Point estimation of the 100 p percent lethal dose using a novel penalised likelihood approach

Estimation of the 100p percent lethal dose (${\text{LD}}_{100p}$) is of great interest to pharmacologists for assessing the toxicity of certain compounds. However, most existing literature focuses on the interval estimation of ${\text{LD}}_{100p}$ and little attention has been paid to its point estimation. Currently, the most commonly used method for estimating the ${\text{LD}}_{100p}$ is the maximum likelihood estimator (MLE), which can be represented as a ratio estimator, with the denominator being the slope estimated from the logistic regression model. However, the MLE can be seriously biased when the sample size is small, a common nature in such studies, or when the dose-response curve is relatively flat (i.e. the slope approaches zero). In this study, we address these issues by developing a novel penalised maximum likelihood estimator (PMLE) that can prevent the denominator of the ratio from being close to zero. Similar to the MLE, the PMLE is computationally simple and thus can be conveniently used in practice. Moreover, with a suitable penalty parameter, we show that the PMLE can (a) reduce the bias to the second order with respect to the sample size and (b) avoid extreme estimates. Through simulation studies and real data applications, we show that the PMLE generally outperforms the existing methods in terms of bias and root mean square error.