  Throughout the life sciences we routinely seek to interpret measurements and
observations using parameterised mechanistic mathematical models. A fundamental
and often overlooked choice in this approach involves relating the solution of
a mathematical model with noisy and incomplete measurement data. This is often
achieved by assuming that the data are noisy measurements of the solution of a
deterministic mathematical model, and that measurement errors are additive and
normally distributed. While this assumption of additive Gaussian noise is
extremely common and simple to implement and interpret, it is often unjustified
and can lead to poor parameter estimates and non-physical predictions. One way
to overcome this challenge is to implement a different measurement error model.
In this review, we demonstrate how to implement a range of measurement error
models in a likelihood-based framework for estimation, identifiability
analysis, and prediction, called Profile-Wise Analysis. This frequentist
approach to uncertainty quantification for mechanistic models leverages the
profile likelihood for targeting parameters and understanding their influence
on predictions. Case studies, motivated by simple caricature models routinely
used in systems biology and mathematical biology literature, illustrate how the
same ideas apply to different types of mathematical models. Open-source Julia
code to reproduce results is available on GitHub.
