 and SNP3 had a pvalue of 0.042 under the Dom × Dom multiplicative model.
Conclusion
The proposed twostage analytical method can be used to search for joint effects of genes in GWAS. The twostage strategy decreased the computational time and the multiple testing burdens associated with GWAS. We have also observed that the loci identified by our twostage strategy can not be detected by singlelocus tests.
Background
Amyotrophic lateral sclerosis (ALS) is a fatal progressive neurodegenerative disease that attacks nerve cells in the brain and spinal cord resulting in muscle weakness and atrophy. Although ALS is listed as a rare disease with a prevalence of approximately 1 per 10,000, it is the most common adult onset form of motor neuron diseases [1,2]. Epidemiological studies have showed that 1.55.3% of cases are familial in nature [36]. The remaining 95% of cases are not associated with a family history of the disease and seem to occur sporadically throughout the community. Several genes that cause familial ALS have been identified [714], especially the SOD1 gene which is believed to be responsible for 20% of familial ALS.
The identification of susceptibility genes of sporadic ALS has been slow in arriving. The search for sporadic ALS genes has generated a large number of candidategene association studies [1519]. To date, we do not have a functional SNP or haplotype that has made a credible contribution to our understanding of disease pathogenesis in the way that the APOEe4 allele does in Alzheimer disease (AD) and the H1 MAPT haplotype does in parkinsonian syndromes [20]. There is an urgent need to understand the genetic architecture of sporadic ALS and ultimately to develop novel drugs for this fatal disease. Sporadic ALS is hypothesized to be a complex disorder in which the disease is modulated by variations in multiple genetic loci interacting with each other and environmental exposures [18]. The lack of major genes may be a reason for the unsuccessful candidate gene studies which investigated one gene at a time.
Recently, Schymick et al. made the first attempt to identify genetic factors that might be relevant in the pathogenesis of sporadic ALS by using a welldesigned GWAS [1]. The first stage singlemarker analysis performed by Schymick et al. showed that 34 SNPs had a pvalue less than 0.0001 with the smallest one being 6.8 × 10^{7}. After adjusted by permutation procedure, none of these SNPs reached the significance level of 0.05. This finding suggests that the ALS phenotype is not driven by a single powerful locus. By testing one marker at a time, the first stage analysis made the implicit assumption that susceptibility loci can be identified through their independent, marginal contributions to the trait variability. More recently, other GWAS in ALS have been conducted by different research groups [2124]. However, all these GWAS used singlemarker analysis. Recent human and animal studies of complex diseases have identified susceptibility genes that marginally contribute to a common trait, to a minor extent only or not at all, but that interact significantly in combined analyses [2532]. Thus, methods that can account for joint effects of genes may be appropriate for analyzing genomewide association data sets.
In this article, we used seventeen twolocus models to analyze the previously published genomewide association data for ALS. We found that three SNPs were significantly associated with sporadic ALS. After we observed the significant twolocus combinations, we further estimated the impact (relative risk and odds ratio) of each of the twolocus combinations on sporadic ALS. It has been recognized that the traditional method will over estimate the odds ratio or relative risk in GWAS [32,33]. Recently, Zollner and Pritchard proposed a new method to estimate penetrance and then odds ratio and relative risk [32]. Through extensive simulation studies, Zollner and Pritchard showed that the estimations of odds ratio and relative risk by their method were not upward biased. By modifying Zollner and Pritchard's method, we proposed a new method to estimate twolocus penetrance, and then estimate the odds ratio, relative risk and sample size needed to replicate the findings for this rare disease.
Methods
In this section, we will give details of the data set and describe a new analytical method to analyze this data set.
The Data Set from GWAS for Sporadic ALS
Schymick et al. have made their data set publicly available through the website of the National Institute of Neurological Disorders and Stroke (NINDS) Human Genetics Resource Center at the Coriell Institute http://ccr.coriell.org/ninds webcite[1]. The data set contained 555,352 unique SNPs across the genome in 276 patients with sporadic ALS and 271 neurologically normal controls. The 555,352 SNPs were carefully chosen tagging SNPs from phase I and II of the HapMap Project. The sampled individuals were all nonHispanic white Americans. There were 102 females and 174 males in cases, and 142 females and 129 males in controls. All sampled individuals had a more than 95% genotype call rate. The average call rate across all samples was 99.6%. Of the 555,352 SNPs studied, the genotype call rate was greater than 99% for 514,088 (representing 92.6% of all SNPs assayed) and greater than 95% for 549,062 (98.9%) SNPs. The phenotype file of this data set contained the status of sporadic ALS, age of onset, site of onset (bulbaronset, upperlimbonset, and lowerlimbonset), gender, and smoking status among other information.
