Pith. sign in

REVIEW 3 major objections 8 minor 55 references

Accounting for overdispersion and clustering in binomial data from N-of-1 trials

T0 review · 3 major / 8 minor · reviewed 2026-07-10 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Two frequentist methods pool binary N-of-1 trial data despite overdispersion

desk verdict Solid adaptation of existing methods to N-of-1 binary data; Method 2 has unresolved coverage issues that need referee attention read the letter →

arxiv 2607.08722 v1 pith:3F3AM2NY submitted 2026-07-09 stat.AP stat.ME

Majnu John , Heejung Bang , Stephanie Winkelbeiner , Philipp Homan This is my paper
classification stat.APstat.ME
keywords N-of-1trialoverdispersionbeta-binomialdistributionquasi-likelihoodrandomeffectsmodelintraclasscorrelationcoefficientpooledestimationbinaryoutcomes
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

N-of-1 trials are single-patient crossover experiments in which two treatments are compared repeatedly within one individual. When the outcome is binary (e.g., did the patient prefer treatment A over B in a given episode?), pooling results across patients is statistically nontrivial because the aggregated binomial counts exhibit extra-binomial variation — more spread than a simple coin-flip model predicts — and because the data have a nested structure (episodes within patients). This paper presents two frequentist methods for estimating the pooled treatment-preference proportion θ across a set of N-of-1 trials. The first method models overdispersion alone: it assumes each patient's count follows a beta-binomial distribution and uses a quasi-likelihood estimator for θ, with the overdispersion parameter ρ estimated by one of several available methods (ANOVA, Fleiss-Cuzick, Pearson, extended quasi-likelihood, pseudo-likelihood, or method of moments). The second method, building on work by Molenberghs and Verbeke, decomposes each patient's success probability into a beta-distributed component (capturing residual overdispersion) and a logistic-transformed normal random effect (capturing within-patient clustering), thereby modeling both sources of extra variation simultaneously. Simulations show that all variants of Method 1 produce nearly identical MSEs for θ, and that Method 2 achieves comparable or sometimes lower MSE but suffers from poor confidence-interval coverage (as low as 78%) when the true proportion is near 0.75, owing to skewed sampling distributions. Across three real datasets — amitriptyline for fibromyalgia, NSAIDs versus paracetamol for osteoarthritis, and theophylline for chronic airflow limitation — all methods yielded nearly identical pooled estimates, typically differing only in the third decimal place.

What carries the argument

Beta-binomial distribution for modeling overdispersed binomial counts; quasi-likelihood estimation with jackknife variance; combined beta and normal random-effects model with numerical integration of the marginal likelihood; sigmoid-CDF approximation for E(κ) using λ = 0.588

What would settle it

If applied to a new N-of-1 dataset with true θ near 0.75 and moderate overdispersion, Method 2 would produce confidence intervals that fail to contain the true θ at rates substantially below the nominal 95% level, while Method 1 variants would maintain approximately nominal coverage.

Watch

Extended reading notes

Core claim

The central claim is that for pooling binary outcomes from N-of-1 trials, a quasi-likelihood framework with beta-binomial overdispersion (Method 1) and a combined beta-normal random-effects model (Method 2) both produce serviceable pooled proportion estimates, with the practical differences lying not in the point estimates — which converge across methods in real data — but in the behavior of the overdispersion parameter estimates and in the coverage properties of confidence intervals, where Method 2 degrades near extreme true proportions. A secondary and somewhat surprising finding is that the extended quasi-likelihood estimator for ρ, despite having the worst MSE among all ρ-estimators, can

Load-bearing premise

Method 2 assumes that the product decomposition of each patient's success probability into a beta-distributed component and a logistic-transformed normal random effect adequately captures the joint structure of overdispersion and clustering — an assumption whose failure is revealed empirically by the method's poor confidence-interval coverage (as low as 78%) when the true pooled proportion is 0.75.

