Erik Olofsen
Leiden University Medical Center, The Netherlands
Objectives:
Sheiner and Melmon defined the “Utility” of therapy as “Benefit – Harm” [1]. When Benefit and Harm are dichotomized to “absent” and “present”, best interpreted as random, and associated with unity costs, Utility becomes the probability of Benefit minus the probability of Harm.
The “true” costs of benefit and harm are hard to specify objectively, and the difference of probabilities (with unity costs) is hard to interpret, because it itself is not a probability.
Roozekrans et al. defined Utility as the probability of “Benefit And No Harm” [2]. This resembles the “Desirability Index” of Renard et al. [3], taking the probability of Benefit as one desirability function, one minus the probability of Harm as a second desirability function, taking their product (weighted geometric mean with weights one), and assuming that the probabilities of Benefit and Harm are independent.
In the study of Roozekrans et al., Benefit and Harm were dichotomized continuous variables, with thresholds that seem to need careful selection. The Utility could then become a function of – in addition to for example effect-site concentration – two thresholds, effectively decreasing its utility.
The first objective of the present study was to reduce the dimensionality of the utility function by finding a way to handle the selection of the thresholds for dichotomization. The second objective was to compare the characteristics of the original, Roozekrans’, and an alternative specification of the Utility Function (UF).
Methods:
Instead of dichotomization, trichotomization was applied, by using the fuzzy logic [4] terms “low”, “moderate”, and “substantial”, where the thresholds for membership still need to be chosen, but are more informative than just “absent” and “present”. With two Desirability functions each associated with a fuzzy set of three members, nine fuzzy rules were defined to have output Utility values of “very low”, “low”, “no”, “high” and “very high”.
Benefit and Harm were simulated using sigmoid functions with lognormal distributions for the pharmacodynamic parameters C100 and gamma, where C100 is the effect-site concentration giving a 100% increase of effect relative to (an arbitrary) baseline, and gamma a shape parameter. Thresholds for “moderate” and “substantial” were set to 25% and 50%. From N=25000 simulated values, their membership to the fuzzy sets were calculated by the number of times the values were within the threshold ranges (0, 25%, 50%, and infinity) divided by N.
Utilities were studied as a function of effect-site concentration, typical values of C100, gamma, and their (co)variances.
Results:
As judged by visual inspection, the UF proposed here had similar characteristics to the one by Roozekrans et al., with respect to their dependence on the values of the pharmacodynamic parameters underlying the intensity of Benefit and Harm, even though the latter was evaluated with one fixed value for the threshold for Harm.
The effect of a difference between the parameters for Benefit and Harm are the most important for decision making. Findings include: 1) Because the thresholds chosen are below the C100, an increase in gamma moves the UF closer to the C100, so an increase in the gamma of Benefit is likely to give higher values for high utility. 2) A higher (population) uncertainty of the value of C100 moves the UF downward for high utility, and upward for low utility.
Conclusions:
While the term “Fuzzy Utility Function” might be proposed for the one studied here, it might actually give a more crisp interpretation than one with just two thresholds for dichotomization, where it provokes discussion on how these should be chosen. However, this depends on how “easy” it is to define “moderate” and “substantial” Benefit and Harm.
As Renard et al. note, the classical UF can have the value of zero for the entire dose or concentration range. In that case, there can indeed still be a value for the effect-site concentration that gives the highest value of “high” or “very high” utility, even if these are matched by simultaneous highest values of “low” and “very low” utility.
Instead of one classical UF, the present approach gives about four UFs to consider when comparing two drugs. It could be possible that one drug gives higher values for “very high” utility, but even much higher values for “very low” utility; it depends on the application how these should be interpreted.
References:
[1] Sheiner LB and Melmon KL, Ann N Y Acad Sci, 304:112, 1978
[2] Roozekrans M et al., Anesthesiology, accepted for publication
[3] Renard D et al., PAGE 18 (2009) Abstr 1506 [www.page-meeting.org/?abstract=1506]
[4] Naranjo CA et al., Clin Pharmacol Ther, 62:209, 1997
Reference: PAGE 27 (2018) Abstr 8644 [www.page-meeting.org/?abstract=8644]
Poster: Methodology - Model Evaluation