Statistical Analysis
Twolocus Analysis Based on Seventeen Twolocus Models
In this article, we used seventeen twolocus models to analyze the genomewide association data. For each SNP, we called one allele a highrisk allele if its frequency in cases was larger than the frequency in controls. For SNP A with alleles A, a and SNP B with alleles B, b, Figure 1 and 2 give eight epistatic twolocus models and nine multiplicative twolocus models with highrisk alleles A and B, respectively. Some of the eight epistatic twolocus models have been used and discussed by Xiong et al. and Zhao et al. [34,35]. The multiplicative models that are good approximations of additive models have been discussed by Hodge and Risch [36,37].
Figure 1. Eight twolocus epistatic models. A and B are the highrisk alleles in the two markers. α and β are the penetrance. ∩: twolocus genotypes with both highrisk genotypes at SNP A and SNP B are highrisk genotypes. ∪: twolocus genotypes with at least one high risk genotype at SNP A or SNP B are highrisk genotypes.
Figure 2. Nine twolocus multiplicative models. A and B are the highrisk alleles in the two markers. The symbol in each cell denotes the relative risk of this cell. φ = θ^{2}, ρ = θ^{3 }and γ = θ^{4}.
Under each of the epistatic models, the nine twolocus genotypes were divided into two groups: highrisk genotype group and lowrisk genotype group. For example, under the model Dom∩Dom, the highrisk group was G_{H }= {aAbB, AAbB, aABB, AABB} and the lowrisk group was G_{L }= {aabb, aAbb, AAbb, aaBB} For the eight epistatic models, we used one degree of freedom (df) χ^{2 }test statistic given by
to test for association of twolocus joint effects, where , , and denote the frequencies of the highrisk genotype group in cases, controls and the pooled sample (cases and controls are pooled together).
For the nine multiplicative models, we constructed a twolocus association test as follows. Let P(Diseaseg) denote the penetrance of twolocus genotype combination g = (g_{1}, g_{2}), where g_{1 }and g_{2 }are the genotypes in the first and second markers, respectively. Let β_{0 }denote the logarithm of the penetrance of genotypes with a relative risk of 1 in the models (see Figure 2) and β_{1 }= logθ, where θ is the relative risk given in Figure 2. Then, the nine multiplicative models can be described by the following log linear model log P(Diseaseg) = β_{0 }+ β_{1}X, where X = x_{1 }+ x_{2}, x_{1 }is the numerical code of g_{1 }and is given by
for a dominant, recessive or multiplicative model, respectively; x_{2 }is similarly defined as the numerical code of g_{2}. Under the log linear model log P(Diseaseg) = β_{0 }+ β_{1}X, β_{1 }= 0 means that all the genotypes have the same penetrance which implies that θ = 1. So a test of the association between the disease and the two loci under the nine multiplicative models is equivalent to a test of the null hypothesis H_{0}: β_{1 }= 0. For the i^{th }individual, let y_{i }denote the trait value (1 for diseased individual and 0 for normal individual) and X_{i }denote the numerical code of the genotype (X in the log linear model). The score test statistic is given by
where N is the sample size, is the average of X_{1},..., X_{N}, and is the average of y_{1},..., y_{N}. Under the null hypothesis, T_{score }follows a χ^{2 }distribution with 1 df. Note that under each of the twolocus epistatic models, if we code X = 1 for a highrisk genotype group and X = 0 for a lowrisk genotype group, then T_{epi }= T_{score}.
The method to search for significant twolocus combinations for each of the seventeen models has the following two steps:
Step 1: For each SNP, let n and m denote the number of individuals in cases and controls (different SNPs may have a different number of cases and controls due to missing genotypes). Let n_{1}, n_{2}, n_{3 }and m_{1}, m_{2}, m_{3 }denote the number of three genotypes in cases and controls, respectively. The 2 df genotypic test statistic is given by
where and . We applied this test statistic to each SNP, calculated the corresponding pvalue, and returned M SNPs with the smallest pvalues (M = 1,000 was used in this article).
