Abstract
Current genome-wide association studies (GWAS) focusing on relatively common single-nucleotide polymorphisms (SNPs) usually adopt a cost-effective multi-staged design in which a proportion of the total samples are genotyped using a commercial SNP array with a reasonably good coverage of the whole genome at the initial stage, and a list of promising SNPs are further genotyped and evaluated on the remaining samples at the second stage. This staged design in principal can also be used for the study of rare genetic variants at the genome-wide scale, but the statistical methods developed for evaluating the relatively common SNPs under the staged design are not appropriate for rare variants due to the invalidity of large sample theorems. Here, we develop a new statistical framework that aims to evaluate rare variants under two-staged (or multi-staged) design. By extensive computer simulations, we evaluate the empirical type I error rate and power of the proposed procedures. A real example from two recent case–control rheumatoid arthritis genetic association studies is also used to demonstrate the performances of the proposed methods.
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Acknowledgements
We would like to thank two anonymous referees for their insightful comments. This work was partially supported by the National Science Foundation of China (10901155, 61134013 to Q.L.).
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Appendix
Appendix
A.1 Lemma
Suppose that X follows a binomial distribution with parameters n and θ. Then for any given α∈(0,1),
Proof. For 
define
Let 
. Define 
.
According to Lemma 1 of Li, Li, & Xiong (2010),28 
So, V(kα,1)=α−o(1), and v(kα,2)=α−o(1). We now consider two cases:
(1) When 
,
(2) When 
, in the similar way, we have the results.
A.2 Derivation of the Threshold for the Joint Testing Statistic
To controll the type I error rate of the joint analysis, we have
Based on the Lemma above, when min(r,s) →∞ and r/s 
, we can get 
means ‘convergence in distribution’ and U(0,1) is the standard uniform distribution. Now, we calculate the threshold c from the above equation under the following two scenarios:
(1) When 
. Then X and Y are independent with the probability density functions 
, respectively. So the joint probability density function of (X,Y)′ is
Then, we calculate
Therefore g(c) is a strictly increasing function on the interval 
. Then the equation g(c)=α has a unique solution of c on 
because of g(−∞)=0 and 
. So, we can use the Bi-section Method to get c.
(2) When π=0.5,
As 
, no solution exists.
And when 
, we have
so the threshold c is the solution of the equation 
. Let 
. Note that 
. So, h(c) is strictly increasing as 
. As 
, h(−∞)=0, 
has a unique solution on 
.
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Li, Q., Pan, D., Yue, W. et al. Evaluating rare variants under two-stage design. J Hum Genet 57, 352–357 (2012). https://doi.org/10.1038/jhg.2012.33
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DOI: https://doi.org/10.1038/jhg.2012.33
