Likelihood-ratio test statistic for the finite-sample case in nonlinear ordinary differential equation models

Likelihood ratios are frequently utilized as basis for statistical tests, for model selection criteria and for assessing parameter and prediction uncertainties, e.g. using the profile likelihood. However, translating these likelihood ratios into p-values or confidence intervals requires the exact form of the test statistic's distribution. The lack of knowledge about this distribution for nonlinear ordinary differential equation (ODE) models requires an approximation which assumes the so-called asymptotic setting, i.e. a sufficiently large amount of data. Since the amount of data from quantitative molecular biology is typically limited in applications, this finite-sample case regularly occurs for mechanistic models of dynamical systems, e.g. biochemical reaction networks or infectious disease models. Thus, it is unclear whether the standard approach of using statistical thresholds derived for the asymptotic large-sample setting in realistic applications results in valid conclusions. In this study, empirical likelihood ratios for parameters from 19 published nonlinear ODE benchmark models are investigated using a resampling approach for the original data designs. Their distributions are compared to the asymptotic approximation and statistical thresholds are checked for conservativeness. It turns out, that corrections of the likelihood ratios in such finite-sample applications are required in order to avoid anti-conservative results. Statistical methods based on the likelihood ratio are ubiquitous in mathematical modelling in systems biology. For example confidence intervals of estimated parameters rely on the statistical properties of the likelihood-ratio test. However, it is often overlooked that these intervals sizes rely on assumptions on the amounts of data, which are regularly violated in typical applications in systems biology. By checking the appropriateness of these assumptions in models from the literature, this study shows that in a surprisingly large fraction of confidence intervals might be too small. Using a geometric interpretation of parameter estimation in the so-called data space, it is motivated why these issues appear and how they depend on the identifiability of the model parameters. In order to avoid such problematic situations, this work makes suggestions on how to adapt the statistical threshold values for likelihood-ratio test. By this, it can be assured that valid statistical conclusions are drawn from the analysis, also in situations where only smaller data sets are available. Such corrections yield for example more conservative confidence interval sizes and thus decrease a potential underestimation of the parameter uncertainty.