Confidence intervals for the Abbott's formula correction of bioassay data for control response

Abbott's formula may be used to correct bioassay data for control response and has become a standard in bioassay evaluation. Although Abbott's formula provides an estimate of (the mean bioassay treatment response corrected for control response), it does not provide a measure of associated variance. The current practice of retaining the variance estimate for (the mean bioassay treatment response not corrected for control response) and applying it to (the mean bioassay treatment response corrected for control response) is invalid. This invalid procedure results in an exaggeration of the reliability of the estimate of (the mean bioassay treatment response corrected for control response) and a confidence interval for (the mean bioassay treatment response corrected for control response) that is centered around an inappropriate value. We present a technique to incorporate a correction for control response into the statistical analysis of bioassays conducted with only a single or small number of treatments, which may be qualitative classes rather than a series of doses. The proposed solution is based upon established techniques for estimating the variance or confidence interval of a ratio of normally distributed variables. The analysis suggests two implications for bioassay experimental design and evaluation: first, the optimal allocation of bioassay replications to control and experimental treatments generally occurs when the number of experimental replications is equal to or slightly greater than the number of control replications, and second, bioassay data should be corrected for control mortality more frequently than is currently recommended. Only if such a correction has negligible effects on both (the mean bioassay treatment response corrected for control response) and variance (the mean bioassay treatment response corrected for control response) can it be safely omitted.