Step 2: Under each of the seventeen twolocus models, we applied a twolocus association test to each of the L twolocus combinations among the M retained SNPs, where L = M(M1)/2. For a twolocus epistatic model given in figure 1, we used the twolocus test T_{epi}. For a multiplicative model given in figure 2, we used the score test T_{score}. In this step, we got a pvalue (called raw pvalue) for each of the L twolocus combinations and each of the seventeen twolocus models.
A permutation procedure was used to adjust for multiple tests and multiple models. In each permutation, we randomly shuffled the cases and controls and repeated step 1 and step 2 based on the permuted data. We performed the permutation procedure B times (B = 1,000 was used in this article). For the i^{th }model and l^{th }twolocus combination (i = 1,...,17; l = 1,..., L), let p_{il }and denote the raw pvalues of the twolocus tests in step 2 based on the original data and on the b^{th }permutated data, respectively. Let
Then, for the i^{th }model and l^{th }twolocus combination, P_{il}, the pvalue adjusted for multiple tests and multiple models, was given by .
A New Method to Estimate Penetrance
When a study identifies a locus or locuscombination that shows evidence of association with a disease, it is common to estimate the impact of this locus or locuscombination on the phenotype of interest. This impact is often expressed as an odds ratio. Estimation of the odds ratio is also helpful for planning successful replication studies.
It is recognized that the traditional estimate of odds ratio is upbiased because it is typically estimated for the locus which was significant for association [32,33]. Recently, Zollner and Pritchard proposed a new method to estimate penetrance (odds ratio can be calculated based on the penetrance) [32]. This new method was based on the likelihood of observed genotypes given that the locus was significant for association. We modified Zollner and Pritchard's method to estimate the penetrance and odds ratio for twolocus combinations under each of the seventeen models given in Figure 1 and Figure 2. We use the Dom∩Dom model given in Figure 1 as an example to describe our method.
We use the following notation:
n, m: the number of cases and controls
the data D = {n_{1},..., n_{9}; m_{1},..., m_{9}}: the counts of nine twolocus genotypes in cases and controls that constitute the significant signal for association
(q_{1},..., q_{9}): the population frequencies of the genotypes
R: the relative risk of highrisk genotype combination to lowrisk genotype combination, R = β/α.
F: the population prevalence of the disease which is assumed to be known.
Because ALS is a rare disease with F = 0.0001, we can estimate q_{i }from the sampled controls. Thus, we assume that q_{i }= (number of i^{th }genotype in controls)/m is known in the following discussion. In the Dom∩Dom model, the 5^{th}, 6^{th}, 8^{th }and 9^{th }genotype combination {(aA, bB), (AA, bB), (aA, BB), (AA, BB)} is the highrisk genotype combination, and the combination of the other genotypes is the lowrisk genotype combination. Let q_{H }= q_{5 }+ q_{6 }+ q_{8 }+ q_{9 }denote the population frequency of the highrisk genotype combination. Then, the penetrance α and β (see Figure 1) can be calculated by
Thus, we have only one unknown parameter R Let S indicate that the twolocus combination of interest shows significant association. As described in the previous section, we use a twostep approach for the twolocus analysis. A significant association of the twolocus combination from our twostep method means that each of the two loci shows significant marginal association at level α_{1 }in step 1 and significant joint association at level α_{2 }in step 2. We calculate the likelihood L(R) using the equation
where the data D = {n_{1},..., n_{9}; m_{1},..., m_{9}}. Since the data D constitutes, by definition, a significant result, so D implies S; hence Pr(SD,R) = 1. If the value of L(R) can be calculated for each given R, we can obtain the MLE of R by using a numerical optimization method (grid search was used in this article). For each R, the numerator can be calculated by the product of two multinomial distributions
where if the k^{th }genotype is a lowrisk genotype; otherwise. The traditional method to estimate the relative risk is to maximize Pr(DR), the numerator in the likelihood function L(R), without considering the fact that the loci were significant for association. There is no simple method to calculate the denominator Pr(SR), the power of our twostep test. We propose to use a simulation method as described below. For a given R, the values of α and β can be calculated by equation (1). When α, β, and q_{i }are known, we can generate the twolocus genotypes for n cases and m controls. Next, we will perform the singlemarker test and the twolocus test on the data set. If the pvalues of the two singlemarker tests are less than α_{1 }and the pvalue of the twolocus test is less than α_{2}, the data set is said to be significant for association. We repeat the process to generate the data sets many times (1 million was used in this article). The proportion of significant data sets is the estimate of Pr(SR).