Editorial extensions

If this is right

  • Researchers conducting pooled analyses of binary N-of-1 trial data now have two ready-to-use frequentist alternatives to the previously available Bayesian approach, broadening accessibility for analysts who prefer or require non-Bayesian methods.
  • The finding that all ρ-estimation variants within Method 1 yield nearly identical θ estimates suggests that for practical N-of-1 pooling, the choice of overdispersion estimator is inconsequential for the main parameter of interest, simplifying methodological decisions.
  • The poor coverage of Method 2 near θ = 0.75 signals that random-effects models for overdispersed binary data may need variance-stabilizing transformations or asymmetric confidence intervals before they can be reliably used in settings where treatment effects are large.
  • The R code provided in the appendix makes both methods immediately deployable, lowering the barrier to adoption for clinical researchers working with N-of-1 trial designs.

Reading between the lines

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

  • If the coverage problem of Method 2 near extreme proportions is a general feature of combined beta-normal random-effects models — not specific to N-of-1 data — then the same issue may arise in other clustered binary-data settings such as cluster randomized trials or longitudinal studies with high event rates, warranting caution in those domains as well.
  • The near-equivalence of all Method 1 variants for θ estimation raises the question of whether the entire apparatus of estimating ρ is necessary for point estimation, or whether ρ matters only for variance estimation and hypothesis testing about overdispersion itself.
  • A natural extension would be to incorporate covariates (e.g., age, sex) into both methods to enable subgroup-specific pooled estimates, which the authors note is possible for Method 2 via its random-effects structure but do not demonstrate.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. The paper presents two frequentist methods for pooling binomial outcomes across N-of-1 trials. Method 1 uses a quasi-likelihood framework with a beta-binomial overdispersion parameter, considering six estimators for the intra-cluster correlation ρ. Method 2 adapts the combined beta-normal random-effects model of Molenberghs et al. to jointly model overdispersion and hierarchical clustering. A simulation study compares the methods across scenarios varying θ and ρ, and three real datasets illustrate the methods. The quasi-likelihood derivation (Appendix A1) is clean, and the provision of R code in Appendix E is a practical strength.

Significance. The paper addresses a genuine methodological gap: frequentist approaches for pooling binary N-of-1 trial data are scarce, with the only prior binary-outcome pooling method being the Bayesian approach of Schluter and Ware (2005). The quasi-likelihood derivation from first principles is a solid contribution, and the systematic comparison of six ρ-estimators in the small-cluster-size regime typical of N-of-1 trials is useful. The finding that the EQL estimator for ρ, despite having poor MSE for ρ itself, yields competitive or superior MSE for θ is an interesting and practically relevant result. The inclusion of reproducible R code is commendable.

major comments (3)
  1. §3, Simulation design: The simulation generates data from Beta-Binomial(n_j, θ, ρ), which is the model underlying Method 1 but not Method 2. Method 2 assumes a combined beta-normal structure (θ_j = π_j·κ_j with π_j ~ Beta and κ_j = sigmoid(ξ_j), ξ_j ~ N(a,1)). Consequently, Method 2 is being evaluated under model misspecification. This could be a primary driver of the poor coverage probabilities at θ=0.75 (Table 2: 78–82%), rather than the asymmetric sampling distributions cited as the explanation. The authors should either (a) include a simulation scenario where data are generated from Method 2's assumed data-generating process, to assess its performance under correct specification, or (b) explicitly discuss the misspecification as a competing explanation for the coverage degradation. Without this, the simulation comparison between the two methods is not on equal footing, which is load-
  2. §2.2, sigmoid-CDF approximation: The constant λ=0.588 is described as obtained by 'trial and error (plotting one function on top of the other)' with no quantification of approximation error. Since E(κ_j) directly enters the final θ estimator for Method 2, the quality of this approximation is load-bearing. A simple numerical comparison of the true integral E[sigmoid(ξ)] for ξ ~ N(a,1) versus the approximation Φ(0.588·a/√(1+0.588²)) across a grid of 'a' values would establish the approximation's range of validity. This is especially important because the coverage problems are most severe at θ=0.75, where the corresponding 'a' values may fall in a regime where the approximation degrades.
  3. §2.2, independence assumption: The decomposition θ = E(π_j)·E(κ_j) assumes independence between π_j and κ_j. While this follows Molenberghs et al.'s framework, no sensitivity analysis or discussion of the potential impact of dependence is provided. Given that Method 2's coverage is suboptimal in certain scenarios, a brief assessment of whether violations of this assumption could contribute to the bias would strengthen the paper's claims about Method 2's viability.
minor comments (8)
  1. §1, paragraph 3: 'the conduct of N-of-1 trials also try to exclude' should be 'tries' or rephrased for subject-verb agreement.
  2. §2.1, Pearson estimator: The estimator is labeled ˆρ_3, but the ANOVA estimator above is also labeled ˆρ_3. This appears to be a numbering error; the Pearson estimator should likely be ˆρ_5 or similar.
  3. §2.1, EQL estimator: The text states 'We used 10^{-8} as the constant that was added in all our simulations.' It would be helpful to note whether this constant was also used in the real data analyses, and whether any X_j values of 0 or n_j occurred in those datasets.
  4. Table 2: The coverage probabilities for Method 1 estimators are all very close (differing by at most ~1.5 percentage points). Given this, the discussion of which ρ-estimator gives 'best' coverage may be over-interpreting simulation noise. A brief note acknowledging this would be appropriate.
  5. Figure 2 and Figure B.1: The boxplots for Method 1 θ-estimates are nearly indistinguishable across ρ-estimators. The separate median plot (Figure B.1) helps, but the differences are still very small. Consider whether a table of median MSEs might communicate the comparison more clearly.
  6. §4.3: The threshold of 0.5 on a 7-point Likert scale is described as 'clinically relevant' per Mahon et al. It would be useful to briefly note whether this threshold has been validated or is a post-hoc choice.
  7. Appendix A1: The notation switches between (X_j, n_j) in the main text and (X_{1j}, X_{2j}, n_j) in the appendix. While the mapping is stated, a brief sentence explaining the motivation for the reparametrization would aid readers following the derivation.
  8. §5, Discussion: The comparison to population-averaged vs. subject-specific interpretations is useful but somewhat general. It would be strengthened by connecting specifically to the N-of-1 context where individual-level inference is the primary goal.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: both methods are independently derived and externally validated by simulation and real data

