pith. sign in

arxiv: 2606.23499 · v1 · pith:2OJXARUWnew · submitted 2026-06-22 · 📊 stat.ME

A generalized multiple-intervention stepped wedge design framework for treatment effect estimation in the presence of non-uniform cluster-period correlation structures

Samantha M. Levy , Jose-Miguel Yamal This is my paper

Pith reviewed 2026-06-26 07:24 UTC · model grok-4.3

classification 📊 stat.ME
keywords stepped wedge designmultiple interventionscluster randomized trialscorrelation structurelinear mixed modelpower calculationtreatment effectcovariance matrix
0
0 comments X

The pith

A covariance framework for multiple-intervention stepped wedge designs separates intracluster correlation from a cluster-period matrix to handle non-uniform structures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a unified covariance framework for multiple-intervention stepped wedge designs (M-SWDs) that factors the overall covariance into a scalar intracluster correlation and an explicit matrix for correlations between different periods within clusters. This allows the model to accommodate exchangeable, autoregressive, and other distance-dependent correlation structures while maintaining closed-form expressions for the variances of treatment effect estimators in linear mixed models. Simulations and analytic results show that assuming a uniform correlation when the true structure varies with time can lead to substantial errors in power calculations, making some designs overly conservative or optimistic. The framework is important because stepped wedge trials often involve rollout over time where correlations naturally decay, and misspecification affects especially the estimation of treatment interactions. By providing practical guidance for power calculation under realistic correlation assumptions, the work supports better design choices in cluster randomized trials.

Core claim

We develop a unified covariance framework for M-SWDs that separates intracluster correlation from an explicit cluster-period correlation matrix. This formulation accommodates exchangeable, autoregressive, and more general distance-dependent correlation structures while preserving closed-form expressions for the variance of treatment effect estimators under linear mixed models.

What carries the argument

The unified covariance framework that separates a scalar intracluster correlation from the cluster-period correlation matrix, enabling flexible structures in variance calculations for treatment effects.

If this is right

  • Misspecification of correlation as exchangeable when it is time-dependent distorts variance estimation and power, particularly for treatment interaction effects.
  • Designs calibrated under independence assumptions may be overly conservative.
  • Compound symmetry assumptions can be either optimistic or conservative depending on the true correlation decay.
  • Explicitly modeling cluster-period correlation at the design stage improves power calculation accuracy in realistic settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Researchers designing stepped wedge trials could routinely perform sensitivity analyses across different correlation structures using this framework.
  • The approach may generalize to other longitudinal cluster trial designs where period-specific correlations matter.
  • Software implementations of power calculators for stepped wedge designs would benefit from incorporating selectable correlation matrices beyond exchangeable.

Load-bearing premise

The observations follow a linear mixed model whose covariance structure can be factored into a scalar intracluster correlation multiplied by a fully specified cluster-period correlation matrix.

What would settle it

Collect data from a stepped wedge trial with known time-dependent correlations and check whether the framework's predicted variances match the observed variability of treatment effect estimates better than standard uniform-correlation models.

Figures

Figures reproduced from arXiv: 2606.23499 by Jose-Miguel Yamal, Samantha M. Levy.

