The generality and easy programmability of modern sampling-based metho
ds for maximisation of likelihoods and summarisation of posterior dist
ributions have led to a tremendous increase in the complexity and dime
nsionality of the statistical models used in practice. However, these
methods can often be extremely slow to converge, due to high correlati
ons between, or weak identifiability of, certain model parameters. We
present simple hierarchical centring reparametrisations that often giv
e improved convergence for a broad class of normal linear mixed models
. In particular, we study the two-stage hierarchical normal linear mod
el, the Laird-Ware model for longitudinal data, and a general structur
e for hierarchically nested linear models. Using analytical arguments,
simulation studies, and an example involving clinical markers of acqu
ired immune deficiency syndrome (AIDS), We indicate when reparametrisa
tion is likely to provide substantial gains in efficiency.