The impact of iterative removal of low-information cluster-period cells from a stepped wedge design

Background Standard stepped wedge trials, where clusters switch from the control to the intervention condition in a staggered manner, can be costly and burdensome. Recent work has shown that the amount of information contributed by each cluster in each period differs, with some cluster-periods contributing a relatively small amount of information. We investigate the patterns of the information content of cluster-period cells upon iterative removal of low-information cells, assuming a model for continuous outcomes with constant cluster-period size, categorical time period effects, and exchangeable and discrete-time decay intracluster correlation structures. Methods We sequentially remove pairs of “centrosymmetric” cluster-period cells from an initially complete stepped wedge design which contribute the least amount of information to the estimation of the treatment effect. At each iteration, we update the information content of the remaining cells, determine the pair of cells with the lowest information content, and repeat this process until the treatment effect cannot be estimated. Results We demonstrate that as more cells are removed, more information is concentrated in the cells near the time of the treatment switch, and in “hot-spots” in the corners of the design. For the exchangeable correlation structure, removing the cells from these hot-spots leads to a marked reduction in study precision and power, however the impact of this is lessened for the discrete-time decay structure. Conclusions Removing cluster-period cells distant from the time of the treatment switch may not lead to large reductions in precision or power, implying that certain incomplete designs may be almost as powerful as complete designs. Supplementary Information The online version contains supplementary material available at 10.1186/s12874-023-01969-7.


A The general variance expression for incomplete designs
We re-state the following general model for a cross-sectional longitudinal cluster randomised trial with a continuous outcome Y kji for subject i = 1, . . . , m in time period j = 1, . . . , T in cluster k = 1, . . . , K as in Kasza and Forbes [1]: where T k is the number of measurement periods in cluster k, and T is the total number of periods in the complete design.
Collapsing the above model to cluster-period means, LettingȲ k = Ȳ k1 , . . . ,Ȳ kT T , and using V k to denote the covariance matrix ofȲ k , cov(Ȳ k ) = The usual estimate of the parameter vector η = (β 1 , . . . , β T , θ) T is then given byη The design matrix Z can be written as where Z k has dimension T k × T , and X k has dimension T k × 1.
We are interested in obtaining an expression for the variance of the treatment effect estimator, var( θ), the (T + 1) × (T + 1)th entry of (Z T V −1 Z) −1 . Then Note: The Z k , V k , and X k matrices may have different dimensions across the K clusters due to incompleteness, but these matrices will be conformable for a particular cluster k.
The (T + 1) × (T + 1)th entry of (Z T V −1 Z) −1 is given by A 22 − A 21 A −1 11 A 12 −1 . We can then write: Then the covariance matrix ofȲ can be decomposed as Since the omitted and included cell means are organised by cluster, V o has a block diagonal structure, with blocks given by V m k , where m k is the number of cells omitted from each cluster k, and V m k is either the variance of the omitted cell or covariance matrix of the omitted pair of cells from cluster k. Note that since we are considering removing two cells at a time, m k will be 0 if no cells are removed from cluster k, 1 if a single cell is removed, or 2 if the pair of cells belongs to cluster k. The matrix V [o] also has a block diagonal structure, with blocks V [m k ] : the covariance matrix of the included cells of cluster k. The matrix W can be decomposed as . . .
and contains the covariances between the omitted observation(s) in cluster k and all included observations. Due to independence between clusters, the only non-zero entries of C m k will correspond to covariances between omitted and included observations in the same cluster: C m k will contain T k − m k non-zero columns. We denote those non-zero columns as C ̸ =0 m k . The inverse of the covariance matrix, V −1 , is then given by The design matrix Z can be decomposed as where Z o is the 2 × (T + 1) sub-matrix of Z corresponding to the omitted cells, and Z [o] is the (T K − 2) × (T + 1) sub-matrix of Z corresponding to the included cells. Standard matrix algebra gives Thus var(θ) [o] is given by the (T + 1) × (T + 1) th entry of are block diagonal matrices, the matrix D will have the form: where again C ̸ =0 m k contains only the non-zero columns of C m k . We can write the matrixZ o as whereT m k is of dimension T ×m k (or m k column(s) vector of length T ) and represents the modified time effect(s) in cluster k andX m k is of dimension 1 × m k and represents the treatment indicator for the omitted cell in cluster k. Since a pair of cells is being omitted, K k=1 m k = 2. Therefore, Z o will be of dimension (T + 1) × 2.
Then, recalling the expression for Z T V −1 Z given in Section A, an expression for o is available, and standard matrix algebra shows that the (T + 1) × (T + 1)th entry of Suppose that the pair of centrosymmetric cells belongs to clusters s and s ′ . These cells could either be in the same or in different clusters, and hence the expression can be written in one of the following two ways: (1) Centrosymmetric pair belonging to different clusters If the pair of centrosymmetric cells belongs to different clusters (s ̸ = s ′ ), then m s = m s ′ = 1 and we can make the following simplifications: The terms V ms = V m s ′ = a, where a is a scalar and represents the variance of each of the omitted cluster-period cells, andX m k is also a scalar, which we will denote byx m k . The terms B 22 can also be written as below by recalling the expression for var( θ) given in Section A: var( θ) Therefore, the information content of the centrosymmetric pair of cluster-period cells denoted by o is given by (2) Centrosymmetric pair belonging to the same cluster: If the pair of centrosymmetric cells belongs to the same cluster (s = s ′ ), then m s = 2 and we can make the following simplifications: The terms V m k is a 2 × 2 matrix and represents the covariance of the omitted pair of cells from cluster k, andX m k is a 1 × 2-dimensional vector, which we will Furthermore, the summations in the second terms of each of B 22 , B 21 , and B 11 can be simplified to terms with cluster index s only. var( θ) Therefore, the information content of the centrosymmetric pair of cluster-period cells denoted by o is given by Figure C1: Information content of the cells in progressively reduced designs, with 50 subjects per cell, assuming a discrete-time decay model with intracluster correlation of 0.05, and cluster autocorrelation of 0.95. The black color indicates that no information content can be calculated for the centrosymmetric cell pair. Figure C2: Information content of the cells in progressively reduced designs, with 10 subjects per cell, assuming a discrete-time decay model with intracluster correlation of 0.01, and cluster autocorrelation of 0.95. The black color indicates that no information content can be calculated for the centrosymmetric cell pair.