Abstract
Multiple testing occurs commonly in genome-wide association studies with dense SNPs map. With numerous SNPs, not only the genotyping cost and time increase dramatically, many family wise error rate (FWER) controlling methods may fail for being too conservative and of less power when detecting SNPs associated with disease is of interest. Recently, several powerful two-stage strategies for multiple testing have received great attention. In this paper, we propose a grid-search algorithm for an optimal design of sample size allocation for these two-stage procedures. Two types of constraints are considered, one is the fixed overall cost and the other is the limited sample size. With the proposed optimal allocation of sample size, bearable false-positive results and larger power can be achieved to meet the limitations. The simulations indicate, as a general rule, allocating at least 80% of the total cost in stage one provides maximum power, as opposed to other methods. If per-genotyping cost in stage two differs from that in stage one, downward proportion of the total cost in earlier stage maintains good power. For limited total sample size, evaluating all the markers on 55% of the subjects in the first stage provides the maximum power while the cost reduction is approximately 43%.
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Appendix
Appendix
Following the work of Wen et al. (2006), the overall FPR, probability of claiming unassociated SNPs significant in both stages, is approximated by \(\hat{\alpha}_{2} (N_{1})\) below,
where the expected value of R, E(R), can be estimated by Mwα1 + M(1−w)(1−β1(N 1)), with \(1 - \beta_{1} (N_{1}) = \Phi\left(\frac{{{\sqrt {N_{1}}}\delta - \sigma_{0} z_{{\alpha_{1} /2}}}}{{\sigma_{1}}}\right)\) as the power in the first stage, Φ is the cumulative density function of standard normal distribution, and z α1/2 represents the 100(1−α1 /2)-th quartile of the standard normal distribution. Similarly, the other index \({\hbox{TPR}} \cong 1 - \beta_{2} = \Phi\left(\frac{{{\sqrt {N_{1} + N_{2}}}\delta - \sigma_{0} z_{{\hat{\alpha}_{2} (N_{1})/2}}}}{{\sigma_{1}}}\right),\) the probability of declaring truly associated SNPs as significant in both stages, is approximately (1−β2) with respect to \(\hat{\alpha}_{2} (N_{1})\) and N 1+N 2. Where σ 0 and σ1 denote the standard deviation of difference in mean allele frequencies under the null and alternative hypothesis, respectively.
Considering fixed total genotyping cost T=MN 1+RN 2 with the same per-genotyping cost in both stages and fixed M and w, the optimal design is to allocate N 1 and N 2 efficiently to achieve maximum TPR. For simplicity, let N 2 = kN 1 and R can be replaced with E(R). Therefore, T can be rewritten as T=MN 1+(Mwα1+M(1−w)(1−β1(N 1)))kN 1. In other words, k=(T−N 1M)/(N 1E(R)) is a function of N 1. The goal is to find the best value of (N 1, k) such that the two-stage method has maximum power. The TPR is defined as
where \(\hat{\alpha}_{2} (N_{1}) = \min(0.05/E(R),\,2 \times [1 - \Phi ({\sqrt {1 + k}} \times z_{{\alpha_{1} /2}})]),\) and Φ, z α_1/2, σ0 and σ1 are defined as previous.
It is not straightforward to derive the analytical form of k, but the maximum TPR can be searched through the range of N 1. The upper bound for N 1 is T/M, which implies all resources are allocated in the first stage. By setting the power in the first stage larger than 0.8, we can obtain a reasonable range for N 1 based on N 1(0) and T/M, where N 1(0) satisfies the equation \(1 - \beta_{1} (N_{{1(0)}}) = \Phi\left(\frac{{{\sqrt {N_{{1(0)}}}}\delta - \sigma_{0} z_{{\alpha_{1} /2}}}}{{\sigma_{1}}}\right) = 0.8.\) Given T, M, w, allele frequency \(\bar{p}\) and the effect size δ, we perform a grid search of (N 1, k) for the optimal design.
When the total number of participants (=N) is limited and when N 1=π N, the TPR is defined as \({\hbox{TPR}} = 1 - \beta_{2} (N_{1}) = \Phi\left(\frac{{{\sqrt N}\delta - \sigma_{0} z_{{\hat{\alpha}_{2} (N_{1})/2}}}}{{\sigma_{1}}}\right) \leq 1 - \beta_{1} (N_{1}),\) where \(\hat{\alpha}_{2} (N_{1}) = \min (0.05/E(R),\,2 \times [1 - \Phi (z_{{\alpha_{1} /2}} /{\sqrt \pi})]).\) Given N, M, and w, the TPR is only affected by \(\hat{\alpha}_{2} (N_{1})\) and is smaller or equal to 1−β1(N 1). If π is smaller, the TPR is bounded from above by 1−β1(N 1), which is small as well. On the other hand, if π is large, the TPR is likely to be large, but the cost increases dramatically. Hence, one needs to strike a balance between genotyping cost and the TPR. Denoting the cost as
where T(π) is also a function of π given N, M, and w. Similarly, a grid search of π can be set up to find the maximum TPR and affordable cost analytically.
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Wen, S.H., Hsiao, C.K. A grid-search algorithm for optimal allocation of sample size in two-stage association studies. J Hum Genet 52, 650–658 (2007). https://doi.org/10.1007/s10038-007-0159-9
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DOI: https://doi.org/10.1007/s10038-007-0159-9
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