Randomized $p$-values in Binomial Models and in Group Testing

When screening for rare diseases in large populations, conducting individual tests can be expensive and time-consuming. In group testing, individuals are pooled and tested together. If a group is tested negative, then all individuals in that group are declared negative. Otherwise, it is concluded that at least one individual in that group is positive. Group testing can be used to classify the individuals with respect to their disease status, to estimate the prevalence in the target population, or to conduct a hypothesis test on the unknown prevalence. In this work, we consider both the case when the population is not stratified and when it is stratified, the latter leading to multiple test problems. We define single- and two-stage randomized $p$-values for a model pertaining to the proportion of positive individuals in binomial distribution and in group testing. Randomized $p$-values are less conservative compared to non-randomized $p$-values under the null hypothesis, but they are stochastically not smaller under the alternative. We show that the proposed $p$-values are valid in the binomial model. Testing individuals in pools for a fixed number of tests improves the power of the tests based on the $p$-values. The power of the tests based on randomized $p$-values as a function of the sample size is also investigated. Simulations and real data analysis are used to compare and analyze the different considered $p$-values.