Information content of stepped-wedge designs when treatment effect heterogeneity and/or implementation periods are present

Optimal designs using generalized estimating equations in cluster randomized crossover and stepped wedge trials

Cluster randomized crossover and stepped wedge cluster randomized trials are two types of longitudinal cluster randomized trials that leverage both the within- and between-cluster comparisons to estimate the treatment effect and are increasingly used in healthcare delivery and implementation science research. While the variance expressions of estimated treatment effect have been previously developed from the method of generalized estimating equations for analyzing cluster randomized crossover trials and stepped wedge cluster randomized trials, little guidance has been provided for optimal designs to ensure maximum efficiency. Here, an optimal design refers to the combination of optimal cluster-period size and optimal number of clusters that provide the smallest variance of the treatment effect estimator or maximum efficiency under a fixed total budget. In this work, we develop optimal designs for multiple-period cluster randomized crossover trials and stepped wedge cluster randomized trials with continuous outcomes, including both closed-cohort and repeated cross-sectional sampling schemes. Local optimal design algorithms are proposed when the correlation parameters in the working correlation structure are known. MaxiMin optimal design algorithms are proposed when the exact values are unavailable, but investigators may specify a range of correlation values. The closed-form formulae of local optimal design and MaxiMin optimal design are derived for multiple-period cluster randomized crossover trials, where the cluster-period size and number of clusters are decimal. The decimal estimates from closed-form formulae can then be used to investigate the performances of integer estimates from local optimal design and MaxiMin optimal design algorithms. One unique contribution from this work, compared to the previous optimal design research, is that we adopt constrained optimization techniques to obtain integer estimates under the MaxiMin optimal design. To assist practical implementation, we also develop four SAS macros to find local optimal designs and MaxiMin optimal designs.

Reconsidering stepped wedge cluster randomized trial designs with implementation periods: Fewer sequences or the parallel-group design with baseline and implementation periods are potentially more efficient

BACKGROUND/AIMS

When designing a cluster randomized trial, advantages and disadvantages of tentative designs must be weighed. The stepped wedge design is popular for multiple reasons, including its potential to increase power via improved efficiency relative to a parallel-group design. In many realistic settings, it will take time for clusters to fully implement the intervention. When designing the HEALing (Helping to End Addiction Long-termSM) Communities Study, implementation time was a major consideration, and we examined the efficiency and practicality of three designs. Specifically, a three-sequence stepped wedge design with implementation periods, a corresponding two-sequence modified design that is created by removing the middle sequence, and a parallel-group design with baseline and implementation periods. In this article, we study the relative efficiencies of these specific designs. More generally, we study the relative efficiencies of modified designs when the stepped wedge design with implementation periods has three or more sequences. We also consider different correlation structures.

METHODS

We compare efficiencies of stepped wedge designs with implementation periods consisting of three to nine sequences with a variety of corresponding designs. The three-sequence design is compared to the two-sequence modified design and to the parallel-group design with baseline and implementation periods analysed via analysis of covariance. Stepped wedge designs with implementation periods consisting of four or more sequences are compared to modified designs that remove all or a subset of 'middle' sequences. Efficiencies are based on the use of linear mixed effects models.

RESULTS

In the studied settings, the modified design is more efficient than the three-sequence stepped wedge design with implementation periods. The parallel-group design with baseline and implementation periods with analysis of covariance-based analysis is often more efficient than the three-sequence design. With respect to stepped wedge designs with implementation periods that are comprised of more sequences, there are often corresponding modified designs that improve efficiency. However, use of only the first and last sequences has the potential to be either relatively efficient or inefficient. Relative efficiency is impacted by the strength of the statistical correlation among outcomes from the same cluster; for example, the relative efficiencies of modified designs tend to be greater for smaller cluster auto-correlation values.

