 Methodology
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Global tests of Pvalues for multifactor dimensionality reduction models in selection of optimal number of target genes
 Hongying Dai^{1}Email author,
 Madhusudan Bhandary^{2},
 Mara Becker^{3},
 J Steven Leeder^{3},
 Roger Gaedigk^{3} and
 Alison A MotsingerReif^{4}
https://doi.org/10.1186/1756038153
© Dai et al.; licensee BioMed Central Ltd. 2012
Received: 17 January 2012
Accepted: 19 April 2012
Published: 22 May 2012
Abstract
Background
Multifactor Dimensionality Reduction (MDR) is a popular and successful data mining method developed to characterize and detect nonlinear complex genegene interactions (epistasis) that are associated with disease susceptibility. Because MDR uses a combinatorial search strategy to detect interaction, several filtration techniques have been developed to remove genes (SNPs) that have no interactive effects prior to analysis. However, the cutoff values implemented for these filtration methods are arbitrary, therefore different choices of cutoff values will lead to different selections of genes (SNPs).
Methods
We suggest incorporating a global test of pvalues to filtration procedures to identify the optimal number of genes/SNPs for further MDR analysis and demonstrate this approach using a ReliefF filter technique. We compare the performance of different global testing procedures in this context, including the KolmogorovSmirnov test, the inverse chisquare test, the inverse normal test, the logit test, the Wilcoxon test and Tippett’s test. Additionally we demonstrate the approach on a real data application with a candidate gene study of drug response in Juvenile Idiopathic Arthritis.
Results
Extensive simulation of correlated pvalues show that the inverse chisquare test is the most appropriate approach to be incorporated with the screening approach to determine the optimal number of SNPs for the final MDR analysis. The KolmogorovSmirnov test has high inflation of Type I errors when pvalues are highly correlated or when pvalues peak near the center of histogram. Tippett’s test has very low power when the effect size of GxG interactions is small.
Conclusions
The proposed global tests can serve as a screening approach prior to individual tests to prevent false discovery. Strong power in small sample sizes and well controlled Type I error in absence of GxG interactions make global tests highly recommended in epistasis studies.
Keywords
Background
Recent advances in genotyping technology have allowed for the rapid and easy interrogation of large numbers of genetic variants for association with common, complex disease. While there have been a number of successes in association mapping studies, the associations found typically explain very little of the overall heritability of the traits being studied. There are several potential reasons for this “missing heritability”, and one of those potential explanations is epistatic interactions (genegene interactions). It is hypothesized that such interactions play an important role in the etiology of complex (nonMendelian) traits, but detecting such interactions presents a number of statistical and computation challenges [1]. In response to these challenges, a number of new datamining approaches have been developed [2].
Multifactor Dimensionality Reduction (MDR) is a popular and highly successful statistical method developed to detect and characterize nonlinear complex genegene or geneenvironment interactions (epistasis) that could be associated with disease susceptibility. The method was first proposed by Ritchie et al. [3] to detect estrogenmetabolism gene interactions associated with sporadic breast cancer. MDR has several advantages over more traditional statistical approaches such as logistic regression modeling: 1) MDR is a nonparametric approach with no requirement to the distribution of data. 2) MDR can analyze nonlinear associations in genotypic combinations. 3) MDR has improved power to detect genegene interaction in small to moderate sample sizes. Since the introduction of the original MDR implementation, many works have been published to improve modeling and prediction accuracy with the MDR method. For more information on the history and development of the method, please refer to the comprehensive review of the MDR and its extended methods by Moore [4].
While the MDR approach is widely used, to make this paper selfcontained, we give a brief description of the method. MDR is often applied to genotypic data to detect genegene (GxG) interactions among single nucleotide polymorphism (SNP) and the original implementation of this method can be extended to detect the interactions in other types of data when the explanatory variables are categorical variables and the outcome variable is binary. As the scale of association studies has expanded (with larger numbers of SNPs), a filtration step is often implemented in the first step of MDR analysis to remove noisy SNPs. In this step, a subset of genes that are unlikely to interact with others is removed by filtration methods such as SURF [5], TuRF [6] etc. ReliefF [7], has become a commonly applied filter, and we will focus on this filter in the current study. After this step, the remaining SNPs are used for the dimensionality reduction and model selection steps of the MDR algorithm. In this step, all variable combinations are considered for kway (k = 2, 3, 4 …) interactions. For each multilocus combination, the ratio of cases to controls within each contingency table cell is calculated, and then each cell is assigned a status of highrisk or lowrisk by comparing this ratio to the ratio of cases: controls in the overall dataset. Cells with a ratio greater than the overall ratio are assigned “highrisk” status, and those with a ratio lower than the overall ratio are assigned “lowrisk” status. Subsequently, a balanced classification accuracy is calculated for each multilocus combination, and the optimal model is selected based on the highest balanced accuracy. This model selection approach is performed in concert with a crossvalidation procedure, usually 10fold, which randomly divides the whole data set into a training set and a validation set. The testing accuracy is the balanced accuracy when the classification rule developed from the training data set is applied to the testing data set. The cross validation count (CVC) summarizes the number of times a model is the top model in each of the crossvalidation splits of the data. The optimal kway (k = 2, 3, 4 …) interaction model with the highest training accuracy and the highest CVC is then selected as the winner model. Finally, the significance of the selected optimal model is assessed by permutation testing (comparing the testing/prediction accuracy against the empirical distribution built by at least 1000 permutations). MDR can be performed by an open source software mdr2.0 and model goodnessoffit and significance can be assessed using software mdrpt1.0 [8] or in the MDR.R R software package [9].
In this work, we seek to address two existing issues in the current MDR analysis. First, current filtration approaches do not evaluate the significance of the SNPs considered (or provide pvalue for their measures) and there is no clear guideline for the cutoff point of such filtration measures. This leads inconsistency in the optimal number of SNPs remaining for the final MDR analysis.
Second, as there is a growing appreciation that the etiology of human diseases is extremely complex, many investigators are using MDR to evaluate many potential interactive effects, and not just a single final best model [10]. In this type of approach, not one but numerous tests can be performed in search of an optimal model in the kway interaction, as the number of partitions for kway interaction over m loci is $\left(\begin{array}{l}m\\ k\end{array}\right)=\frac{m!}{k!(mk)!}$. For instance, if an investigator is interested to detect significant 2way interactions among 50 SNPs, 1225 tests will be performed which will inflate the familywise Type I error rate to $1{(1\alpha )}^{\left(\begin{array}{l}m\\ k\end{array}\right)}=1{(10.05)}^{1225}\approx 1$, where $\alpha =0.05$ is the nominal error rate for an individual test without proper control.
False discoveries and losing power to detect the signal after the multiplicity adjustment are two concurrent issues in analyzing high dimensional data. Instead of replacing all the existing methods to control the false discovery, we propose to add the global tests to the current MDR framework as an adhoc screening process to prevent false discoveries. We will explain the rationale and utility of global tests in Section Global tests.
Incorporating global tests within the filtration procedures can reveal a trend of gene interactive patterns when noisy genes (SNPs) are removed step by step using ReliefF or other filtration techniques. In the current study, we demonstrate this approach in a candidate gene study of drug response in Juvenile Idiopathic Arthritis, using the ReleifF filter. Additionally, we perform a simulation study comparing several different global testing approaches for a range of genetic etiologies to compare the power of different approaches.
Methods
Global tests
