The Comparison of Dosage‐mortality Data

S ummary . The theoretical and practical applications of the dosage‐mortality curve usually involve either a comparison of similar series of records to determine whether they differ significantly, or the estimation of the dosage corresponding to a selected or observed mortality and the error of this estimate. In an earlier paper we have considered the methods appropriate for computing the dosage‐mortality curve as a straight line and for measuring its error of estimation. The present paper is an extension of these methods to cover some of the more frequent applications of the curve. In measuring the degree of agreement between different series of dosage‐mortality data, it is convenient to refer to one series, which has been transformed into a straight regression line of known accuracy, as the standard, and to determine the agreement of similar or smaller groups of data with this standard. When the standard is of absolute accuracy, experimentally the exceptional case, only the second group of observations is subject to sampling error and the X 2 test is applied in a form appropriate for the binomial distribution. When the standard curve has been determined from homogeneous data that involve no errors other than those of the random sampling of test animals which vary in their inherent susceptibility, two forms of the X 2 test are available, both based primarily upon the weighting coefficients used in computing the dosage‐mortality regression lines. In one case the additional observations are combined with the original ones for a recomputation of the standard curve and of the X 2 measure for the agreement of this second curve with the pooled data, the X 2 for the discrepancy of the additional observations (or of their separately fitted curve) being derived by difference. In the second case the X 2 s for these discrepancies are computed directly and, with two or more observations in the second series, separate the difference between the two series of data in position from that in slope. When the standard curve has been determined from heterogeneous data and the observed variation is statistically in excess of that attributable to errors of sampling, the X 2 test is no longer applicable, and the discrepancy between the standard and one or more other observations is compared with the observed errors of estimation by means of the statistic t. Several of these procedures are illustrated by the numerical example taken from the preceding paper. In expressing the relative susceptibilities of different biological races or species, or the relative potencies of toxic agents, the comparisons are in terms of dosages required to produce selected or observed mortalities rather than of mortalities produced by specified dosages. It is then important to meaaure the accuracy of such estimated dosages. When the mortality is chosen by the experimenter, the dosage estimated from the standard curve is subject only to the errors involved in its determination, and the relative merits of selecting different mortalities, such as at 5.0 and at 7.0 probits, as a basis for these comparisons is considered in the light of the use that is to be made of them. As an illustration, numerical data are given of one such application. When the mortality at which the dosages in a standard and a second series are to be related is determined by experiment, we have the conditions observed in the biological assay of drugs or other toxic materials. There is not only the error of the standard curve to be considered, but also the error in the mortality observed with the unknown which is being standardised. When the stock of test animals used in a particular biological assay is constant in respect to the standard preparation, both with regard to its average susceptibility and its relative susceptibility within the population, an unknown material can be standardised by direct comparison with a single curve of comparatively high accuracy, determined from many experiments with the standard preparation. When the population of test animals is not stable in these respects, each separate assay must be based upon parallel tests between the standard preparation and the unknown, and require, therefore, more observations to secure the same accuracy. A method is described for determining the ratio of potencies and the error of this ratio in both instances.