Fig. 2. A comparison of the four methods prevalence estimates and confidence intervals for varying proportion of cases and for three sample sizes.

Mixture distributions of non-cases and T1D patients from WTCCC⁸ were constructed with $p_{\mathrm{C}}=\left\{0.1,0.4,0.8\right\}$ (shown in blue, grey and red respectively) and $n=\mathrm{\{}500,1000,5000\mathrm{\}}$ (shown in panels (e–f), (c–d), (a–b) respectively). a, c, e The constructed mixture distributions and reference distributions (${\mathrm{R}}_{\mathrm{C}}$, shaded red and ${\mathrm{R}}_{\mathrm{N}}$, shaded blue) from which they were constructed. b, d, f Prevalence estimates, ${\widehat{p}}_{\mathrm{C}}$ (bullseye) obtained by each of the four methods for varying $p_{\mathrm{C}}$ (x-axis) and cohort size, $n$ (rows). Each estimated ${\widehat{p}}_{\mathrm{C}}$ value is shown together with a violin plot illustrating the distribution of the 100,000 estimates of prevalence ($p_{\mathrm{C}}^{\prime }$) in the bootstrap samples and with confidence intervals ($\alpha =0.05$) shown as horizontal lines with vertical bars at the ends. Dashed vertical lines indicate reference prevalence values $p_{\mathrm{C}}$. In all the cases, for the Excess method we observe a large offset between the violin plots (including confidence intervals) and the ${\widehat{p}}_{\mathrm{C}}$ value. This offset is a result of the systematic bias of the Excess method. The other three methods generally show much less bias. Sample sizes: ${\mathrm{R}}_{\mathrm{C}}$ – cases WTCCC T1D ($n=982$), ${\mathrm{R}}_{\mathrm{N}}$ – non-cases WTCCC T2D ($n=962$), mixtures – sampled with replacement from a holdout half of the ${\mathrm{R}}_{\mathrm{C}}$ ($n=981$) and ${\mathrm{R}}_{\mathrm{N}}$ ($n=962$) samples.