Supporting Information

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Inference on Haplotype Phases from Multi-Locus Unordered Genotypes. After the previous imputation step, parental data were free of missing values. The inference of gametic haplotypes given multi-locus unphased genotypes was then carried out by determining the transmitted and untransmitted alleles for each parent-child pair at each marker locus. Here, the "haplotype" is of the generalized definition, that "the alleles (at different genes) received by an individual from one parent." For genome scan, the alleles that constitute a haplotype [used by backward haplotype transmission association (BHTA)] may locate on different chromosomes. The inference was implemented under five different scenarios as follows.

• No missing, no error, and no ambiguity. This is a scenario that haplotypes can be perfectly inferred from genotypes (details omitted).
• No missing, no error, but with ambiguity. This scenario happens when both parents and the affected child are all heterozygous for the same alleles at the same marker. Although one can impute the missing transmission phases using more sophisticated methods, and possibly more efficiently by taking into account the information from nearby markers, we had chosen and implemented here a simple solution to resolve this problem: we randomly assigned either of the two possible transmission phases by flipping a coin. By doing so, the marginal results at each locus were not changed whereas transmission information concerning the functional interaction might be slightly affected.
• The child's genotype is partly missing with no error. For this scenario, given there is no disagreement between the observed child's genotype and the parental genotypes, one can calculate the transmissions partially and impute the missing part using population parameters.
• The child's genotype is completely missing. Of all possible transmission scenarios, we chose one randomly as imputed haplotypes.
• There is discrepancy between the parental genotypes and the child's genotype (in only a few of the markers). To avoid possible bias, here we random selected one allele of the given marker and used it for all four transmission values (transmitted from mother, mother's untransmitted, transmitted from father, and father's untransmitted).