Abstract
The cost of large-scale association studies may be reduced substantially by analysis of pooled DNA from multiple individuals. Here we examine the optimal symmetric and asymmetric designs for pooling experiments for quantitative traits under a range of assumptions about the underlying genetic model and the sources of experimental errors in allele frequency estimation. The results indicate that, in the absence of experimental errors and for common alleles with additive effects, a symmetric pooling scheme comparing the top 27% with the bottom 27% of the trait distribution is optimal, extracting 80% the total information available. A symmetric design is not optimal for rare or recessive alleles, which require asymmetric (or other) pooling strategies. Allele frequency measurement errors reduce the optimal pooling fraction as well as the overall efficiency of the pooling design. In contrast, random variation in the amount of DNA contributed by individuals to a pool reduces only the overall efficiency of the pooling design. Our results emphasize the importance of minimising experimental errors and suggest a pooling fraction of around 20%.
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Acknowledgements
We would like to thank Jing Hua Zhao for helpful comments. This research was supported in part by a UK MRC research studentship to A Jawaid, UK MRC component grant G9700821, Wellcome Trust grant 055379, and National Institutes of Health grant EY-12562.
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Appendices
Appendix A
Variance due to unequal contribution of DNA samples
Restricting the terminology to this appendix, let Xi represent the total number of alleles contributed by individual i in a pool made up of n individuals, with Xi ∼ N(μ,τ2μ2). Let Yi represent the number of A1 alleles contributed by individual i. For genotype A1A1 with population frequency p2, Yi=Xi; for A1A2 with frequency p1, Yi ∼Bin(Xi,1/2); and for A2A2 with frequency p0, Yi=0. The population frequency of allele A1 is p=p1/2+p2, and the frequency of allele A1 in the pool is
being the approximate variance for a quotient of correlated random variables.21 The required terms are
after simplification. Assuming Hardy-Weinberg equilibrium and large μ, this reduces to
Appendix B
Optimal symmetric design in thepresence of experimental error
Let G be the proportion of A1 alleles in a genotype, so that G = 0, 1/2 or 1, and Var(G) = pq/2. According to an additive genetic model, the expected value of the trait X given G is 
. Using the implied covariance, Cov(X,G)=pqa, and a linear approximation, the expected value of G given X 
. In the lower pool, E(X)≈−Φ(Φ−1(f))/f, where Φ is the standard normal density function, Φ−1 is the inverse standard normal distribution function, and f is the lower pooling fraction.22 The expected values of G is in the lower pool and, by symmetry, the upper pool are therefore
from before. The NCP is therefore
Appendix C
Analytical approximation for theoptimal symmetric design in the presence ofexperimental error
The design is optimised by maximising the value of the NCP, which is equivalent to maximising the value of y2/(f+f2κ2), where y = Φ(z) and f = Φ(z) for normal deviate z. Taking the derivative with respect to z and multiplying by non-zero terms yields
as the equation specifying the minimum. When κ = 0, the solution to this equation occurs at z0 = −0.61, with f0 = 0.27 and y0 = 0.33 (Bader et al.12). For small κ, we write z = z0 + δ. To lowest order in δ, the above equation is
When κ is large, we use the asymptotic expansion f = −(y/z) + (y/z3), and the equation specifying the optimum reduces to −2yκ2/z3 = 1. Taking the natural logarithm of both sides and equating exponents,
Writing x = − + δ yields δ = (1/8)−(1/4)ln[(2/π)1/2κ2] to lowest order in δ. The result of this perturbation theory expansion for large κ is
An appropriate crossover between the small-κ formula and the large-κ formula is κ = 1.
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Jawaid, A., Bader, J., Purcell, S. et al. Optimal selection strategies for QTL mapping using pooled DNA samples. Eur J Hum Genet 10, 125–132 (2002). https://doi.org/10.1038/sj.ejhg.5200771
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DOI: https://doi.org/10.1038/sj.ejhg.5200771
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