Profile-Wise Analysis: A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Developing mechanistic insight by combining mathematical models and experimental data is especially critical in mathematical biology as new data and new types of data are collected and reported. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally-efficient workflow we call Profile-Wise Analysis (PWA) that addresses all three steps in a unified way. Recently-developed methods for constructing ‘profile-wise’ prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters. We then demonstrate how to combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well. Our three case studies illustrate practical aspects of the workflow, focusing on ordinary differential equation (ODE) mechanistic models with both Gaussian and non-Gaussian noise models. While the case studies focus on ODE-based models, the workflow applies to other classes of mathematical models, including partial differential equations and simulation-based stochastic models. Open-source software on GitHub can be used to replicate the case studies.Author summary: Parameter estimation and model prediction are essential steps when mathematical models are used to provide biological insight or to make practical predictions about future scenarios. We present an efficient, unified workflow that addresses parameter identifiability, parameter estimation and model prediction from a likelihood-based frequentist perspective. Our workflow, called Profile-Wise Analysis (PWA), involves constructing ‘profile-wise’ predictions that propagate profile-likelihood-based confidence sets for model parameters to predictions, explicitly isolating how different parameter combinations affect model predictions. Combining profile-wise prediction confidence sets gives an overall prediction confidence set that efficiently approximates the full likelihood-based prediction confidence set. Three case studies, focusing on canonical mathematical models used in biology and ecology, illustrate various aspects of the workflow for commonly-encountered ODE-based mechanistic models with both Gaussian and non-Gaussian measurement error.