Achievement tests are used to characterize the proficiency of higher-education students. Item response theory (IRT) models are applied to these tests to estimate the ability of students (as latent variable in the model). In order for quality IRT parameters to be estimated, especially ability parameters, it is important that the appropriate number of dimensions is identified. Through a case study, based on a statistics exam for students in higher education, we show how dimensions and other model parameters can be chosen in a real situation. Our model choice is based both on empirical and on background knowledge of the test. We show that dimensionality influences the estimates of the item-parameters, especially the discrimination parameter which provides information about the quality of the item. We perform a simulation study to generalize our conclusions. Both the simulation study and the case study show that multidimensional models have the advantage to better discriminate between examinees. We conclude from the simulation study that it is safer to use a multidimensional model compared to a unidimensional if it is unknown which model is the correct one.

Item calibration is a technique to estimate characteristics of questions (called items) for achievement tests. In computerized tests, item calibration is an important tool for maintaining, updating and developing new items for an item bank. To efficiently sample examinees with specific ability levels for this calibration, we use optimal design theory assuming that the probability to answer correctly follows an item response model. Locally optimal unrestricted designs have usually a few design points for ability. In practice, it is hard to sample examinees from a population with these specific ability levels due to unavailability or limited availability of examinees. To counter this problem, we use the concept of optimal restricted designs and show that this concept naturally fits to item calibration. We prove an equivalence theorem needed to verify optimality of a design. Locally optimal restricted designs provide intervals of ability levels for optimal calibration of an item. When assuming a two-parameter logistic model, several scenarios with D-optimal restricted designs are presented for calibration of a single item and simultaneous calibration of several items. These scenarios show that the naive way to sample examinees around unrestricted design points is not optimal.

The importance of large scale achievement tests, like national tests in school, eligibility tests for university, or international assessments for evaluation of students, is increasing. Pretesting of questions for the above mentioned tests is done to determine characteristic properties of the questions by adding them to an ordinary achievement test. If computerized tests are used, it has been shown using optimal experimental design methods that it is efficient to assign pretest questions to examinees based on their abilities. We can consider the specific distribution of abilities of the available examinees and apply restricted optimal designs.A previously used algorithm optimizes the criterion directly. We develop here a new algorithm which builds on an equivalence theorem. It discretizises the design space with the possibility to change the grid during the run, makes use of an exchange idea and filters computed designs. We illustrate how the algorithm works in some examples and how convergence can be checked. We show that this new algorithm can be used flexibly even if different models are assumed for different questions.

Multidimensional achievement tests are recently gaining more importance in educational and psychological measurements. For example, multidimensional diagnostic tests can help students to determine which particular domain of knowledge they need to improve for better performance. To estimate the characteristics of candidate items (calibration) for future multidimensional achievement tests, we use optimal design theory. We generalize a previously developed exchange algorithm for optimal design computation to the multidimensional setting. We also develop an asymptotic theorem saying which item should be calibrated by examinees with extreme abilities. For several examples, we compute the optimal design numerically with the exchange algorithm. We see clear structures in these results and explain them using the asymptotic theorem. Moreover, we investigate the performance of the optimal design in a simulation study.