CONCLUSION

If a three-sequence stepped wedge design with implementation periods is being considered for a future cluster randomized trial, then a corresponding modified design using only the first and last sequences should be considered if sole focus is on efficiency. However, a parallel-group design with baseline and implementation periods and analysis of covariance-based analysis can be a practical, efficient alternative. For stepped wedge designs with implementation periods and a larger number of sequences, modified versions that remove 'middle' sequences should be considered. Due to the potential sensitivity of design efficiencies, statistical correlation should be carefully considered.

Information content of stepped wedge designs under the working independence assumption

The stepped wedge design is increasingly popular in pragmatic trials and implementation science research studies for evaluating system-level interventions that are perceived to be beneficial to patient populations. An important step in planning a stepped wedge design is to understand the efficiency of the treatment effect estimator and hence the power of the study. We develop several novel analytical results for designing stepped wedge cluster randomized trials analyzed through generalized estimating equations under a misspecified working independence correlation structure. We first contribute a general variance expression of the treatment effect estimator when data collection is scheduled for each cluster-period. Because resource and patient-centered considerations may intentionally call for an incomplete design with outcome data being omitted for certain cluster-periods, we further derive the information content based on the robust sandwich variance to identify data elements that may be preferentially omitted with minimum loss of precision in estimating the treatment effect. We prove that centrosymmetric pairs of cluster-periods, treatment sequences and periods have identical information content and thus contribute equally to the treatment effect estimation, as long as the true covariance structure for the cluster-period means remains centrosymmetric. Finally, we provide an example of how to obtain an incomplete stepped wedge design that admits a more efficient independence GEE estimator but requires less data collection effort. Our results elegantly extend existing ones from linear mixed models coupled with model-based variances to accommodate a misspecified independence working correlation structure through the robust sandwich variances.

Generalizing the information content for stepped wedge designs: A marginal modeling approach

Stepped wedge trials are increasingly adopted because practical constraints necessitate staggered roll-out. While a complete design requires clusters to collect data in all periods, resource and patient-centered considerations may call for an incomplete stepped wedge design to minimize data collection burden. To study incomplete designs, we expand the metric of information content to discrete outcomes. We operate under a marginal model with general link and variance functions, and derive information content expressions when data elements (cells, sequences, periods) are omitted. We show that the centrosymmetric patterns of information content can hold for discrete outcomes with the variance-stabilizing link function. We perform numerical studies under the canonical link function, and find that while the patterns of information content for cells are approximately centrosymmetric for all examined underlying secular trends, the patterns of information content for sequences or periods are more sensitive to the secular trend, and may be far from centrosymmetric.

Optimal allocation of clusters in stepped wedge designs with a decaying correlation structure

The cluster randomized stepped wedge design is a multi-period uni-directional switch design in which all clusters start in the control condition and at the beginning of each new period a random sample of clusters crosses over to the intervention condition. Such designs often use uniform allocation, with an equal number of clusters at each treatment switch. However, the uniform allocation is not necessarily the most efficient. This study derives the optimal allocation of clusters to treatment sequences in the cluster randomized stepped wedge design, for both cohort and cross-sectional designs. The correlation structure is exponential decay, meaning the correlation decreases with the time lag between two measurements. The optimal allocation is shown to depend on the intraclass correlation coefficient, the number of subjects per cluster-period and the cluster and (in the case of a cohort design) individual autocorrelation coefficients. For small to medium values of these autocorrelations those sequences that have their treatment switch earlier or later in the study are allocated a larger proportion of clusters than those clusters that have their treatment switch halfway the study. When the autocorrelation coefficients increase, the clusters become more equally distributed across the treatment sequences. For the cohort design, the optimal allocation is almost equal to the uniform allocation when both autocorrelations approach the value 1. For almost all scenarios that were studied, the efficiency of the uniform allocation is 0.8 or higher. R code to derive the optimal allocation is available online.

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.

Assessing the implementation and effectiveness of early integrated palliative care in long-term care facilities in France: an interventional mixed-methods study protocol

BACKGROUND

Majority of residents in long-term care facilities (LTCF) have limited and delayed access to palliative care even though many suffer from incurable chronic illnesses that will likely require the provision of palliative care. We present the study protocol of "PADI-Palli", an intervention aims to advance early integrated palliative care into standard care delivered in LTCF. This study will assess the effectiveness of early integrated palliative care on palliative care accessibility for older persons in LTCF, and identify the key factors for the successful implementation of early integrated palliative care and its sustainability in the LTCF context.

