Multipoint approximations of identity-by-descent probabilities for accurate linkage analysis of distantly related individuals

Abstract

We propose an analytical approximation method for the estimation of multipoint identity by descent (IBD) probabilities in pedigrees containing a moderate number of distantly related individuals. We show that in large pedigrees where cases are related through untyped ancestors only, it is possible to formulate the hidden Markov model of the Lander-Green algorithm in terms of the IBD configurations of the cases. We use a first-order Markov approximation to model the changes in this IBD-configuration variable along the chromosome. In simulated and real data sets, we demonstrate that estimates of parametric and nonparametric linkage statistics based on the first-order Markov approximation are accurate. The computation time is exponential in the number of cases instead of in the number of meioses separating the cases. We have implemented our approach in the computer program ALADIN (accurate linkage analysis of distantly related individuals). ALADIN can be applied to general pedigrees and marker types and has the ability to model marker-marker linkage disequilibrium with a clustered-markers approach. Using ALADIN is straightforward: It requires no parameters to be specified and accepts standard input files.

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 G₆^p to individuals 1 and 2. Thus, the paternal allele G₁^p of individual 1 and the paternal allele G₂^p of individual 2 are IBD. The dashed gray arrows correspond to the p/m entries in Table 1.

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 G_T, 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 s_T and s_Dnf have been replaced by the IBD variable Π, which defines the IBD configuration of the alleles contained in G_T. The ordered genotypes G_A of the ancestors (A) are not explicitly modeled.

Figure 3

Prostate Cancer Pedigree I Affected individuals are represented by a black symbol, and genotyped individuals are indicated with an asterisk.

Figure 4

Pedigree II The pedigree was taken from a pituitary adenoma study of Vierimaa et al. Affected individuals are represented by a black symbol, and genotyped individuals are indicated with an asterisk.

Figure 5

Comparison of ALADIN and MORGAN with Exact Method The figure shows exact, approximate, and true simulated value of NPL_pairs 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, Δ_mean of ALADIN and MORGAN were 0.0913 and 0.1015, respectively. The maximum errors Δ_max were respectively 4.439 and 4.571, and were both attained at 142.4 cM, a region where NPL_pairs changed rapidly. In two replicates, the situation was as in (B): ALADIN and MORGAN produced similar scores that both overestimated NPL_pairs as compared to the value obtained with MERLIN but did not overestimate the value of NPL_pairs 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.

Figure 6

Empirical Type I Error Rate of ALADIN (A) shows the empirical type I error rate for NPL_pairs in pedigree I. (B) shows empirical type I error rate for NPL_pairs in pedigree II. Note that the p value corresponding to a given significance threshold on NPL_pairs is given by the solid curve.

Figure 7

Evaluation of ALADIN in a Large Pedigree The ALADIN and MORGAN estimates of NPL_pairs are compared to the NPL_pairs of the true inheritance vector for eight autosomal chromosomes (1, 3, 5, 7, 11, 13, and 17) for pedigree II. Exact multipoint computation of NPL_pairs 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.

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 NPL_pairs 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.

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.