When the relative risk R has been estimated, the corresponding estimates of α and β can be obtained from equation (1). The estimate of odds ratio of the highrisk genotype group is given by .
Following Zollner and Prichard, when there are more than two genotype groups in the models such as these in Figure 2, we define the odds ratio of one group to be the odds of this group divided by the odds of the combination of the others. For example, there are three genotype groups in the Dom × Dom model: low risk genotype group G_{L }= {aabb}, middle risk genotype group G_{M }= {aabB, aaBB, aAbb, AAbb}, and high risk genotype group G_{H }= {aAbB, aABB, AAbB, AABB}. The odd ratio of the high risk group OR^{H }is the odds of G_{H }divided by the odds of G_{M }∪ G_{L }= {aabb, aabB, aaBB, aAbb, AAbb}. The odd ratio of the low risk genotype group OR^{L }is the odds of G_{L }divided by the odds of G_{M }∪ G_{H }= {aabB, aaBB, aAbb, AAbb, aAbB, aABB, AAbB, AABB}. The odds ratio estimation method will be the same as the case of two genotype groups.
We used this new proposed method to estimate the odds ratio for each of the twolocus combinations that showed significant association with ALS in our twolocus analysis. Based on the estimated penetrance, we used a simulation method to estimate the sample size required to replicate the findings with 80% power.
Results
We applied the twolocus analysis with two steps to the genomewide association data set for sporadic ALS. The analysis was done for all genotypes with a call rate greater than or equal to 95% (549,062 SNPs left). SNPs on the sex chromosome were excluded in the analysis. In the first step, we returned 1,000 SNPs with the smallest pvalues which corresponded to use a pvalue cutoff α_{1 }= 0.0023. Then we tested all of the L = 499,500 twolocus combinations under each of the seventeen models and used 1,000 permutations to evaluate the adjusted pvalue for each of the twolocus combinations. After adjusting for multiple tests and multiple SNPs, we found two twolocus combinations with pvalues less than 0.05. There were three SNPs involved in the two twolocus combinations. The details of the three SNPs are given in Table 1. The combination of SNP1 and SNP2 followed the Dom∩Dom model with a pvalue of 0.032 and SNP1 and SNP3 followed the Dom × Dom model with a pvalue of 0.042. Table 2 gives the number of cases and controls in each of the nine genotypes for the two twolocus combinations. This table shows that the two twolocus combinations fit the two models, Dom∩Dom and Dom × Dom. For example, for SNP1 and SNP2, there were more cases than controls for genotypes with at least one C allele at SNP1 and at least one G allele at SNP2 and there were more controls than cases for the other genotypes, which indicated that SNP1 and SNP2 followed the Dom∩Dom model. In Schymick et al.'s 2 df singlegene analysis [1], SNP1 was ranked 1^{st }with a pvalue of 6.8 × 10^{7}, SNP 2 was ranked 10^{th }with a pvalue of 2.2 × 10^{5}, and SNP 3 was ranked 2^{nd }with a pvalue of 1.7 × 10^{6}.
Table 1. Information of the three SNPs. HRA: highrisk allele.
Table 2. (number of cases)/(number of controls) in each of the twolocus genotypes.
To estimate the impact of the two twolocus combinations on sporadic ALS, we first estimated the penetrance of the twolocus genotypes for each of the two twolocus combinations under the corresponding model. Based on the estimated penetrance, we estimated the relative risk, odds ratio and sample size required to replicate the significant findings with 80% power. We followed what is in Zollner and Pritchard to obtain the 95% CI of the estimates [32], that is, we generated 95% CI by comparing the likelihood of all initial parameter points with the likelihood of the point estimate. We included all points for which twice the difference of loglikelihoods was < 95th percentile of a χ^{2 }distribution with 1 df. The estimations using both the proposed method (adjusted estimates) and the traditional method (unadjusted estimates) are summarized in Table 3. From this table, we can see that the unadjusted relative risk, odds ratio were higher than the adjusted ones, and the unadjusted sample size was smaller than the adjusted one. These results were consistent with the finding of others that the traditional estimates of relative risk and odds ratio are upbiased [33,34].
