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

Abstract

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Key steps in using mechanistic mathematical models to interpret data often include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. We present a computationally efficient workflow that addresses these steps in a general likelihood-based framework. We first summarise the theoretical background of our workflow for efficient parameter identifiability, parameter estimation, and predictive confidence set construction. A central aspect of the workflow is using profile likelihood to construct profile-wise prediction intervals that propagate confidence sets for model parameters forward to predictions in a way that explicitly isolates how different parameter combinations affect model predictions. We also show how to combine these prediction confidence sets to give an overall prediction confidence set that accounts for all parameters and approximates the full likelihood-based prediction confidence set well. We demonstrate practical aspects of the workflow via three case studies focusing on ordinary differential equation (ODE) mechanistic models with both Gaussian and non-Gaussian noise models. While the three case studies focus on ODE-based models, the workflow applies to other classes of mathematical models, such as partial differential equations and simulation-based stochastic models. We provide open-source software on GitHub to replicate the case studies

Modelling count data with partial differential equation models in biology

“…individual data realisations) for either of the measurement error models . This observation motivates us to consider prediction intervals for data realisations [29,39].…”

“…, S , we evaluate the 5% and 95% quantile of the associated measurement error distribution, which we denote u (s) 0.05 (x i , t j ) and u (s) 0.95 (x i , t j ), respectively. One way of interpreting this is that instead of considering a single mean prediction for each θ like we did in Figure 4, we now compute an individual prediction interval for each parameter value, and we refer to this as a prediction for data realisations [39]. For each x i and t j we take the union over the individual 5% and 95% quantiles, [min(u (s) 0.05 (x i , t j )), max(u (s) 0.95 (x i , t j ))] which gives the (90%) prediction intervals for data realisations in Figure 5.…”

“…Technically, these are a form of tolerance interval [44] which take into account both parameter estimation uncertainty and future data realisation uncertainty and, here, with high (at least 95%) confidence, provide 90% prediction intervals. A Bonferroni approach for turning these into pure (though conservative) prediction intervals is also possible [29,39]. Each panel in Figure 5 The prediction intervals for data realisations in Figure 5 for the additive Gaussian measurement error model are biologically unrealistic since they predict extensive regions where the density is negative across the central region of the experiment at t = 12 and 24 hours.…”

“…While the Fisher-Kolmogorov model and generalisations thereof are widely used to interpret cell biology experiments [4,34], exploring the impact of using different measurement error models is poorly understood for these applications. We consider likelihood-based parameter estimation, parameter identifiability and prediction [30,38,45] using two different measurement error models. In the first instance we implement the standard additive Gaussian noise model, and we then repeat the same procedure using an experimentally-motivated binomial measurement error model.…”

“…More detailed applications of reaction-diffusion PDEs involves keeping track and counting individuals [20,37], often expressing these counts in terms of observed densities by estimating the number of individuals per unit area (or per unit volume, depending on the geometry of the application). With such data it is possible to estimate parameters in the reaction-diffusion PDE, interpret biological processes in terms of those parameter values, and then make predictions and forecasts of future scenarios with the parameterised PDE model [39].…”