When a proportion of m loci have k way interactions (H _{ α }), it is expected to observe the pvalues shifting towards 0. To see this, let F(t) be the CDF under H _{ α } and $F\left(t\right)>{F}_{0}\left(t\right)$ for $t\in \mathfrak{R}$, then $\mathrm{Pr}(P\le p)=\mathrm{Pr}\left({F}_{0}^{1}\right(P\left)\right)\le {F}_{0}^{1}\left(p\right))=F({F}_{0}^{1}\left(p\right))>p$ for $p\in \left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$ (Pattern 2 in Figure 1). Due to correlations/linkage disequilibrium among SNPs and the redundancy of SNPs in high order models, sometimes pvalues shift toward 1, i.e. $\mathrm{Pr}(P\le p)<p$ for $p\in \left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$(Pattern 3 in Figure 1). When pvalues are correlated, they might peak near center of histogram (Pattern 4 in Figure 1). Patterns 3 and 4 are deviated from uniformity but they do not indicate potential k way interactions among m loci.
The rationale of global testing is to ensure pvalues are not randomly and uniformly distributed (Pattern 1) before we investigate each single pvalue. Correlated pvalues without significant effects (H _{0}) might even shift toward 1 or peak near the center (Patterns 3 and 4). The goals are to rule out Patterns 1, 3 and 4 and only move forward to the final MDR analysis when pvalues are in Pattern 2.
If the entire set of pvalue follows a uniform distribution, then it is very likely for a small pvalue to be a false discovery by chance. As shown in Figure 1, the entire set of pvalues might have four different Patterns: uniform, shifting to 0, shifting to 1 or peak near the center. In all four cases, we notice that the minimum pvalues are less than 0.05 (0.0001111 in Pattern 1, 2.65e6 in Pattern 2, 0.00617 in Pattern 3 and 0.003734 in Pattern 4). If we take the distribution of the entire set of pvalues into account, then the minimum pvalues in Patterns 1, 3 and 4 are false discoveries by chance.
Combined global testing and filtration technique
Rejecting H _{0} indicates significant GxG interactions in some target genes.
We propose incorporating global testing of pvalues with ReliefF [7] gene filtration technique to detect the patterns of kway GxG interaction among m genes (SNPs) and remove noisy genes (SNPs) with little interactive effects to determine the optimal number of SNPs for the final MDR analysis. The ReliefF algorithm estimates weights to measure the potential accuracy of attributes in prediction of phenotype. The redundant attribute will be assigned a lower score. When applied in genegene interactions, a higher ReliefF score indicates a stronger interactive effect for the corresponding gene (SNP). ReliefF algorithm first uses x nearest neighborhood approach $(x=1\text{,}\phantom{\rule{0.12em}{0ex}}2\text{,}\cdots \text{,}\phantom{\rule{0.12em}{0ex}}m)$ to match a selected subject with x subjects in neighborhood (with shortest distances across all SNPs) from the control group and from test group respectively. An attribute (SNP) will be assigned score 1 (−1) if the attribute from the selected subject matches (mismatches) one of x nearest subjects from the same phenotype group. Similarly, an attribute will be assigned score −1 (1) if the attribute from the selected subject matches (mismatches) one of the nearest subjects from the different phenotype group. The score will be aggregated for all subjects and normalized (divided) by the total number of subjects and neighbors. Detailed description of ReliefF algorithm for filtering genotyping data can be found in Section 3 of [4].
Global tests of Pvalues
Here we introduce 6 global tests that can be applied to test hypothesis (1). These six tests are based on different approaches to detect deviation from uniformity. We will survey these methods and compare their power using a case study and MonteCarlo simulations.
Test 1 one sided KolmogorovSmirnov test [KS]
KS test is a nonparametric test that can be applied to compare the distance between an empirical distribution of i.i.d. pvalues and Uniform(0, 1). For hypothesis test (1), define the onesided KS statistic as ${D}_{n}^{+}=\underset{p}{sup}\left(\frac{1}{n}\sum _{i=1}^{n}{I}_{\{{p}_{i}\le p\}}p\right)$, where ${I}_{\{{p}_{i}\le p\}}$is an indicator function which equals 1 if ${p}_{i}\le p$and 0 if ${p}_{i}>p$. According to [11], the pvalue of onesided KS test follows $\mathrm{Pr}\left({D}_{n}^{+}>t\right)=t\sum _{i=0}^{\left[n\right(1t\left)\right]}\left(\begin{array}{l}n\\ i\end{array}\right){\left(1ti/n\right)}^{ni}{\left(t+i/n\right)}^{i1}$ where $\left[n\right(1t\left)\right]$ is the largest integer not greater than $n(1t)$ and $t\in \left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$.
Test 2 onesided inverse chisquare test [inverse chi]
Fisher [12] shows that if ${p}_{i}\stackrel{i.i.d.}{~}Uniform(0,1)$ for $i=1\text{,}\phantom{\rule{0.12em}{0ex}}2\text{,}\cdots \text{,}\phantom{\rule{0.12em}{0ex}}n$, then $2\sum _{i=1}^{n}ln\left({p}_{i}\right)~{\chi}_{2n}^{2}$where ${\chi}_{2n}^{2}$is chisquare distribution with 2n degrees of freedom. For a one sided test (1), reject H _{0} if $2\sum _{i=1}^{n}ln\left({p}_{i}\right)>{\chi}_{2n,1\alpha}^{2}$where ${\chi}_{2n,1\alpha}^{2}$ is $(1\alpha )*100\%$percentile of ${\chi}_{2n}^{2}$.
Test 3 one sided inverse normal test [inverse norm]
Transform pvalue to normal z score by letting ${z}_{i}={\Phi}^{1}\left({p}_{i}\right)$where ${\Phi}^{1}$ is inverse cumulative normal distribution. Under H _{ 0 }, $Z=\left(\sum _{i=1}^{n}{z}_{i}\right)/\sqrt{n}~N\left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$. For one sided test (1), reject H _{0} if Z < Z _{ α } where Z _{ α } is $\alpha *100\%$ percentile of the standard normal distribution.
Test 4 one sided logit test [logit]
Logit transform pvalue by letting $L=\sum _{i=1}^{n}ln({p}_{i}/(1{p}_{i}\left)\right)$. [13] shows that under H _{0}, the distribution of L can be closely approximated by Student’s tdistribution with $5n+4$degrees of freedom, namely ${L}^{*}=L\sqrt{\frac{3(5n+4)}{{\pi}^{2}n(5n+2)}}\approx {t}_{5n+4}$. Therefore, for onesided test (1), we can reject H _{0} if ${L}^{*}<{t}_{5n+4,\alpha}$where ${t}_{5n+4,\alpha}$ is $\alpha *100\%$ percentile of the tdistribution.
Test 5 one sided Wilcoxon test [Wilcoxon]
Order n pvalues from MDR testing along with n _{ 2 } observations randomly drawn from Uniform(0,1) from least to greatest and denote them by ${S}_{1}\text{,}\phantom{\rule{0.12em}{0ex}}{S}_{2}\text{,}\cdots \text{,}\phantom{\rule{0.12em}{0ex}}{S}_{N}$ with $N=n+{n}_{2}$. Let W be the sum of the ranks corresponding to n pvalues from MDR testing. For onesided test (1), we can reject H _{0} if $W\le n(N+1){\omega}_{\alpha}$where the constant ω _{ α } is chosen to make the Type I error probability equal α. Values of ω _{ α } are given in Table A6 by [14]. For large sample sizes, i.e. min(n,n _{2}) going to infinity, one can apply normal approximation on the standardized W.
Test 6 Tippett and Wilkinson’s test [Tippett]
Tippett’s Test [15] is based on the property of the minimal pvalue in multiple testing. Let ${p}_{\left(1\right)}\text{,}\phantom{\rule{0.12em}{0ex}}{p}_{\left(2\right)}\text{,}\cdots \text{,}\phantom{\rule{0.12em}{0ex}}{p}_{\left(n\right)}$be the ordered pvalues in an ascending order. When pvalues identically and independently follow Uniform(0,1) distribution, Tippett’s test will reject H _{0} if ${p}_{\left(1\right)}<1{(1\alpha )}^{1/n}$. The pvalue of Tippett’s test equals $1{(1{p}_{\left(1\right)})}^{n}$. Tippett’s test is very easy to perform but it only takes the smallest pvalue into account.
Wilkinson [16] extended Tippett’s procedure to the r ^{ th } smallest pvalues where $r=1\text{,}\phantom{\rule{0.12em}{0ex}}2\text{,}\cdots \text{,}\phantom{\rule{0.12em}{0ex}}n$. By expanding ${(\alpha +(1\alpha \left)\right)}^{n}$, Wilkinson tabulated the probability, denoted by C _{ γ,α } of obtaining r significant statistics by chance in a group of n tests. Suppose there are r tests with pvalues less than α, Wilkinson’s test rejects H _{0} if ${c}_{r,\alpha}<\alpha $[17]. Because P _{ (r) } has been distributed with parameters r and nr + 1, tables of the incomplete beta function can be used to obtain critical values of P _{ (r) } directly. In our work, we will not include Wilkinson’s test in case study and power simulation because this method does not provide pvalue for the testing results.
Case study
We used a real dataset to illustrate how to apply our proposed global testing to prevent false discovery and to determine the optimal number of SNPs for the final MDR analysis. Juvenile Idiopathic Arthritis (JIA) is one of the most common chronic diseases of childhood, affecting an estimated 300,000 children in the U.S. alone, and is an important cause of morbidity and disability in children [18]. Although methotrexate (MTX) is the most commonly used secondline agent used to treat JIA worldwide, this antifolate drug has shown considerable interindividual variability in clinical response and adverse reactions [19]. The polyglutamation of methotrexate (MTXglu) is an intracellular mechanism that retains the drug and enhances target enzyme inhibition within the folate pathway [20], and high concentrations of “long chain” methotrexate polyglutamates (MTXglu_{35}) have been associated with improved response to the drug in adults with rheumatoid arthritis [21]. Studies have reported the extensive variability in intracellular MTXglu concentrations in JIA, and an association of long chain MTXglu with toxicity (but not efficacy) in children [22]. Due to the complexity of the folate cycle as well as the extensive variability in response to the drug in clinical practice, it is hypothesized that genetic factors may contribute to differences seen in distinct patterns of MTXglu concentrations intracellularly, which might further impact patients’ responses to MTX.
List of 25 SNPs from 17 candidate genes in the folate pathway
SNP  RS #  MAF* 

