Optimal designs of two-stage studies for estimation of sensitivity, specificity and positive predictive value

The cost efficiency of estimation of sensitivity, specificity and positive predictive value from two-stage sampling designs is considered, assuming a relatively cheap test classifies first-stage subjects into several categories and an expensive gold standard is applied at stage two. Simple variance formulae are derived and used to find optimal designs for a given cost ratio. The utility of two-stage designs is measured by the reduction in variances compared with one-stage simple random designs. Separate second-stage design is also compared with proportional allocation (PA). The maximum percentage reductions in variance from two-stage designs for sensitivity, specificity and positive predictive value estimation are P per cent, (1-P) per cent and W, respectively, where P is the population prevalence of disease and W the population percentage of test negatives. The optimum allocation of stage-two resources is not obvious: the optimum proportion of true cases at stage two may even be less than under PA. PA is near optimal for sensitivity estimation in most cases when prevalence is low, but inefficient compared with the optimal scheme for specificity.