Figure 3.

Performance of the maximum likelihood estimator in simulations, using the two assay designs highlighted in Table 1. Left column: assay on row 4 (confidence interval [CI] < 1.5 logs); right column: assay on row 9 (CI < .5 logs). (A) Maximum likelihood estimate $\hat{\theta }$ and 95% CI plotted in blue; the diagonal line shows the case of a perfect unbiased estimator. Asymptotic CIs are reported, using likelihood of log-transformed θ. (B) Bias $\hat{\theta }/\theta$ plotted in blue; the horizontal line at 1 shows the case of a perfect unbiased estimator. (C) Size of the estimated 95% CI plotted in black (note different y-axis scales). (D) Binomial probability expression used to estimate infection frequency, assuming n replicate wells of c cells apiece. According to this expression, the probability that all wells are negative equals e^−Cθ, where C is the total number of cells across all replicate wells. A–C plot the actual infection frequency $\theta$ used in simulations on the x-axis. Each point on the curves is the geometric mean (A and B) or arithmetic mean (C) of 20 000 replicate simulations using the same $\theta$ (step size 0.025 logs). Curves are solid where <5% of simulated assays yield all-negative or all-positive results, dashed at 5%–50%, and not shown at >50%. Blue (A and B) or gray (C) shaded regions show the middle 95% of simulations; jaggedness results from the discrete nature of the dilution assay. Left to right in each panel, the thin vertical lines show the LLD, the LLQ, and the ULQ = ULD. Note that the assay shown at right is more sensitive (20-fold lower LLD), more precise (narrower shaded regions in A and B, smaller CI in C), and more accurate (curve in A better tracks the diagonal; curve in B better tracks the horizontal).