Nonparametric methods for group testing data, taking dilution into account

Group testing methods are used widely to assess the presence of a contaminant, based on measurements of the concentration of a biomarker, for example to test the presence of a disease in pooled blood samples. The test would be perfect if it produced a positive result whenever the contaminant was present, and a negative result otherwise. However, in practice the test is always at least somewhat imperfect, for example because it is sensitive to the proportion of contaminated items in the group, rather than to the sheer existence of one or more contaminated items. We develop a nonparametric method for accommodating this dilution effect. Our approach allows us to estimate, under minimal assumptions, the probability $m(x)$ that an item is contaminated, conditional on the value $x$ of an explanatory variable, and to estimate the probability, $q$, that an individual chosen at random is disease free, and the specificity Sp, and the sensitivity Se, of the test. These are all ill-posed problems, where poor convergence rates are usually encountered, but despite this, our estimators of $q$, Sp and Se are root-$N$ consistent, where $N$ denotes the total number of individuals in all the groups, and our estimator of $m(x)$ converges at the rate it would enjoy if $q$, Sp and Se were known.