8.2 LM-sampler

MORGAN's Autozyg and Lodscore programs use MCMC to estimate ibd probabilities and multilocus LOD scores, respectively, in pedigrees. The latent (unobserved) parameters of interest in MCMC estimation of ibd probabilities and LOD scores are the meiosis indicators at marker and/or trait loci for each non-founder in the pedigree. Observed data are trait values and unphased marker genotypes for some or all pedigree members. With unphased genotypes, it may or may not be possible to determine the grandparental source (i.e., the inheritance indicator) of each allele unambiguously. MORGAN uses MCMC to sample meiosis indicators (S) conditional on observed data (Y).

MORGAN implements two different block Gibbs samplers, a locus- and a meiosis-sampler, for sampling from S conditional on Y. Each method updates a subset, S_u, of S conditional on Y and on the rest of S (S_f). The difference between the two methods is the choice of S_u.

The locus-sampler (or L-sampler) chooses S_u to be S.j for some j. In other words, a single locus is selected and inheritance indicators at that locus are updated based on the genotype data at all loci and on the current realization of inheritance indicators at all loci other than j. The MORGAN user can determine whether a locus is to be selected at random each time or if loci are taken in a pre-determined random order, as described in the next section. This method is a modification of the Elston-Stewart algorithm (Elston RC and Stewart J (1971) Human Heredity 21:523-542) and it can be used whenever single locus pedigree peeling is possible. If inter-locus recombination fractions are strictly positive, the L-sampler is irreducible. On the downside, mixing is poor if loci are tightly linked. The meiosis-sampler (or M-sampler) chooses S_u to be Si. for some i. It is , in a sense, perpendicular to the L-sampler in that at each iteration a single meiosis is selected and inheritance indicators for that meiosis are updated conditional on the genotype data at all loci and the current realization of inheritance indicators for all other meioses. The M-sampler is a modification of the Lander-Green algorithm (Lander ES and Green P (1987) PNAS 84:2363-2367) for peeling along a chromosome using the Baum algorithm (Baum LE (1972) in O. Shea, ed., `Inequalities-III; Proceedings of the Third Symposium on Inequalities, UCLA, 1969', Academic Press, NY pp. 1-8). At each iteration, a single meiosis is randomly selected or meioses can be updated sequentially. As with locus selection in the L-sampler, MORGAN allows the user to choose the meiosis selection The M-sampler mixes well in the presence of tightly linked loci, but it can perform poorly in large pedigrees with missing data.

MORGAN's Autozyg and Lodscore programs use a combination of the L- and M-samplers, referred to as the LM-sampler. The user may choose the fraction of updates that are of each type; the default is 20% L-sampler, 80% M-sampler.

For mathematical details on the L-, M- and LM-samplers, see Thompson EA (2000) Statistical Inference from Genetic Data on Pedigrees, in `NSF-CBMS Regional Conference Series in Probability and Statistics, Volume 6. Institute of Mathematical Sciences, Beachwood OH and American Statistical Association, Alexandria VA.