The importance of proper designs for benchmark studies in computational biology has been discussed in several publications. The absence of high-performing software implementations seems to be a major reason why ODE-based modeling is not yet a routinely applied computational approach for analyzing experimental data. Although parameter estimation is a central task of modeling, the lack of reliable computational approaches for fitting is still a bottleneck in systems biology. In addition, the optimization problem needs to be defined in terms of initialization, search space, termination criteria, and hyperparameters that set up and configure the numerical algorithms. Such optimization-based fitting of a model requires the selection of a generic numerical optimization approach as a core algorithm. Both approaches coincide with normally distributed measurement errors. In most cases, parameter estimation is performed by the optimization of a suitable objective function such as minimization of the sum of squared residuals for least squares estimation or maximization of the likelihood for maximum likelihood estimation. Application-specific calibration of the models is therefore required which corresponds to the estimation of these unknown parameters based on experimental data. Hence, they are represented as unknown parameters in mathematical models. Typical parameters in systems biology such as the abundances of compounds or the strengths and velocities of biochemical interactions are typically context-dependent, i.e., they vary between species, tissues, and cell types. ![]() In this article, I focus on the optimization-based fitting of these models although many aspects are general and also apply to other modeling types and approaches. In the BioModels Database, currently, 83% of all models which are uniquely assigned to a modeling approach are ODE models. Most frequently, ordinary differential equation models (ODEs) are applied because they enable a non-discretized description of the dynamics of a system and allow for quantitative evaluation of experimental data including statistical interpretations in terms of confidence and significance. Depending on the questions of interest and on the amount of available data, the type of models and the level of detail vary. A broad range of mathematical models is applied in systems biology. Detailed information on the creation of the benchmarks is provided in Chapter 14 of Methods and Procedures in TIMSS 2015 at and of Methods and Procedures in TIMSS Advanced 2015 at. See Exhibit 1 for TIMSS and Exhibit 2 for TIMSS Advanced in the Highlights of TIMSS and TIMSS Advanced 2015. The experts then provide a summary description of performance at each anchor point leading to a content-referenced interpretation of the achievement results. These experts focus on the content of each item and describe the kind of knowledge demonstrated by students answering the item correctly. To interpret the content of anchored items, these items are grouped by content area within benchmarks and reviewed by mathematics and science experts. The content of these items describes what students at each benchmark level of achievement know and can do. Once benchmark scores have been chosen, items are identified that students are likely to score highly on. ![]() Scale anchoring involves selecting benchmarks (scale points) on the TIMSS and TIMSS Advanced achievement scales to be described in terms of student performance. TIMSS Advanced established similar benchmarks for advanced mathematics and physics, but did not include the Low International Benchmark because TIMSS Advanced assesses a highly select population of students. To describe student performance at various points along the TIMSS mathematics and science achievement scales, TIMSS uses scale anchoring to summarize and describe student achievement at four points on the mathematics and science scales- Advanced (625), High (550), Intermediate (475), and Low (400) international benchmarks. Similarly, international benchmarks were developed for the TIMSS Advanced physics and advanced mathematics scales. International benchmarks for achievement were developed in an attempt to provide a concrete interpretation of what the scores on the TIMSS mathematics and science achievement scales mean (for example, what it means to have a scale score of 555 or 480).
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