 Research
 Open Access
 Published:
The impact of iterative removal of lowinformation clusterperiod cells from a stepped wedge design
BMC Medical Research Methodology volume 23, Article number: 160 (2023)
Abstract
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 clusterperiods contributing a relatively small amount of information. We investigate the patterns of the information content of clusterperiod cells upon iterative removal of lowinformation cells, assuming a model for continuous outcomes with constant clusterperiod size, categorical time period effects, and exchangeable and discretetime decay intracluster correlation structures.
Methods
We sequentially remove pairs of “centrosymmetric” clusterperiod 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 “hotspots” in the corners of the design. For the exchangeable correlation structure, removing the cells from these hotspots leads to a marked reduction in study precision and power, however the impact of this is lessened for the discretetime decay structure.
Conclusions
Removing clusterperiod 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.
Background
Stepped wedge designs are a particular type of longitudinal cluster randomised trial design that are being increasingly used to evaluate interventions in public health and related fields [1]. All clusters commence in the control condition and cross over to the intervention condition in a staggered manner until all clusters implement the intervention condition by the final period. An example of a schematic of a stepped wedge design is provided in the top panel of Fig. 1. The requirement that clusters measure participants’ outcomes in each period can make these trials particularly burdensome and expensive to clusters, and, when a cohort sampling structure is employed, to participants as well [2]. However, stepped wedge designs have several beneficial characteristics: they are useful for testing interventions that cannot be undone or that must be gradually rolled out across a system, and they ensure that all clusters receive the intervention during the course of the trial.
Recent work has shown that the clusterperiod cells in the stepped wedge design differ in the amount of information they contribute to estimation of the treatment effect [3]. The “information content” of a clusterperiod cell was defined as the ratio of the variance of the treatment effect estimator for the resulting design when that cell is removed to the variance of the treatment effect estimator for the complete design [3,4,5]. A key finding from these papers is that some clusterperiods contribute a relatively small amount of information about the treatment effect. This strongly suggests that “incomplete” designs where not all clusters contribute measurements in all periods (e.g., bottom panel of Fig. 1), may still provide sufficient power to detect effects of interest. The observed patterns of information content of individual cells have previously been used to provide a basis for the selection of incomplete stepped wedge designs; in Kasza et al. [5], four incomplete designs where clusters would provide measurements in a limited number of pre and posttreatment switch periods were considered. However, how the information content changes when sets of two or more cells are removed from the design has not been considered. Further investigation of the properties of the information content of clusterperiod cells of stepped wedge designs is needed to guide the selection of incomplete stepped wedge designs; such incomplete designs may be more feasible for trialists than a complete stepped wedge.
Although incomplete stepped wedge designs have been considered in the statistical literature, e.g. Hemming et al. [6] and Unni et al. [7], like Kasza et al. [5], these have focused on a limited selection of prespecified designs and were not based on any information content considerations. For example, Hemming et al. [6] and Unni et al. [7] considered a design with one control period followed by two intervention periods in each sequence and stepped wedge designs with a transition or learning period in which no measurements are taken between the control and intervention periods. A more general investigation of a range of incomplete stepped wedge designs was conducted by Hooper et al. [8] for trials with a continuous outcome and continuous recruitment of participants. The authors sought to identify incomplete designs obtained by removing participants yielding the lowest reduction in the precision of the treatment effect estimator. They developed an algorithm to minimise the number of participants in a design by removing two participants from a complete design at a time (where the two participants were selected so that the resulting design maintained a “skewsymmetric” structure [9]).
In this paper we will use the information content metric to guide the exploration of a range of progressively reduced stepped wedge designs, starting with a complete design and removing clusterperiod cells until a minimally viable incomplete design is obtained (i.e. the smallest design that can provide an estimate of the treatment effect). We progress through incomplete designs by removing clusterperiod cells with low information content. We then assess the patterns of information content across these designs, how much precision is lost as we remove cells, and how study power correspondingly reduces. We have two aims in this work. First, we wish to obtain a better understanding of how the pattern of information content of clusterperiod cells of a stepped wedge design changes as cells with low information content are removed. Second, we provide an approach to help researchers identify less burdensome designs that maintain power similar to complete stepped wedge designs.
