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What is estimated in cluster randomized crossover trials with informative sizes? -- A survey of estimands and common estimators

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Pith's one-line read This paper asks what common estimators in two-period cluster randomized crossover (CRXO) trials actually estimate when cluster, period, or cluster-period sizes carry information about treatment effects, and derives which estimators…

desk verdict A careful, useful mapping of CRXO estimators to causal estimands under informative sizes; the consistency table is the real contribution, and the paper deserves serious peer review. read the letter →

arxiv 2505.00925 v1 pith:5EGMV4OC submitted 2025-05-02 stat.ME

Kenneth M. Lee , Andrew B. Forbes , Jessica Kasza , Andrew Copas , Brennan C. Kahan , Paul J. Young , Michael O. Harhay , Fan Li This is my paper · ORCID
classification stat.ME MSC 62F1262K10
keywords clusterrandomizedcrossovertrialsinformativesizesestimandsindependenceestimatingequationmixedeffectsmodelsfixedconsistencytreatmenteffectheterogeneity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what common estimators in two-period cluster randomized crossover (CRXO) trials actually estimate when cluster, period, or cluster-period sizes carry information about treatment effects. It defines four interpretable estimands—the individual-average, cluster-period-average, cluster-average, and period-average treatment effects—as special cases of a weighted average treatment effect. Its central result is that the unweighted independence estimating equation (IEE) estimator is always consistent for the individual-average estimand, and its inverse cluster-period, inverse cluster, and inverse period size weighted versions are always consistent for the other three estimands. The unweighted and weighted nested exchangeable mixed effects estimators instead converge to data-dependent weighted averages that are generally not any of the four estimands. This gives trial analysts a map for choosing an estimator when informative sizes are plausible, and a warning that some widely used mixed models can answer no clear causal question.

What carries the argument

The central object is the family of weighted average treatment effect estimands, $wATE=\sum_{i,j,k}w_{ijk}(Y_{ijk}(1)-Y_{ijk}(0))/\sum_{i,j,k}w_{ijk}$, with weights $1$, $1/K_{ij}$, $1/\sum_j K_{ij}$, and $1/\sum_i K_{ij}$ selecting the iATE, cpATE, cATE, and pATE. The consistency proofs expand each estimator's probability limit under the i.i.d. cluster superpopulation assumption and check which limit equals one of these weighted forms. For the nested exchangeable mixed model, the limiting per-cluster weight is proportional to $1/[1+(K_{i-}-1)\rho_{wp}-K_{i-}\rho_{bp}]$, so whenever this factor varies with cluster-period size the limit is a data-dependent weighted average outside the named estimand family.

What would settle it

Under the paper's own assumptions, simulate a 2-period CRXO with treatment effects that depend on cluster-period size, e.g. $Y_{ijk}(1)-Y_{ijk}(0)=a+bK_{ij}$, and compute the unweighted IEE estimate as $I$ grows; for the central claim to hold, it must converge to $\mathbb{E}[\sum_{j,k}(a+bK_{ij})]/\mathbb{E}[\sum_j K_{ij}]$, the individual-average limit, not to the equally weighted cluster-period average. A data-generating process satisfying cluster i.i.d. sampling and independent randomization that produces a different probability limit would refute the always-consistency result.

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Extended reading notes

Core claim

The paper claims that, in a 2-period, 2-sequence cross-sectional CRXO trial with continuous outcomes and clusters drawn as i.i.d. superpopulation units, the unweighted IEE estimator is always consistent for the iATE, $\mathbb{E}[\sum_{j=1}^2\sum_{k=1}^{K_{ij}}(Y_{ijk}(1)-Y_{ijk}(0))]/\mathbb{E}[\sum_{j=1}^2 K_{ij}]$, and the IEEcpw, IEEcw, and IEEpw estimators are always consistent for the cpATE, cATE, and pATE, respectively, regardless of informative sizes. The same is true of the FEcpw estimator for cpATE. The exchangeable mixed effects model is consistent for iATE (and its inverse-size weighted forms for cpATE/cATE) only when cluster-period sizes are equal within clusters, $K_{i1}=K_{i2}$ for all $i$; the fixed effects model generally converges to pATE rather than iATE unless a constant period-size ratio $\lambda_i=\lambda$ holds. The nested exchangeable mixed effects model, unweighted or weighted, converges to limits like $\mathbb{E}\{ [1+(K_{i-}-1)\rho_{wp}-K_{i-}\rho_{bp}]^{-1} \sum_{j,k}(Y_{ijk}(1)-Y_{ijk}(0))\}/\mathbb{E}\{2K_{i-}[1+(K_{i-}-1)\rho_{wp}-K_{i-}\rho_{bp}]^{-1}\}$, which depends on cluster sizes and on estimated within- and between-period ICCs and is generally not iATE, cpATE, cATE, or pATE. Simulations confirm the derived relative biases, and a reanalysis of a 49-cluster CRXO trial shows NEME estimates drifting from the IEE-based estimates.