ABCG2 C > T  rs7699188  0.13 
ABCG2 15846 A > C  no rs [35]  0.01 
ABCG2 G > A  rs35252139  0.13 
ABCG2 A > G  rs35229708  0.13 
ABCG2 C > T  55930652  0.27 
ATIC C > T  rs12995526  0.3 
BHMT A > G  rs3733890  0.33 
DHFR A > T  rs7387  0.3 
GGH C > T  rs3758149  0.27 
MTHFD2 indel  rs71391718  0.31 
MTHFR C > T  rs1801133  0.3 
MTHFR A > C  rs1801131  0.33 
MTHFR G > A  rs2274976  0.06 
MTR A > G  rs1805087  0.19 
MTRR A > G  rs1801394  0.57 
SHMT1 C > T  rs1979277  0.37 
TYMS *2/*3  rs34743033  0.49 
TYMS indel  rs11280056  0.32 
FOLH1 C > T  rs61886492  0.03 
GART A > G  rs8788  0.21 
GART A > G  rs8971  0.19 
SLC25A32 G > A  rs17803441  0.07 
ADORA2a C > T  rs2298383  0.61 
ITPA T > C  rs2295553  0.52 
SLCO1B1 T > C  rs4149056  0.12 
There were 30 subjects in Cluster 1 and 74 subjects in Cluster 2. The MTXglu clustering phenotype was coded 1 and 0 for MDR analysis. Genotypes, coded 0 for common homozygote, 1 for heterozygote and 2 for rare homozygote for 25 SNPs, were measured. The overall goal of the analysis was to assess whether interactions among SNPs are associated with MTXglu clustering. While the scale of this study is not so large that an exhaustive search of all SNPs is computationally limited, this data is used to demonstrate the proposed approach.
FDR adjusted Pvalue in global testing
# SNPs  Remaining  SNP  ReliefF  Pvalues of global testing  