full rationale

The paper presents two frequentist methods for pooling binomial N-of-1 trial data. Method 1's quasi-likelihood estimator (Eq. 1) is derived from first principles in Appendix A1, starting from the beta-binomial model and solving the quasi-score equations. The estimator does not reduce to a fitted quantity by construction. Method 2 adapts the combined beta-normal random-effects model of Molenberghs et al. [12-15], which is an externally published framework. The final theta estimator uses a sigmoid-CDF approximation with lambda=0.588 chosen by trial-and-error, and an independence assumption between pi_j and kappa_j. While these are unvalidated modeling choices that may affect correctness (as the skeptic notes regarding poor coverage at theta=0.75), they are not circular: the estimator is not defined in terms of the quantity it claims to estimate, and the simulation study generates data from known beta-binomial parameters and evaluates the methods against those known truths. The real data analyses apply the methods to external datasets. No step in the derivation chain reduces to its own inputs by construction, and no self-citation is load-bearing for the central methodological claims.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. The free parameters are practical implementation choices (λ, ε) rather than theoretical constructs. The axioms are standard modeling assumptions for overdispersed binomial data, with the independence assumption in Method 2 being the most structurally important.

free parameters (2)
  • λ (sigmoid-CDF approximation constant) = 0.588
    Chosen by trial-and-error plotting to approximate Sigmoid(x) ≈ Φ(λx). Not derived from first principles.
  • ε (zero-count constant for EQL) = 1e-8
    Added to X_j values of 0 or n_j to prevent infinite deviance in EQL estimation. Authors acknowledge better methods exist.
assumptions (3)
  • domain assumption X_j ~ BetaBinom(n_j, θ, ρ) adequately models overdispersed binomial N-of-1 data
    Section 2.1 assumes beta-binomial distribution for the aggregated binary data from each patient.
  • domain assumption π_j and κ_j are independent in Method 2
    Section 2.2 states 'Assuming π_j and κ_j are independent' to obtain θ = E(π_j)E(κ_j). This independence is not tested.
  • standard math Jackknife methods consistently estimate variances for all estimators
    Section 2.1 and throughout: jackknife is used for variance estimation of both θ and ρ estimates where closed forms are unavailable or unreliable.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Accounting for overdispersion and clustering in binomial data from N-of-1 trials." pith.science (2026). https://pith.science/paper/3F3AM2NY

