Estimation of the number of studies with positive trends when studies with negative trends are present

A two-point estimator is proposed for the proportion of studies with positive trends among a collection of studies, some of which may demonstrate negative trends. The proposed estimator is the y-intercept of the secant line joining the points (a, (F) over cap(a)) and (b, (F) over cap b)), where (F) over cap(p) is the empirical distribution function of p-values from one-railed tests for positive trend derived from the individual studies. Although this estimator is negatively biased for any choice of the points 0 less than or equal to a < b less than or equal to 1, the bias is less than that of the previously proposed one-point estimator defined by setting b = 1. The bias of the two-point estimator is smallest when a and b approach the inflection point of the true distribution function, E [(F) over cap(p)]. The utility of the two-point estimator is demonstrated by using it to estimate the number of male-mouse liver carcinogens among carcinogenicity studies conducted by the National Toxicology Program.