Load-bearing premise

All stated probability limits assume clusters are independent, identically distributed draws from an infinite superpopulation and that the treatment sequence is independent of potential outcomes and all cluster-period sizes; the GLS limits also assume the variance components converge to fixed values rather than being re-estimated in a way that shifts the weights.

Editorial extensions

If this is right

  • When informative sizes are plausible, the unweighted IEE estimator gives a defensible estimate of the individual-average treatment effect without requiring a no-informative-sizes assumption.
  • Weighted IEE estimators (IEEcpw, IEEcw, IEEpw) give defensible estimates of cluster-period, cluster, and period average effects, so a pre-specified analysis can use them to answer each level of hypothesis.
  • NEME-based estimates should not be interpreted as targeting any of the four estimands if treatment effects vary with cluster, period, or cluster-period size; a notable gap between NEME and IEE estimates may itself signal informative sizes.
  • Fixed effects and exchangeable mixed models are usable for their intended estimands only under restrictions such as $K_{i1}=K_{i2}$ (or a constant period-size ratio for fixed effects); otherwise an FE analysis may be estimating the period-average rather than the individual-average effect.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is a sensitivity diagnostic that reports IEE, IEEcw, IEEpw, and IEEcpw estimates side by side; under the paper's consistency map, material differences among them indicate which sizes are informative, although the paper explicitly notes it does not provide a formal statistical test.
  • The FE result suggests that protocols pre-specifying fixed-effects analyses in CRXO trials should also pre-specify the target as pATE (or verify a constant period-size ratio), since the FE estimator's estimand depends on the balance of period sizes rather than on the analyst's intent.
  • One can quantify how far a NEME estimate is from any named estimand by plugging estimated ICCs and observed cluster sizes into the limiting weight $1/[1+(K_{i-}-1)\hat{\rho}_{wp}-K_{i-}\hat{\rho}_{bp}]$ and reporting the implied weighted average as a sensitivity target.
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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. This paper studies four target estimands for 2-period cross-sectional cluster randomized crossover (CRXO) trials with continuous outcomes: the individual-average (iATE), cluster-period-average (cpATE), cluster-average (cATE), and period-average (pATE) treatment effects. It formalizes informative cluster, period, and cluster-period sizes, and derives the probability limits of unweighted and inverse-size-weighted versions of the independence estimating equation (IEE), fixed effects (FE), exchangeable mixed effects (EME), and nested exchangeable mixed effects (NEME) estimators under an iid superpopulation model in which sequence assignment is independent of potential outcomes and cluster-period sizes. The central results are: the unweighted IEE is always consistent for the iATE, and its inverse cluster-period, inverse cluster, and inverse period size weighted counterparts are always consistent for the cpATE, cATE, and pATE; EME and weighted EME are consistent for iATE, cpATE, and cATE when Ki1=Ki2; NEME and its weighted versions generally converge to weighted estimands with data-dependent weights and are not consistent for any of the four estimands; and FE results require lambda_i=lambda for all i and often target the pATE or cpATE instead of the iATE or cATE. A simulation study and a reanalysis of the PEPTIC CRXO trial illustrate and apply these results.

Significance. If the results hold, the paper provides a practical and theory-grounded map from common estimators to well-defined causal estimands in CRXO trials, and it sharpens warnings about the use of NEME in the presence of informative sizes. The algebra in Appendices C through F is detailed, and the simulation results are consistent with the stated theory in the specified DGP. The finite-population versus superpopulation distinction is explicitly acknowledged as a scope condition, and the derivations rely on clearly stated assumptions. I could not identify a load-bearing error in the central consistency mapping. The main caveat is that the exact NEME limits are derived for GLS weights with fixed variance components; the manuscript acknowledges the dependence on probability limits of estimated ICCs, which is sufficient for the qualitative conclusion but should be stated more precisely.