Our paper is organised as follows: in the Methods Section, we describe our statistical model for continuous outcomes, intracluster correlation structures, and provide a general expression for the variance of the treatment effect estimator valid for complete and incomplete designs. We also provide a more general definition of information content than has been considered previously, outline our procedure for removing low information content cells, and define a metric for comparing the precision of designs. In the Results Section, we implement our procedure for some specific trial design examples and describe the information content patterns and corresponding changes in design precision and power. We also assess the impact on design precision of removing prespecified proportions of clusterperiod cells across a range of trial configuration parameters. We conclude with a discussion of our findings in the last section.
Methods
A statistical model for continuous outcomes
We consider stepped wedge designs with different sets of participants in each period, often termed repeated crosssectional sampling. We define \(Y_{kji}\) as the continuous measured outcome for participant \(i=1,\dots ,m\) at time \(j=1,\dots ,T\) in cluster \(k=1,\dots ,K\). The model that we consider applies to all longitudinal cluster randomised designs. For simplicity we assume that one cluster is randomised to each sequence so that \(K=T1\). We consider two intracluster correlation structures: the exchangeable structure and the discretetime decay correlation structure. The exchangeable correlation structure assumes that the correlation between the outcomes of any pair of participants in the same cluster is identical regardless of the distance in time between their measurements. This was first implemented via a linear mixed model in a seminal paper on analysis of stepped wedge designs [10]. The discretetime decay correlation structure allows the correlation between participants’ outcomes to depend on their periods of measurement, with decreasing correlation as the time between their periods of measurement increases [4]. Underlying linear mixed effects models yielding these two correlation structures can be represented as:
where \(\beta _{j}\) is the fixed time effect for period j, with \(\beta _{1}=0\) for identifiability; \(\mu\) is the overall mean outcome in period 1; \(X_{kj}\) is the intervention indicator variable, equal to 0 when cluster k at period j is in the control condition, and 1 when cluster k at period j is in the treatment condition; \(\theta\) is the treatment/intervention effect of interest, \(\alpha _{k}\) is the random effect for cluster k for the exchangeable model (1), and \(\mathbf {\gamma }_{k}=\left( \gamma _{k1},\dots ,\gamma _{kT} \right) ^{T}\) is the vector of clusterperiod random effects for the discretetime decay model (2) with covariance matrix \(\tau ^2 \textbf{R}\) (elements of \(\textbf{R}\) described below), and \(\epsilon _{kji}\) is the subjectspecific random error. In model (2), the correlation between different subjects’ outcomes measured in periods j and s within a cluster is assumed to depend on the time between these periods: \(cov\left( \gamma _{kj},\gamma _{ks}\right) =\tau ^2 r^{\left js\right }\) and so \(corr\left( Y_{kji},Y_{ksl}\right) =\frac{\tau ^2}{\tau ^2+\sigma ^{2}_{\epsilon }} r^{\left js\right }\) = \(\rho r^{\left js\right }\), \(0 < r \le 1\). That is, the \(\left( j,s\right)\) element of \(\textbf{R}\) is given by \(r^{\left js\right }\). We refer to \(\rho\) as the withinperiod intracluster correlation (ICC), representing the correlation between two subjects’ outcomes measured within the same cluster in the same period. We refer to the parameter r as the cluster autocorrelation (CAC), representing the proportionate reduction in correlation from one period to the next. Note that model (1) is a special case of model (2), and is returned when \(r=1\): \(corr(Y_{kji},Y_{kjl})=corr\left( Y_{kji},Y_{ksl}\right) =\rho .\)
Variance of the treatment effect estimator for incomplete designs
When the correlation structure and correlations are known (as is assumed when calculating study power), a common approach for estimating the treatment effect \(\theta\) uses an estimator \(\hat{\theta }\) obtained via generalised least squares. The variance of the treatment effect estimator, \(var(\hat{\theta })\), is a key ingredient in calculating the required sample size and power of the trial and we therefore focus on this quantity at the trial design stage.
