Consequences of Mistyping a Null Allele for Case-Control Association Studies
It is assumed allele A is the true risk allele for various values of γ (relative risk of AA homozygote) and frequencies of the null allele (p0). The overestimation of the effect under a dominant model (top figure) and the underestimation of the effect under a recessive model (bottom figure) are shown.

Distribution of Triallelic-Locus Size and Null-Allele Frequency
(A) The triallelic loci for which no polymorphism within the primer is known in dbSNP (build 127) are plotted against the maximum potential size of each locus, assuming these can reflect deletions (by taking the physical position of the immediately adjacent SNPs that look normal on the Illumina Human Hap550 platform).
(B) The number of triallelic loci is plotted against the null-allele frequency.

Overview of 1655 Triallelic Loci Identified on Autosomes and Chromosome X
Immediately to the left of each chromosome are depicted all the SNPs present on the Illumina Human Hap550 platform. CNVs known in the Database of Genomic Variants are shown to the right of each chromosome. Next to this, the triallelic loci are shown for which the length of each bar denotes the null-allele frequency. Blue indicates a single triallelic SNP locus, orange indicates a locus in which multiple adjacent triallelic SNPs have been identified, and gray indicates a single triallelic SNP locus for which polymorphisms are known within the region complementary to the primer of the triallelic SNP (dbSNP build 127).

Distribution of Distance of Secondary Polymorphisms Present within Primers of Human Hap550 SNPs
Distribution plot of the distance of secondary polymorphisms present within primers of Human Hap550 SNPs to the actual SNP. Polymorphisms are known in dbSNP (build 127) for 85,045 of the SNPs present on the Illumina Hap550 platform within the 50 bp long primers. For the 1880 fitted triallelic SNPs, this is the case for 437 of the SNPs (expected 235, Fisher's exact p value < 10⁻¹⁸). When investigating how far away these secondary polymorphisms are from the actual SNPs, it turns out that within the triallelic SNPs, these secondary polymorphisms usually map much closer to the actual SNP than for the nontriallelic SNPs (Wilcoxon-Mann-Whitney p value < 10⁻⁷⁶).

Imputation Scenarios
When assuming all alleles have a nonzero frequency for both the triallelic SNP and the neighboring biallelic SNP and that some LD is present (i.e., at least one haplotype has not been observed), there are 24 different imputation scenarios possible. For ten of these scenarios, the biallelic SNP can help to discriminate between B0 and BB and/or between A0 and AA for the triallelic SNP. For the first imputation scenario, a detailed description of this procedure is provided: With this set of observed and unobserved haplotypes, a limited number of genotype combinations exist. This allows for deducing a set of genotypic rules that can help to discriminate between B0 and BB genotypes for the triallelic SNP, on the basis of the genotype of the neighboring biallelic SNP.

HWE Test Statistics, when Analyzing Triallelic SNPs, Called under Biallelic Assumptions
These calculations show the HWE test statistic for various sample sizes and different frequencies of the 0 allele. If we incorrectly assume triallelic SNPs are biallelic, analysis of sample sizes that are representative of current genome-wide association studies will result in significant HWE deviations, even when the 0 allele has a fairly low frequency, e.g., when testing 3000 samples, and assuming a call rate of 100% for samples having one or two copies of the A or B allele, triallelic SNPs with a null-allele frequency above 2% have an expected deviation from HWE with p < 0.05.

Association Analysis with Triallelic SNPs and Marginal Effect on SNPs, Used for Imputation
(A) Marginal association signals of SNPs, used for imputing triallelic SNPs, with disease. Fixed associations for the null allele of 600 triallelic SNPs were defined in such a way that each of the triallelic SNPs approximated a Fisher's exact test p value of 10⁻⁴, 10⁻⁶, or 10⁻⁸. We then assessed whether a marginal association signal was present within the SNPs that had been used to impute the triallelic genotypes. The median marginal effect and the cumulative distribution of the marginal association p value for each of these SNPs are shown, ranked on significance (see text for details).
(B) Quantile-quantile plot of observed versus expected p values in a triallelic SNP null-allele association analysis in celiac disease, for which cases and controls had been typed on different platforms. Eight triallelic SNPs with a Fisher's exact test p value < 0.01 are indicated. The λ_inflation factor is 0.96, suggesting no inflation of the test statistic. Three SNPs map within the major histocompatibility complex region (indicated in red).

Distribution of A and B Allele Intensities of 11,652 Chromosome X SNPs, Present on the Illumina Human HapMap550 Platform
Each dot represents the median coordinate of the A0 (males, green), AA (females, blue), B0 (males, yellow), BB (females, red), or AB (females, gray) cluster for a single SNP. The A and B intensities have been scaled in such a way that for each SNP, the median AB cluster center is identical for all chromosome X SNPs. As shown in (A), it is evident that single-copy genotypes (A0 and B0) clearly show different intensity characteristics than AA and BB genotypes. Another observation is that the A and B probes have slightly different characteristics, because the A0 and AA distributions overlap slightly less than the B0 and BB distributions, indicating that on average, A0 and AA samples can be better distinguished from each other. To correct for these differences in intensity characteristics, parameter β is calibrated on these chromosome X SNP distributions. As shown in (B), the genotype-calling algorithm uses parameter β to distinguish between A0 and AA and between B0 and BB. For a given β, an angle for the A0 distribution is determined where the A0 distribution percentile equals β. The same holds for the angle of the B0 distribution. In the present example, increasing β increases the angle of the A line slightly more than it increases the angle of the B line. Examples are shown where β is 25 ([B], left) and where β is 75 ([B], right), resulting in different genotype assignments (A0, AA, B0, and BB genotypes assignments are indicated in yellow, red, green, and blue, respectively).

Genotyping Methodology for SNPs with a Third, Untyped Allele
The graphs show the intensities of the A labeled probe (x axis) and B labeled probe (y axis) of both a theoretical SNP with an third, untyped allele (top figures) and a real SNP (rs7571895, bottom figures). (A) shows that six genotypes for a triallelic SNP exist. The A0 and AA, and B0 and BB, genotype clusters usually overlap somewhat. (B) shows that initially 00 genotypes are assigned to samples that have an intensity lower than threshold α. The remaining samples are designated an initial A0/AA, AB, or B0/BB genotype with an existing calling algorithm. As shown in (C), parameter β is then used to discriminate between A0 and AA and between B0 and BB genotypes (see text). This allows for determining whether Hardy-Weinberg equilibrium is observed. As shown in (D), parameters α and β are then optimized (with a maximum-likelihood-estimation procedure) until the SNP does adhere to Hardy-Weinberg equilibrium conditions. Triallelic-genotype assignments, based on the MLE procedure for SNP rs7571895, are shown in (E). (F) shows that subsequent analysis of neighboring SNPs results in the identification of biallelic SNP rs654797, which is in strong LD with the null allele of rs7571895. Although LD does not seem to be perfect (r² < 1), we assume that this is probably because of imperfections in the initial genotype assignments and that some of the haplotypes (indicated with an asterisk) are not actually present. This allows for identifying a set of triallelic genotype imputation rules that are applied to the data and result in (G) improved genotype assignments for rs7571895, as is clearly visible when distinguishing C0 from CC samples (green arrow) and 00 from A0 samples (black arrow).