Interval Estimation of the LD 50 Based on an Up-and-down Experiment

It is well known that an up-and-down method can be more efficient than fixed-sample methods in estimating the LD50 of a quantal response curve. A problem that has not been addressed by many is that of obtaining a confidence interval for the LD50 from the up-and-down method. Dixon and Mood (1948, Journal of the American Statistical Association 43, 109-126) proposed a confidence interval using a maximum likelihood approach, but not much is known about its properties. In this paper, a new confidence interval for the LD50 based on turning points is obtained, which uses the concept of phi-mixing. Simulation results indicate that the coverage probabilities of both methods tend to be less than the nominal level unless the sample size is large. Even so, when the tolerance distribution is normal, the proposed confidence interval is found to be superior to Dixon's interval in terms of the coverage, the width, and stability. The advantages of the method do not appear to hold in the presence of nonnormal tolerance distribution.