Letting \(\bar{Y}_{kj}=\frac{1}{m} \sum _{i=1}^{m}Y_{kji}\) be the mean outcome in cluster k in period j and \(\bar{\textbf{Y}}_{k}=\left( \bar{Y}_{k1},\dots ,\bar{Y}_{kT} \right) ^{T}\) be the vector of clusterperiod means for cluster k, then the covariance matrix for a cluster, assumed common across clusters, is given by \({\textbf {V}}_{\bar{Y}}=\frac{\sigma ^{2}_{\epsilon }}{m}\textbf{I}_{T \times T}+\tau ^{2} \textbf{R}\) where \(\textbf{I}_{T \times T}\) is the \(T\times T\) identity matrix. When \(r=1\), the matrix \(\textbf{R}=\textbf{J}_{T \times T}\), a \(T\times T\) matrix of ones, and this reduces to model (1). Letting \({\textbf {X}}_{k}=\left( X_{k1},\dots ,X_{kT}\right) ^{T}\) be the vector of treatment indicators for cluster k, the variance of the treatment effect estimator can then be represented as [3]:
This expression is valid for complete stepped wedge designs but a more general expression is required for incomplete designs. Letting \(T_{k}\) represent the number of measurement periods in the sequence assigned to cluster k, \({\textbf {Z}}_{k}\) be the \(T_{k} \times T\)dimensional matrix encoding the parameterisation of the time effects corresponding to cluster k, resembling a \(T \times T\) identity matrix with rows corresponding to unobserved periods deleted, \({\textbf {V}}_{k}\) be the \(T_{k} \times T_{k}\) covariance matrix for cluster k and \({\textbf {X}}_{k}\) be the \(T_{k} \times 1\)dimensional column vector of treatment indicators for the measurement periods for cluster k, then a more general expression for the variance of the treatment effect estimator is given by:
This expression is valid for both complete and incomplete stepped wedge designs. Note that the standard variance expression for complete designs, equation (3), is returned if all clusterperiods are measured, which would mean that the \({\textbf {Z}}_{k}\) matrices are all identity matrices, and all clusters would have the same covariance matrix for a cluster, \({\textbf {V}}_{\bar{Y}}\). The derivation for equation (4) is provided in Section A of Additional file 1.
Obtaining and evaluating progressively reduced designs
Information content of pairs of cells
The information content metric that was introduced by Kasza et al. [3] was used to identify the amount of information that each individual cell, period or cluster of a stepped wedge design contributes to estimation of the treatment effect. It was defined as the ratio of the variance of the treatment effect estimator for the design when a cell is removed to that of the variance of the treatment effect estimator for the complete design. However, since Kasza et al. [3] also proved that the information content is equal for cells in a centrosymmetric pair for the intracluster correlation structures we assume, we will instead consider the information content of centrosymmetric pairs of cells. Informally, a centrosymmetric pair of cells is one in which the cells are at the same location in the design with interchange of the \(01\) labelling of the cells and a reversal of both time and cluster order. Formally, the location of the partner cell of a specific cell in a centrosymmetric pair can be obtained by the destination of the cell after reflecting the design schematic along the central horizontal and vertical axes: for a standard stepped wedge over T time periods, and \(K=T1\) sequences, the cell in cluster k and period j, with indices \(\left( k,j\right)\), is the centrosymmetric partner of the cell with indices \(\left( K+1k, T+1j \right)\) [3] (see Fig. 2). In this paper we will obtain progressively reduced designs by removing a centrosymmetric pair of clusterperiod cells at each iteration, thus maintaining the centrosymmetry of the reduced designs.
We modify the information content definition from Kasza et al. [3] in two ways: by (1) replacing the numerator with the variance of the treatment effect estimator for designs with a centrosymmetric pair of cells removed rather than a single cell, and (2) replacing the denominator with an expression for \(var\hat{(\theta )}\) that accommodates incomplete designs.
