Abstract
Correlated multiple testing is widely performed in genetic research, particularly in multilocus analyses of complex diseases. Failure to control appropriately for the effect of multiple testing will either result in a flood of falsepositive claims or in true hits being overlooked. Cheverud proposed the idea of adjusting correlated tests as if they were independent, according to an â€˜effective numberâ€™ (M_{eff}) of independent tests. However, our experience has indicated that Cheverud's estimate of the M_{eff} is overly large and will lead to excessively conservative results. We propose a more accurate estimate of the M_{eff}, and design M_{eff}based procedures to control the experimentwise significant level and the false discovery rate. In an evaluation, based on both real and simulated data, the M_{eff}based procedures were able to control the error rate accurately and consequently resulted in a power increase, especially in multilocus analyses. The results confirm that the M_{eff} is a useful concept in the errorrate control of correlated tests. With its efficiency and accuracy, the M_{eff} method provides an alternative to computationally intensive methods such as the permutation test.
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Introduction
Complex diseases such as diabetes, schizophrenia and cancer lay a heavy burden on public health, with frequencies over 1%. As they are caused by interactions among genes, an increasing number of studies (Hoh et al, 2001; Nelson et al, 2001; Ritchie et al, 2001; Cordell and Clayton, 2002) have employed multilocus methods to detect the causative genes. Despite the abundance of such studies, only 16â€“30% of the reports of significant associations have been consistently replicated (Page et al, 2003). One reason for the low replication rate is the failure in controlling for the effect of multiple testing (Lander and Kruglyak, 1995).
Multilocus methods search loci in combinations, which inevitably tests a great number of hypotheses simultaneously: a klocus analysis of n loci would have C_{n}^{k} locuscombinations to inspect. In addition to C_{n}^{k} being a large and cumbersome number, the tests are correlated with each other. For example, locuscombination AB is obviously correlated with combination BC for they share a mutual locus B. Even when loci are completely independent, their combinations are still correlated. These characteristics complicate the control of the error rate in multilocus analyses. Appropriate adjustment for multiple testing is very important to prevent a flood of falsepositive claims and to prevent true associations being overlooked.
A traditional criterion for multiple testing is the experimentwise significant level (EWSL, Î±_{e}), which is the probability that one or more hypotheses would be falsely rejected. To control the EWSL, the corresponding pointwise significant level (PWSL, Î±_{p}), which is the probability that an individual hypothesis would be falsely rejected, should be calculated appropriately. For independent tests, the Bonferroni (or Sidak) correction provides a simple and accurate control: Î±_{p}=1âˆ’(1âˆ’Î±_{e})^{1/M}â‰ˆÎ±_{e}/M, where M is the number of tests. However, this adjustment is overly conservative for correlated tests. Lander and Botstein (1989) and Lander and Kruglyak (1995) offered analytical adjustments for correlated tests in singlelocus analyses, but it is hard to apply them to multilocus analyses.
In practice, most researchers have therefore employed resamplingbased methods to control the EWSL of correlated tests. Hoh et al (2001) and Ritchie et al (2001) used the permutation test; Nelson et al (2001) used the crossvalidation test. Although resamplingbased methods are accurate, they demand intensive computation, which constrains them to cases with a limited number of tests. Taking the permutation test as an example, Churchill and Doerge (1994) recommended at least 1000 shuffles to estimate a 0.05 EWSL and at least 10â€‰000 shuffles to estimate a 0.01 EWSL.
An alternative criterion for multiple testing is the false discovery rate (FDR) (Benjamini and Hochberg, 1995), which is the expected proportion of falsely rejected hypotheses among all those rejected. When all hypotheses are true, the FDR is equal to the EWSL, but otherwise it is smaller. For independent tests, Benjamini and Hochberg (1995) and Benjamini and Liu (1999) proposed a stepup procedure and a stepdown procedure to control the FDR. For correlated tests, Benjamini and Yekutieli (2001) proposed a simple but highly conservative procedure. Storey (2002) and Storey and Tibshirani (2003) proposed a variation of the FDR: the positive false discovery rate (pFDR), which is the conditional FDR given that at least one hypothesis is rejected. These authors gave direct estimates of the pFDR for independent tests, and suggested resamplingbased methods such as the permutation test for correlated tests. Correlation among tests is still a problem for FDR methods.