Removed  SNP (GxG)  Removed  Score  KS  Inverse chi  Inversenorm  Logit  Wilcoxon  Tippett 
0  25(300)  0.29121  0.22008  1.00000  1.00000  0.80991  0.33927  
1  24(276)  rs35252139  −0.0308  0.21644  0.12424  1.00000  1.00000  0.55397  0.33340 
2  23(253)  rs35229708  −0.0308  0.16697  0.06603  1.00000  1.00000  0.43198  0.32718 
3  22(231)  rs2298383  −0.0279  0.13417  0.03087  1.00000  1.00000  0.25182  0.32062 
4  21(210)  rs12995526  −0.0250  0.16987  0.05667  1.00000  1.00000  0.45717  0.31374 
5  20(190)  rs7699188  −0.0202  0.11033  0.01793  1.00000  1.00000  0.31680  0.30658 
6  19(171)  rs1805087  −0.0183  0.03887  0.00320  1.00000  1.00000  0.18665  0.29916 
7  18(153)  rs4149056  −0.0067  0.02048  0.00106  1.00000  1.00000  0.09019  0.29154 
8  17(136)  55930652  −0.0038  0.01051  0.00080  1.00000  1.00000  0.10254  0.28376 
9  16(120)  * no rs [35]  −0.0029  0.00455  0.00016  1.00000  1.00000  0.02642  0.27592 
10  15(105)  rs17803441  0.0010  0.00312  0.00011  1.00000  1.00000  0.00567  0.26811 
11  14(91)  rs34743033  0.0087  0.00022  0.00001  1.00000  1.00000  0.00015  0.26049 
12  13(78)  rs61886492  0.0096  0.00022  0.00000  0.00001  0.00000  0.00006  0.25328 
13  12(66)  rs71391718  0.0183  0.00008  0.00000  0.00000  0.00000  0.00001  0.25328 
14  11(55)  rs3758149  0.0192  0.00008  0.00000  0.00000  0.00000  0.00006  0.25328 
15  10(45)  rs1801131  0.0240  0.00026  0.00004  0.00004  0.00004  0.00016  0.25328 
16  9(36)  rs1801394  0.0365  0.00022  0.00004  0.00003  0.00003  0.00015  0.25328 
17  8(28)  rs8788  0.0375  0.00008  0.00003  0.00002  0.00002  0.00015  0.25328 
18  7(21)  rs7387  0.0452  0.01051  0.00255  0.00547  0.00442  0.00799  0.25328 
19  6(15)  rs1801133  0.0481  0.05212  0.06679  0.10791  0.12185  0.05337  0.25328 
20  5(10)  rs8971  0.0481  
21  4(6)  rs2274976  0.0644  
22  3(3)  rs1979277  0.0673  
23  2(1)  rs11280056  0.0702  
24  rs3733890  0.0750  
25  rs3733890  0.1163  
Optimal number of SNPs removed  5  4  11  11  8  Not Found 
To circumvent this limitation, we incorporated global testing of pvalues and ReliefF algorithm using the method proposed in Section Combined global testing and filtration technique. We first generated pvalues for all 2way interactions among 25 SNPs through permutation testing. Then we applied global testing, including KS test, Inverse chi test, Inverse norm test, Logit test, Wilcoxon test and Tippett’s tests on $\left(\begin{array}{l}25\\ 2\end{array}\right)=300$ pvalues of 2way interactions among 25 SNPs. The global tests were performed to evaluate whether the distribution of pvalues deviated from uniformity (null hypothesis) and shifted towards 0 (alternative hypothesis  Pattern 2 in Figure 1). Then we removed one SNP with the lowest ReliefF score step by step and repeated the global testing process until only 5 SNPs were remained. We stopped the global testing procedure at 5 SNPs because it is not meaningful or necessary to perform global testing when the number of SNPs is less than 5 in any case study.
The entire procedure of filtration and global testing of pvalues are summarized in Table 2. The optimal number of SNPs is the largest number of SNPs with global testing pvalue <0.05. In this case study, KS test and Inverse chi test were more sensitive to deviation from uniformity as the tests became significant after 5 and 4 SNPs removed respectively (Table 2). Inverse norm test and Logit test were more conservative, suggesting removal of 11 SNPs. Wilcoxon test, removing 8 SNPs, was moderate as compared to the other tests. Tippett’s test failed to detect significant GxG interactions with all FDR corrected pvalues > 0.05. Our further simulation studies (discussed in Section Power simulation) indicate that Tippett’s test, which only takes the smallest pvalue into account, might not be appropriate for global testing of pvalues.
After the filtration and global testing, we removed 4 SNPs as suggested by Inverse chi test with FDR correction (Table 2) and performed MDR analysis on the remaining 21 SNPs. The results of MDR analysis indicated that there was significant twoway interaction between DHFR (rs7387) and ITPA (rs2295553) with testing balance accuracy = 0.7374 (p = 0.0045). The MDR analysis was performed by an open source software mdr2.0 and model goodnessoffit and significance was assessed by permutation using software mdrpt1.0 [8].
The dihydrofolate reductase (DHFR) enzyme is a well known important target of MTX action. When DHFR is inhibited by MTX the subsequent production of reduced folates such as tetrahydrofolate (THF) 5,10 methenylTHF and 5methylTHF are altered, affecting not only total cellular folate concentrations but also the downstream effects from one carbon donation including homocysteine remethylation and pyrimidine and purine synthesis thought to result in an antiproliferative effect [23]. The inhibition of DHFR also results in a buildup of the precursor dihydrofolate (DHF) which in its polyglutamated state has inhibitory effects upon enzymes within the pathway as well [24]. Inosine triphosphate pyrophosphatase (ITPA) plays a role in de novo purine synthesis, and is closely related to adenosine metabolism, which is thought to contribute to MTX response via its antiinflammatory effect [25]. Variations in ITPA interestingly have been shown by other authors to contribute to MTX response as part of a candidate gene study in JIA [26], as well as “predisposing genetic attribute” in studies utilizing MDR in adults with rheumatoid arthritis [27, 28]. How these 2 genes directly affect MTXglu patterns remains difficult to determine, as the direct understanding of how MTXglu patterns are associated with response is yet to be elucidated. However, both genes encode enzymes closely linked to or directly affected by MTX, thus as we gain a more detailed knowledge of cellular folate metabolism and its disruption by antifolate agents such as MTX, we will then develop a better understanding of this complex system, and how alterations in the folate pathway affect response to the drug.
Power simulation
for $p\in \left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$. The proportion of tests where H _{ α } holds is denoted by the mixing weight $\mathrm{\text{Pr}}({Z}_{i}=1)=\pi $ where $\pi \in \left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$.
The marginal distribution of combined pvalues becomes $P~(1\pi )\text{Uniform}(0,1)+\pi \text{Beta}\left(a\text{,}\phantom{\rule{0.12em}{0ex}}b\right)$, which indicates that with $(1\pi )\times 100\%$ of chance, a pvalue is drawn from Uniform(0,1) and with $\pi \times 100\%$ of chance, a pvalue is drawn from Beta(α,b). Beta distribution is very flexible to characterize the patterns of pvalues (Figure 1) where Uniform (0,1) is a special case of Beta (1,1). One can also adjust the shape and scale parameters a and b to model the deviation from uniformity.
The pvalues from MDR analysis are correlated due to linkage disequilibrium among SNPs and sharing the SNPs among GxG interactions. The dependence among pvalues might cause inflation of Type I errors or lead to bias in global tests. As a result, it is critical to extensively simulate pvalues with varying correlation structures and assess the robustness of global tests for correlated pvalues. In this work, we simulated correlated Uniform variables with random correlation matrix Σ and Beta random variables with correlation coefficient $\rho =0.2\text{,}\phantom{\rule{0.12em}{0ex}}0.8\text{,}\phantom{\rule{0.12em}{0ex}}\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)\text{,}\phantom{\rule{0.12em}{0ex}}\text{Uniform}\left(0.1\phantom{\rule{0.12em}{0ex}}\text{,}\phantom{\rule{0.12em}{0ex}}0.9\right)$ respectively. The details of generating correlated uniform [29] and beta distributions [30] are summarized in Appendix 1.