Let A represent a pair of centrosymmetric cells: \(A=\{(k,j), (K+1k, T+1j)\}\) for some cluster k and period j. Suppose that D is a skewsymmetric design. Definition 1 of Bowden et al. [9] states that a longitudinal cluster randomised trial design is skewsymmetric if the clusters can be ordered so that \(X_{kj}=1X_{K+1k,T+1j}\) or no measurements are taken in both \(X_{kj}\) and \(X_{K+1k,T+1j}\) . Both designs in Fig. 1 are skewsymmetric designs, for example. This design D may be a complete stepped wedge design or an incomplete stepped wedge design, where centrosymmetric cell pairs have been removed from a complete stepped wedge design. Then let \(var_{D}\hat{(\theta )}\) be the variance of the treatment effect estimator for trial design D. We denote the variance of the treatment effect estimator when we delete A from D as \(var_{D[A]}\hat{(\theta )}\). We then define the information content of the cells A within the design D as:
A derivation of the analytical form of the information content of pairs of cells is provided in Section B of the Additional file 1. Although we will calculate the variances and information content numerically, we show in the derivation of the analytical form that the precision of the reduced design can be represented as the sum of the precision of the previous design and a constant. This means that the information content can be represented as a function of the variance of the previous design and the constant term which is similar to the work in Kasza and Forbes [3].
Removal of pairs of cells
To obtain progressively reduced designs, we remove centrosymmetric pairs of cells with low information content in an iterative manner starting from a complete stepped wedge design. For the initial design we calculate the information content for each of the \(KT/2 = T(T1)/2\) centrosymmetric clusterperiod cell pairs. The next step is to identify the cell pairs with the lowest information content, and then remove them from the initial design. Where multiple pairs of cells have the same information content, we remove the pair with the smallest cluster and period index (the cell closest to the toplefthand corner of the design and its centrosymmetric cell counterpart) so that only one pair of cells is removed at each iteration. We then calculate the information content for each of the remaining pairs and remove the pair with the lowest information content for this reduced design. This process continues until the treatment effect cannot be estimated for the reduced design. The algorithm is written out below.
Let \(D_{0}\) denote the initial complete design. For the design \(D_{l}\) at iteration l, \(l\ge 1\), let \(A_{l}^*\) denote the centrosymmetric pair of cells with the lowest information content corresponding to design \(D_{l}\), and then define \(D_{l}=D_{l1}[A_{l1}^*]\) to be the design at iteration l obtained by omitting the cellpair \(A_{l1}^*\) from design \(D_{l1}\). We then define the information content of a centrosymmetric pair of cells A in design \(D_{l}\) as the ratio of the variance of the treatment effect estimator for the design \(D_{l}[A]\) to the variance for design \(D_{l}\): \(IC_{D_{l}}(A)=var_{D_{l}[A]}\hat{(\theta )}/var_{D_{l}}\hat{(\theta )}\), where \(D_{l}=D_{l1}[A_{l1}^*]\) for \(l = 1, 2, \dots\) up to a maximum of \(T(T1)/2 1\). The algorithm proceeds as follows:

1
Calculate the information content of each centrosymmetric pair of cells A in design \(D_{0}\): \(IC_{D_{0}}(A)\).

2
Omit the pair of cells \(A_{0}^*\) with the lowest information content. If more than one pair has the same information content, we remove the pair containing the cell with the smallest cluster and period index so that only one pair is removed at each iteration.

3
Generate the reduced design \(D_{1}= D_{0}[A_{0}^*]\).
For iterations \(l = 2,\dots\) up to a maximum of \(T(T1)/2 1\):

4
Calculate the information content of each pair of cells A in the reduced design \(D_{l1}\): \(IC_{D_{l1}}(A)\).

5
Remove the pair of cells \(A_{l1}^*\) with the lowest information content.

