Spatial Confounding in Hurdle Multilevel Beta Models: the Case of the Brazilian Mathematical Olympics for Public Schools

Abstract

Summary Among the many disparities for which Brazil is known is the difference in performance across students who attend the three administrative levels of Brazilian public schools: federal, state and municipal. Our main goal is to investigate whether student performance in the Brazilian Mathematical Olympics for Public Schools is associated with school administrative level and student gender. For this, we propose a hurdle hierarchical beta model for the scores of students who took the examination in the second phase of these Olympics, in 2013. The mean of the beta model incorporates fixed and random effects at the student and school levels. We explore different distributions for the random school effect. As the posterior distributions of some fixed effects change in the presence, and distribution, of the random school effects, we also explore models that constrain random school effects to the orthogonal complement of the fixed effects. We conclude that male students perform slightly better than female students and that, on average, federal schools perform substantially better than state or municipal schools. However, some of the best municipal and state schools perform as well as some federal schools. We hypothesize that this is due to individual teachers who successfully motivate and prepare their students to perform well in the mathematical Olympics.

Robust Modeling for Continuous Bounded Spatial Data

“…The second example compares the robustness of the proposed GKR and GJtR models in estimation and prediction against skewness and outliers to the conventional GBR model, using the Brazilian Mathematical Olympiad of Public Schools (OBMEP) data presented by Pereira et al (2020).…”

“…In the second phase, students take a discursive test containing six questions worth 20 points each, resulting in a final score within the closed interval of [0, 120]. Pereira et al (2020) examined the performance of students from the state of Minas Gerais in the OBMEP 2013, analyzing data at the student and school levels. Their study involved a sample of 2063 students from 212 schools.…”

“…Prates et al (2015) and Bevilacqua et al (2024) employed copula-based approaches and introduced spatial beta regression models. Pereira et al (2020) proposed a hurdle multilevel beta model, while Tang et al (2023) introduced a zero-inflated spatial beta regression model.…”

On the Effects of Spatial Confounding in Hierarchical Models

“…Paddock et al (2016) fitted standard conditional autoregression and RSR models to longitudinal data and observed that RSR provided more precise estimates of regression coefficients of the cluster‐level covariates when compared with standard conditional autoregression. However, as pointed out by Pereira et al (2020), in the RSR model, they considered the cluster spatial random effects to be constrained only to the orthogonal complement of the cluster‐level fixed effects. There is no reason to expect that, in this case, the resultant random effect will have variation restricted to the orthogonal complement of all covariates available.…”

A spectral adjustment for spatial confounding

“…Indeed, in recent work, Khan and Calder (2020) showed that the variants of RSR often times are inferior to regression models that completely ignore spatial dependence when coefficient estimation is of interest. Therefore, proper adjustment for spatial confounding continues to be an important and open problem as evidenced by the many contexts in which it is being addressed (Hefley et al 2017;Pereira et al 2020;Azevedo et al 2020;Nobre et al 2020).…”