Independent Uniform(0,1),

Correlated $\text{Uniform}\left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$,

Correlated $0.9\text{Uniform}\left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)+0.1\text{Beta}\left(5\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$, (4.1)

Correlated $0.5\text{Uniform}\left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)+0.5\text{Beta}\left(5\text{,}\phantom{\rule{0.12em}{0ex}}1\right)$, (4.2)

Correlated $0.9\text{Uniform}\left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)+0.1\text{Beta}\left(6\text{,}\phantom{\rule{0.12em}{0ex}}3\right)$, (4.3)

Correlated $0.5\text{Uniform}\left(0\text{,}\phantom{\rule{0.12em}{0ex}}1\right)+0.5\text{Beta}\left(6\text{,}\phantom{\rule{0.12em}{0ex}}3\right)$. (4.4)
Type I error of six global tests of pvalues when pvalues are independent or strongly correlated (The nominal Type I error rate is 0.05 and the severe inflation of Type I error with simulated error rate > 0.1 is written in bold italic)
Independent Uniform (0,1)  

n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.052  0.053  0.049  0.051  0.052  0.050 
50  0.051  0.052  0.051  0.048  0.051  0.050 
100  0.046  0.049  0.050  0.049  0.049  0.051 
200  0.047  0.048  0.047  0.052  0.049  0.048 
300  0.051  0.054  0.053  0.051  0.053  0.053 
400  0.042  0.046  0.046  0.048  0.047  0.046 
500  0.051  0.050  0.049  0.051  0.050  0.050 
Correlated Uniform (0,1)  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.061  0.059  0.063  0.050  0.065  0.062 
50  0.058  0.060  0.063  0.049  0.061  0.062 
100  0.060  0.066  0.068  0.049  0.066  0.067 
200  0.063  0.069  0.073  0.050  0.072  0.072 
300  0.069  0.071  0.074  0.052  0.073  0.074 
400  0.064  0.066  0.073  0.048  0.071  0.072 
500  0.064  0.070  0.071  0.049  0.068  0.069 
Correlated Uniform (0,1) ± 0.1Beta (5,1), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.061  0.047  0.058  0.035  0.061  0.057 
50  0.081  0.037  0.054  0.046  0.06  0.053 
100  0.080  0.039  0.061  0.045  0.056  0.057 
200  0.156  0.031  0.062  0.039  0.059  0.059 
300  0.180  0.017  0.04  0.039  0.049  0.042 
400  0.270  0.025  0.055  0.052  0.053  0.056 
500  0.278  0.017  0.052  0.046  0.060  0.052 
Correlated 0.5 Uniform (0,1) ± 0.5Beta (5,1), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.239  0.007  0.009  0.035  0.013  0.009 
50  0.652  0.002  0.022  0.025  0.02  0.022 
100  0.911  0.001  0.018  0.023  0.018  0.019 
200  0.998  0  0.021  0.034  0.025  0.021 
300  1  0  0.029  0.033  0.039  0.027 
400  1  0  0.027  0.024  0.034  0.023 
500  1  0  0.031  0.024  0.036  0.025 
Correlated 0.9 Uniform (0,1) ± 0.1Beta (6,3), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.072  0.041  0.05  0.038  0.052  0.05 
50  0.107  0.039  0.065  0.051  0.068  0.062 
100  0.148  0.048  0.086  0.04  0.089  0.086 
200  0.227  0.038  0.088  0.048  0.101  0.086 
300  0.296  0.024  0.097  0.042  0.119  0.095 
400  0.353  0.04  0.108  0.053  0.131  0.105 
500  0.439  0.041  0.12  0.033  0.148  0.111 
Correlated 0.5 Uniform (0,1) ± 0.5Beta (6,3), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.421  0.01  0.038  0.025  0.063  0.029 
50  0.837  0.008  0.089  0.019  0.147  0.071 
100  0.989  0.008  0.16  0.024  0.253  0.122 
200  1  0.001  0.268  0.023  0.411  0.207 
300  1  0  0.392  0.022  0.581  0.32 
400  1  0.004  0.492  0.031  0.679  0.422 
500  1  0.003  0.551  0.024  0.761  0.486 
Type I error of six global tests of pvalues when pvalues are moderately correlated (The nominal Type I error rate is 0.05 and the severe inflation of Type I error with simulated error rate > 0.1 is written in bold italic)
Correlated Uniform (0,1) ± 0.1Beta (5,1),$\rho =0.2$  

n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.018  0.031  0.022  0.048  0.021  0.026 
50  0.014  0.019  0.008  0.046  0.01  0.013 
100  0.014  0.014  0.013  0.05  0.014  0.014 
200  0.003  0.01  0.006  0.044  0.004  0.007 
300  0.002  0.005  0.002  0.05  0.001  0.002 
400  0.001  0.003  0  0.045  0  0 
500  0.002  0.002  0.003  0.042  0.001  0.003 
Correlated 0.5 Uniform (0,1) ± 0.5Beta (5,1), $\rho =0.2$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0  0  0  0.023  0  0 
50  0  0  0  0.023  0  0 
100  0  0  0  0.029  0  0 
200  0  0  0  0.025  0  0 
300  0  0  0  0.024  0  0 
400  0  0  0  0.021  0  0 
500  0  0  0  0.024  0  0 
Correlated 0.9 Uniform (0,1) ± 0.1Beta (6,3), $\rho =0.2$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.041  0.051  0.047  0.047  0.042  0.049 
50  0.032  0.049  0.043  0.042  0.034  0.048 
100  0.033  0.032  0.035  0.043  0.035  0.036 
200  0.017  0.009  0.023  0.052  0.017  0.024 
300  0.017  0.016  0.015  0.039  0.011  0.016 
400  0.028  0.006  0.014  0.044  0.01  0.017 
500  0.021  0.007  0.013  0.048  0.009  0.015 
Correlated 0.5 Uniform (0,1) ± 0.5Beta (6,3), $\rho =0.2$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.008  0.004  0.003  0.025  0.001  0.003 
50  0.021  0.002  0.001  0.024  0.001  0.002 
100  0.046  0.001  0.001  0.028  0  0.001 
200  0.165  0  0  0.023  0  0 
300  0.337  0  0  0.025  0  0 
400  0.558  0  0  0.017  0  0 
500  0.66  0  0  0.018  0  0 
Correlated Uniform (0,1) ± 0.1Beta (5,1), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.034  0.043  0.036  0.047  0.034  0.04 
50  0.019  0.022  0.02  0.05  0.018  0.021 
100  0.012  0.018  0.011  0.038  0.008  0.012 
200  0.002  0.008  0.006  0.045  0.006  0.006 
300  0.005  0.009  0.005  0.037  0.004  0.007 
400  0.004  0.005  0.002  0.038  0.002  0.002 
500  0  0.003  0.001  0.05  0  0.002 
Correlated 0.5 Uniform (0,1) ± 0.5Beta (5,1), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.001  0.002  0.001  0.021  0  0.001 
50  0  0  0  0.026  0  0 
100  0  0  0  0.016  0  0 
200  0  0  0  0.02  0  0 
300  0  0  0  0.031  0  0 
400  0.002  0  0  0.02  0  0 
500  0.001  0  0  0.028  0  0 
Correlated 0.9 Uniform (0,1) ± 0.1Beta (6,3), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.049  0.046  0.054  0.044  0.055  0.05 
50  0.039  0.031  0.042  0.042  0.036  0.043 
100  0.027  0.027  0.036  0.05  0.033  0.037 
200  0.033  0.023  0.03  0.041  0.025  0.032 
300  0.042  0.018  0.029  0.038  0.023  0.032 
400  0.041  0.012  0.02  0.046  0.017  0.021 
500  0.053  0.013  0.026  0.054  0.02  0.028 
Correlated 0.5 Uniform (0,1) ± 0.5Beta (6,3), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.02  0.006  0.008  0.03  0.006  0.009 
50  0.051  0  0  0.031  0  0 
100  0.123  0  0.003  0.023  0.001  0.004 
200  0.271  0  0  0.021  0  0 
300  0.414  0  0  0.033  0  0 
400  0.552  0  0  0.024  0  0 
500  0.663  0  0  0.028  0  0 
Correlated Uniform (0,1) ± 0.1Beta (5,1), $\rho =\text{Uniform}(0.1,0.9)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.036  0.041  0.034  0.048  0.031  0.038 
50  0.042  0.039  0.043  0.054  0.039  0.048 
100  0.028  0.03  0.029  0.054  0.028  0.029 
200  0.037  0.018  0.023  0.045  0.021  0.025 
300  0.016  0.012  0.013  0.025  0.015  0.014 
400  0.034  0.008  0.017  0.045  0.015  0.019 
500  0.031  0.006  0.014  0.046  0.005  0.016 
Correlated 0.5 Uniform (0,1) ± 0.5Beta (5,1), $\rho =\text{Uniform}(0.1,0.9)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.011  0.005  0.002  0.017  0.001  0.002 
50  0.029  0.002  0.003  0.022  0.001  0.003 
100  0.048  0  0  0.023  0.001  0.001 
200  0.099  0  0  0.023  0.000  0 
300  0.161  0  0  0.026  0.000  0 
400  0.176  0  0  0.016  0.000  0 
500  0.218  0  0  0.024  0.001  0 
Correlated 0.9 Uniform (0,1) ± 0.1Beta (6,3), $\rho =\text{Uniform}(0.1,0.9)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.05  0.036  0.038  0.033  0.044  0.039 
50  0.054  0.031  0.047  0.043  0.045  0.044 
100  0.044  0.031  0.048  0.05  0.045  0.056 
200  0.091  0.034  0.057  0.047  0.058  0.058 
300  0.117  0.024  0.054  0.049  0.054  0.05 s8 
400  0.097  0.022  0.048  0.048  0.048  0.05 
500  0.116  0.014  0.042  0.035  0.043  0.043 
Correlated 0.5 Uniform (0,1) ± 0.5Beta (5,1), $\rho =\text{Uniform}(0.1,0.9)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.108  0.009  0.016  0.027  0.017  0.016 
50  0.298  0.002  0.014  0.025  0.013  0.014 
100  0.459  0.002  0.014  0.024  0.016  0.014 
200  0.774  0  0.016  0.017  0.022  0.017 
300  0.908  0  0.016  0.023  0.023  0.015 
400  0.933  0  0.024  0.024  0.029  0.019 
500  0.975  0  0.016  0.02  0.032  0.013 