METHODS

This multicentre interventional study utilises a pragmatic research design with a convergent parallel mixed-methods approach. The qualitative study will use a case study design and the quantitative study will use a stepped wedge cluster randomised trial. In total, 21 participating LTCF from three French regions will be randomly allocated to one of seven clusters. The clusters will cross over from the usual care to the active intervention condition over the course of the study. The primary outcome relates to the accurate identification of palliative care needs and early access to palliative care for LTCF residents. Secondary outcomes are quality of care, quality of life for residents and their families, and quality of life at work for professionals. Measurements will be performed before and after the intervention. Implementation and evaluation of PADI-Palli intervention is grounded in the Consolidated Framework for Implementation Research.

DISCUSSION

Existing evidence demonstrates that early integrated palliative care in cancer care leads to a significant improvement in patient outcomes and processes of care. Little is known, however, about early integrated palliative care in the context of LTCF for older persons. This study has the potential to fill this gap in the literature by providing evidence on the effectiveness of early integrated palliative care for older persons in LTCF. Moreover, this study will provide a better understanding of the relevant contextual elements that facilitate or hinder early integrated palliative care implementation and transferability. If proven effective, this intervention can be scaled to other care settings in which older persons require palliative care.

TRIAL REGISTRATION

ClinicalTrials.gov ID: NCT04708002; National registration: ID-RCB number: 2020-A01832-37.

Analysis of stepped wedge cluster randomized trials in the presence of a time-varying treatment effect

Stepped wedge cluster randomized controlled trials are typically analyzed using models that assume the full effect of the treatment is achieved instantaneously. We provide an analytical framework for scenarios in which the treatment effect varies as a function of exposure time (time since the start of treatment) and define the "effect curve" as the magnitude of the treatment effect on the linear predictor scale as a function of exposure time. The "time-averaged treatment effect" (TATE) and "long-term treatment effect" (LTE) are summaries of this curve. We analytically derive the expectation of the estimatorδ ^ $$ \hat{\delta} $$ resulting from a model that assumes an immediate treatment effect and show that it can be expressed as a weighted sum of the time-specific treatment effects corresponding to the observed exposure times. Surprisingly, although the weights sum to one, some of the weights can be negative. This implies thatδ ^ $$ \hat{\delta} $$ may be severely misleading and can even converge to a value of the opposite sign of the true TATE or LTE. We describe several models, some of which make assumptions about the shape of the effect curve, that can be used to simultaneously estimate the entire effect curve, the TATE, and the LTE. We evaluate these models in a simulation study to examine the operating characteristics of the resulting estimators and apply them to two real datasets.

Influence of cluster-period cells in stepped wedge cluster randomized trials

Stepped wedge cluster randomized trials (SWCRT) are increasingly used for the evaluation of complex interventions in health services research. They randomly allocate treatments to clusters that switch to intervention under investigation at variable time points without returning to control condition. The resulting unbalanced allocation over time periods and the uncertainty about the underlying correlation structures at cluster-level renders designing and analyzing SWCRTs a challenge. Adjusting for time trends is recommended, appropriate parameterizations depend on the particular context. For sample size calculation, the covariance structure and covariance parameters are usually assumed to be known. These assumptions greatly affect the influence single cluster-period cells have on the effect estimate. Thus, it is important to understand how cluster-period cells contribute to the treatment effect estimate. We therefore discuss two measures of cell influence. These are functions of the design characteristics and covariance structure only and can thus be calculated at the planning stage: the coefficient matrix as discussed by Matthews and Forbes and information content (IC) as introduced by Kasza and Forbes. The main result is a new formula for IC that is more general and computationally more efficient. The formula applies to any generalized least squares estimator, especially for any type of time trend adjustment or nonblock diagonal matrices. We further show a functional relationship between IC and the coefficient matrix. We give two examples that tie in with current literature. All discussed tools and methods are implemented in the R package SteppedPower.