@misc{pith2026260708722,
  author       = {Pith},
  title        = {Pith review of: Accounting for overdispersion and clustering in binomial data from N-of-1 trials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3F3AM2NY}},
  note         = {Machine review of arXiv:2607.08722}
}
read the original abstract

N-of-1 trials are patient centered randomized controlled trials. Although the primary goal of N-of-1 trials is to obtain the results for each patient separately, pooling the results across patients also has relevance. In this paper, we present two analytical strategies to pool the results across N-of-1 trials, when the main outcome for each patient is a binomial variable. Our first method takes into account the extra-binomial variation, while as the second approach takes into account hierarchical clustering in addition to overdispersion. We illustrate the methods using real data analysis and compare the methods using simulations.

Figures

Figures reproduced from arXiv: 2607.08722 by the authors.

Figure 1
Figure 1. Boxplots for the MSEs of ρˆ estimates in various scenarios. Rows, top to bottom, correspond to true ρ values 0.01, 0.05, 0.10 and 0.20. Columns, left to right, correspond to true θ values 0.25, 0.50 and 0.75. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Boxplots for the MSEs of ˆθ estimates in various scenarios. Rows, top to bottom, correspond to true ρ values 0.01, 0.05, 0.10 and 0.20. Columns, left to right, correspond to true θ values 0.25, 0.50 and 0.75 The above considerations would have helped us to understand the reasons for low values of MSE for ˆθQL if there were certain scenarios where variance of ρˆEQL was lower that for the other estimators. However, as… view at source ↗
Figure 3
Figure 3. Histograms of ˆθ estimates based on method 2, for the cases with true θ = 0.75 Next we consider the effects of increasing the sample size on the ˆθ estimators based on method 1. We did not consider method 2 because of the low coverage seen above for this method in certain scenarios above. For all the simulations so far the overall number of patients (i.e. J) that we considered was 30, split as 10 patients each with … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Plot to illustrate the effect of sample size increase on estimators based on method 1. Boxplots for [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Single-patient (n-of-1) trials: a pragmatic clinical decision methodology for patient-centered comparative effectiveness research.J Clin Epidemiol.2013; 66(8 Suppl):S21-8

    Duan N, Kravitz RL., Schmid CH. Single-patient (n-of-1) trials: a pragmatic clinical decision methodology for patient-centered comparative effectiveness research.J Clin Epidemiol.2013; 66(8 Suppl):S21-8

    work page 2013

  2. [2]

    The n-of-1 clinical trial: the ultimate strategy for individualizing medicine?Per Med.2011;8(2): 161-173

    Lillie EO, Patay B, Diamant J, Issell B, Topol EJ, Schork NJ. The n-of-1 clinical trial: the ultimate strategy for individualizing medicine?Per Med.2011;8(2): 161-173

    work page 2011

  3. [3]

    Combining single patient (N-of-1) trials to estimate population treatment effects and to evaluate individual patient responses to treatment.J Clin Epidemiol.1997;50(4):401-10

    Zucker DR, Schmid CH, McIntosh MW, D’Agostino RB, Selker HP, Lau J. Combining single patient (N-of-1) trials to estimate population treatment effects and to evaluate individual patient responses to treatment.J Clin Epidemiol.1997;50(4):401-10

    work page 1997

  4. [4]

    Individual (N-of-1) trials can be combined to give popu- lation comparative treatment effect estimates: methodologic considerations.J Clin Epidemiol

    Zucker DR, Ruthazer R, Schmid CH. Individual (N-of-1) trials can be combined to give popu- lation comparative treatment effect estimates: methodologic considerations.J Clin Epidemiol. 2010;63(12):1312-23

    work page 2010

  5. [5]

    A statistical model for the “N-of-1” study.J Clin Epidemiol.1990;43(5):499-508

    Rochon J. A statistical model for the “N-of-1” study.J Clin Epidemiol.1990;43(5):499-508

    work page 1990

  6. [6]

    A comparison of four methods for the analysis of N-of-1 Trials.PLoS ONE 2014;9(2):e87752

    Chen X, Chen P. A comparison of four methods for the analysis of N-of-1 Trials.PLoS ONE 2014;9(2):e87752. 28

    work page 2014

  7. [7]