minor comments (5)
  1. [Section 5.2 and Appendix H] The DGP description states delta_2 = 0.6, but the formulas and numerical values immediately afterward use delta_2 = 0.5 (for example, the iATE formula yielding 0.53 uses delta_2 = 0.6, while the following parenthetical says delta_2 = 0.5); please reconcile the two values, since the reported bias results correspond to only one specification.
  2. [Section 4.2.1, Eq. (26)] The probability limit is derived for GLS weights with fixed variance components, while in practice the NEME weights use estimated ICCs; please state explicitly that Eq. (26) is the limit under variance components fixed at their probability limits and that the feasible GLS estimator is assumed to share this limit.
  3. [Appendix B] The proof of Proposition 1.2 is described as a proof by induction, but it proceeds by constructing an example; please correct the wording to 'by example' or 'by construction.'
  4. [Data Availability Statement] The statement 'no new data were created or analyzed' conflicts with the reanalysis of the PEPTIC trial dataset in Section 6; please amend the statement to reflect that existing trial data were reanalyzed.
  5. [Section 3.3] The NEME covariance matrix block is defined only under Ki1 = Ki2, and this restriction is later used for the NEME results; please flag the equal-within-cluster-period-size assumption at the first definition to avoid confusion for readers who encounter the restriction only in Section 4.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified; consistency results are derived from explicitly stated randomization and superpopulation assumptions rather than from fitted constants or self-citation.

full rationale

The paper's central claims are derivations, not definitions in disguise. The estimands iATE, cpATE, cATE, and pATE are defined in Section 2 and Table 1 directly from potential outcomes Y_ijk(1)-Y_ijk(0) with explicit individual-level weights w_ijk, independently of any estimator. The estimators in Section 3 are standard OLS/GLS/weighted estimating-equation estimators, and the consistency claims in Section 4 and Appendices C-F follow by taking probability limits under the stated superpopulation model with independent and identically distributed clusters and sequence independence S_i ⊥ Ω = {Y_ijk(0), Y_ijk(1), K_ij}. For example, Equation (17) is the probability limit of the explicit IEE estimator in Equation (16), not a fitted parameter renamed as a prediction; similarly, (19), (22), and (24) are limits of the explicitly weighted IEE estimators. The NEME limits in (26) and (28) are stated for GLS weights and the paper explicitly notes that the data-dependent weights involve the probability limits of model-based ICC estimators, which is a technical caveat about misspecification rather than circularity. Self-citations to prior P-CRT and PB-CRT work [17, 18] and to the wATE family [10] are used as background and motivation; the CRXO-specific results are derived in the paper's own appendices and do not rest on an unverified self-citation chain or an imported uniqueness theorem. No step in the derivation reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims do not introduce free parameters fitted to data. The estimands are externally defined from potential outcomes; the consistency results rely on the stated superpopulation and randomization assumptions and on standard regularity for GLS plug-in estimators.

assumptions (6)
  • domain assumption Clusters are independent and identically distributed draws from an infinite superpopulation of clusters.
    Invoked in Section 2 when defining the estimands and used throughout Section 4 to take probability limits as I grows.
  • domain assumption Cluster-level Stable Unit Treatment Value Assumption: Y_ijk = S_i Y_ijk(1) + (1 - S_i) Y_ijk(0), no interference between units.
    Given in Equation (1) in Section 2; connects observed outcomes to potential outcomes.
  • domain assumption Randomization: the sequence indicator S_i is independent of potential outcomes and cluster-period sizes, S_i independent of Omega.
    Stated in Section 4 and used in all probability limit derivations, e.g., Section C.1.
  • domain assumption For GLS estimators (EME, NEME), the estimated variance components converge in probability to fixed limits (rho_wp, rho_bp) so plug-in weights yield the stated limits.
    Used in Section 4.2 and Appendix E for NEME limits; standard M-estimation consistency for variance parameters.
  • standard math Regularity conditions for laws of large numbers applied to i.i.d. cluster-level quantities (K_i1, K_i2, potential outcomes), with finite expectations.
    Implicit in the convergence statements throughout Section 4.
  • domain assumption A 2-period, 2-sequence CRXO design with equal allocation of clusters to sequences (I/2 per sequence) and cross-sectional sampling of individuals.
    The FE derivation (Appendix F.1) explicitly assumes I/2 clusters per sequence; asymptotic results rely on equal randomization probabilities.

