RE: PK simulation and replicates

Dear Marie, There is a lot of confusion when it comes to replication in pop pk stochastic simulations. If you are referring to subjects then simulations of population predictions from your model will require a large sample size so that the sample statistic converges to the population parameter (e.g., sample mean from the simulations converges to the population mean as the sample size increases to infinity, this is known as the Law of Large Numbers in Statistics). If you want to quantify uncertainty in the population predictions via a confidence interval, then you will also want a sufficiently large number of replicate trials with the large sample size for each trial accounting for parameter uncertainty which can be thought of as trial-to-trial uncertainty. As Jacob indicates, the number of replicate trials needed increases with higher coverage probability for the CI. For a 90% CI, I typically go with 1000 replicated trials but if you wanted say a 99% CI you would want a considerably higher number of replicate trials. If the interest in the pop PK/PD stochastic simulations is to conduct clinical trial simulations for a proposed study design of a fixed (finite) sample size, then replicate trials accounting for the parameter uncertainty would be used to obtain prediction intervals (PI) for the sample statistic of the finite sample size of the proposed study design. The number of replicate trials for quantification of a PI like a CI will also depend on the selected coverage probability. Note for most study designs the fixed/finite sample size is usually too small for the sample statistic to converge to the population parameter and so prediction intervals essentially reflect sampling variation from one trial to the next as well as parameter uncertainty whereas confidence intervals do not because of the large (infinite) sample size the sample statistic will converge to the population parameter which is a fixed number, i.e., it has no sample-to-sample variation. One can think of a CI as a PI but with the sample size going to infinity. Thus, PIs are always wider than CIs because of the smaller sample size used in the simulations for each trial. In other words a PI will converge to a CI as the sample size gets larger. As Jacob indicates, the internal VPC is a special case where the prediction interval is not to make inference for a future trial but to assess the predictive performance of the current trial/data used to develop the model. In this setting the machinery for the stochastics simulations for VPCs is the same except we don't account for parameter uncertainty because we are not using these intervals to make inference for a future trial. The number of replicated trials of your observed data for the VPCs should be guided by the percentiles you want to summarize for these prediction (VPC) intervals. The more extreme percentiles you want to quantify the larger the number of replicate trials. Again, for VPC intervals constructed based on the 5th and 95th percentiles (i.e., inner 90% range) 1000 replicated trials would be reasonable. For more information about sample size and trial replication, and the distinction between CIs, PIs, and VPCs, see the following articles: Hu, C. "Variability and uncertainty: interpretation and usage of pharmacometrics simulations and intervals." JPP 2022;49:487-481. Kowalski, K.G. "Integration of Pharmacometric and Statistical Analyses Using Clinical Trial Simulations to Enhance Quantitative Decision Making in Clinical Drug Development." Stats in Biopharm Res 2019;11:85-103. Kind regards, Ken Kenneth G. Kowalski President Kowalski PMetrics Consulting, LLC Email: <mailto:[email protected]> [email protected] Cell: 248-207-5082