    Understanding Variation in Sets of N-of-1 Trials.PLoS ONE 2016;11(12):e0167167

    Araujo A, Julious S, Senn S. Understanding Variation in Sets of N-of-1 Trials.PLoS ONE 2016;11(12):e0167167

    work page 2016

  8. [8]

    Single patient (n-of-1) trials with binary treatment preference.Stat Med.2005;24(17):2625-2636

    Schluter PJ, Ware RS. Single patient (n-of-1) trials with binary treatment preference.Stat Med.2005;24(17):2625-2636

    work page 2005

Show all 55 references
  1. [9]

    Beta-binomial model for meta-analysis of odds ratios.Stat Med

    Bakbergenuly I, Kulinskaya E. Beta-binomial model for meta-analysis of odds ratios.Stat Med. 2017;36(11):1715-1734

    2017

  2. [10]

    Overdispersion models for correlated multinomial data: Applications to blinding assessment.Stat Med.2019;38(25):4963-4976

    Landsman V, Landsman D, Li CS, Bang H. Overdispersion models for correlated multinomial data: Applications to blinding assessment.Stat Med.2019;38(25):4963-4976

    2019

  3. [11]

    Longitudinal data analysis for discrete and continuous outcomes.Bio- metrics.1986;42(1):121-30

    Zeger SL, Liang KY. Longitudinal data analysis for discrete and continuous outcomes.Bio- metrics.1986;42(1):121-30

    1986

  4. [12]

    An extended random-effects approach to modeling repeated, overdispersed count data.Lifetime Data Anal.2007;13(4):513–531

    Molenberghs G, Verbeke G, Demetrio CGB. An extended random-effects approach to modeling repeated, overdispersed count data.Lifetime Data Anal.2007;13(4):513–531

    2007

  5. [13]

    A family of generalized linear models for repeated measures with normal and conjugate random effects.Stat Sci.2010;3:325-347

    Molenberghs G, Verbeke G, Demetrio CGB, Vieira AMC. A family of generalized linear models for repeated measures with normal and conjugate random effects.Stat Sci.2010;3:325-347

    2010

  6. [14]

    A combined beta and normal random- effects model for repeated, over-dispersed binary and binomial data.J Multivar Anal.2012; 111:94-109

    Molenberghs G, Verbeke G, Iddib S, Demetrio CGB. A combined beta and normal random- effects model for repeated, over-dispersed binary and binomial data.J Multivar Anal.2012; 111:94-109

    2012

  7. [15]

    Modeling overdispersed longitudi- nal binary data using a combined beta and normal random-effects model.Arch Public Health

    Kassahun W, Neyens T, Molenberghs G, Faes C, Verbeke G. Modeling overdispersed longitudi- nal binary data using a combined beta and normal random-effects model.Arch Public Health. 2012;70(1):7

    2012

  8. [16]

    Jackknife Estimators of Variance for Parameter Estimates from Esti- mating Equations with Applications to Clustered Survival Data.Biometrics.1994;50:842–846

    Lipsitz S, Dear K, Zhao L. Jackknife Estimators of Variance for Parameter Estimates from Esti- mating Equations with Applications to Clustered Survival Data.Biometrics.1994;50:842–846

    1994

  9. [17]

    Practical considerations of the jackknife estimator of variance for generalized esti- mating equations.Statistical Papers.1997;38(3):363–369

    Ziegler A. Practical considerations of the jackknife estimator of variance for generalized esti- mating equations.Statistical Papers.1997;38(3):363–369

    1997

  10. [18]

    Estimating intraclass correlation for binary data.Biomet- rics1999;55: 137–48

    Ridout MS, Demetrio CG, Firth D. Estimating intraclass correlation for binary data.Biomet- rics1999;55: 137–48

  11. [19]

    The intra-cluster correlation coefficient in cluster randomized trials: a review of definitions.Int

    Eldridge SM, Ukoumunne OC, Carlin JB. The intra-cluster correlation coefficient in cluster randomized trials: a review of definitions.Int. Stat Rev.2009;77: 378-394

    2009

  12. [20]

    Comparison of methods for estimating the intraclass correlation coefficient for binary responses in cancer prevention cluster randomized trials.Contemp

