Abstract
This chapter also (as does Chap. 5) discusses an extension of association analyses to include a larger set of hypotheses beyond just the single markers that have been genotyped in a particular study. Imputed SNP analysis is in certain respects identical to haplotype analysis since the imputed SNPs are on specific haplotypes or haplotype combinations. Testing imputed SNPs as well as genotyped SNPs is thus simply a more focused kind of haplotype analysis and serves to extend the set of hypotheses that are tested to encompass known but ungenotyped variants. Imputed SNP analysis plays a special role during a post-GWAS phase when during meta-analysis many studies are combined in efforts to find associations that are too small to be detectable in any one study. Imputation is necessarily relied upon when (as is generally the case) not all studies used the same genotyping platform or chip version.
This chapter discusses the basic statistical method, namely, Hidden Markov Model (HMM) that is used for fast and very large-scale SNP imputation in a number of high-performance programs. A brief introduction to the HMM methods is provided and the basic principles behind estimating the parameters of an HMM are illustrated with R code. The basics of a particular algorithm, patterned loosely after that implemented in the program MACH, are described.
Since nearly all large-scale SNP imputation methods require that phased haplotypes be provided for SNPs to be imputed (or measured for the purpose of imputation), a discussion of the use of phasing algorithms is also provided, with the details also modeled after the MACH program.
The use of imputed SNPs as independent variables in regression analysis is introduced with the discussion mostly focused on the same approach (expectation substitution) used in haplotype analysis. Use of imputed SNPs in association analyses for a single study is described, although the use of imputed SNPs in meta-analysis is deferred until Chap. 8.
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Notes
- 1.
Actually the EM algorithm generally will only find a local maximum; running the EM repeatedly using multiple starting values for the parameters in λ is recommended for assessment of whether a global maximum has been achieved.
- 2.
In other settings, the EM algorithm requires calculating expectations of sufficient statistics or of log likelihood functions in toto, rather than just the unobserved data; see [19]. It is because the HMM is a model for multinomial data (i.e., number of transitions), and since the log of the likelihood of a multinomial is linear in the observed or unobserved counts, calculating the expectation of unobserved transitions, given observed data (signals), is sufficient for the estimation of model parameters.
- 3.
The number of unique states, N, can be reduced to h(1 + h)/2 since haplotype order is not being considered here. The R code above becomes slightly less intelligible when mapping the state number j to the pair of haplotypes referred to by j which is why this redundancy has not been removed.
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Stram, D.O. (2014). SNP Imputation for Association Studies. In: Design, Analysis, and Interpretation of Genome-Wide Association Scans. Statistics for Biology and Health. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9443-0_6
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