A novel bootstrap procedure for assessing the relationship between class size and achievement

There is on-going concern about the relationship between class size and achievement for children in their first years of schooling. The Institute of Education's class size project was set up to address this issue and began recruiting in the autumn of 1996. However, because of the non-normality of achievement measures, especially in mathematics, the results have hitherto been presented by using transformed achievement measures. This makes the interpretation difficult for non-statisticians. Ideally, the data would be modelled on the original scale and a bootstrap procedure used to ensure that inferences are robust to non-normality. However, the data are multilevel. In the paper we therefore propose a nonparametric residual bootstrap procedure that is suitable for multilevel models, show that it is consistent and present a simulation study which demonstrates its potential to yield substantial reductions in the difference between nominal and actual confidence interval coverage, compared with a parametric bootstrap, when the underlying distribution of the data is non-normal. We then apply our approach to estimate the relationship between class size and achievement for children in their reception year, after adjusting for other possible determinants.