6
Generate the reduced design \(D_{l}=D_{l1}[A_{l1}^*]\).
Iterate steps \(46\) until the treatment effect can no longer be estimated. To allow for estimation of the treatment effect, there must be at least one period which contains at least one intervention and at least one control cell. The final design reached by the algorithm may contain more than two cells to allow for this.
Relative precision metric
We define a measure of “precision loss” to compare the precision of each reduced design with the precision of the complete design. The precision loss of design \(D_{l}\) relative to the complete design \(D_{0}\) is defined as:
Higher values correspond to greater loss of precision compared to the complete design, and \(0\%\) corresponds to no loss of precision.
Results
Illustrative examples
The Hill et al. trial
We first obtain progressively reduced designs for a trial configuration motivated by the Hill et al. stepped wedge trial [11] that was explored in Kasza et al. [5]. We consider a simplification of the original trial, with a complete stepped wedge design with 4 clusters and 5 periods, and an equal number of participants per clusterperiod (90 participants per clusterperiod, the average clusterperiod size for this trial). The corresponding trial design schematic is illustrated by the complete design given in Fig. 1. We consider both exchangeable and discretetime decay correlation structures, and assume the same correlation parameter values as in Kasza et al. [5]: \(\rho =0.14\) for the exchangeable structure, and for the discretetime decay structure, \(\rho =0.15\) together with a decay in correlation of \(5\%\) per period, corresponding to a CAC of \(r=0.95\).
Figures 3(ah) and 4(ah) display the progressively reduced designs at each iteration obtained from applying the algorithm in the Removal of pairs of cells subsection to this design, assuming exchangeable correlation and discretetime decay correlation structures, respectively. The designs in both figures display the information content of each centrosymmetric cell pair through the information content value and cell colour, with darker colours indicating more informationrich cellpairs. Progressive removal of low information content cellpairs shows that clusterperiod cells distant from the time of the treatment switch are generally removed first. Cells immediately before and after the time of the treatment switch remain until near the end. A comparison between Figs. 3 and 4 shows that the patterns of cell removal are slightly different between the two correlation structures. For the exchangeable correlation structure, informationrich cells are concentrated in the offdiagonal corners of the design in addition to the main diagonal. But, for the discretetime decay correlation structure, more information is concentrated around the treatment switches while cells away from the main diagonal contain less information. The final design shown in Fig. 3(h) indicates that the middle pair of cells could be deleted. However, after deletion of that pair of cells, the information content of the remaining cells in the design with only four remaining clusterperiod cells cannot be calculated.
Using the metrics of precision loss and power to assess the range of designs in Figs. 3 and 4, we find that changes in precision and power are nonlinear functions of the proportion of cells that have been removed. We consider a standardised effect size of 0.25 when considering a model with an exchangeable correlation structure, and an effect size of 0.35 for the model with a discretetime decay correlation structure. This implies study power of the complete design of around \(90\%\) with a twosided significance level of 0.05 for each complete design. Figure 5 displays the precision loss (top panel) and power (bottom panel) for the two correlation structures for each of the designs in Figs. 3 and 4 according to the proportion of the total number of clusterperiod cells that have been removed. Precision loss increases slightly but remains low until around half of clusterperiod cells have been removed under both models; there is a sharp increase in precision loss just after the midway point, with a slightly steeper increase under the exchangeable model than the discretetime decay model. This jump in precision loss appears to arise from removing cells from the corners, which we refer to as “hotspot” corners, of the incomplete design when half of the clusterperiod cells are removed, with a greater impact for the exchangeable correlation structure than for the discretetime decay structure. Power, being directly related to the variance of the treatment effect estimator for a particular effect size, displays the same trajectory as precision loss, only mirrored (i.e. increases in the precision loss metric correspond to reductions in power). For the exchangeable model, the complete design has \(88.23\%\) power to detect an effect size of 0.25, reducing to \(82.83\%\) with a corresponding precision loss of \(14.60\%\) for the incomplete design obtained once \(50\%\) of the clusterperiod cells were removed. This corresponds to the design shown in Fig. 3(f). For the discretetime decay model, the complete design has \(88.78\%\) power to detect a standardised effect size of 0.35, reducing to \(84.24\%\) with a corresponding precision loss of \(12.84\%\) for the incomplete design obtained once \(50\%\) of the clusterperiod cells were removed, corresponding to the design shown in Fig. 4(f).
