On the calculation of the error of biological assays

Abstract

It is now realized by a number of workers that the formula which it has been customary to use hitherto for calculating the limits of error of the dose corresponding to a given response (e.g. the median effective dose) or of the result of a biological assay is only approximate. The usual method gives an approximation to the standard error of the logarithm of the dose in question, or of the result, and then uses it in conjunction with the table of the normal probability integral to find the limits of error. The approximation is obtained by the usual statistical method of treating, as Karl Pearson used to say, statistical differentials as mathematical differentials. Bliss (1935) was, I think, the first to realize that exact fiducial limits could be obtained and actually calculated them in a numerical example—but without discussing their theoretical and practical implications. Their mathematical derivation was first published for particular cases by C. Eisenhart (1939) and by E. C. Fieller (1940) in the appendix to his important paper on the biological standardization of insulin. Fieller also gave the general result with a promise (if circumstances allow) to discuss it more fully elsewhere. Later a method for calculating exact fiducial limits was given in the ‘British Standard Method for the biological assay of vitamin D₃ by the chick test’ (1940).