    Wu S, Crespi CM, Wong WK. Comparison of methods for estimating the intraclass correlation coefficient for binary responses in cancer prevention cluster randomized trials.Contemp. Clin. Trials2012;33(5): 869-80

  13. [21]

    Confidence interval estimation of the intraclass correlation coefficient for binary outcome data.Biometrics2004;60: 807-811

    Zou G, Donner A. Confidence interval estimation of the intraclass correlation coefficient for binary outcome data.Biometrics2004;60: 807-811

  14. [22]

    Profile likelihood-based confidence interval of the intraclass correlation for binary outcome data sampled from clusters.Stat Med.2018;31(29):3982-4002

    Saha KK. Profile likelihood-based confidence interval of the intraclass correlation for binary outcome data sampled from clusters.Stat Med.2018;31(29):3982-4002

    2018

  15. [23]

    Cochran W.G. (1939). The Use of the Analysis of Variance in Enumeration by Sampling. Journal of the American Statistical Association2012;34: 492-510. 29

    1939

  16. [24]

    heritability

    Elston RC. Response to query: Estimating "heritability" of a dichotomous trait.Biometrics 1977;33: 232-233

    1977

  17. [25]

    A coefficient of agreement for nominal scales.Educ

    Cohen J. A coefficient of agreement for nominal scales.Educ. and Psych. Meas.1960;20(1): 37–46

    1960

  18. [26]

    A review of inference procedures for the intraclass correlation coefficient in the one-way random effects model.Int

    Donner A. A review of inference procedures for the intraclass correlation coefficient in the one-way random effects model.Int. Stat. Rev.1986;5467-82

    1986

  19. [27]

    Sibling and parent-offspring correlation estimation with variable family size.PNAS

    Karlin S, Cameron EC, Williams PY. Sibling and parent-offspring correlation estimation with variable family size.PNAS. 1981;78: 2664-2668

    1981

  20. [28]

    An extended quasi-likelihood function.Biometrika1987;74, 221–232

    Nelder JA, Pregibon D. An extended quasi-likelihood function.Biometrika1987;74, 221–232

  21. [29]

    Quasi-likelihood functions, generalized linear models, and the Gauss—Newton method.Biometrika1974;61(3): 439–447

    Wedderburn RWM. Quasi-likelihood functions, generalized linear models, and the Gauss—Newton method.Biometrika1974;61(3): 439–447

  22. [30]

    A note on extended quasi-likelihood estimation.JRSS-B

    Davidian M, Carroll RJ. A note on extended quasi-likelihood estimation.JRSS-B. 1988;50: 74-82

    1988

  23. [31]

    Likelihood, Quasi-Likelihood and Pseudolikelihood: Some Comparisons

    Nelder JA, Lee Y. Likelihood, Quasi-Likelihood and Pseudolikelihood: Some Comparisons. Journal of the Royal Statistical Society. Series B (Methodological)1992;54: 273-284

    1992

  24. [32]

    Simpson’s paradox from adding constants in contingency tables as an example of Bayesian noncollapsibility.Amer Statistician2010;64: 340-344

    Greenland S. Simpson’s paradox from adding constants in contingency tables as an example of Bayesian noncollapsibility.Amer Statistician2010;64: 340-344

  25. [33]

    Overdispersion: models and estimation.Comput

    Hinde J, Demetrio CGB. Overdispersion: models and estimation.Comput. Stat. Data Anal. 1998;27: 151–170

    1998

  26. [34]

    Biometrika1986;73583-588

    MooreD.Asymptoticpropertiesofmomentestimatesforoverdispersedcountsandproportions. Biometrika1986;73583-588

  27. [35]

    On Linear and Quadratic Estimating Functions.Biometrika1987;74(3): 591-597

    Crowder M. On Linear and Quadratic Estimating Functions.Biometrika1987;74(3): 591-597

  28. [36]

    Joint estimation of the mean and dispersion parameters in the analysis of proportions: a comparison of efficiency and bias.The Canadian Journal of Statistics1998;26: 83–94

    Paul SR, Islam AS. Joint estimation of the mean and dispersion parameters in the analysis of proportions: a comparison of efficiency and bias.The Canadian Journal of Statistics1998;26: 83–94

  29. [37]

    Quadratic estimating equations for the estimation of regression and dispersion pa- rameters in the analysis of proportions.Sankhya B2001;63: 43–55

