Statistical efficiency and optimal design for stepped cluster studies under linear mixed effects models

Abstract

In stepped cluster designs the intervention is introduced into some (or all) clusters at different times and persists until the end of the study. Instances include traditional parallel cluster designs and the more recent stepped-wedge designs. We consider the precision offered by such designs under mixed-effects models with fixed time and random subject and cluster effects (including interactions with time), and explore the optimal choice of uptake times. The results apply both to cross-sectional studies where new subjects are observed at each time-point, and longitudinal studies with repeat observations on the same subjects. The efficiency of the design is expressed in terms of a 'cluster-mean correlation' which carries information about the dependency-structure of the data, and two design coefficients which reflect the pattern of uptake-times. In cross-sectional studies the cluster-mean correlation combines information about the cluster-size and the intra-cluster correlation coefficient. A formula is given for the 'design effect' in both cross-sectional and longitudinal studies. An algorithm for optimising the choice of uptake times is described and specific results obtained for the best balanced stepped designs. In large studies we show that the best design is a hybrid mixture of parallel and stepped-wedge components, with the proportion of stepped wedge clusters equal to the cluster-mean correlation. The impact of prior uncertainty in the cluster-mean correlation is considered by simulation. Some specific hybrid designs are proposed for consideration when the cluster-mean correlation cannot be reliably estimated, using a minimax principle to ensure acceptable performance across the whole range of unknown values.