A larger design
In this subsection we consider a design with a larger number of periods to further illustrate the algorithm: a stepped wedge design with 9 clusters, 10 periods, and 50 participants per clusterperiod. We consider the discretetime decay correlation structure, assuming a withinperiod ICC of \(\rho =0.05\) with a decay in correlation of \(5\%\) per period, corresponding to a CAC of \(r=0.95\). The schematic of the complete stepped wedge design is shown in the topleft panel of Fig. 6, with the information content of pairs of clusterperiod cells of the complete design illustrated below. The righthand side of Fig. 6 displays the incomplete design after approximately \(50\%\) of clusterperiod cells have been removed, with the information content of clusterperiod cell pairs shown below. This incomplete design appears similar to a “staircase design” where clusters contribute measurements immediately before and after the treatment switch, but with some additional measurements in the first and final periods of the design, and different sequences contain different numbers of control and intervention periods. As in the The Hill et al. trial subsection, information is once again concentrated near the time of the treatment switch. The complete set of reduced designs commencing with the complete design and terminating with the minimally viable design is displayed in the Additional file 1 (Fig. C1).
Figure 7 displays the precision loss (top panel) and power (bottom panel) for the discretetime decay correlation structure of the series of progressively reduced designs in the Additional file 1 (Fig. C1) according to the proportion of the total number of clusterperiod cells that have been removed. We consider a standardised effect size of 0.2, yielding \(90.2\%\) power for the complete design. The reduced design at the midway point, i.e. when approximately \(50\%\) of clusterperiod cells are removed, is only \(6\%\) less efficient than the complete design and the power is only \(2\%\) lower than for the complete design.
Researchers can explore progressively reduced designs for this and other trial settings, along with plots of study power and precision loss, with our web app available at https://monashbiostat.shinyapps.io/iterativeinfcontent/.
More general results
To assess broad patterns of precision loss across a range of trial configuration parameters likely to be seen in practice, we explore the precision loss of progressively reduced designs for several combinations of correlation and trial configuration parameters (See Table 1). We consider stepped wedge designs with 5 and 10 periods, with 10 or 100 participants per clusterperiod, with withinperiod ICC values of 0.01, 0.05, and 0.15, with cluster autocorrelation values of 1, 0.95, and 0.8. We apply the iterative removal algorithm to each of these sets of configurations, and calculate the precision loss for the designs obtained by removing \(20\%\), \(50\%\) and \(80\%\) of the clusterperiod cells. For the 5period designs, this corresponded to designs with \(80\%\), \(50\%\) and \(20\%\) of cells remaining; for the 10period designs, this corresponded to designs with \(80\%\), \(48.89\%\) and \(20\%\) of cells remaining.
Figure 8 displays the precision loss for the incomplete designs with the predefined removal percentages across all combinations of the trial configurations. The broad patterns of precision loss are fairly consistent across all the designs considered: designs with \(20\%\) of clusterperiod cells removed generally have very little precision loss; designs with around \(50\%\) of clusterperiod cells removed have slightly higher loss of precision, but not exceeding \(25\%\); and designs with \(80\%\) of clusterperiod cells removed have much higher loss of precision. In general, less precision is lost at the predefined checkpoints in the removal process for designs with more periods compared to designs with fewer periods, for equivalent correlation parameter values. Across all trial configurations, precision loss tends to be slightly lower under the discretetime decay correlation structure \((r=0.95\) or 0.8) than the exchangeable correlation structure \(\left( r=1\right)\). When there are a large number of participants and a small number of periods (bottomleft panel), more precision is lost for various correlation parameters as compared to the same number of periods but with a small number of participants (topleft panel). When \(20\%\) of clusterperiod cells are removed, the reduced designs have almost the same precision as the complete designs. The minimum and maximum precision loss when \(20\%\) of cells are removed are \(0.01\%\) and \(2.86\%\), respectively and when approximately \(50\%\) of cells are removed are \(0.99\%\) and \(21.21\%\), respectively. This indicates that incomplete designs are likely to provide useful alternatives to complete stepped wedge designs across a broad range of designs and intracluster correlation structures.
