Multilevel models with random residual variances for joint modelling school value-added effects on the mean and variance of student achievement

School value-added models are widely applied to study the effects of schools on student achievement and to monitor and hold schools to account for their performances. The traditional model is a multilevel linear regression of student current achievement on student prior achievement, background characteristics, and a school random intercept effect. The predicted random effect aims to measure the mean academic progress students make in each school. In this article, we argue that much is to be gained by additionally studying the variance in student progress in each school. We therefore extend the traditional model to allow the residual variance to vary as a log-linear function of the student covariates and a new school random effect to predict the influence of schools on the variance in student progress. We illustrate this new model with an application to schools in London. Our results show the variance in student progress varies substantially across schools - even after adjusting for differences in the variance in student progress associated with different student groups - and that this variation is predicted by school characteristics. We discuss the implications of our work for research and school accountability.