Correlated $0.9\text{Uniform}(0,1)+0.1\text{Beta}(0.4,6)$,

Correlated $0.6\text{Uniform}(0,1)+0.4\text{Beta}\left(0.4\text{,}\phantom{\rule{0.12em}{0ex}}6\right)$,

Correlated $0.9\text{Uniform}(0,1)+0.1\text{Beta}\left(0.5\text{,}\phantom{\rule{0.12em}{0ex}}4.5\right)$,

Correlated $0.6\text{Uniform}(0,1)+0.4\text{Beta}\left(0.5\text{,}\phantom{\rule{0.12em}{0ex}}4.5\right)$,

Correlated $0.9\text{Uniform}(0,1)+0.1\text{Beta}(1,5)$, and

Correlated $0.6\text{Uniform}(0,1)+0.4\text{Beta}(1,5)$.
Power of six global tests of correlated Pvalues (The correlation coefficient for Beta random variables is ρ . Uniform distributions have random correlation matrices)
Correlated 0.9Uniform (0,1) ± 0.1Beta (0.4,6),$\rho =0.2$  

n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.176  0.359  0.259  0.332  0.191  0.294 
50  0.27  0.576  0.418  0.505  0.303  0.464 
100  0.407  0.824  0.599  0.652  0.456  0.672 
200  0.645  0.967  0.827  0.791  0.657  0.877 
300  0.808  0.995  0.92  0.861  0.785  0.941 
400  0.896  0.997  0.966  0.896  0.861  0.979 
500  0.949  0.999  0.984  0.933  0.926  0.993 
Correlated 0.6 Uniform (0,1) ± 0.4Beta (0.4,6), $\rho =0.2$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.797  0.957  0.896  0.784  0.815  0.918 
50  0.987  1  0.997  0.939  0.983  1 
100  1  1  1  0.972  1  1 
200  1  1  1  0.998  1  1 
300  1  1  1  1  1  1 
400  1  1  1  0.999  1  1 
500  1  1  1  0.999  1  1 
Correlated 0.9 Uniform (0,1) ± 0.1Beta (0.5,4.5), $\rho =0.2$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.151  0.266  0.207  0.225  0.183  0.221 
50  0.227  0.466  0.336  0.326  0.264  0.369 
100  0.343  0.664  0.493  0.437  0.397  0.534 
200  0.545  0.863  0.709  0.537  0.595  0.757 
300  0.706  0.944  0.832  0.587  0.719  0.863 
400  0.811  0.982  0.913  0.632  0.81  0.933 
500  0.89  0.992  0.951  0.695  0.887  0.966 
Correlated 0.6 Uniform (0,1) ± 0.4Beta (0.5,4.5), $\rho =0.2$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.161  0.291  0.212  0.225  0.173  0.237 
50  0.251  0.475  0.362  0.302  0.292  0.385 
100  0.322  0.64  0.504  0.397  0.39  0.528 
200  0.517  0.882  0.701  0.527  0.58  0.743 
300  0.726  0.952  0.845  0.568  0.731  0.873 
400  0.806  0.976  0.894  0.607  0.805  0.917 
500  0.895  0.998  0.95  0.676  0.883  0.965 
Correlated 0.9 Uniform (0,1) ± 0.1Beta (1,5), $\rho =0.2$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.127  0.118  0.139  0.056  0.144  0.136 
50  0.203  0.199  0.211  0.063  0.235  0.206 
100  0.248  0.272  0.28  0.064  0.271  0.269 
200  0.406  0.448  0.453  0.06  0.446  0.438 
300  0.535  0.559  0.541  0.073  0.542  0.535 
400  0.619  0.657  0.649  0.07  0.657  0.631 
500  0.725  0.743  0.729  0.071  0.731  0.715 
Correlated 0.6 Uniform (0,1) ± 0.4Beta (1,5), $\rho =0.2$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.572  0.56  0.599  0.117  0.611  0.587 
50  0.893  0.864  0.904  0.104  0.917  0.891 
100  0.991  0.981  0.986  0.114  0.991  0.981 
200  1  1  0.999  0.091  1  0.999 
300  1  1  1  0.109  1  1 
400  1  1  1  0.111  1  1 
500  1  1  1  0.085  1  1 
Power of six global tests of correlated Pvalues (The correlation coefficient for Beta random variables is ρ Uniform distributions have random correlation matrices)
Correlated 0.9 Uniform (0,1) ± 0.1Beta (0.4,6),$\rho =0.8$  

