School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
Department of Engineering Science and Biomedical Engineering, University of Auckland, Auckland, New Zealand.
Bull Math Biol. 2024 May 27;86(7):80. doi: 10.1007/s11538-024-01294-0.
Many commonly used mathematical models in the field of mathematical biology involve challenges of parameter non-identifiability. Practical non-identifiability, where the quality and quantity of data does not provide sufficiently precise parameter estimates is often encountered, even with relatively simple models. In particular, the situation where some parameters are identifiable and others are not is often encountered. In this work we apply a recent likelihood-based workflow, called Profile-Wise Analysis (PWA), to non-identifiable models for the first time. The PWA workflow addresses identifiability, parameter estimation, and prediction in a unified framework that is simple to implement and interpret. Previous implementations of the workflow have dealt with idealised identifiable problems only. In this study we illustrate how the PWA workflow can be applied to both structurally non-identifiable and practically non-identifiable models in the context of simple population growth models. Dealing with simple mathematical models allows us to present the PWA workflow in a didactic, self-contained document that can be studied together with relatively straightforward Julia code provided on GitHub . Working with simple mathematical models allows the PWA workflow prediction intervals to be compared with gold standard full likelihood prediction intervals. Together, our examples illustrate how the PWA workflow provides us with a systematic way of dealing with non-identifiability, especially compared to other approaches, such as seeking ad hoc parameter combinations, or simply setting parameter values to some arbitrary default value. Importantly, we show that the PWA workflow provides insight into the commonly-encountered situation where some parameters are identifiable and others are not, allowing us to explore how uncertainty in some parameters, and combinations of parameters, regardless of their identifiability status, influences model predictions in a way that is insightful and interpretable.
许多在数学生物学领域中常用的数学模型都涉及到参数不可识别性的挑战。即使是相对简单的模型，也经常会遇到实际的不可识别性问题，即数据的质量和数量无法提供足够精确的参数估计。特别是，经常会遇到一些参数可识别而另一些参数不可识别的情况。在这项工作中，我们首次将一种新的基于似然的工作流程，称为 Profile-Wise Analysis（PWA），应用于不可识别的模型。PWA 工作流程在一个简单实现和解释的统一框架中解决了可识别性、参数估计和预测问题。之前的工作流程实现仅处理理想化的可识别问题。在本研究中，我们说明了如何将 PWA 工作流程应用于简单种群增长模型中结构不可识别和实际不可识别的模型。处理简单的数学模型使我们能够在一个教学式的、自包含的文档中呈现 PWA 工作流程，该文档可以与提供在 GitHub 上的相对简单的 Julia 代码一起学习。使用简单的数学模型可以使 PWA 工作流程的预测区间与黄金标准全似然预测区间进行比较。总之，我们的例子说明了 PWA 工作流程如何为我们提供了一种系统的方法来处理不可识别性问题，特别是与其他方法相比，例如寻找特定的参数组合，或者简单地将参数值设置为一些任意的默认值。重要的是，我们表明 PWA 工作流程提供了对常见情况的深入了解，即一些参数是可识别的，而另一些参数是不可识别的，这使我们能够探索一些参数的不确定性，以及参数的组合，无论其可识别性状态如何，如何以有洞察力和可解释的方式影响模型预测。
