Conducting and Evaluating Multilevel Studies: Recommendations, Resources, and a Checklist

Multilevel methods allow researchers to investigate relationships that expand across levels (e.g., individuals, teams, and organizations). The popularity of these methods for studying organizational phenomena has increased in recent decades. Methodologists have examined how these methods work under different conditions, providing an empirical base for making sound decisions when using these methods. In this article, we provide recommendations, tools, resources, and a checklist that can be useful for scholars involved in conducting or assessing multilevel studies. The focus of our article is on two-level designs, in which Level-1 entities are neatly nested within Level-2 entities, and top-down effects are estimated. However, some of our recommendations are also applicable to more complex multilevel designs.

On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model

The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike stochastic differential equation, the stochastic Volterra equation is extremely computationally expensive to simulate. In other words, it is difficult to compute option prices under the rough Heston model by conventional Monte Carlo simulation. In this paper, we prove that Euler’s discretization method for the stochastic Volterra equation with non-Lipschitz diffusion coefficient error[|Vt−Vtn|p] is finitely bounded by an exponential function of t. Furthermore, the weak error |error[Vt−Vtn]| and convergence for the stochastic Volterra equation are proven at the rate of O(n−H). In addition, we propose a mixed Monte Carlo method, using the control variate and multilevel methods. The numerical experiments indicate that the proposed method is capable of achieving a substantial cost-adjusted variance reduction up to 17 times, and it is better than its predecessor individual methods in terms of cost-adjusted performance. Due to the cost-adjusted basis for our numerical experiment, the result also indicates a high possibility of potential use in practice.

Sino–German Computational and Applied Mathematics

This short article serves as an epilog of the thirteen preceding papers in this special issue of CMAM. All contributions are authored by participants of the 7th Sino–German Workshop on Computational and Applied Mathematics at the Kiel University. The topics cover fourth-order problems, solvers and multilevel methods, a posteriori error control and adaptivity, and data science.

An Optimal Multilevel Method with One Smoothing Step for the Morley Element

In this paper, a class of multilevel preconditioning schemes is presented for solving the linear algebraic systems resulting from the application of Morley nonconforming element approximations to the biharmonic Dirichlet problem. Based on an appropriate space splitting of the finite element spaces associated with the refinements and the abstract Schwarz framework, we prove that the proposed multilevel methods with one smoothing step are optimal, i.e., the convergence rate is independent of the mesh sizes and mesh levels. Moreover, the computational complexity is also optimal since the smoothers are performed only once on each level in the algorithm. Numerical experiments are provided to confirm the optimality of the suggested methods.

Joint inversion of high-frequency induction and lateral logging sounding data in earth models with tilted principal axes of the electrical resistivity tensor

In this article, we are the first to formulate the direct and inverse problems of resistivity logging on determining the components of the electrical resistivity tensor of rocks from a set of high-frequency induction and lateral logging sounding measurements. Using a finite element approximation, high-order hierarchical basis functions, computationally efficient multilevel methods and a multistart algorithm with the DFO-LS local optimization method, we investigate the capability of reconstructing the horizontal and vertical resistivity components, as well as the tilt of the resistivity tensor principal axes with regard to the study of complex geological objects. A separate consideration is given to a realistic generalized geoelectric model of the unique hydrocarbon source with hard-to-recover reserves, the Bazhenov Formation.

Multilevel Methods and Statistics: The Next Frontier

The purpose of this article is to take stock of extant multilevel methodological and statistical work and highlight needed areas for future research. A basic overview of the history and progression of multilevel methods and statistics in the organizational sciences is provided, as well as a discussion of recent developments to summarize the current state of the science. The eight articles in the current feature topic are also summarized and integrated to depict several themes and directions for the next wave of multilevel methods and statistics. Last, to highlight what still needs to be accomplished in the field, several unresolved issues and future research topics are noted and an agenda related to future multilevel work is discussed.

Fear of crime examined through diversity of crime, social inequalities, and social capital: An empirical evaluation in Peru

Latin America is a violent region where fear of crime is well spread but still not fully understood. Using multilevel methods for a large and subnational representative household survey (N = 271,022), we assess the determinants of fear of crime in Peru, the country with the highest fear of crime and crime victimization in the region. Our results show that body-aimed victimization (physical or sexual abuse from a member of their household, and sexual offenses) is the strongest driver of fear of crime, even higher than armed victimization. Moreover, safety measures based on social capital are negatively related to fear of crime, suggesting that they are palliatives rather than real protections. Finally, our study shows that people in a higher socioeconomic status are more likely to fear more because they have more (resources) to lose. Policy implications address Latin America as a whole and punitive policies against crime are common in the region, while evidence-based decisions are scarce.