n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.19  0.402  0.285  0.329  0.204  0.323 
50  0.293  0.654  0.458  0.547  0.324  0.514 
100  0.41  0.833  0.644  0.665  0.48  0.7 
200  0.705  0.963  0.856  0.785  0.693  0.901 
300  0.864  0.989  0.943  0.861  0.834  0.956 
400  0.934  0.996  0.967  0.878  0.862  0.978 
500  0.962  0.999  0.989  0.914  0.928  0.994 
Correlated 0.6 Uniform (0,1) ± 0.4Beta (0.4,6), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.848  0.968  0.913  0.787  0.837  0.924 
50  0.994  0.999  0.998  0.922  0.989  0.998 
100  1  1  0.999  0.956  0.999  0.999 
200  1  1  1  0.987  1  1 
300  1  1  1  0.993  1  1 
400  1  1  1  0.992  1  1 
500  1  1  1  0.994  1  1 
Correlated 0.9 Uniform (0,1) ± 0.1Beta (0.5,4.5), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.145  0.282  0.199  0.243  0.158  0.237 
50  0.281  0.491  0.376  0.345  0.296  0.409 
100  0.374  0.686  0.532  0.425  0.418  0.573 
200  0.598  0.874  0.734  0.531  0.627  0.771 
300  0.774  0.951  0.866  0.579  0.767  0.89 
400  0.855  0.975  0.914  0.641  0.836  0.936 
500  0.938  0.993  0.962  0.635  0.91  0.977 
Correlated 0.6 Uniform (0,1) ± 0.4Beta (0.5,4.5), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.149  0.287  0.221  0.217  0.178  0.242 
50  0.265  0.518  0.4  0.334  0.323  0.438 
100  0.421  0.693  0.539  0.434  0.438  0.575 
200  0.604  0.856  0.726  0.485  0.644  0.754 
300  0.778  0.951  0.858  0.582  0.761  0.886 
400  0.871  0.978  0.931  0.612  0.858  0.942 
500  0.921  0.981  0.948  0.658  0.901  0.957 
Correlated 0.9 Uniform (0,1) ± 0.1Beta (1,5), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.157  0.147  0.158  0.066  0.154  0.154 
50  0.192  0.234  0.231  0.069  0.222  0.22 
100  0.32  0.355  0.347  0.064  0.344  0.34 
200  0.491  0.521  0.501  0.061  0.502  0.488 
300  0.655  0.698  0.663  0.068  0.651  0.652 
400  0.776  0.784  0.754  0.068  0.751  0.741 
500  0.847  0.844  0.82  0.072  0.823  0.802 
Correlated 0.6 Uniform(0,1) ± 0.4Beta(1,5), $\rho =0.8$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.698  0.676  0.695  0.124  0.703  0.682 
50  0.966  0.921  0.938  0.131  0.948  0.922 
100  0.999  0.988  0.991  0.11  0.997  0.988 
200  1  1  1  0.116  1  1 
300  1  1  1  0.122  1  1 
400  1  1  1  0.122  1  1 
500  1  1  1  0.114  1  1 
Correlated 0.9 Uniform(0,1) ± 0.1Beta(0.4,6), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.147  0.349  0.245  0.332  0.181  0.288 
50  0.263  0.59  0.422  0.529  0.298  0.481 
100  0.396  0.804  0.627  0.648  0.447  0.687 
200  0.675  0.963  0.831  0.804  0.662  0.89 
300  0.809  0.99  0.927  0.836  0.808  0.951 
400  0.899  0.998  0.966  0.907  0.865  0.979 
500  0.958  0.999  0.98  0.928  0.912  0.99 
Correlated 0.6 Uniform(0,1) ± 0.4Beta(0.4,6), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.827  0.974  0.921  0.789  0.808  0.941 
50  0.994  1  0.999  0.945  0.977  0.999 
100  1  1  1  0.983  1  1 
200  1  1  1  0.997  1  1 
300  1  1  1  1  1  1 
400  1  1  1  1  1  1 
500  1  1  1  0.999  1  1 
Correlated 0.9 Uniform(0,1) ± 0.1Beta(0.5,4.5), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.141  0.269  0.205  0.229  0.162  0.231 
50  0.235  0.444  0.33  0.321  0.257  0.367 
100  0.359  0.661  0.526  0.408  0.423  0.562 
200  0.543  0.879  0.709  0.516  0.577  0.747 
300  0.726  0.95  0.849  0.562  0.731  0.878 
400  0.81  0.977  0.902  0.632  0.81  0.928 
500  0.914  0.993  0.959  0.683  0.895  0.967 
Correlated 0.6 Uniform (0,1) ± 0.4Beta (0.5,4.5), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.138  0.26  0.199  0.238  0.169  0.212 
50  0.241  0.469  0.356  0.335  0.288  0.385 
100  0.347  0.662  0.506  0.402  0.408  0.55 
200  0.547  0.885  0.703  0.519  0.585  0.761 
300  0.694  0.949  0.852  0.592  0.727  0.88 
400  0.802  0.987  0.914  0.627  0.804  0.935 
500  0.882  0.991  0.948  0.641  0.871  0.959 
Correlated 0.9 Uniform(0,1) ± 0.1Beta(1,5), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.121  0.118  0.133  0.071  0.132  0.129 
50  0.194  0.21  0.214  0.064  0.217  0.211 
100  0.277  0.311  0.305  0.071  0.306  0.299 
200  0.449  0.502  0.486  0.068  0.485  0.477 
300  0.585  0.643  0.612  0.058  0.623  0.599 
400  0.66  0.72  0.697  0.075  0.696  0.682 
500  0.767  0.8  0.775  0.059  0.759  0.759 
Correlated 0.6 Uniform(0,1) ± 0.4Beta(1,5), $\rho =\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.594  0.591  0.632  0.124  0.637  0.616 
50  0.913  0.9  0.921  0.139  0.922  0.914 
100  0.996  0.992  0.992  0.111  0.995  0.99 
200  1  1  1  0.102  1  1 
300  1  1  1  0.115  1  1 
400  1  1  1  0.116  1  1 
500  1  1  1  0.095  1  1 
Correlated 0.9 Uniform(0,1) ± 0.1Beta(0.4,6), $\rho =\text{Uniform}\left(0.1\text{,}\phantom{\rule{0.12em}{0ex}}0.9\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.183  0.397  0.292  0.363  0.217  0.324 
50  0.291  0.639  0.459  0.545  0.34  0.508 
100  0.418  0.821  0.625  0.653  0.463  0.694 
200  0.669  0.958  0.855  0.78  0.68  0.881 
300  0.818  0.993  0.924  0.861  0.782  0.947 
400  0.919  0.997  0.97  0.908  0.89  0.985 
500  0.974  1  0.988  0.926  0.932  0.994 
Correlated 0.6 Uniform(0,1) ± 0.4Beta(0.4,6), $\rho =\text{Uniform}\left(0.1\text{,}\phantom{\rule{0.12em}{0ex}}0.9\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.815  0.964  0.928  0.825  0.836  0.945 
50  0.99  1  0.997  0.92  0.988  0.997 
100  1  1  1  0.971  1  1 
200  1  1  1  0.993  1  1 
300  1  1  1  0.997  1  1 
400  1  1  1  0.998  1  1 
500  1  1  1  1  1  1 
Correlated 0.9 Uniform(0,1) ± 0.1Beta(0.5,4.5), $\rho =\text{Uniform}\left(0.1\text{,}\phantom{\rule{0.12em}{0ex}}0.9\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.154  0.283  0.216  0.228  0.171  0.24 
50  0.272  0.465  0.352  0.331  0.296  0.391 
100  0.358  0.675  0.512  0.421  0.414  0.561 
200  0.528  0.871  0.706  0.526  0.578  0.749 
300  0.749  0.958  0.857  0.578  0.752  0.89 
400  0.857  0.986  0.924  0.632  0.841  0.938 
500  0.906  0.99  0.951  0.679  0.877  0.962 
Correlated 0.6 Uniform(0,1) ± 0.4Beta(0.5,4.5), $\rho =\text{Uniform}\left(0.1\text{,}\phantom{\rule{0.12em}{0ex}}0.9\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.137  0.282  0.2  0.228  0.159  0.234 
50  0.237  0.484  0.359  0.306  0.293  0.392 
100  0.345  0.679  0.503  0.411  0.416  0.544 
200  0.577  0.858  0.719  0.512  0.6  0.757 
300  0.725  0.946  0.848  0.584  0.743  0.873 
400  0.868  0.981  0.92  0.621  0.85  0.931 
500  0.9  0.989  0.947  0.676  0.869  0.959 
Correlated 0.9 Uniform(0,1) ± 0.1Beta(1,5), $\rho =\text{Uniform}\left(0.1\text{,}\phantom{\rule{0.12em}{0ex}}0.9\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.141  0.166  0.16  0.078  0.155  0.155 
50  0.213  0.224  0.235  0.054  0.251  0.232 
100  0.318  0.362  0.364  0.073  0.365  0.362 
200  0.477  0.525  0.498  0.047  0.503  0.485 
300  0.621  0.651  0.637  0.071  0.622  0.621 
400  0.718  0.747  0.721  0.071  0.708  0.707 
500  0.804  0.818  0.794  0.075  0.785  0.78 
Correlated 0.6 Uniform(0,1) ± 0.4Beta(1,5), $\rho =\text{Uniform}\left(0.1\text{,}\phantom{\rule{0.12em}{0ex}}0.9\right)$  
n  KS  Inverse chi  Inverse norm  Tippett  Wilcoxon  Logit 
20  0.694  0.656  0.705  0.109  0.721  0.685 
50  0.938  0.901  0.929  0.125  0.94  0.915 
100  0.994  0.989  0.991  0.114  0.995  0.99 
200  1  1  1  0.143  1  1 
300  1  0.999  1  0.113  1  1 
400  1  1  1  0.122  1  1 
500  1  1  1  0.119  1  1 
The global tests have been implemented in R. The R code is available at http://www.childrensmercy.org/Content/view.aspx?id=22812.
Discussion and conclusions
Multifactor Dimensionality Reduction (MDR) is a novel statistical method developed to characterize and detect nonlinear complex genegene interactions (epistasis) that could be associated with disease susceptibility. We suggest incorporating global test to filtration procedures to reveal a trend of gene interactive patterns when noisy genes are removed step by step using ReliefF or other filtration techniques. The optimal number of genes for further MDR analysis can be identified by pvalues of global testing. A real data applications and empirical assessment of our proposed methods reveal strong trends in global testing of pvalues and clear patterns of distribution of pvalues in three scenarios: 1) presence of GxG interactions, 2) absence of GxG interactions, 3) weak GxG interactions that needs filtration to remove noisy genes. The proposed global tests can serve as a screening approach before individual tests to prevent false discovery. Strong power in small sample sizes and well controlled Type I error in absence of GxG interactions makes these tests highly recommended in epistasis studies.