Figure 1
Figure 1. Figure 1: Sample (a) Concurrent and (b) Factorial multiple-intervention stepped wedge design studies with 8 clusters [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Heatmaps of four correlation structures (Independence, Compound Symmetry, AR(1), and Toeplitz) for T=6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: The corresponding correlation matrix, R, is R = RTz =   1 δ1 δ2 . . . δT −1 δ1 1 δ1 . . . δT −2 δ2 δ1 1 . . . δT −3 . . . . . . . . . . . . . . . δT −1 δT −2 δT −3 . . . 1   such that rjm = δh with h = |j − m| is the lag. We assume δ0 = 1 for identifiability, corresponding to perfect within-period correlation. Under the Toeplitz structure, all time pairs that are the same distance apart share t… view at source ↗
Figure 3
Figure 3. Figure 3: Factorial design with 8 cluster and 6 periods utilized for subsequent analytic and simulation studies. White [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Analytic power curves as a function of standardized effect size for main and interaction effects under correct [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sensitivity of theoretical power to ICC composition under fixed correlation structures. Power is shown as [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Theoretical power as a function of individual autocorrelation ( [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Empirical bias of fixed-effect estimators across standardized effect sizes, stratified by the true cluster-period [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Ratio of estimated standard errors to true standard errors across standardized effect sizes, stratified by the true [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Analytic power curves across standardized effect sizes for a fixed multiple-intervention stepped wedge design. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of theoretical power curves and simulated power curves. Results for an 8 cluster, 6 period, [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of theoretical power curves and simulated power curves under small sample size ( [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of theoretical power curves and simulated power curves under small sample size ( [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Power curves shown for main effects (left) and interaction effect (right) across number of individuals per [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Theoretical power curves show the marginal increase in power from adding 10 individuals per cluster-period, [PITH_FULL_IMAGE:figures/full_fig_p029_14.png] view at source ↗
read the original abstract

Existing power and design methods for multiple-intervention stepped wedge designs (M-SWDs) typically assume exchangeable cluster-period correlation, despite evidence that correlation often decays over time. Misspecification of this correlation structure can substantially distort variance estimation and power, particularly for treatment interaction effects. We develop a unified covariance framework for M-SWDs that separates intracluster correlation from an explicit cluster-period correlation matrix. This formulation accommodates exchangeable, autoregressive, and more general distance-dependent correlation structures while preserving closed-form expressions for the variance of treatment effect estimators under linear mixed models. Using analytic results and simulation studies, we demonstrate that assuming uniform correlation when the true structure is time-dependent can lead to substantial power mischaracterization. Specifically, we find that designs calibrated under independence assumptions may be overly conservative and compound symmetry can be either optimistic or conservative. These findings demonstrate the importance of explicitly modeling cluster-period correlation at the design stage of M-SWDs and provide practical guidance for power calculation and design selection in realistic settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper develops a unified covariance framework for multiple-intervention stepped wedge designs (M-SWDs) that separates a scalar intracluster correlation from an explicit cluster-period correlation matrix. This accommodates exchangeable, autoregressive, and distance-dependent structures while retaining closed-form variance expressions for treatment-effect estimators under linear mixed models. Analytic derivations and simulation studies are used to demonstrate that misspecifying the correlation structure (e.g., assuming exchangeability when the true structure is time-dependent) can substantially distort power calculations, particularly for interaction effects.

Significance. If the derivations and simulations hold, the framework addresses a documented limitation in existing M-SWD power methods by enabling realistic, non-uniform correlation modeling at the design stage without sacrificing closed-form variance expressions. The explicit separation of ICC from the cluster-period matrix and the provision of analytic results plus simulations constitute clear strengths for practical trial design.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'more general distance-dependent correlation structures' is used without naming the specific families (e.g., Toeplitz, exponential decay) that are actually implemented; a brief enumeration would improve clarity.
  2. [Methods] The manuscript states that closed-form expressions are preserved, but the precise matrix inversion or Woodbury identity steps used to obtain the variance of the treatment estimator are not previewed; adding a short outline in the methods section would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our unified covariance framework for M-SWDs. We note the recommendation for minor revision but observe that no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; framework is a deliberate modeling choice with independent analytic validation

full rationale

The paper presents a covariance factorization (scalar ICC times explicit cluster-period correlation matrix) as an explicit modeling decision that extends existing LMM variance formulas to non-exchangeable structures. This choice is not derived from or reduced to its own fitted outputs; the closed-form variance expressions follow directly from standard linear mixed model algebra once the covariance structure is posited. Validation occurs via separate analytic derivations plus simulation studies that compare power under misspecified versus correctly specified structures, providing external checks. No self-citation chain is invoked to justify uniqueness or to substitute for the derivation, and no step renames a fitted quantity as a prediction. The central contribution therefore remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities are enumerated beyond the general allowance for arbitrary cluster-period correlation matrices.