Divergent confidence intervals among pre-specified analyses in the HiSTORIC stepped wedge trial: An exploratory post-hoc investigation

BACKGROUND

The high-sensitivity cardiac troponin on presentation to rule out myocardial infarction (HiSTORIC) study was a stepped-wedge cluster randomised trial with long before-and-after periods, involving seven hospitals across Scotland. Results were divergent for the binary safety endpoint (type 1 or type 4b myocardial infarction or cardiac death) across certain pre-specified analyses, which warranted further investigation. In particular, the calendar-matched analysis produced an odds ratio in the opposite direction to the primary logistic mixed-effects model analysis.

METHODS

Several post-hoc statistical models were fitted to each of the co-primary outcomes of length of hospital stay and safety events, which included adjusting for exposure time, incorporating splines, and fitting a random time effect. We improved control of patient characteristics over time by adjusting for multiple additional covariates using different methods: direct inclusion, regression adjustment for propensity score, and weighting. A data augmentation approach was also conducted aiming to reduce the effect of sparse data bias. Finally, the raw data was examined.

RESULTS

The new statistical models confirmed the results of the pre-specified trial analysis. In particular, the observed divergence between the calendar-matched and other analyses remained, even after performing the covariate adjustment methods, and after using data augmentation. Divergence was particularly acute for the safety endpoint, which had an event rate of 0.36% overall. Examining the raw data was particularly helpful to assess the sensitivity of the results to small changes in event rates and identify patterns in the data.

CONCLUSIONS

Our experience reveals the importance of conducting multiple pre-specified sensitivity analyses and examining the raw data, particularly for stepped wedge trials with low event rates or with a small number of sites. Before-and-after analytical approaches that adjust for differences in patient populations but avoid direct modelling of the time trend should be considered in future stepped wedge trials with similar designs.

The batched stepped wedge design: A design robust to delays in cluster recruitment

Stepped wedge designs are an increasingly popular variant of longitudinal cluster randomized trial designs, and roll out interventions across clusters in a randomized, but step-wise fashion. In the standard stepped wedge design, assumptions regarding the effect of time on outcomes may require that all clusters start and end trial participation at the same time. This would require ethics approvals and data collection procedures to be in place in all clusters before a stepped wedge trial can start in any cluster. Hence, although stepped wedge designs are useful for testing the impacts of many cluster-based interventions on outcomes, there can be lengthy delays before a trial can commence. In this article, we introduce "batched" stepped wedge designs. Batched stepped wedge designs allow clusters to commence the study in batches, instead of all at once, allowing for staggered cluster recruitment. Like the stepped wedge, the batched stepped wedge rolls out the intervention to all clusters in a randomized and step-wise fashion: a series of self-contained stepped wedge designs. Provided that separate period effects are included for each batch, software for standard stepped wedge sample size calculations can be used. With this time parameterization, in many situations including when linear models are assumed, sample size calculations reduce to the setting of a single stepped wedge design with multiple clusters per sequence. In these situations, sample size calculations will not depend on the delays between the commencement of batches. Hence, the power of batched stepped wedge designs is robust to unexpected delays between batches.

Power analysis for stepped wedge trials with multiple interventions

Stepped wedge design (SWD) trials are cluster randomized trials that feature staggered, unidirectional cross-over between treatment conditions. Existing literature on power for SWDs focuses primarily on designs with two conditions, typically a control and an intervention condition. However, SWDs with more than one treatment condition are being proposed and conducted. We present a linear mixed model for SWDs with two or more interventions, including both multiarm and factorial designs. We derive standard errors of the intervention effect coefficients, and present power calculation methods. We consider both repeated cross-sectional and cohort designs. Design features, with a focus on treatment allocations, are examined to determine their impact on power.

