Pharmacology, medicine, psychology, and the animal sciences all employ repeated measurement designs (RMDs). However, RMDs may experience carryover effects, which are the primary cause of bias in treatment effect estimation. In order to eliminate the carryover effects for odd v (the number of treatments), minimal circular balanced and strongly balanced repeated measurement designs (RMDs) are the ones that should be used. The minimal circular partially balanced and weakly balanced RMDs are used for even v. In order to obtain these important classes of minimal circular RMDs in periods of equal, two, and three different sizes, an R-based algorithm is developed in this article. The newly developed algorithm has made so simple the construction of balanced RMDs and their generalized classes. As a result, it is a novel piece of research.

In experiments related to agriculture, psychology, medicine, animal sciences, pharmacology, and other fields, repeated measurements designs (RMDs) are utilized which are economical design. However, the use of these designs may result in carryover effects, which are the primary cause of bias. Balanced or strongly balanced RMDs can control these effects. Strongly balanced RMDs estimate the direct effects and carryover effects independently; therefore, these designs have an edge over the balanced RMDs. In this article, therefore, some general procedures are presented to obtain minimal circular strongly and nearly strongly balanced RMDs in periods of k different sizes with high efficiency of Separability and of carryover effects.

Minimal circular balanced and strongly balanced neighbor designs (MCBNDs & MCSBNDs) are popular to control neighbor effects and to estimate neighbor and direct effects independently. In this article, an R-coded algorithm is presented to obtain efficient MCBNDs and MCSBNDs for odd v (number of treatments), and its four generalized classes for v even. Developed algorithm has made the construction of the efficient circular designs robust to neighbor effects so easy in blocks of equal and unequal sizes.

The experiments where response of a treatment (direct effect) is affected by the treatment(s) applied in neighboring units, neighbor designs are used to balance the neighbor effects. Being the economical, minimal neighbor designs are preferred by the experimenters. Minimal circular neighbor designs could not be constructed for almost every case of v even, where v is number of treatments. For v even, minimal circular generalized neighbor designs are preferred. In this article, algorithms are developed to obtain minimal circular generalized neighbor designs in which (a) v/2 of the unordered pairs, and (b) 3v/2 of the unordered pairs, do not appear as neighbor whereas the remaining ones appear once. These algorithms are also coded with R-language. 

Residual effects in repeated measurements design (RMDs) leads to wrong estimation of direct treatment effects. Minimal strongly balanced RMDs are preferred to balance out the residual effects. The partially strongly balanced designs form an important family of RMDs which provide designs where minimal strongly balanced RMDs do not exist. In this article, a general construction of efficieint circular partially strongly-balanced RMDs is given in periods of k different sizes which produces these designs in periods of equal sizes, two different sizes, three different sizes, …, by putting k = 1, 2, 3, …, respectively. 

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. 

Minimal neighbor designs are useful to balance out neighbor effects economically. The method of cyclic shifts provides the construction of these minimal designs in circular blocks only for v odd, where v is the number of treatments to be compared. Minimal circular weakly balanced neighbor designs are used for v even. In this article, two classes of minimal circular weakly balanced neighbor designs are constructed for v even. In class I, v/2 of all unordered pairs of two distinct treatments appear twice as neighbors while the remaining ones appear once. In class II, 3v/2 of all unordered pairs appear twice as neighbors.

Minimal neighbor balanced designs are economical, therefore, these are preferred by the experimenters to minimize the bias due to neighbor effects. Minimal circular balanced neighbor designs cannot be constructed for almost every case of v even, where v is number of the treatments to be compared. For v even, the circular GN₂-designs in which each treatment appears exactly once as neighbors with all other treatments except the one with which it appears twice, are considered the better alternate to the minimal balanced neighbor designs. In this article, an algorithm is developed to generate the circular GN₂-designs for v even which can be converted directly into minimal circular balanced and strongly balanced neighbor designs. This algorithm is also coded with R.

The response of a treatment (direct effect) applied on a given unit may be affected by the treatments applied to its neighboring units (neighbor effects). Neighbor designs are considered robust to neighbor effects. Minimal neighbor designs are economical, therefore, these are preferred by the experimenters. Method of cyclic shifts (Rule I) provides the minimal neighbor designs for odd v (number of treatments). Method of cyclic shifts (Rule II) provides the minimal circular Quasi Rees neighbor designs for v even which are considered to be the good alternate to the minimal neighbor designs. In this article, for every case of v even, minimal circular Quasi Rees neighbor designs are constructed in such a way that these designs can also be converted directly into minimal circular balanced and strongly balanced neighbor designs.

Repeated measurements designs (RMDs) are very useful and economical but unfortunately, with the use of RMDs, a major source of bias is arisen, that is the carry over effect. Minimal designs which are strongly and nearly strongly balanced, are preferred to estimate the direct and carry over effects independently. In this article, some new classes of minimal circular nearly strongly balanced RMDs are constructed in periods of two and three different sizes which can be converted directly into minimal circular balanced and minimal strongly balanced which are highly efficient.