Multipoint Approximations of Identity-by-Descent Probabilities for Accurate Linkage Analysis of Distantly Related Individuals  Cornelis A. Albers, Jim Stankovich, Russell Thomson, Melanie Bahlo, Hilbert J. Kappen  The American Journal of Human Genetics  Volume 82, Issue 3, Pages 607-622 (March 2008) DOI: 10.1016/j.ajhg.2007.12.016 Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 1 Descent Graph of the Top Configuration in Table 1 Squares represents alleles of males, and circles represent alleles of females. The solid black arrows indicate the transmission of founder allele G6p to individuals 1 and 2. Thus, the paternal allele G1p of individual 1 and the paternal allele G2p of individual 2 are IBD. The dashed gray arrows correspond to the p/m entries in Table 1. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 2 Graphical Model Representation The graphical model reflects the conditional independencies of the single-point likelihood. White circles represent unobserved variables, gray circles represent model parameters assumed to be fixed and known, and gray rectangles represent observed variables, i.e., the marker genotypes. Panel (A) shows the graphical model corresponding to the single-point likelihood defined in terms of genotype variables and segregation indicators. Conditioned on the ordered genotypes GT, the variables shown in the dashed rectangle are independent of the variables outside the dashed rectangle. The graphical model can be alternatively constructed as in panel (B). As the group of ancestors (A) is defined to have no genotype or phenotype information, the corresponding ordered genotype variables (indicated by dotted lines) can be removed from the model, yielding the graphical model shown in panel (C). The HMM used by ALADIN is based on the model shown in panel (D). Here, the unobserved segregation indicators sT and sDnf have been replaced by the IBD variable Π, which defines the IBD configuration of the alleles contained in GT. The ordered genotypes GA of the ancestors (A) are not explicitly modeled. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 3 Prostate Cancer Pedigree I Affected individuals are represented by a black symbol, and genotyped individuals are indicated with an asterisk. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 4 Pedigree II The pedigree was taken from a pituitary adenoma study of Vierimaa et al.2 Affected individuals are represented by a black symbol, and genotyped individuals are indicated with an asterisk. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 5 Comparison of ALADIN and MORGAN with Exact Method The figure shows exact, approximate, and true simulated value of NPLpairs for two typical replicates of pedigree IIa in the condition of linked chromosomes and linkage equilibrium. The location where linkage was simulated is indicated with the asterisk. Forty-two of the 44 simulated replicates were similar to (A). For this replicate, Δmean of ALADIN and MORGAN were 0.0913 and 0.1015, respectively. The maximum errors Δmax were respectively 4.439 and 4.571, and were both attained at 142.4 cM, a region where NPLpairs changed rapidly. In two replicates, the situation was as in (B): ALADIN and MORGAN produced similar scores that both overestimated NPLpairs as compared to the value obtained with MERLIN but did not overestimate the value of NPLpairs of the true inheritance vector. Here, the maximum errors of ALADIN and MORGAN were 14.05 and 14.15, respectively, and were attained at 23.96 cM by both methods. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 6 Empirical Type I Error Rate of ALADIN (A) shows the empirical type I error rate for NPLpairs in pedigree I. (B) shows empirical type I error rate for NPLpairs in pedigree II. Note that the p value corresponding to a given significance threshold on NPLpairs is given by the solid curve. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 7 Evaluation of ALADIN in a Large Pedigree The ALADIN and MORGAN estimates of NPLpairs are compared to the NPLpairs of the true inheritance vector for eight autosomal chromosomes (1, 3, 5, 7, 11, 13, and 17) for pedigree II. Exact multipoint computation of NPLpairs was not feasible. The inheritance vector at the middle marker, indicated by the asterisk at the horizontal axis, was fixed such that all six cases shared one allele IBD at that location. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 8 Evaluation of ALADIN with Real Data The approximation error of ALADIN and MORGAN is compared on the real data set for subpedigree Ia for varying number of MCMC scans of MORGAN. The figure shows boxplots of the absolute difference between the approximate and exact NPLpairs of all points on chromosomes 1–10. The number of MCMC scans is shown between parentheses, and the computation times are denoted by t on the horizontal axis. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions

Figure 9 Scaling of Computation Time (A) shows on a log-log scale the computation time as a function of the number of markers used in the multipoint analysis. Computation time of ALADIN scaled linearly with the number of markers, whereas that of MORGAN scaled quadratically with the number of markers. (B) shows computation time as a function of A, the number of untyped ancestors, for fixed number of individuals T = 4 and 100 markers. The American Journal of Human Genetics 2008 82, 607-622DOI: (10.1016/j.ajhg.2007.12.016) Copyright © 2008 The American Society of Human Genetics Terms and Conditions