pith-pipeline@v0.9.1-grok · 5712 in / 1013 out tokens · 38216 ms · 2026-06-26T07:24:34.325678+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

118 extracted references · 104 canonical work pages

  1. [1]

    Statistics in Medicine , author =

    Power analysis for stepped wedge trials with multiple interventions , volume =. Statistics in Medicine , author =. 2022 , pages =. doi:10.1002/sim.9301 , abstract =

    work page doi:10.1002/sim.9301 2022

  2. [2]

    Journal of Clinical Epidemiology , author =

    Proposed variations of the stepped-wedge design can be used to accommodate multiple interventions , volume =. Journal of Clinical Epidemiology , author =. 2017 , pmid =. doi:10.1016/j.jclinepi.2017.04.004 , abstract =

    work page doi:10.1016/j.jclinepi.2017.04.004 2017

  3. [3]

    Cancer Research , author =

    The. Cancer Research , author =. 1987 , pages =

    1987

  4. [4]

    American journal of epidemiology , author =

    Intraclass. American journal of epidemiology , author =. 1995 , pages =. doi:10.1093/oxfordjournals.aje.a117194 , abstract =

    work page doi:10.1093/oxfordjournals.aje.a117194 1995

  5. [5]

    American Journal of Public Health , author =

    Design and. American Journal of Public Health , author =. 2004 , pmid =

    2004

  6. [6]

    Statistics in Medicine , author =

    Analysis of cluster randomized cross-over trial data: a comparison of methods , volume =. Statistics in Medicine , author =. 2007 , note =. doi:10.1002/sim.2537 , abstract =

    work page doi:10.1002/sim.2537 2007

  7. [7]

    Contemporary Clinical Trials , author =

    Design and analysis of stepped wedge cluster randomized trials , volume =. Contemporary Clinical Trials , author =. 2007 , pmid =. doi:10.1016/j.cct.2006.05.007 , abstract =

    work page doi:10.1016/j.cct.2006.05.007 2007

  8. [8]

    Biometrika , author =

    Longitudinal data analysis using generalized linear models , volume =. Biometrika , author =. 1986 , pages =. doi:10.1093/biomet/73.1.13 , abstract =

    work page doi:10.1093/biomet/73.1.13 1986

  9. [9]

    Biometrical Journal , author =

    Relative efficiency of unequal versus equal cluster sizes in cluster randomized trials using generalized estimating equation models , volume =. Biometrical Journal , author =. 2018 , note =. doi:10.1002/bimj.201600262 , abstract =

    work page doi:10.1002/bimj.201600262 2018

  10. [10]

    Journal of Consulting and Clinical Psychology , author =

    Random-effects regression models for clustered data with an example from smoking prevention research , volume =. Journal of Consulting and Clinical Psychology , author =. 1994 , note =. doi:10.1037/0022-006X.62.4.757 , abstract =

    work page doi:10.1037/0022-006x.62.4.757 1994

  11. [11]

    Biometrics , author =

    Random-effects models for longitudinal data , volume =. Biometrics , author =. 1982 , pmid =

    1982

  12. [12]

    Implementation Science , author =

    Evaluating the impact of multilevel evidence-based implementation strategies to enhance provider recommendation on human papillomavirus vaccination rates among an empaneled primary care patient population: a study protocol for a stepped-wedge cluster randomized trial , volume =. Implementation Science , author =. 2018 , keywords =. doi:10.1186/s13012-018-...

    work page doi:10.1186/s13012-018-0778-x 2018

  13. [13]

