Abstract
Modern association studies often involve a large number of markers and hence may encounter the problem of testing multiple hypotheses. Traditional procedures are usually over-conservative and with low power to detect mild genetic effects. From the design perspective, we propose a two-stage selection procedure to address this concern. Our main principle is to reduce the total number of tests by removing clearly unassociated markers in the first-stage test. Next, conditional on the findings of the first stage, which uses a less stringent nominal level, a more conservative test is conducted in the second stage using the augmented data and the data from the first stage. Previous studies have suggested using independent samples to avoid inflated errors. However, we found that, after accounting for the dependence between these two samples, the true discovery rate increases substantially. In addition, the cost of genotyping can be greatly reduced via this approach. Results from a study of hypertriglyceridemia and simulations suggest the two-stage method has a higher overall true positive rate (TPR) with a controlled overall false positive rate (FPR) when compared with single-stage approaches. We also report the analytical form of its overall FPR, which may be useful in guiding study design to achieve a high TPR while retaining the desired FPR.
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Appendices
Appendix 1
TPR is defined as
Similarly, by the same argument for conditional probability, the overall FPR is
Define T(X) as the test statistic used in the first stage where X denotes the data from MN1 genotypings, and T(X*) for the second stage where X* contains the information based on the R(N1+N2) genotypings. The test statistic can be the difference in allele frequency between the case and control groups, or a chi-square statistic derived from contingency tables. These two statistics are correlated because partial data are used in both X and X*. Without loss of generality, we assume here that both statistics converge asymptotically to normality, providing N1 is large, and that the allele frequencies obtained from N1 and N1+N2 are roughly equal. Let ρ denote the correlation between T(X) and T(X*), the overall FPR at α1 and α2 is then:
where \(\sum = \left(\begin{array}{*{20}c} 1 & \rho\\ \rho & 1\\ \end{array}\right)\) and \(\sum\nolimits ^{{-1/2}} = \left(\begin{array}{*{20}c} s & t\\ t & s\\ \end{array}\right)\) whose elements depend on ρ. Applying their asymptotic normality with zero mean vector and the covariance matrix ∑, the above becomes
where \(a=z_{1-\alpha_{1}/2}\) and \(b=z_{1-\alpha_{2}/2}=z_{1-\alpha_{1}/2R}\) are the percentiles of standard normal, and Z1 and Z2 are independent standard normal distributions. This FPR will be bounded closely from above by \( \Pr (Z_{2}\ge t \cdot a + s \cdot b)\) if ρ is large. This boundary is determined by α1, α2, R, and ρ, and is smaller than 0.001 when R ≥10; while a quick guess for R would be M α1. If ρ=0, this boundary becomes Pr(Z2≥ b), which is simply α2.
To investigate the effect of sample size, note that T(X) is close to \(\sqrt{N_{1}/(N_{1}+N_{2})}\times T({\mathbf{X}}^{*})\) if the average allele frequencies in X and X* are similar, therefore,
where Φ is the cumulative density function of the standard normal. If the proportion N1/(N1+N2) is larger than \((z_{1-{\alpha_{1}}/2}/z_{1-{\alpha_{2}}/2})^{2}\), the overall FPR is approximately α2. Otherwise, it becomes \(2 \times \left[ 1-\Phi \left( \sqrt{N/N_{1}}\times z_{1-\alpha_{1}/2} \right) \right]\). Similarly, the TPR at fixed α1 and α2 is approximately (1−β2).
To derive the expected number of success and error counts, it suffices to calculate the path probability Q2 in the second stage. Conditional on α2=α1/R, its upper bound becomes α2/α1, and is approximately 1/[E(R0)+E(Ra)]. Similarly, the conditional probability U2, at fixed α1 and α2, is approximately (1−β2)/(1−β1).
Appendix 2
Let pc and pn represent the allele frequencies for the case and control groups, respectively, and δ their absolute difference. Let \(\bar{p}\) be their average, weighted by corresponding sample sizes Nc and Nn. Following the assumption that the allele frequency ranges from 10% to 90%, it can be derived that
Therefore, the relationship between \(\bar{p}\) and δ can be formulated. In the general form, when p0 stands for the detectable minimal allele frequency, the above equation becomes
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Wen, SH., Tzeng, JY., Kao, JT. et al. A two-stage design for multiple testing in large-scale association studies. J Hum Genet 51, 523–532 (2006). https://doi.org/10.1007/s10038-006-0393-6
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DOI: https://doi.org/10.1007/s10038-006-0393-6
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