PLINK 1.9 accelerates this in several ways: Estimation of diplotype frequencies and maximum-likelihood D′ has been streamlined. Bit population counts are used to fill the contingency table; then we use the analytic solution to Hill’s diplotype frequency cubic equation [24,25] and only compute and compare log likelihoods in this step when multiple solutions to the equation are in the valid range.90% confidence intervals were originally estimated by computing relative likelihoods at 101 points (corresponding to D′=0,D′=0.01,…,D′=1) and checking where the resulting cumulative distribution function (cdf) crossed 5% and 95%. However, the likelihood function rarely has more than one extreme point in (0,1) (and the full solution to the cubic equation reveals the presence of additional extrema); it is usually possible to exploit this unimodality to establish good bounds on key cdf values after evaluating just a few likelihoods. In particular, many confidence intervals can be classified as “recombination” after inspection of just two of the 101 points; see Figure 3.Figure 3Rapid classification of “recombination” variant pairs. This is a plot of 101 equally spaced D’ log-likelihoods for (rs58108140, rs140337953) in
