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8.1 Introduction to lm_lods, lm_markers and lm_bayes

lm_lods estimates location LOD scores by working along the chromosome, estimating likelihood ratios between adjacent locations of the trait locus, starting from unlinked, and proceeding through the linkage group to unlinked again. We have three methods of combining these local likelihood ratios into an overall LOD score method. One reduces to an eigenvalue method used by Thompson (2000: sec 9.2, P.118). Other alternatives are simply to combine the ratios from the left, or from the right. Weighted combinations do a better job (William Stewart), but we do not pursue this here as better methods are available in lm_bayes and lm_markers.

lm_markers is an implementation of the Lange-Sobel estimator, using our LM-sampler. It is so-named because only the meiosis indicators at marker loci are sampled, and only conditional on the marker data. The Lange-Sobel estimate works reasonably well in reasonable time, provided a good MCMC sampler is used, and provided the trait data do not have strong impact on the conditional distribution of meiosis indicators. Recall that the method samples meiosis indicators conditionally only on the marker data. Because of this the method can produce quite accurate LOD scores in the absence of linkage, but can be inaccurate in estimating the strength of linkage signals. As well as producing the LOD score, our current method provides a batch-means pointwise estimate of the Monte Carlo standard error of the LOD-score estimate.

lm_bayes is currently our preferred method. It samples trait locations from a posterior distribution, and then divides it by the prior to produce the likelihood and hence the LOD score. Estimation is in two phases. A preliminary run with discrete uniform prior gives order-of-magnitude relative likelihoods. Then, using the inverse of these likelihoods as prior weights (to produce an approximately uniform posterior) a second run is made to estimate the likelihood. It is important that the initial run is long enough for all points to be sampled, and for the unlinked trait position to have a reasonable number of realizations. For locations at which LOD scores are very negative, or for the unlinked position when there is some location with strong positive LOD score this can be problematic.

Our current implementation of lm_bayes provides two LOD score estimates. The first is a crude estimate which counts realizations of locations sampled to estimate the posterior: as can be seen from the output this can be quite erratic. The Rao-Blackwellized estimator is much preferred, and produces good estimates in reasonable time.


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