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Pith. "Pith review of What is estimated in cluster randomized crossover trials with informative sizes? -- A survey of estimands and common estimators." pith.science (2026). https://pith.science/paper/5EGMV4OC

@misc{pith2026250500925,
  author       = {Pith},
  title        = {Pith review of: What is estimated in cluster randomized crossover trials with informative sizes? -- A survey of estimands and common estimators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5EGMV4OC}},
  note         = {Machine review of arXiv:2505.00925}
}
read the original abstract

In the analysis of cluster randomized trials (CRTs), previous work has defined two meaningful estimands: the individual-average treatment effect (iATE) and cluster-average treatment effect (cATE) estimand, to address individual and cluster-level hypotheses. In multi-period CRT designs, such as the cluster randomized crossover (CRXO) trial, additional weighted average treatment effect estimands help fully reflect the longitudinal nature of these trial designs, namely the cluster-period-average treatment effect (cpATE) and period-average treatment effect (pATE). We define different forms of informative sizes, where the treatment effects vary according to cluster, period, and/or cluster-period sizes, which subsequently cause these estimands to differ in magnitude. Under such conditions, we demonstrate which of the unweighted, inverse cluster-period size weighted, inverse cluster size weighted, and inverse period size weighted: (i.) independence estimating equation, (ii.) fixed effects model, (iii.) exchangeable mixed effects model, and (iv.) nested exchangeable mixed effects model treatment effect estimators are consistent for the aforementioned estimands in 2-period cross-sectional CRXO designs with continuous outcomes. We report a simulation study and conclude with a reanalysis of a CRXO trial testing different treatments on hospital length of stay among patients receiving invasive mechanical ventilation. Notably, with informative sizes, the unweighted and weighted nested exchangeable mixed effects model estimators are not consistent for any meaningful estimand and can yield biased results. In contrast, the unweighted and weighted independence estimating equation, and under specific scenarios, the fixed effects model and exchangeable mixed effects model, can yield consistent and empirically unbiased estimators for meaningful estimands in 2-period CRXO trials.

Figures

Figures reproduced from arXiv: 2505.00925 by the authors.

Figure 1
Figure 1. A standard 2-period, 2-sequence, 4 cluster cross-sectional CRXO trial design, with individuals [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Different individual treatment effects (assuming a constant [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. 1.) Simulation relative bias (%) results in scenarios with homogeneous treatment effects (non [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (12 more)
Figure 3
Figure 3. Figure 3: , alongside the empirical variance. The model-based and jackknife variance estimators explicitly [PITH_FULL_IMAGE:figures/full_fig_p027_3.png]
Figure 4
Figure 4. Figure 4: The coverage probability of the 95% confidence interval and the power of the different unweighted [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]
Figure 5
Figure 5. Figure 5: Simulation relative bias (%) results in scenarios with informative cluster sizes. Dashed lines show [PITH_FULL_IMAGE:figures/full_fig_p029_5.png]
Figure 6
Figure 6. Figure 6: Simulation relative bias (%) results in scenarios with informative period sizes. Dashed lines show [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]
Figure 7
Figure 7. Figure 7: Histograms showing the distribution of hospital LOS and log-LOS among patients in a cross [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]
Figure 8
Figure 8. Figure 8: The efficiency of the different unweighted and weighted models as captured by the average of [PITH_FULL_IMAGE:figures/full_fig_p067_8.png]
Figure 9
Figure 9. Figure 9: The coverage probability of the 95% confidence interval and the power of the different unweighted [PITH_FULL_IMAGE:figures/full_fig_p068_9.png]
Figure 10
Figure 10. Figure 10: Simulation bias results in scenarios with heterogeneous treatment effects (informative cluster sizes) [PITH_FULL_IMAGE:figures/full_fig_p070_10.png]
Figure 11
Figure 11. Figure 11: 1.) Simulation relative bias (%) results in scenarios with homogeneous treatment effects (non [PITH_FULL_IMAGE:figures/full_fig_p071_11.png]
Figure 12
Figure 12. Figure 12: The coverage probability of the 95% confidence interval and the power of the different unweighted [PITH_FULL_IMAGE:figures/full_fig_p072_12.png]
Figure 13
Figure 13. Figure 13: Simulation relative bias (%) results in scenarios with informative cluster sizes. Dashed lines show [PITH_FULL_IMAGE:figures/full_fig_p072_13.png]
Figure 14
Figure 14. Figure 14: Simulation relative bias (%) results in scenarios with informative period sizes. Dashed lines show [PITH_FULL_IMAGE:figures/full_fig_p073_14.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.