Mixed-effects models for the design and analysis of stepped wedge cluster randomized trials: An overview

The stepped wedge cluster randomized design has received increasing attention in pragmatic clinical trials and implementation science research. The key feature of the design is the unidirectional crossover of clusters from the control to intervention conditions on a staggered schedule, which induces confounding of the intervention effect by time. The stepped wedge design first appeared in the Gambia hepatitis study in the 1980s. However, the statistical model used for the design and analysis was not formally introduced until 2007 in an article by Hussey and Hughes. Since then, a variety of mixed-effects model extensions have been proposed for the design and analysis of these trials. In this article, we explore these extensions under a unified perspective. We provide a general model representation and regard various model extensions as alternative ways to characterize the secular trend, intervention effect, as well as sources of heterogeneity. We review the key model ingredients and clarify their implications for the design and analysis. The article serves as an entry point to the evolving statistical literatures on stepped wedge designs.

Information content of stepped wedge designs with unequal cluster-period sizes in linear mixed models: Informing incomplete designs

In practice, stepped wedge trials frequently include clusters of differing sizes. However, investigations into the theoretical aspects of stepped wedge designs have, until recently, typically assumed equal numbers of subjects in each cluster and in each period. The information content of the cluster-period cells, clusters, and periods of stepped wedge designs has previously been investigated assuming equal cluster-period sizes, and has shown that incomplete stepped wedge designs may be efficient alternatives to the full stepped wedge. How this changes when cluster-period sizes are not equal is unknown, and we investigate this here. Working within the linear mixed model framework, we show that the information contributed by design components (clusters, sequences, and periods) does depend on the sizes of each cluster-period. Using a particular trial that assessed the impact of an individual education intervention on log-length of stay in rehabilitation units, we demonstrate how strongly the efficiency of incomplete designs depends on which cells are excluded: smaller incomplete designs may be more powerful than alternative incomplete designs that include a greater total number of participants. This also serves to demonstrate how the pattern of information content can be used to inform a set of incomplete designs to be considered as alternatives to the complete stepped wedge design. Our theoretical results for the information content can be extended to a broad class of longitudinal (ie, multiple period) cluster randomized trial designs.

The hunt for efficient, incomplete designs for stepped wedge trials with continuous recruitment and continuous outcome measures

BACKGROUND

We consider the design of stepped wedge trials with continuous recruitment and continuous outcome measures. Suppose we recruit from a fixed number of clusters where eligible participants present continuously, and suppose we have fine control over when each cluster crosses to the intervention. Suppose also that we want to minimise the number of participants, leading us to consider "incomplete" designs (i.e. without full recruitment). How can we schedule recruitment and cross-over at different clusters to recruit efficiently while achieving good precision?

METHODS

The large number of possible designs can make exhaustive searches impractical. Instead we consider an algorithm using iterative improvements to hunt for an efficient design. At each iteration (starting from a complete design) a single participant - the one with the smallest impact on precision - is removed, and small changes preserving total sample size are made until no further improvement in precision can be found.

RESULTS

Striking patterns emerge. Solutions typically focus recruitment and cross-over on the leading diagonal of the cluster-by-time diagram, but in some scenarios clusters form distinct phases resembling before-and-after designs.

CONCLUSIONS

There is much to be learned about optimal design for incomplete stepped wedge trials. Algorithmic searches could offer a practical approach to trial design in complex settings generally.

Design, rationale and analysis plan for the Stand Up for Health trial in contact centres: a stepped wedge feasibility study

Background

Contact centres are one of the most sedentary workplaces, with employees spending a very high proportion of their working day sitting down. About a quarter of contact centre staff regularly experience musculoskeletal health problems due to high levels of sedentary behaviour, including lower back pain. There have been no previous randomised studies specifically aiming to reduce sedentary behaviour in contact centre staff. To address this gap, the Stand Up for Health (SUH) study aims to test the feasibility and acceptability of a complex theory-based intervention to reduce sedentary behaviour in contact centres.