Discussion
In this paper, we have considered an iterative approach to obtaining progressively reduced designs by removing pairs of cells with low information content for standard stepped wedge designs with repeated crosssectional sampling. Where previous work focused on the information content of individual cells within a complete stepped wedge, we continually updated the information content of cell pairs across a range of designs, from the complete design to a minimally viable design. Our findings show that for many trial configurations, incomplete stepped wedge designs with up to \(50\%\) of clusterperiod cells removed are still nearly as efficient as the complete stepped wedge, with less than \(25\%\) precision loss and typically only a small reduction in study power. Removing hotspot corners generally has a larger impact for the exchangeable intracluster correlation structure than for the discretetime decay correlation structure. This is due to the reduction in the similarity of observations from the same cluster as the time between their measurements increases when the discretetime decay correlation structure is assumed: measurements in these corners do not offer as much information about the treatment effect when these subjects’ outcomes are less similar to subjects’ outcomes near the time of the treatment switch in that cluster.
In Section C of Additional file 1 we provide an additional example where both the withinperiod ICC and clusterperiod size are small: \(\rho = 0.01\) and \(m=10\). As illustrated in Fig. C2(x) for this setting, the incomplete stepped wedge design where hotspot corners are not present, is still able to provide sufficient power and high precision (power reduction: \(3.26\%\), precision loss: \(10.20\%\)). For this value of m and \(\rho\), the mixed model uses mainly vertical comparisons rather than horizontal [12]. Therefore, the first and last period contribute nothing to these comparisons because they are all 0’s or all 1’s. There is still some contribution from horizontal comparisons so the cells just before and after the intervention switch are still useful horizontally. For this design, measurements in the first and final periods are removed during the early iterations of the algorithm. The intuition for this finding arises from the work of Matthews and Forbes [12] in which settings with a small clusterperiod size and small withinperiod ICC lead to the treatment effect estimator being dominated by vertical comparisons. With the first and final periods containing all control and all intervention conditions, respectively, there is no information within these periods upon which to estimate the treatment effect, and correspondingly these cells are removed early in the iterative removal process (Fig. C2).
Our work indicates that designs which resemble staircase designs, where the number of pre and postswitch measurement periods differ across the sequences of the design, can be highly efficient. Further, depending on the design parameters, these reduced designs may include measurement periods in the first and final periods of the design. That is, were such designs adopted, some clusters would need to provide measurements at the beginning and end of the trial, without needing to provide measurements in the middle of the trial. The feasibility and usefulness of such designs in practice would require discussion with trialists in the specific subject matter context. This work indicates that staircase designs may often be efficient alternatives to complete stepped wedge designs, but more research is required to investigate the statistical properties of these designs. For example, we are currently investigating the ways in which observations within a complete stepped wedge design could be rearranged or redistributed to provide staircase designs with as much power as the complete design.
The patterns of information content we obtained across a range of incomplete designs are generally consistent with those found previously for complete stepped wedge designs [3,4,5]: cells closest to the time of the treatment switches contain by far the most information for estimation of the treatment effect and the offdiagonal corners may also contribute a great deal of information. Moreover, we found that incomplete designs such as those depicted in Figs. 3(f) and 4(f) were highly efficient: these designs resemble Design B in Section 4 of Kasza et al. [5], chosen for its retention of cells with the highest information content under their definition. Another algorithmic search for selecting an efficient design has previously been used to remove two participants at each iteration using iterative improvements with continuous recruitment; i.e. time was considered a continuous phenomenon in that paper [8]. Although none of the reduced designs in that paper were exact staircase designs (where each cluster contributes measurements in a restricted number of consecutive pre and postswitch periods) [3], in certain scenarios that algorithm found designs that closely approximated staircase designs. Our work can be considered as a discretetime version of Hooper et al. [8]: our algorithm searches for cells with the lowest information content at each iteration while considering time as a discrete phenomenon (as is common in the design and analysis of stepped wedge trials).