Table 3. Penetrence, relative risk and odds ratio of the twolocus combinations.
Discussion
In this study we proposed a new analytical method that considered joint effects of genes to analyze a data set from the GWAS in sporadic ALS previously performed by Schymick et al. [1]. Our analysis showed that the combination of SNP1 and SNP2 and the combination of SNP1 and SNP3 had significant effects on sporadic ALS.
Population stratification may lead to falsepositive results. We had also checked the population stratification problem in this data set using the following method. We randomly chose 5,000 SNPs and got their pvalues by a single marker test. If population stratification did exist in this data set, among the 5,000 pvalues, there should be more small pvalues than expected under the uniform distribution. We used the oneside Kolmorgorov test statistic to test if the 5,000 pvalues followed a uniform distribution. We repeated the procedure 10 times. The Kolmorgorov test results showed that the pvalues followed a uniform distribution for all 10 replications, which indicated that there was no population stratification in this data set. The lack of population stratification in the data set was consistent with the results of Schymick et al. [1]. Schymick et al. studied the potential population structure in this data by using STRUCTURE program [38]. The analysis with STRUCTURE showed that there was no discernible difference in the population substructure between cases and controls.
Significant associations claimed by association studies often fail to be replicated. One possible reason is the overestimation of the effect in terms of the odds ratio or relative risk of the claimed variants. The overestimation of the effect leads to the underestimation of the sample size required to replicate the finding. In this article, we proposed a new method to estimate the effect of claimed variants. Based on the study of Zollner and Pritchard [32], we expected that the estimates of odds ratio and relative risk based on our proposed method would be nearly unbiased. Thus we provided a useful tool to estimate the sample size for the follow up studies. For example, in order to replicate the finding of SNP1 and SNP2 (the adjusted pvalue less than 0.05 under the Dom ∩ Dom model) with 80% power, the sample size required is 800 estimated using our proposed method instead of 680 estimated using the traditional method.
Currently, several methods are available to test associations by taking joint effects of genes into account, such as combinatorial searching method (CSM) and the multifactor dimensionality reduction (MDR) method [39,40]. We used the twostep CSM and MDR, replacing the twolocus analysis test in step 2 by the CSM or MDR, to perform the twolocus analysis. For the twostep MDR, we returned 50 SNPs instead of 1, 000 SNPs in the first step due to the computational intensity. Both of the twostep CSM and MDR found rs4363506 (SNP1) and rs12680546 (on chromosome 8) as the best twolocus combination. However, the adjusted pvalues of the twostep CSM and MDR were 0.2 and 0.156. This means that the twostep CSM and MDR did not find any twolocus combinations that had significant association with sporadic ALS. The possible reasons are as follows: The genotypes of the twolocus combinations we found (such as those given in Table 3) are ordered. For example, penetrance of H_{1}H_{2 }≥ penetrance of H_{1}h_{2 }≥ penetrance of h_{1}h_{2}, where H_{1}(h_{1}) and H_{2}(h_{2}) are the highrisk (lowrisk) genotypes in the first and second marker, respectively. The CSM and MDR ignore the order of genotypes and therefore can group any two genotypes togetherin essence searching for the "best" one among 21,146 different partitions of the twolocus genotypes. By searching for irrelevant twolocus genotype combinations, the CSM and MDR did not gain more information but increased the noise level, and thus lost power.
Conclusion
The proposed twostage analytical method can be used to search for twolocus joint effects of genes in GWAS. The twostage strategy significantly decreased the computational time and the multiple testing burdens associated with GWAS. We have also observed that the three SNPs identified by our twostage strategy can not be detected by singlelocus tests.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
QS and SZ designed the study. ZZ contributed the twolocus data analysis under the direction of SZ. SZ performed the penetrance estimation. JCS & BJT assisted in data interpretation and approved the final manuscript. QS and SZ contributed to the writing of the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Institute of Health (NIH) grants R01 GM069940 and the OverseasReturned Scholars Foundation of Department of Education of Heilongjiang Province (1152 HZ01). This work was supported in part by the Intramural Research Program of the National Institute on Aging (project Z01 AG00094902).
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