Shrinkage Estimation in Multilevel Normal Models

This review traces the evolution of theory that started when Charles Stein in 1955 [In Proc. 3rd Berkeley Sympos. Math. Statist. Probab. I (1956) 197--206, Univ. California Press] showed that using each separate sample mean from $k\ge3$ Normal populations to estimate its own population mean $\mu_i$ can be improved upon uniformly for every possible $\mu=(\mu_1,...,\mu_k)'$. The dominating estimators, referred to here as being "Model-I minimax," can be found by shrinking the sample means toward any constant vector. Admissible minimax shrinkage estimators were derived by Stein and others as posterior means based on a random effects model, "Model-II" here, wherein the $\mu_i$ values have their own distributions. Section 2 centers on Figure 2, which organizes a wide class of priors on the unknown Level-II hyperparameters that have been proved to yield admissible Model-I minimax shrinkage estimators in the "equal variance case." Putting a flat prior on the Level-II variance is unique in this class for its scale-invariance and for its conjugacy, and it induces Stein's harmonic prior (SHP) on $\mu_i$.Comment: Published in at http://dx.doi.org/10.1214/11-STS363 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Trachemys gaigeae

Number of Pages: 6Integrative BiologyGeological Science

Examining and contrasting the cognitive activities engaged in undergraduate research experiences and lab courses

While the positive outcomes of undergraduate research experiences (UREs) have been extensively categorized, the mechanisms for those outcomes are less understood. Through lightly structured focus group interviews, we have extracted the cognitive tasks that students identify as engaging in during their UREs. We also use their many comparative statements about their coursework, especially lab courses, to evaluate their experimental physics-related cognitive tasks in those environments. We find there are a number of cognitive tasks consistently encountered in physics UREs that are present in most experimental research. These are seldom encountered in lab or lecture courses, with some notable exceptions. Having time to reflect and fix or revise, and having a sense of autonomy, were both repeatedly cited as key enablers of the benefits of UREs. We also identify tasks encountered in actual experimental research that are not encountered in UREs. We use these findings to identify opportunities for better integration of the cognitive tasks in UREs and lab courses, as well as discussing the barriers that exist. This work responds to extensive calls for science education to better develop students' scientific skills and practices, as well as calls to expose more students to scientific research.Comment: 11 pages, 3 figure