    BMC Health Services Research , author =

    Effectiveness of sensor monitoring in an occupational therapy rehabilitation program for older individuals after hip fracture, the. BMC Health Services Research , author =. 2017 , keywords =. doi:10.1186/s12913-016-1934-0 , abstract =

    work page doi:10.1186/s12913-016-1934-0 2017

  14. [14]

    BMC Psychiatry , author =

    Enhancing doctor-patient relationships in community health care institutions: the. BMC Psychiatry , author =. 2023 , pmid =. doi:10.1186/s12888-023-04948-w , abstract =

    work page doi:10.1186/s12888-023-04948-w 2023

  15. [15]

    Journal of Clinical Epidemiology , author =

    Systematic review of stepped wedge cluster randomized trials shows that design is particularly used to evaluate interventions during routine implementation , volume =. Journal of Clinical Epidemiology , author =. 2011 , keywords =. doi:10.1016/j.jclinepi.2010.12.003 , abstract =

    work page doi:10.1016/j.jclinepi.2010.12.003 2011

  16. [16]

    Age and Ageing , author =

    Effectiveness of sensor monitoring in a rehabilitation programme for older patients after hip fracture: a three-arm stepped wedge randomised trial , volume =. Age and Ageing , author =. 2019 , pages =. doi:10.1093/ageing/afz074 , abstract =

    work page doi:10.1093/ageing/afz074 2019

  17. [17]

    Scientific Reports , author =

    Assessment, management, and incidence of neonatal jaundice in healthy neonates cared for in primary care: a prospective cohort study , volume =. Scientific Reports , author =. 2022 , note =. doi:10.1038/s41598-022-17933-2 , abstract =

    work page doi:10.1038/s41598-022-17933-2 2022

  18. [18]

    BMJ Open , author =

    Screening and treatment to reduce severe hyperbilirubinaemia in infants in primary care (. BMJ Open , author =. 2019 , pmid =. doi:10.1136/bmjopen-2018-028270 , abstract =

    work page doi:10.1136/bmjopen-2018-028270 2019

  19. [19]

    Trials , author =

    Understanding the cluster randomised crossover design: a graphical illustration of the components of variation and a sample size tutorial , volume =. Trials , author =. 2017 , keywords =. doi:10.1186/s13063-017-2113-2 , abstract =

    work page doi:10.1186/s13063-017-2113-2 2017

  20. [20]

    The Lancet Infectious Diseases , author =

    Stepped-wedge trial design to evaluate. The Lancet Infectious Diseases , author =. 2015 , pmid =. doi:10.1016/S1473-3099(15)00078-X , language =

    work page doi:10.1016/s1473-3099(15)00078-x 2015

  21. [21]

    The BMJ , author =

    Reporting of stepped wedge cluster randomised trials: extension of the. The BMJ , author =. 2018 , pmid =. doi:10.1136/bmj.k1614 , abstract =

    work page doi:10.1136/bmj.k1614 2018

  22. [22]

    Clinical trials (London, England) , author =

    Statistical. Clinical trials (London, England) , author =. 2016 , pmid =. doi:10.1177/1740774516646578 , abstract =

    work page doi:10.1177/1740774516646578 2016

  23. [23]

    Statistical Methods in Medical Research , author =

    Mixed-effects models for the design and analysis of stepped wedge cluster randomized trials:. Statistical Methods in Medical Research , author =. 2021 , pmid =. doi:10.1177/0962280220932962 , abstract =

    work page doi:10.1177/0962280220932962 2021

  24. [24]

    BMJ , author =

    The stepped wedge cluster randomised trial: rationale, design, analysis, and reporting , volume =. BMJ , author =. 2015 , pmid =. doi:10.1136/bmj.h391 , abstract =

    work page doi:10.1136/bmj.h391 2015

  25. [25]

    Statistics in Medicine , author =

    Statistical efficiency and optimal design for stepped cluster studies under linear mixed effects models , volume =. Statistics in Medicine , author =. 2016 , note =. doi:10.1002/sim.6850 , abstract =

    work page doi:10.1002/sim.6850 2016

  26. [26]

