Set-valued methods for estimation of parameters in population models
Alexander Danis(1), Sebastian Ueckert(2), Andrew C. Hooker(2), Warwick Tucker(1)
(1) Department of Mathematics, Uppsala University, Sweden. (2) Department of Pharmaceutical Biosciences, Uppsala University, Sweden
Objectives: The estimation of parameters with statistical methods, well-known in the pharmacometric community, often rely on assumptions about parameter distributions. Less well-known are the deterministic methods based on set-valued computations  which originate from the work of Moore in the 1960's. Here we present a new method that combine two complementary set operations, expansion and contraction. We apply it to estimate parameters in mixed effects models under simulated noisy time series data and compare the results with those obtained by the well-known NONMEM software package.
Methods: In set-valued methods, the values that a variable can assume is enclosed by a real interval, and all computations are performed with interval arithmetic, ensuring rigorous results. Intervals of possible values of parameters are split into subintervals and the consistency of each such subinterval with the data is examined. Contraction of the parameter and data intervals is obtained by constraint propagation  and expansion of data intervals occurs when the whole parameter search space is inconsistent. This procedure continues until there is a set of values in parameter and data space that are consistent. The sets resulting from the estimation procedure can be transformed into point clouds from which statistical properties such as means and covariances can be retrieved.
Results: The performance of the presented methods were tested on three population models, each containing one individual parameter and two population parameters. One of the models was constructed to be non-identifiable. Cases with poor and rich data, with varying type and amount of noise, and, various population sizes, were considered. For normally distributed noises, the results were compared with those obtained by NONMEM. For all three models, the set valued estimator performed well. NONMEM produced results with comparable accuracy on the identifiable problems, but, could not produce adequate estimates for the non-identifiable model. The consistency approach has the advantage that it does not need distributional information to work properly.
Conclusions: Solution strategies based on set-valued computations can be a complement to traditional estimation methods. They apply directly to raw data and do not require a priori information about parameter distributions. Being set based, they naturally solve non-identifiable estimation problems. Their scope of application include model selection and experimental design .
 Parameter Reconstruction for Biochemical Networks Using Interval Analysis, Warwick Tucker and Vincent Moulton, Reliable Computing, volume 12, 2006.  Parameter estimation in non-linear mixed effects models using interval analysis, Andrew C. Hooker and Warwick Tucker, PAGE 17 (2008) Abstr 1369.  PAGE (2011) Abstr P. Gennemark.