In 2001, Cheverud proposed a new idea to adjust correlated tests as though they were independent according to an effective number (M_{eff}) of independent tests. He also gave an estimation of the M_{eff} for singlelocus analyses. In Nyholt (2004)'s evaluation, the M_{eff} method approximated the permutation test accurately, and was more than 100 times faster. It is highly desirable to apply this method to control the EWSL and FDR in multilocus analyses. However, our experience suggests that Cheverud's (2001) estimate of M_{eff} is overly large, especially when there are a substantial number of moderately correlated tests. This situation is, unfortunately, typical of multilocus analyses, leading to overly conservative results.
In this study, we propose a more accurate estimate of M_{eff}, and design M_{eff}based procedures to control the EWSL and FDR. An assessment based on both real and simulated data shows that the procedures can control the error rate accurately in multilocus analyses.
Approach
The M_{eff} method
When loci are in linkage disequilibrium (LD), hypothesis tests in singlelocus analyses are not independent. Cheverud (2001) proposed the method below to adjust the correlated tests:
Step 1: Calculate the correlation matrix for the loci.
Step2: Estimate the effective number (M_{eff}) of independent tests from the eigenvalues of the correlation matrix with Equation (1), where M is the number of tests (equal to the number of loci in singlelocus analyses) and Î»_{i} (i=1,â€¦, M) are the eigenvalues
Step 3: Adjust the test criteria as though there were M_{eff} independent tests with the Sidak (1967) correction below
Step 4: Test the association between genotypes and phenotypes locus by locus. If the pvalue of any test is lower than Î±_{p}, the nonassociation hypothesis is rejected.
A new M_{eff} equation
The key step of the M_{eff} method is Step 2, estimating the M_{eff} from the eigenvalues. As Cheverud (2001) explained, Equation (1) considers two extreme situations. When the tests are completely independent, the eigenvalues are all equal to 1 and Equation (1) gives M_{eff}=M. When the tests are completely identical, the first eigenvalue is M, while all the others are zero, and Equation (1) gives M_{eff}=1. Although Equation (1) gives correct estimates at the two extremes, it overestimates the M_{eff} in other situations. If the M tests are composed of c (1â©½câ©½M) copies of M/c independent tests, the eigenvalues are Sequence (3)
According to Equation (1), M_{eff}=M+1â€“c, but actually there are only M/c independent tests. The ratio r is
In multilocus analyses, M is large and c is moderate, so r tends to be considerably larger than 1.
Intuitively, c identical tests will result in an eigenvalue Î»_{i}=c, while a test partially correlated with others will result in an eigenvalue 0<Î»_{i}<1. Thus, an eigenvalue can be decomposed into two parts: the integral part and the nonintegral part. The integral part represents identical tests that should be counted as one in the M_{eff}. The nonintegral part represents a partially correlated test that should be counted as a fractional number between 0 and 1. From this intuitive consideration, we propose Equation (5) to estimate the M_{eff}:
where I(xâ©¾1) is the indicator function, which gives 1 when xâ©¾1 and gives 0 otherwise, and is the floor function, which gives the largest integer less than or equal to x. Thus, completely correlated tests will be counted as I(xâ©¾1) and partially correlated tests will be counted as . Equation (5) gives the right M_{eff} for integral eigenvalue Sequence (6), where t is the true number of independent tests and c_{1}, c_{2},â€¦, c_{t} are integers
Multilocus analyses
By regarding locuscombinations as â€˜generalized lociâ€™ (GLs), we can apply the M_{eff} method to multilocus analyses. For example, if two loci have g_{1} and g_{2} genotypes, respectively, their combination can be regarded as a new â€˜locusâ€™ with g_{1}g_{2} genotypes. Consequently, the calculation of the correlation coefficients in Step 1 needs adaptation. As GLs are multinomial variables, neither the Pearson correlation (used by Cheverud, 2001) nor the LD measure (used by Nyholt, 2004) is suitable.