    International Journal of Epidemiology , author =

    Methods for sample size determination in cluster randomized trials , volume =. International Journal of Epidemiology , author =. 2015 , pmid =. doi:10.1093/ije/dyv113 , abstract =

    work page doi:10.1093/ije/dyv113 2015

  27. [27]

    , volume =

    Simple sample size calculation for cluster-randomized trials. , volume =. International Journal of Epidemiology , author =. 1999 , pages =. doi:10.1093/ije/28.2.319 , abstract =

    work page doi:10.1093/ije/28.2.319 1999

  28. [28]

    Contemporary clinical trials , author =

    Ethical and epistemic issues in the design and conduct of pragmatic stepped-wedge cluster randomized clinical trials , volume =. Contemporary clinical trials , author =. 2022 , pmid =. doi:10.1016/j.cct.2022.106703 , abstract =

    work page doi:10.1016/j.cct.2022.106703 2022

  29. [29]

    Clinical Trials , author =

    Inadequacy of ethical conduct and reporting of stepped wedge cluster randomized trials:. Clinical Trials , author =. 2017 , note =. doi:10.1177/1740774517703057 , abstract =

    work page doi:10.1177/1740774517703057 2017

  30. [30]

    Trials , author =

    Stepped wedge cluster randomized controlled trial designs: a review of reporting quality and design features , volume =. Trials , author =. 2017 , pmid =. doi:10.1186/s13063-017-1783-0 , abstract =

    work page doi:10.1186/s13063-017-1783-0 2017

  31. [31]

    International Journal of Epidemiology , author =

    Reflection on modern methods: when is a stepped-wedge cluster randomized trial a good study design choice? , volume =. International Journal of Epidemiology , author =. 2020 , pmid =. doi:10.1093/ije/dyaa077 , abstract =

    work page doi:10.1093/ije/dyaa077 2020

  32. [32]

    Annals of Family Medicine , author =

    What. Annals of Family Medicine , author =. 2004 , pmid =. doi:10.1370/afm.141 , abstract =

    work page doi:10.1370/afm.141 2004

  33. [33]

    1981 , pages =

    American Journal of Epidemiology , author =. 1981 , pages =. doi:10.1093/oxfordjournals.aje.a113261 , abstract =

    work page doi:10.1093/oxfordjournals.aje.a113261 1981

  34. [34]

    International Journal of Epidemiology , author =

    Cluster randomized trials with a small number of clusters: which analyses should be used? , volume =. International Journal of Epidemiology , author =. 2018 , pages =. doi:10.1093/ije/dyx169 , abstract =

    work page doi:10.1093/ije/dyx169 2018

  35. [35]

    BMC Medical Research Methodology , author =

    The stepped wedge trial design: a systematic review , volume =. BMC Medical Research Methodology , author =. 2006 , pmid =. doi:10.1186/1471-2288-6-54 , abstract =

    work page doi:10.1186/1471-2288-6-54 2006

  36. [36]

    Design and

    DONNER, ALLAN and Klar, Neil , year =. Design and

  37. [37]

    Trials , author =

    Stepped wedge randomised controlled trials: systematic review of studies published between 2010 and 2014 , volume =. Trials , author =. 2015 , keywords =. doi:10.1186/s13063-015-0839-2 , abstract =

    work page doi:10.1186/s13063-015-0839-2 2010

  38. [38]

    Eldridge, Sandra and Kerry, Sally , year =. A

  39. [39]

    Statistics in Medicine , author =

    Relative efficiency of unequal cluster sizes in stepped wedge and other trial designs under longitudinal or cross‐sectional sampling , volume =. Statistics in Medicine , author =. 2018 , pmid =. doi:10.1002/sim.7943 , abstract =

    work page doi:10.1002/sim.7943 2018

  40. [40]

