Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications

Abstract

Nonlinear mixed effects models have become a standard platform for analysis when data is in the form of continuous and repeated measurements of subjects from a population of interest, while temporal profiles of subjects commonly follow a nonlinear tendency. While frequentist analysis of nonlinear mixed effects models has a long history, Bayesian analysis of the models has received comparatively little attention until the late 1980s, primarily due to the time-consuming nature of Bayesian computation. Since the early 1990s, Bayesian approaches for the models began to emerge to leverage rapid developments in computing power, and have recently received significant attention due to (1) superiority to quantify the uncertainty of parameter estimation; (2) utility to incorporate prior knowledge into the models; and (3) flexibility to match exactly the increasing complexity of scientific research arising from diverse industrial and academic fields. This review article presents an overview of modeling strategies to implement Bayesian approaches for the nonlinear mixed effects models, ranging from designing a scientific question out of real-life problems to practical computations.

A review of mechanistic learning in mathematical oncology

“…In oncology, data-driven approaches have previously contributed substantially to scientific progress and process automation ( 35 ). To name just a few examples, (un-)supervised machine learning has greatly supported areas of drug response prediction ( 36 , 37 ) and molecular tumor subtype identification ( 38 , 39 ), whereas generative models and deep learning have revolutionized computer vision tasks such as volumetric tumor segmentation ( 40 , 41 ), image-based outcome predictions ( 42 , 43 ) and automated intervention planning.…”

“…Interestingly, hierarchical models have the potential to benefit from more sophisticated data-driven approaches to integrate high-throughput data, such as omics or imaging ( 8 ). This can be done by changing the linear covariate model with more complex machine learning algorithms able to capture complex relations between the parameters of the individual-level model and the high dimensional covariates ( 124 , 125 ), and/or by using Bayesian inference ( 38 ).…”

Using Bayesian statistics in confirmatory clinical trials in the regulatory setting: a tutorial review

“…Possible sources of prior information include: clinical trials conducted overseas, patient registries, clinical data on very similar products, and pilot studies. Recently, there has been breakthrough development of informative prior distribution that enables incorporating the information from previous trials, and eventually reducing sample size of a new trial, while providing appropriate mechanism of discounting [ 81 – 84 ]. We provide details on the formulation of an informative prior and relevant regulatory considerations in External data borrowing section.…”

“…Here, we illustrate the simplest form of the Bayesian multiplicity adjustment method using Bayesian hierarchical modeling. [ 83 , 84 , 146 , 148 ]. Bayesian hierarchical modeling is a specific Bayesian methodology that combines results from multiple arms or studies to obtain estimates of safety and effectiveness parameters [ 149 ].…”

A review of the Bayesian approach with the MCMC and the HMC as a competitor of classical likelihood statistics for pharmacometricians

“…Statistical inference on the clinical trial data collected from subjects among vast populations is a routine task in pharmacometrics. Even a simple ‘PK analysis’ with only one compartment involves a complicated nonlinear mixed effects model with multiple parameters expressing the human body in probability [ 1 2 3 ]. To fit the non-linear mixed effects models, the classical statistical inference linearizes the non-linear models based on the first-order or second-order Taylor approximations and looks for the estimators based on the restricted likelihood estimation (REML) which involves complicated high dimensional integration [ 2 3 4 5 ].…”