Similar to the Pearson correlation for numerical variables, the correlation for multinomial variables can be defined as in Equation (7a), but the variance (Var) and the covariance (Cov) should be measured differently. Let X_{1} and X_{2} denote two GLs, that is, two multinomial variables. Their variance and covariance can be measured with the entropy and mutual information (Cover and Thomas, 1991), respectively, as in Equation (7b) and (7c).
where P(X_{i}) is the probability function of X_{i} and E[x] is the expectation of a random variable x.
Besides its compatibility with both single and multilocus analysis, this correlation coefficient has a property:
where A, B and C are independent loci or GLs.
An M_{eff}based FDR procedure
Let R denote the number of rejected hypotheses and V the number of those falsely rejected. Benjamini and Hochberg (1995) defined the FDR as
where I(R>0) is the indicator function, which gives 1 when R>0 and gives 0 otherwise. The FDR is equal to the EWSL when all null hypotheses are true and is smaller otherwise. They also proposed the following stepup procedure to control the FDR for independent tests. Let p_{(1)}â©½â‹¯â©½p_{(M)} be the ordered pvalues of M independent hypotheses H_{(1)},â€¦,â€‰H_{(M)}. To control the FDR at level q (usually 5%), hypotheses H_{(1)},â€¦,â€‰H_{(k)} should be rejected, where
The ordered pvalues are compared with an arithmetic sequence from q/M to q. The start of the sequence, q/M, ensures that the FDR is equal to the EWSL when all the hypotheses are true. The end of the sequence, q, ensures that the FDR is controlled at q when all the hypotheses are rejected. To involve the effect of correlation, we can replace the start q/M with q/M_{eff}, keep the end as q, and consequently replace the step q/M with (qâˆ’q/M_{eff})/(Mâˆ’1). The M_{eff}based stepup procedure is to reject H_{(1)},â€¦, H_{(k)}, where
Performance
Singlelocus analyses
The M_{eff} method has previously been applied to two singlelocus analyses. Cheverud (2001) applied it to the LG/J and SM/J mouse data (Cheverud et al, 2001), and Nyholt (2004) applied it to the angiotensinIconverting enzyme (ACE) gene data (Keavney et al, 1998). In these analyses, Cheverud (2001) and Nyholt (2004) derived two eigenvalue sequences from the two data sets, respectively. In this study, we calculated different M_{eff}'s with both Equations (1) and (5) from the two eigenvalue sequences, and compared the results. As shown in Table 1, the two equations gave similar M_{eff}'s, so they should have similar performance on the two data sets. Setting Î±_{e}=5% for the Keavney data set, Nyholt (2004) obtained Î±_{p}=0.015 from a permutation test with 50â€‰000 shuffles, and obtained Î±_{p}=0.011 with the M_{eff}=4.59 from Equation (1). Using M_{eff}=4.01 from Equation (5), we obtained Î±_{p}=0.013, a value closer to the permutation test's 0.015 than the value of 0.011 obtained from Equation (1).
Multilocus analyses
To compare the performance of Equations (1) and (5) in multilocus analyses, we used four subdata sets extracted from a schizophrenia data set (Xu et al, 2004). The schizophrenia data set concerns 12 genes of the dopamine pathway, COMT (7), ALDH3 (11), ADH (14), TYR (1), TPO (2), MAO (3), DAO (2), AOX1 (2), DDC (4), HDG (1), DBH (2) and GSTZ1 (1), with 83 cases and 108 controls. (The integers in the parentheses are the numbers of SNPs genotyped for each gene.) To investigate the performance on correlated loci, we extracted the SNPs of gene COMT, ALDH3 and ADH as three subdata sets COMT, ALDH3 and ADH, respectively. The three genes were reported as potential effective SNPs by Xu et al (2004), and they have enough SNPs (more than five) to ensure the multiplicity of tests in the analyses. To investigate the performance on independent loci, we selected one SNP per gene as the fourth subdata set and called it DOP (short for dopamine). For each data set, we performed single, two and threelocus analyses.