    Journal of Clinical Epidemiology , author =

    Stepped wedge designs could reduce the required sample size in cluster randomized trials , volume =. Journal of Clinical Epidemiology , author =. 2013 , keywords =. doi:10.1016/j.jclinepi.2013.01.009 , abstract =

    work page doi:10.1016/j.jclinepi.2013.01.009 2013

  41. [41]

    Journal of Clinical Epidemiology , author =

    The stepped wedge cluster randomized trial always requires fewer clusters but not always fewer measurements, that is, participants than a parallel cluster randomized trial in a cross-sectional design , volume =. Journal of Clinical Epidemiology , author =. 2013 , pages =. doi:10.1016/j.jclinepi.2013.07.008 , number =

    work page doi:10.1016/j.jclinepi.2013.07.008 2013

  42. [42]

    cluster randomized trials:

    The efficiency of stepped wedge vs. cluster randomized trials:. Journal of Clinical Epidemiology , author =. 2013 , pages =. doi:10.1016/j.jclinepi.2013.07.007 , number =

    work page doi:10.1016/j.jclinepi.2013.07.007 2013

  43. [43]

    Journal of Clinical Epidemiology , author =

    Sample size calculations for stepped wedge and cluster randomised trials: a unified approach , volume =. Journal of Clinical Epidemiology , author =. 2016 , pmid =. doi:10.1016/j.jclinepi.2015.08.015 , abstract =

    work page doi:10.1016/j.jclinepi.2015.08.015 2016

  44. [44]

    Statistics in Medicine , author =

    Design and analysis of three-arm parallel cluster randomized trials with small numbers of clusters , volume =. Statistics in Medicine , author =. 2021 , note =. doi:10.1002/sim.8828 , abstract =

    work page doi:10.1002/sim.8828 2021

  45. [45]

    Statistics in medicine , author =

    Constrained randomization and statistical inference for multi-arm parallel cluster randomized controlled trials , volume =. Statistics in medicine , author =. 2022 , pmid =. doi:10.1002/sim.9333 , abstract =

    work page doi:10.1002/sim.9333 2022

  46. [46]

    JAMA , author =

    Reporting of. JAMA , author =. 2019 , pages =. doi:10.1001/jama.2019.3087 , abstract =

    work page doi:10.1001/jama.2019.3087 2019

  47. [47]

    Clinical Cancer Research , author =

    Multi-. Clinical Cancer Research , author =. 2008 , pages =. doi:10.1158/1078-0432.CCR-08-0325 , abstract =

    work page doi:10.1158/1078-0432.ccr-08-0325 2008

  48. [48]

    The Lancet , author =

    More multiarm randomised trials of superiority are needed , volume =. The Lancet , author =. 2014 , pages =. doi:10.1016/S0140-6736(14)61122-3 , number =

    work page doi:10.1016/s0140-6736(14)61122-3 2014

  49. [49]

    International Journal of Epidemiology , author =

    Sample size calculators for planning stepped-wedge cluster randomized trials: a review and comparison , volume =. International Journal of Epidemiology , author =. 2022 , pmid =. doi:10.1093/ije/dyac123 , abstract =

    work page doi:10.1093/ije/dyac123 2022

  50. [50]

    Statistical Methods in Medical Research , author =

    Sample size determination for stepped wedge cluster randomized trials in pragmatic settings , volume =. Statistical Methods in Medical Research , author =. 2021 , note =. doi:10.1177/09622802211022392 , abstract =

    work page doi:10.1177/09622802211022392 2021

  51. [51]

    BMJ Open , author =

    Systematic review finds major deficiencies in sample size methodology and reporting for stepped-wedge cluster randomised trials , volume =. BMJ Open , author =. 2016 , pmid =. doi:10.1136/bmjopen-2015-010166 , abstract =

    work page doi:10.1136/bmjopen-2015-010166 2016

  52. [52]

