Why are rare variants hard to impute? Coalescent models reveal theoretical limits in existing algorithms

Genotype imputation is an indispensable step in human genetic studies. Large reference panels with deeply sequenced genomes now allow interrogating variants with minor allele frequency < 1% without sequencing. Although it is critical to consider limits of this approach, imputation methods for rare variants have only done so empirically; the theoretical basis of their imputation accuracy has not been explored. To provide theoretical consideration of imputation accuracy under the current imputation framework, we develop a coalescent model of imputing rare variants, leveraging the joint genealogy of the sample to be imputed and reference individuals. We show that broadly used imputation algorithms include model misspecifications about this joint genealogy that limit the ability to correctly impute rare variants. We develop closed-form solutions for the probability distribution of this joint genealogy and quantify the inevitable error rate resulting from the model misspecification across a range of allele frequencies and reference sample sizes. We show that the probability of a falsely imputed minor allele decreases with reference sample size, but the proportion of falsely imputed minor alleles mostly depends on the allele count in the reference sample. We summarize the impact of this error on genotype imputation on association tests by calculating the r2 between imputed and true genotype and show that even when modeling other sources of error, the impact of the model misspecification has a significant impact on the r2 of rare variants. To evaluate these predictions in practice, we compare the imputation of the same dataset across imputation panels of different sizes. Although this empirical imputation accuracy is substantially lower than our theoretical prediction, modeling misspecification seems to further decrease imputation accuracy for variants with low allele counts in the reference. These results provide a framework for developing new imputation algorithms and for interpreting rare variant association analyses.