The application of extreme value theory to pharmacometrics
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ORIGINAL PAPER
The application of extreme value theory to pharmacometrics Peter L. Bonate1 Received: 1 May 2020 / Accepted: 21 September 2020 Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Clinical trials are often analyzed by examining the means, e.g., what is the mean treatment effect or what is the mean treatment difference, but there are times when analysis of the maximums (or minimums) are of interest. For instance, what is the highest heart rate that could be observed or what the smallest treatment effect that could be expected? While inference on the means is based on the central limit theorem, the corresponding theorem for maximums or minimums is the Fisher–Tippett theorem, also called the extreme value theorem (EVT). This manuscript will introduce EVT to pharmacometricians, particularly block maxima analysis and peak over threshold analysis, and provide examples for how it can be applied to pharmacometric data, particularly the analysis of pharmacokinetics and ECG safety data, like QTcF intervals. Keywords Statistics Safety analysis QT interval Exposure–response Gumbel Weibull Frechet
Introduction Emil Gumbel, one of the founding fathers of extreme value theory (EVT), once said ‘‘It is improbable for the impossible to never happen’’ [1]. What he meant by that was rare or extreme events will rarely happen, but they will happen eventually (given enough time), because rare events happen every day. Deepwater Horizon was not supposed to blow-up in the Gulf of Mexico, but it did in 2010. The Fukushima Daiichi nuclear plant was never supposed to meltdown, but it did in 2011 when a 15 ft tsunami swept over the seawall protecting it following a 9.0 magnitude earthquake. Ironically, the company met just 4 days earlier to discuss building a higher sea wall to shield the facility from such a catastrophe, but the idea was dismissed by a senior manager as unlikely to ever happen [2]. StevensJohnson syndrome is a very rare, idiopathic, autoimmune response to a drug that is potentially life-threatening. Estimated to have an incidence of 3 to 7 cases per million in the U.S. [3], in total there are about 300 cases per year in the U.S. [4]. The list of rare events that happens daily goes on and on.
& Peter L. Bonate [email protected] 1
When analyzing data from clinical trials, it is often the means that are of interest. Common statistical endpoints are mean treatment effect, mean change from baseline, mean difference between active treatment and placebo, and mean survival. To draw inferences, such as ‘‘what is the probability of observing a mean treatment difference of X by chance’’, the central limit theorem (CLT) is used. The CLT basically states if X is an independent and identically distributed (iid) random variable, the sampling distribution for the sample mean X approximates a normal distribution with mean l and variance r2/n as the sample size n gets larger. The CLT allows us to understand the underlying distribution of the sample mean when repeated samples are d
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