Methods

The Stand Up for Health study has a stepped wedge cluster randomised trial design, which is a pragmatic design whereby clusters (contact centres) are randomised to time points at which they will begin to receive the intervention. All contact centre staff have the opportunity to experience the intervention. To minimise the resource burden in this feasibility study, data collection is not continuous, but undertaken on a selective number of occasions, so the stepped wedge design is “incomplete”. Eleven contact centres in England and Scotland have been recruited, and the sample size is approximately 27 per centre (270 in total). The statistical analysis will predominantly focus on assessing feasibility, including the calculation of recruitment rates and rates of attrition. Exploratory analysis will be performed to compare objectively measured sedentary time in the workplace (measured using an activPAL™ device) between intervention and control conditions using a linear mixed effects regression model.

Discussion

To our knowledge, this is the first stepped wedge feasibility study conducted in call centres. The rationale and justification of our novel staircase stepped wedge design has been presented, and we hope that by presenting our study design and statistical analysis plan, it will contribute to the literature on stepped wedge trials, and in particular feasibility stepped wedge trials. The findings of the study will also help inform whether this is a suitable design for other settings where data collection is challenging.

Trial registration

The trial has been registered on the ISRCTN database: http://www.isrctn.com/ISRCTN11580369

Statistical Considerations for Embedded Pragmatic Clinical Trials in People Living with Dementia

There is overwhelming need for nonpharmacological interventions to improve the health and well-being of people living with dementia (PLWD). The National Institute on Aging Imbedded Pragmatic Alzheimer's Disease (AD) and AD-Related Dementias Clinical Trials (IMPACT) Collaboratory supports clinical trials of such interventions embedded in healthcare systems. The embedded pragmatic clinical trial (ePCT) is ideally suited to testing the effectiveness of complex interventions in vulnerable populations at the point of care. These trials, however, are complex to conduct and interpret, and face challenges in efficiency (i.e., statistical power) and reproducibility. In addition, trials conducted among PLWD present specific statistical challenges, including difficulty in outcomes ascertainment from PLWD, necessitating reliance on reports by caregivers, and heterogeneity in measurements across different settings or populations. These and other challenges undercut the reliability of measurement, the feasibility of capturing outcomes using pragmatic designs, and the ability to validly estimate interventions' effectiveness in real-world settings. To address these challenges, the IMPACT Collaboratory has convened a Design and Statistics Core, the goals of which are: to support the design and conduct of ePCTs directed toward PLWD and their caregivers; to develop guidance for conducting embedded trials in this population; and to educate quantitative and clinical scientists in the design, conduct, and analysis of these trials. In this article, we discuss some of the contemporary methodological challenges in this area and develop a set of research priorities the Design and Statistics Core will undertake to meet these goals. J Am Geriatr Soc 68:S68-S73, 2020.

Sample size and power calculations for open cohort longitudinal cluster randomized trials

When calculating sample size or power for stepped wedge or other types of longitudinal cluster randomized trials, it is critical that the planned sampling structure be accurately specified. One common assumption is that participants will provide measurements in each trial period, that is, a closed cohort, and another is that each participant provides only one measurement during the course of the trial. However some studies have an "open cohort" sampling structure, where participants may provide measurements in variable numbers of periods. To date, sample size calculations for longitudinal cluster randomized trials have not accommodated open cohorts. Feldman and McKinlay (1994) provided some guidance, stating that the participant-level autocorrelation could be varied to account for the degree of overlap in different periods of the study, but did not indicate precisely how to do so. We present sample size and power formulas that allow for open cohorts and discuss the impact of the degree of "openness" on sample size and power. We consider designs where the number of participants in each cluster will be maintained throughout the trial, but individual participants may provide differing numbers of measurements. Our results are a unification of closed cohort and repeated cross-sectional sample results of Hooper et al (2016), and indicate precisely how participant autocorrelation of Feldman and McKinlay should be varied to account for an open cohort sampling structure. We discuss different types of open cohort sampling schemes and how open cohort sampling structure impacts on power in the presence of decaying within-cluster correlations and autoregressive participant-level errors.