    American Journal of Preventive Medicine , author =

    Food. American Journal of Preventive Medicine , author =. 2023 , pages =. doi:10.1016/j.amepre.2023.03.022 , abstract =

    work page doi:10.1016/j.amepre.2023.03.022 2023

  53. [53]

    BMJ , author =

    Impact of. BMJ , author =. 2011 , pmid =. doi:10.1136/bmj.d5886 , abstract =

    work page doi:10.1136/bmj.d5886 2011

  54. [54]

    Trials , author =

    Designing a stepped wedge trial: three main designs, carry-over effects and randomisation approaches , volume =. Trials , author =. 2015 , keywords =. doi:10.1186/s13063-015-0842-7 , abstract =

    work page doi:10.1186/s13063-015-0842-7 2015

  55. [55]

    Statistics in Medicine , author =

    Sample size calculation for stepped wedge and other longitudinal cluster randomised trials , volume =. Statistics in Medicine , author =. 2016 , pmid =. doi:10.1002/sim.7028 , abstract =

    work page doi:10.1002/sim.7028 2016

  56. [56]

    Statistical Methods in Medical Research , author =

    Impact of non-uniform correlation structure on sample size and power in multiple-period cluster randomised trials , volume =. Statistical Methods in Medical Research , author =. 2019 , note =. doi:10.1177/0962280217734981 , abstract =

    work page doi:10.1177/0962280217734981 2019

  57. [57]

    Statistics in Medicine , author =

    Accounting for a decaying correlation structure in cluster randomized trials with continuous recruitment , volume =. Statistics in Medicine , author =. 2019 , note =. doi:10.1002/sim.8089 , abstract =

    work page doi:10.1002/sim.8089 2019

  58. [58]

    Statistics in Medicine , author =

    Bias and inference from misspecified mixed-effect models in stepped wedge trial analysis , volume =. Statistics in Medicine , author =. 2017 , note =. doi:10.1002/sim.7348 , abstract =

    work page doi:10.1002/sim.7348 2017

  59. [59]

    Statistical Methods in Medical Research , author =

    Inference for the treatment effect in multiple-period cluster randomised trials when random effect correlation structure is misspecified , volume =. Statistical Methods in Medical Research , author =. 2019 , note =. doi:10.1177/0962280218797151 , abstract =

    work page doi:10.1177/0962280218797151 2019

  60. [60]

    Trials , author =

    Allocation techniques for balance at baseline in cluster randomized trials: a methodological review , volume =. Trials , author =. 2012 , keywords =. doi:10.1186/1745-6215-13-120 , abstract =

    work page doi:10.1186/1745-6215-13-120 2012

  61. [61]

    Statistics in Medicine , author =

    An evaluation of constrained randomization for the design and analysis of group-randomized trials with binary outcomes , volume =. Statistics in Medicine , author =. 2017 , pmid =. doi:10.1002/sim.7410 , abstract =

    work page doi:10.1002/sim.7410 2017

  62. [62]

    Statistical design of

  63. [63]

    2010 , note =

    Annals of Internal Medicine , author =. 2010 , note =. doi:10.7326/0003-4819-152-11-201006010-00232 , abstract =

    work page doi:10.7326/0003-4819-152-11-201006010-00232 2010

  64. [64]

    The BMJ , author =

    Improving the reporting of pragmatic trials: an extension of the. The BMJ , author =. 2008 , pmid =. doi:10.1136/bmj.a2390 , abstract =

    work page doi:10.1136/bmj.a2390 2008

  65. [65]

    BMJ , author =

    Consort 2010 statement: extension to cluster randomised trials , volume =. BMJ , author =. 2012 , pmid =. doi:10.1136/bmj.e5661 , abstract =

    work page doi:10.1136/bmj.e5661 2010

  66. [66]

    Statistics in Medicine , author =

    Cohort versus cross-sectional design in large field trials:. Statistics in Medicine , author =. 1994 , note =. doi:10.1002/sim.4780130108 , abstract =

    work page doi:10.1002/sim.4780130108 1994

  67. [67]