Global testing has not been implemented in MDR analyses in the literature we have reviewed. Currently, researchers rely on adjustment of individual pvalues such as false discovery rate (FDR) as suggested by [31]. Due to high dimensionality in genetic interactions, the FDR and other multiple testing adjustments often lose power in MDR analyses. Some MDR studies [27] have utilized the false positive report probability proposed by [32] but this method has been pointed out by [33] to be heuristic and wrong in formulation. In contrast, the global tests proposed by this paper are based on rigorous statistical theories and inferences.
Through extensive simulation on correlated pvalues, our study shows that the Inverse chi test is the most powerful approach to be incorporated with the filtration techniques to determine the optimal number of SNPs for the final MDR analysis. The KS test might have high inflation of Type I errors when pvalues are highly correlated or when pvalues peak near the center of histogram (Pattern 4). The Tippett’s test has very low power when the effect size of Pattern 2 is small.
We observe mild inflation of Type I error (<0.07) when pvalues are Uniform with a random correlation matrix. Our global tests are implemented for screening SNPs and investigators can continue to use multiplicity adjustment algorithms such as FDR to adjust individual pvalues in the final MDR analysis to prevent false discoveries. As a result, slight inflation in Type I error (<0.07) is acceptable in practice. Moreover, in our case study, we show that one can utilize the decreasing trend of global test results (Figure 4) to facilitate decision making. If global tests provided false discoveries, then the trend of global tests results would randomly fluctuate up and down. Figure 4 with a decreasing trend for global testing results as well Figure 3 with histograms systematically switching to Pattern 2 can also serve as diagnostic tools to prevent false discoveries or selection bias in global tests.
It is worthwhile to point out the proposed global tests can effectively prevent false discovery without losing the power to detect significant GxG interactions. To prevent the false discovery, current MDR applications typically choose one optimal model for each kway interaction. This method has two major drawbacks: firstly, the false positive discovery is not reduced by choosing one optimal model; secondly, choosing one optimal models may overlook other potential GxG interactions that also contributes to the disease susceptibility.
The major contribution of our manuscript is to incorporate global testing procedures to MDR framework. Our proposed global tests will provide pvalues to help practitioners determine the appropriate number of SNPs to be remained in the final analysis. The current filtration process does not provide pvalues. Therefore, using arbitrary cutoff value in the current process might lead to overfiltering or underfiltering of SNPs.
All 6 global tests are based on statistical inference instead of permutation. These 6 tests run very fast in a single computer. The major computational challenges are in the generation of pvalues for MDR through permutation tests but this is not the major focus of our work. Several works have been devoted to improve the efficiency and shorten the computing time in MDR analysis in highthroughput data. We will defer interested readers to the corresponding citations for computing issues in highthroughput MDR analysis. These computational limitations make our strategy appropriate in large scale candidate gene studies, but may be limited in application to genomewide association studies until further improvements in computing speed are realized or very largescale computing resources are available.
MDR permutation computing time is largely dependent on the dimension of data sets. In other words, the computing time increases as the number of SNPs and/or the number of subjects increases. Interestingly, the dimension of data does little impact on the computing time of global tests. The computing time for global tests of 1000 pvalues is very close to tests of 10 pvalues. Several filtration approaches have been proposed and some (ReliefF, SURF and TuRF etc.) have been implemented in the MDR software (http://www.epistasis.org). In this work, we utilize ReliefF for filtration. There have been other filtration techniques proposed in literature. For instance, [34] introduced entropybased information gain to search and evaluate interactions among risk factors. The current MDR software [8] provides ReliefF, entropy, chisquare test, etc. about 10 filtration methods. The global tests could be integrated in the workflow with other filtration techniques, although the comparison and evaluation of all filtration technique requires more research attention.
Appendix 1
Simulation of correlated pvalues
We generate correlated Beta variables using the method proposed by [30]. According to Bayesian theory, random variables from a Beta prior and a BetaBinomial conjugate function will yield correlated random deviates whose marginal distribution is also Beta. Firstly, randomly generate a variable K from $K~\text{Beta}\text{Binomial}\left(v\text{,}\phantom{\rule{0.12em}{0ex}}\alpha \text{,}\phantom{\rule{0.12em}{0ex}}\beta \right)$ where α and β are the shape parameters. Conditioning on $K=k$, generate P deviates from $\text{Beta}(\alpha +k\text{,}\phantom{\rule{0.12em}{0ex}}v+\beta 1)$. By integrating on K, the P deviates have unconditional marginal distribution as $\text{Beta}\left(\alpha \text{,}\phantom{\rule{0.12em}{0ex}}\beta \right)$ and the correlation coefficient among P deviates is $\rho =v/(v+\alpha +\beta )$. In this paper, we simulated different correlation coefficient with constant $\rho =0.2\text{,}\phantom{\rule{0.12em}{0ex}}08$ or $\rho $as a random variable from $\rho ~\text{Beta}\left(2\text{,}\phantom{\rule{0.12em}{0ex}}5\right)$ and $\rho ~\text{Uniform}\left(0.1\text{,}\phantom{\rule{0.12em}{0ex}}0.9\right)$ respectively.
In the above method, $\rho =v/(v+\alpha +\beta )$ can be written as $v=(\alpha +\beta )\rho /(1\rho )$ but algorithm to generate BetaBinomial with noninteger v is not widely available. As a result, we use an alternative method to generate correlated uniform distributions . In essence, correlated uniform variables, U, with a random correlation matrix Σ can be generated by transforming multivariate normal variables X using formula $U=F\left(X\right)$ where F is the CDF of the standard normal distribution. First, we generated a positive definite covariance matrix, Σ_{0} with randomly selected eigenvalues and randomly generated orthogonal matrix as eigenvectors(R clusterGeneration package). Let σ_{ij} be the component in Σ_{0}. We can convert the covariance matrix, Σ_{0} to correlation matrix Σ with components ${r}_{\mathit{ij}}=\frac{{\sigma}_{\mathit{ij}}}{\sqrt{{\sigma}_{\mathit{ii}}{\sigma}_{\mathit{jj}}}}$. To ensure the correlation is invariant to transformation, we need to adjust correlation matrix Σ into ${\sum}_{}^{\mathit{adj}}=2sin(\pi \sum /6)$. Perform Choleski factorization to generate $C={\left({\sum}^{\mathit{adj}}\right)}^{\frac{1}{2}}$. Generate a vector of i.i.d. standard normal variables, X _{0}. Let $X={X}_{0}\ast C$ and $U=F\left(X\right)$. As a result, the variables U are correlated uniform variables with correlation matrix Σ.
Declarations
Acknowledgements
This work is supported for collaboration between HD and AMR by Bursary Award of the 1^{st} Short Course on Statistical Genetics and Genomics from University of Alabama at Birmingham from the National Institute of Health R25GM093044 (PI: Tiwari). We are grateful to two reviewers whose comments have helped us improve the manuscript.
Authors’ Affiliations
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