Benchmarking plays a relevant role in performance analysis, and statistical methods can be fruitfully exploited for its aims. While clustering, regression, and frontier analysis may serve some benchmarking purposes, we propose to consider archetypal analysis as a suitable technique. Archetypes are extreme points that synthesize data and that, in our opinion, can be profitably used as benchmarks. That is, they may be viewed as key reference performers in the comparison process. We suggest a three-step data driven benchmarking procedure, which enables users: (i) to identify some reference performers, (ii) to analyze their features, (iii) to compare observed performers with them. An exploratory point of view is preferred, and graphical devices are adopted throughout the procedure. Finally, our approach is presented by means of an illustrative example based on The Times league table of the world top 200 universities

Directional data are constrained to lie on the unit sphere of ℝq for some q ≥ 2. To address the lack of a natural ordering for such data, depth functions have been defined on spheres. However, the depths available either lack flexibility or are so computationally expensive that they can only be used for very small dimensions q. In this work, we improve on this by introducing a class of distance‐based depths for directional data. Irrespective of the distance adopted, these depths can easily be computed in high dimensions too. We derive the main structural properties of the proposed depths and study how they depend on the distance used. We discuss the asymptotic and robustness properties of the corresponding deepest points. We show the practical relevance of the proposed depths in two applications, related to (i) spherical location estimation and (ii) supervised classification. For both problems, we show through simulation studies that distance‐based depths have strong advantages over their competitors. The Canadian Journal of Statistics 46: 593–609; 2018 © 2018 Société statistique du Canada

Performance data are usually collected in order to build well-defined performance indicators. Since such data may conceal additional information, which can be revealed by secondary analysis, we believe that mining of performance data may be fruitful. We also note that performance databases usually contain both qualitative and quantitative variables for which it may be inappropriate to assume some specific (multivariate) underlying distribution. Thus, a suitable technique to deal with these issues should be adopted. In this work, we consider nonlinear principal component analysis (PCA) with optimal scaling, a method developed to incorporate all types of variables, and to discover and handle nonlinear relationships. The reader is offered a case study in which a student opinion database is mined. Though generally gathered to provide evidence of teaching ability, they are exploited here to provide a more general performance evaluation tool for those in charge of managing universities. We show how nonlinear PCA with optimal scaling applied to student opinion data enables users to point out some strengths and weaknesses of educational programs and services within a university. MINING PERFORMANCE DATA THROUGH NLPCA WITH OPTIMAL SCALING 97 Figure 3. Category point biplot of the NLPCA component loadings and the category points of the University Program variable. Note: Variables are identified as follows: Program Organization = Program organization of teaching; Program Workload = Program Workload; Adequacy of lecture hall = Adequacy of lecture hall; Work/Credits = Workload-credit ratio; Keep sched hours = Keep scheduled hours; Readings = Prescribed reading list; Prev = Student's previous knowledge of the topic; Ex = Clear exam rules; Sched = On schedule with program; Ov = Overall Class Satisfaction; Int = Student interest in topic; Mot = Teacher ability to motivate; In = Availability of lecturer inside class; Out = Availability of lecturer outside class; Clarity = Clarity of teaching; and Class Size = Class Size. The categories of the supplementary variable University Program are identified as follows: ECON = Economics; PSY = Psychology; LAW = Law; ENG=Engineering; LANG=Foreign Languages; ARTS=Arts; and MATH=Mathematics.groups of variables in Figure 3, together with Workload-credit Ratio and Keep Scheduled Hours. These variables seem to indicate the more organizational aspects of the lectures, whereas the others regard more the content of the lecture or teacher quality. This difference may explain why these three variables do not clearly belong to one of the two groups of variables. The supplementary variable Class Size appears to be negatively related to the teacher/lecture evaluation variables, indicating that students are to some extent less satisfied when classrooms are crowded. This is in line with the literature, which indicates that better teaching evaluation is associated with a smaller class size (see e.g.[47], among many others). We note, in addition, that there is a strong negative relationship between this vari...

The box-and-whiskers plot is an extraordinary graphical tool that provides a quick visual summary of an observed distribution. In spite of its many extensions, a really suitable boxplot to display circular data is not yet available. Thanks to its simplicity and strong visual impact, such a tool would be especially useful in all fields where circular measures arise: biometrics, astronomy, environmetrics, Earth sciences, to cite just a few. For this reason, in line with Tukey's original idea, a Tukey-like circular boxplot is introduced. Several simulated and real datasets arising in biology are used to illustrate the proposed graphical tool.

The paper investigates the link between student relations and their performances at university. A social influence mechanism is hypothesized as individuals adjusting their own behaviors to those of others with whom they are connected. This contribution explores the effect of peers on a real network formed by a cohort of students enrolled at a graduate level in an Italian University. Specifically, by adopting a network effects model, the relation between interpersonal networks and university performance is evaluated assuming that student performance is related to the performance of the other students belonging to the same group. By controlling for individual covariates, the network results show informal contacts, based on mutual interests and goals, are related to performance, while formal groups formed temporarily by the instructor have no such effect