    Statistics in Medicine , author =

    A simple sample size formula for analysis of covariance in cluster randomized trials , volume =. Statistics in Medicine , author =. 2012 , note =. doi:10.1002/sim.5352 , abstract =

    work page doi:10.1002/sim.5352 2012

  68. [68]

    BJOG : an international journal of obstetrics and gynaecology , author =

    Randomised controlled trials—the gold standard for effectiveness research , volume =. BJOG : an international journal of obstetrics and gynaecology , author =. 2018 , pmid =. doi:10.1111/1471-0528.15199 , number =

    work page doi:10.1111/1471-0528.15199 2018

  69. [69]

    Korean Journal of Anesthesiology , author =

    Randomization in clinical studies , volume =. Korean Journal of Anesthesiology , author =. 2019 , pmid =. doi:10.4097/kja.19049 , abstract =

    work page doi:10.4097/kja.19049 2019

  70. [70]

    , volume =

    Randomisation. , volume =. BMJ : British Medical Journal , author =. 1991 , pmid =

    1991

  71. [71]

    Neuropsychiatric Disease and Treatment , author =

    Randomized controlled trials – a matter of design , volume =. Neuropsychiatric Disease and Treatment , author =. 2016 , pmid =. doi:10.2147/NDT.S101938 , urldate =

    work page doi:10.2147/ndt.s101938 2016

  72. [72]

    BMJ , author =

    Treatment allocation in controlled trials: why randomise? , volume =. BMJ , author =. 1999 , pmid =. doi:10.1136/bmj.318.7192.1209 , abstract =

    work page doi:10.1136/bmj.318.7192.1209 1999

  73. [73]

    Statistics in Medicine , author =

    Stepped-wedge cluster randomised controlled trials: a generic framework including parallel and multiple-level designs , volume =. Statistics in Medicine , author =. 2015 , note =. doi:10.1002/sim.6325 , abstract =

    work page doi:10.1002/sim.6325 2015

  74. [74]

    Intensive Care Medicine , author =

    Introducing. Intensive Care Medicine , author =. 2004 , keywords =. doi:10.1007/s00134-004-2268-7 , abstract =

    work page doi:10.1007/s00134-004-2268-7 2004

  75. [75]

    Annals of Epidemiology , author =

    Regression. Annals of Epidemiology , author =. 2005 , keywords =. doi:10.1016/j.annepidem.2004.08.007 , abstract =

    work page doi:10.1016/j.annepidem.2004.08.007 2005

  76. [76]

    Epidemiology , author =

    The. Epidemiology , author =. 2007 , pages =. doi:10.1097/EDE.0b013e3181200199 , abstract =

    work page doi:10.1097/ede.0b013e3181200199 2007

  77. [77]

    Journal of Clinical Epidemiology , author =

    Explanatory and. Journal of Clinical Epidemiology , author =. 2009 , pages =. doi:10.1016/j.jclinepi.2009.01.012 , abstract =

    work page doi:10.1016/j.jclinepi.2009.01.012 2009

  78. [78]

    New England Journal of Medicine , author =

    Pragmatic. New England Journal of Medicine , author =. 2016 , note =. doi:10.1056/NEJMra1510059 , abstract =

    work page doi:10.1056/nejmra1510059 2016

  79. [79]

    Clinical trials (London, England) , author =

    Exploring the ethical and regulatory issues in pragmatic clinical trials , volume =. Clinical trials (London, England) , author =. 2015 , pmid =. doi:10.1177/1740774515598334 , abstract =

    work page doi:10.1177/1740774515598334 2015

  80. [80]

    JAMA , author =

    Practical. JAMA , author =. 2003 , pages =. doi:10.1001/jama.290.12.1624 , abstract =

    work page doi:10.1001/jama.290.12.1624 2003

Showing first 80 references.