The mobile Boolean model: an overview and further results

This paper offers an overview of the mobile Boolean stochastic geometric model which is a time-dependent version of the ordinary Boolean model in a Euclidean space of dimension $d$. The main question asked is that of obtaining the law of the detection time of a fixed set. We give various ways of thinking about this which result into some general formulas. The formulas are solvable in some special cases, such the inertial and Brownian mobile Boolean models. In the latter case, we obtain some expressions for the distribution of the detection time of a ball, when the dimension $d$ is odd and asymptotics when $d$ is even. Finally, we pose some questions for future research.Comment: 19 page

A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity

We give a short proof of the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity {\em not} involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.Comment: 9 pages, 2 diagram

Incorporating Cost in Power Analysis for Three-Level Cluster Randomized Designs

In experimental designs with nested structures entire groups (such as schools) are often assigned to treatment conditions. Key aspects of the design in these cluster randomized experiments include knowledge of the intraclass correlation structure and the sample sizes necessary to achieve adequate power to detect the treatment effect. However, the units at each level of the hierarchy have a cost associated with them and thus researchers need to decide on sample sizes given a certain budget, when designing their studies. This paper provides methods for computing power within an optimal design framework (that incorporates costs of units in all three levels) for three-level cluster randomized balanced designs with two levels of nesting. The optimal sample sizes are a function of the variances at each level and the cost of each unit. Overall, larger effect sizes, smaller intraclass correlations at the second and third level, and lower cost of level-3 and level-2 units result in higher estimates of power.experimental design, statistical power, optimal sampling

Fixed Effects and Variance Components Estimation in Three-Level Meta-Analysis

Meta-analytic methods have been widely applied to education, medicine, and the social sciences. Much of meta-analytic data are hierarchically structured since effect size estimates are nested within studies, and in turn studies can be nested within level-3 units such as laboratories or investigators, and so forth. Thus, multilevel models are a natural framework for analyzing meta-analytic data. This paper discusses the application of a Fisher scoring method in two- and three-level meta-analysis that takes into account random variation at the second and at the third levels. The usefulness of the model is demonstrated using data that provide information about school calendar types. SAS proc mixed and HLM can be used to compute the estimates of fixed effects and variance components.meta-analysis, multilevel models, random effects

How Consistent Are Class Size Effects?

Evidence from Project STAR has suggested that on average small classes increase student achievement. However, thus far researchers have focused on computing mean differences in student achievement between smaller and larger classes. In this study I focus on the distribution of the small class effects at the school level and compute the inconsistency of the treatment effects across schools. I use data from Project STAR and estimated small class effects for each school on mathematics and reading scores from kindergarten through third grade. The results revealed that school-specific small class effects are both positive and negative and that although students benefit considerably from being in small classes in some schools, in other schools being in small classes is a disadvantage. Small class effects were inconsistent and varied significantly across schools. Full time teacher aide effects were also inconsistent across schools and in some schools students benefit considerably from being in regular classes with a full time aide, while in other schools being in these classes is a disadvantage.small classes, treatment variability, meta-analysis