We used the permutation test as the golden standard to evaluate the performance. In each permutation shuffle, we applied the M_{eff} method to control the EWSL (Î±_{e}). As a result, we obtained a decision of whether or not to reject the nonassociation hypothesis. After repeating this procedure 3000 times, we counted the proportion of rejection (P_{r}). Ideally, P_{r} should be equal to the preset Î±_{e}. The association between a GL and the phenotype was tested with a combination of the Ï‡^{2} test and the Fisher's exact test (When Cochran (1954)'s condition was satisfied, the association was tested with the Ï‡^{2} test. When the condition was not satisfied, it was tested with the Fisher's exact test. Cochran's condition: in a contingency table, every cell has an expected value no less than 1, and 80% of the cells have expected values of no less than 5. This algorithm was implemented in ACM algorithm 643 (Clarkson et al, 1993).)
Figure 1 shows the comparison between P_{r} and Î±_{e}. In singlelocus analyses, both Equations (1) and (5) controlled P_{r} at Î±_{e}. In two and threelocus analyses, the M_{eff}'s obtained from Equation (1) were much greater than those obtained from the permutation test (see Table 2), and consequently the P_{r} of Equation (1) was much lower than Î±_{e}. In contrast, the P_{r} of Equation (5) approximated Î±_{e} fairly well in most cases.
Effects on FDR
We performed twolocus analyses on simulated caseâ€“control data to evaluate the effects of the M_{eff} on the FDR. The simulated data contains 10 genomic regions with five observed biallelic SNPs in each region. All SNPs are in the Hardyâ€“Weinberg equilibrium and the minor alleles have the same frequency. SNPs in different regions are in linkage equilibrium, and SNPs in the same region are in LD with r^{2}=0.8 (Hill and Robertson, 1968) between two successive SNPs. The disease is caused by the epistatic interaction of two unobserved loci, which hide in the first region and the second region, respectively. The LD (r^{2}) between a causative loci and its nearest marker SNP is 0.8. As neither of the causative loci has a marginal effect, a twolocus analysis is necessary to detect the epistatic interaction. Consequently, there are C_{50}^{2}=1225 SNP pairs to inspect, and only the 25 pairs composed of the loci from the first and second regions are associated with the disease. The data were simulated from the six epistasis models in Table 3 (Ritchie et al, 2003). For each model, 10â€‰000 data sets were generated with 200 cases and 200 controls in each of them.
Both Benjamini and Hochberg (1995)'s direct stepup procedure and the new M_{eff}based stepup procedure were applied to control the FDR. Although Benjamini and Yekutieli (2001) designed an FDR procedure for correlated multiple testing, we did not employ it because it is much more conservative than the direct stepup procedure. The M_{eff} was estimated with Equation (5). As shown in Figure 2, the M_{eff}based stepup procedure controlled the FDR more accurately than the direct method, and was considerably more powerful.
Discussion and conclusion
In 2001, Cheverud proposed the idea of adjusting correlated tests as if they were independent according to the effective number (M_{eff}) of independent tests. Piepho (2001) has also proposed a quick method for computing approximate threshold levels that control the genomewise type I error rate of tests for quantitative trait locus detection in interval mapping and composite interval mapping. In the multilocus analyses presented in this paper, our extended concept of M_{eff} succeeded in controlling both the EWSL and the FDR. This result supports the use of M_{eff} in errorrate control of correlated multiple testing. As a datadriven algorithm based on a correlation matrix, the M_{eff} method can be applied to a wide range of problems. Nowadays, dense and linked markers are widely used in genetic study, and it is hard to model their genetic behavior precisely, especially in multilocus analyses. The diversity of populations also prohibits models obtained from one population from being applied to another. Avoiding the error of an imprecise model, the datadriven M_{eff} method is suitable for different situations. In addition to the datadriven property, the M_{eff} can be calculated for various types of data by generalizing the definition of correlation coefficient. With Equation (7), it can be applied to categorical data. With the Pearson correlation, it can be applied to numerical data such as microarray data. With King and Chinchilli (2001)'s generalized correlation coefficient, it can be applied to a mixture of numerical and categorical data.
