Design and analysis of serial limiting dilution assays with small sample sizes

We study the design of serial limiting dilution assays (SLDAs) for estimation of effector cell frequency in settings that restrict severely the amount of material available for assay, and for which the assumption of a single-hit Poisson model is reasonable. For different designs (that is, dilution factor, number of dilution levels, and number of replicates at each level), we evaluate maximum likelihood and minimum chi-square estimators, using four quantitative criteria: probability of a non-informative assay (PNI), bias, coefficient of variation (CV), and mean squared error (MSE). Our results are exact, not simulation-based, nor approximations. We found that differences between these two estimators were insubstantial by comparison with the effect on the estimators of the experimental design. Bias was severe in small assays, and could lead, in extremely small designs, to overestimation of effector frequency by as much as 100%. These four criteria provide a rigorous basis for determining the most effective apportionment of total effort among replicates and dilution levels. By increasing the dilution factor and favoring the number of replicates over the number of dilution levels, the PNI is reduced while bias, CV and MSE are increased; and vice versa. Our study is easily extended to estimation of the density of biologically active particles per volume. It can also be extended to alternatives to the single-hit Poisson model for a serial limiting dilution assay.