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. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.

Keywords: cluster studies; intra-cluster correlation; optimal design; stepped-wedge designs.

© 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.

Figures

Figure 1

Schematic for some designs with eight clusters. The horizontal direction represents time and the total duration of the study (which is the same for each design) is equally divided between the columns. The rows represent clusters. So each cell represents the observation‐times in a single cluster over one fixed time period. Treated cells are denoted by 1, and Controls by 0.

Figure 2

The cluster‐mean correlation (R) as a function of cluster‐size (M = Tm) in cross‐sectional studies with individual‐observation ICC (ηC) = 0.001, 0.01, 0.05, 0.10, 0.20.

Figure 3

Precision–ratio plots. The relative precision of some stepped designs compared to the cross‐over design is plotted against the cluster‐mean correlation parameter R. Designs 1a to 1e are those illustrated for eight clusters in Figure 1. The Modified SW design corresponds to two copies of the middle four clusters in Figure 1e.

Figure 4

A best balanced design. The boundary line (of slope R) divides the treated (●) from the untreated (○) points, and contains the uptake points in the ‘stepped‐wedge’ clusters. If T/(K − 2P) is an even integer, the design is a Hybrid design as defined above in section 2.

Figure 5

BBDs for 10 clusters over six time‐points. The boxed and circled points in (b) and (f) relate to overall optimal designs for particular R‐values as discussed in the text.

Figure 6

The relative efficiency of cluster designs for large studies as a function of the cluster‐mean correlation R. Three feasible (and admissible) stepped designs are shown: PD, SW (solid lines) and Minimax – i.e. 0.634H∞ – (broken line). The shaded region is prohibited by the irreversibility constraint and bounded by the quadratic curve 1−R+13R2.

Figure 7

A near‐Minimax design with 11 (groups of) clusters (row 4 from Table 2). It achieves at least 99.25% of the precision of the large‐study minimax design (0.634H∞) for any value of R.