(A) shows distribution of the outcome of coin tosses generated using an 80∶20 mixture of neutral coins and biased coins (probability of a head  = .9), compared with the outcome of a series of biased coin tosses (probability of a head  = .58); the mixed coin toss (blue) has the same mean bias (p = .58) as the biased coin toss (black). (B) shows distribution of a 10∶80∶10 mixture of a biased coin (probability of a head  = .1), neutral coin, and a biased coin (probability of a head  = .9), compared with the outcome of a series of neutral coin tosses. In both simulations, coins are selected and flipped 10 times and the resulting number of heads, ranging from 0 through 10, are shown. The increased variance of the outcomes in the mixture setting carries information about the presence of some non-neutral coins in the experiment.

In (A) 100 high and 100 low extremes of triglyceride levels drawn from the Malmo Diet and Cancer Study – Cardiovascular Arm in APOB and (B) 350 cases of Crohn's disease and 350 controls collected by the NIDDK IBD Genetics Consortium in NOD2, identified from pooled data and then individually genotyped. The background (gray) represents the binomial probability distribution while the foreground (red points) shows observed data from NOD2 and APOB sequencing, in which, for example, APOB (A) the n = 3 row indicates three observed variants, one seen in 3 cases and 0 controls, one seen in 2 cases and 1 control, and one seen in 0 cases and 3 controls.

(A) shows power comparisons for the population genetics model simulations. Power comparisons are for C-alpha, Madsen-Browning (MB), Variable threshold (VT), and Li-Leal's approach (presence/absence Li-Leal_p and count of rare variants Li-Leal_c). These simulations reflect the presence of selection on the variation which predisposes to phenotype. As we increase the mixing proportions between risk and protective variants (moving from mixtures 1 to 6, which reflects 0, 10, 20, 30, 40 and 50% chance of any of the phenotypically relevant variants are protective, rather than risk), C-alpha maintains power, while other tests lose power. In (B), the each of 6 variants explains 0.1% of the variance of the phenotype. All three approaches have high power when all the effects are detrimental. For burden tests, the power drops markedly when 3 variants are protective and 3 are detrimental. “Selected” controls are chosen from the lower 1% of the liability distribution. The solid (dashed) lines represent power for selected (unselected) controls.