Abstract
There is much recent interest in finding rare genetic variants associated with various diseases. Owing to the scarcity of rare mutations, single-variant analyses often lack power. To enable pooling of information across variants, we use a random effect formulation within a retrospective modeling framework that respects the retrospective data collecting mechanism of case–control studies. More concretely, we model the control allele frequencies of the variants as random effects, and the systematic differences between the case and control frequencies as fixed effects, resulting in a mixed model. The use of Poisson approximation and gamma-distributed random effects results in a generalized negative binomial distribution for the joint distribution of the control and case frequencies. Variants are selected by conducting stepwise likelihood ratio tests. The superiority of the proposed method over two existing variant selection methods is demonstrated in a simulation study. The effects of non-gamma random effects and correlated variants are also found to be not too detrimental in the simulation study. When the proposed procedure is applied to identify rare variants associated with obesity, it identifies one additional variant not picked up by existing methods.
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We would like to thank the referees for their helpful comments and suggestions. The research of the third author was supported by the National Science Foundation of China Grant 11271346.
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Appendix
Appendix First and second derivatives of the log-likelihood function
It can be assumed without loss of generality that δj=0 for J⩾j>k, hence the model involves only the parameters v, μ and δ1,…,δk. The log-likelihood function can be written as l=l1+l2, where
and
The first and second derivatives of l1 can be obtained readily by symbolic differentiation. Note in particular that ∂2l1/∂δi∂δj=0 for i≠j. The only thing left to do is to find the derivatives of l2. As l2 is a function of v only, the only non-zero derivatives are
and
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Kuk, A., Nott, D. & Yang, Y. A stepwise likelihood ratio test procedure for rare variant selection in case–control studies. J Hum Genet 59, 198–205 (2014). https://doi.org/10.1038/jhg.2014.1
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DOI: https://doi.org/10.1038/jhg.2014.1