While broad patterns of information content appear to be fairly consistent across a range of trial configuration parameters, we provide an online app at https://monashbiostat.shinyapps.io/iterativeinfcontent/ so that readers can explore incomplete designs for userdefined trial configurations. This app enables trialists to specify their desired trial configurations including the number of periods, the number of participants in each clusterperiod, withinperiod intracluster correlation ICC, cluster autocorrelation CAC, type of correlation structure, and the effect size of interest. Selecting ‘Yes’ in the ‘allow for decay correlation’ option and specifying a CAC lower than 1 in the app enables a discretetime decay correlation structure. However, blockexchangeable correlation structures can also be accommodated, by selecting ‘No’ in the ‘allow for decay correlation’ option, and providing a CAC lower than 1. Finally, to choose an exchangeable correlation structure, again select ‘No’ in the ‘allow for decay correlation’ option while setting the CAC to 1.
There are a number of aspects of this work that can be extended. Our results pertain to settings where continuous outcomes are analysed with linear mixed models, but settings with binary outcomes are also common. While it is possible that different patterns might emerge when considering binary outcomes, given the similarity of the results in Li et al. [13] for binary outcomes and marginal models with those of Kasza and Forbes [3] for linear mixed models, we would expect this similarity to carry through to incomplete designs. We also assumed equal clusterperiod sizes in this paper, but further work could consider unequal clusterperiod sizes arising from different numbers of individuals recruited per clusterperiod, or with cohort sampling structures with dropout or loss to followup. Our work could also be extended to consider designs with transition periods, meaning no data collection in the period(s) just prior to commencement of the intervention. Additionally, we assumed no treatment effect heterogeneity is present, otherwise the centrosymmetric properties would not hold for the information content of cells [4], and therefore the extent of the impact of treatment effect heterogeneity requires further research. Finally, in further work we intend to evaluate design precision while incorporating the associated costs as considered by Grantham et al. [14].
Conclusions
In summary, obtaining incomplete designs guided by the information content of pairs of clusterperiod cells ensures that any measurements taken are going to be highly informative for estimating the treatment effect. We have shown that certain incomplete stepped wedge designs with measurements concentrated around the main diagonal (and possibly in the hotspot corners) may result in only a small precision loss relative to the complete stepped wedge design and hence may be nearly as powerful as the full stepped wedge design.
Availability of data and materials
Data sharing is not applicable to this article as no new data were created or analyzed in this study. Project code is available at https://github.com/EhsanRD/IterativeRemovalsSW.
Abbreviations
 ICC:

Intracluster correlation
 CAC:

Cluster autocorrelation
 IC:

Information content
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This work was supported by the Australian Research Council Discovery Project DP210101398, an Australian Government Research Training Program (RTP) Scholarship, and an Australian Trials Methodology (AusTriM) Research Network supplementary scholarship.
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All authors conceived the study and contributed to the design of the study. ER developed the web app, wrote the first draft of the manuscript. KLG, ABF and JK provided critical review of the manuscript. All authors read and approved the final manuscript.
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RezaeiDarzi, E., Grantham, K.L., Forbes, A.B. et al. The impact of iterative removal of lowinformation clusterperiod cells from a stepped wedge design. BMC Med Res Methodol 23, 160 (2023). https://doi.org/10.1186/s12874023019697
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DOI: https://doi.org/10.1186/s12874023019697
Keywords
 Longitudinal cluster randomised trials
 Correlation structure
 Highly efficient design
 Incomplete design
 Discretetime decay