Confidence Limits for the LD 50 using the Moving Average-Angle Method

The use of moving averages as a smoothing procedure is well known to statisticians. Moving average interpolation for LD50 estimates was proposed by Thompson 11947] and recently extended by Bennett [1952] to include the angular transformation. The purpose of this transformation is, of course, to make the variance of an observed proportion largely independent of the unknown true proportion. Thompson and Weil [19521 and Weil [1952] have published formulas and tables useful in the calculation of approximate confidence limits for the LD50 without reference to the angular transformation. The following sections present exact* confidence limits and significance tests under the moving average-angle procedure, when the doses are separated by a constant interval (almost always a log interval in bioassay) and an equal number of animals has been exposed to each dose. Under the moving average-angle method, formulas to handle differing intervals and/or numbers of animals may be easily derived when needed (cf. Bennett, [1952]). The observed proportional response at each dose is transfoimecd to an angle, 0(p) = aresin -Vp, using, for example, Table XII in Fisher and Yates [1948]. When p = 0 or 1, Bartlett's adjustments (cf. Eisenhart, [1947]) should be applied, namely 0(0) = arcsin V/1/4n and 0(1) = 90 0(0), where n is the number of animals at each dose. Then simple averages of A; successive angles are computed, where A; is usually 3 or 5. Each average angle is associated with the middle dose of the respective set of A; doses. The LD50 is estimated by linear interpolation between the two