Cheverud (2001) proposed Equation (1) to estimate the M_{eff}. As he explained, Equation (1) considers two extreme situations: hypothesis tests are completely independent or they are completely identical. However, Equation (4) shows that Equation (1) tends to overestimate the M_{eff} when there are a large number of moderately correlated tests. Unfortunately, in multilocus analyses, this situation is usually encountered, and Equation (1) will lead to conservative reports. In this paper, we propose Equation (5) to estimate the M_{eff} more accurately. It considers a more general situation than the two extremes Equation (1) concerns, where tests are composed of copies of independent tests. Therefore, it outperforms Equation (1) in multilocus analyses. Figure 1 clearly shows that both Equations (1) and (5) perform excellently in singlelocus analyses, but only the latter is effective in multilocus analyses. The higher performance of Equation (5) over Equation (1) is not because it better fits the correlation coefficient defined by Equation (7). As Equation (4) has shown, no matter what correlation coefficient is employed to calculate the correlation matrix, Equation (1) will overestimate the M_{eff} for eigenvalue Sequence (3), while Equation (5) will give an appropriate estimate. We also used the Pearson correlation in singlelocus analyses, but as shown in Table 2, its performance was similar to that of Equation (7).
If n tests need adjustment, the computation complexity of a permutation test is O(stn), where s is the number of shuffles and t is the time needed to test a hypothesis. The computation cost for the M_{eff} method concentrates on eigenvalues' calculation whose complexity is O(mn^{3}), where m is the time needed to multiply two real numbers. Theoretically, when n is large, the permutation test is more efficient than the M_{eff} method. However, because m is tiny in comparison with st, the M_{eff} method is much faster than the permutation test until n is very large. The largest n we encountered in this study was 1225 in Section Effects on FDR. It took about 10â€‰000â€‰s to repeat the calculation of the 1225 pvalues 10â€‰000 times, and took about 3â€‰s to solve the eigenvalues with the numeric package Lapack. (The computation time was measured on a 2.8â€‰GHz Pentium with Linux 2.6.10.) When there are about 1000 tests, the M_{eff} method is at least 1000 times faster than the permutation test. This would meet the need of a genome scan with about 1000 SNPs per chromosome. As SNPs in different chromosomes are in linkage equilibrium, the correlation matrix of a singlelocus analysis is diagonally blocked. We can calculate the M_{eff} for each chromosome individually and then sum them up as the M_{eff} for the scan.
Freed from the heavy computation burden, statisticians can realize many sophisticated multilocus methods. For example, we can split a classification tree's nodes according to the EWSL or the FDR rather than the PWSL. In a tree's growth procedure, a node splits when the pvalue of the optimal variable exceeds a threshold. The more the variables are, the easier it is for the optimal variable to exceed the threshold, and the more easily the tree overfits. Zhang and Bonney (2000) considered this problem in splitting, but they arbitrarily set the PWSL<0.001 as the threshold. With the M_{eff} method, the threshold can be estimated efficiently. If the threshold is calculated with the permutation test, the computation will be very demanding because every node needs thousands of shuffles.
Multiple testing is not only important in data analysis but also in experiment design. Neglecting the issue would lead to an underpowered design, while overemphasizing it would waste resources. Owing to the expense of genetic experiments, it is highly desirable to apply the M_{eff} method to experiment designs. By calculating the M_{eff} from a pilot study, we can estimate the appropriate PWSL and the sample size for an experiment involving multiple testing.
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Acknowledgements
We are grateful to Qi Xu et al for their generous sharing of the published schizophrenia data. We thank Ying Huang, Yulong Zhang and the journal reviewers for their helpful comments. This work was supported by the Ministry of Science and Technology of People's Republic of China under Contract No. 2003CB517106.
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Li, J., Ji, L. Adjusting multiple testing in multilocus analyses using the eigenvalues of a correlation matrix. Heredity 95, 221â€“227 (2005). https://doi.org/10.1038/sj.hdy.6800717
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DOI: https://doi.org/10.1038/sj.hdy.6800717
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