    Paul SR. Quadratic estimating equations for the estimation of regression and dispersion pa- rameters in the analysis of proportions.Sankhya B2001;63: 43–55

  30. [38]

    Paul SR, Saha KK, Balasooriya U. An empirical investigation of different operating character- istics of several estimators of the intraclass correlation in the analysis of binary data.Journal of Statistical Computation and Simulation2003;73: 507–523

  31. [39]

    Proportions with extraneous variance: Single and independent samples.JASA (1973).6846-54

    Kleinman JC. Proportions with extraneous variance: Single and independent samples.JASA (1973).6846-54

    1973

  32. [40]

    R packagedirmultversion 0.1.3-4

    Tvedebrink T. R packagedirmultversion 0.1.3-4. Estimation in Dirichlet-Multinomial distri- bution. (2013)

    2013

  33. [41]

    Confidence intervals for the common intraclass correlation in the analysis of clustered binary responses.Journal of Biopharmaceutical Statistics2018;28(4): 682-697

    Saha KK, Wang S. Confidence intervals for the common intraclass correlation in the analysis of clustered binary responses.Journal of Biopharmaceutical Statistics2018;28(4): 682-697. 30

  34. [42]

    Asymptotic Statistics

    van der Vaart AW. Asymptotic Statistics. Cambridge Series Statistical and Probabilistic Math- ematics (2012)

    2012

  35. [43]

    Generalized, linear, and mixed models

    McCulloch CE, Searle SR. Generalized, linear, and mixed models. John Wiley and Sons, New York (2001)

    2001

  36. [44]

    lme4: Mixed-effects modeling with R Springer (2010)

    Bates DM. lme4: Mixed-effects modeling with R Springer (2010). Available online at http://lme4.r-forge.r-project.org/book/

    2010

  37. [45]

    Clinical usefulness of amitriptyline in fibromyalgia: the results of 23 N-of-1 randomized controlled trials.J Rheumatol1991;18:447- 451

    Jaeschke R, Adachi J, Guyatt G, Keller J, Wong B. Clinical usefulness of amitriptyline in fibromyalgia: the results of 23 N-of-1 randomized controlled trials.J Rheumatol1991;18:447- 451

  38. [46]

    Switching from NSAIDs to paracetamol: a series of n of 1 trials for individual patients with osteoarthritis.Ann Rheum Dis.2003;62: 1156-1161

    Wegman AC, van der Windt DA, de Haan M Deville WL, Fo CT, de Vries TP. Switching from NSAIDs to paracetamol: a series of n of 1 trials for individual patients with osteoarthritis.Ann Rheum Dis.2003;62: 1156-1161

    2003

  39. [47]

    Theo- phylline for irreversible chronic airflow limitation: a randomized study comparing n of 1 trials to standard practice.Chest.1999;115(1):38-48

    Mahon JL, Laupacis A, Hodder RV, McKim DA, Paterson NA, Wood TE, Donner A. Theo- phylline for irreversible chronic airflow limitation: a randomized study comparing n of 1 trials to standard practice.Chest.1999;115(1):38-48

    1999

  40. [48]

    Comparison of population-averaged and subject-specific approaches for analyzing repeated binary outcomes.Am J Epidemiol.1998; 147(7):694-703

    Hu FB, Goldberg J, Hedeker D, Flay BR, Pentz MA. Comparison of population-averaged and subject-specific approaches for analyzing repeated binary outcomes.Am J Epidemiol.1998; 147(7):694-703

    1998

  41. [49]

    A comparison of cluster-specific and population- averaged approaches for analyzing correlated binary data.Int Stat Rev.199159:25–35

    Neuhaus JM, Kalbfleisch JD, Hauck WW. A comparison of cluster-specific and population- averaged approaches for analyzing correlated binary data.Int Stat Rev.199159:25–35

  42. [50]

    To GEE or not to GEE: comparing population average and mixed models for estimating the associations between neighborhood risk factors and health.Epidemiology

    Hubbard AE, Ahern J, Fleischer NL, van der Laan M, Lippman SA, Jewell N, Bruckner T, Satariano WA. To GEE or not to GEE: comparing population average and mixed models for estimating the associations between neighborhood risk factors and health.Epidemiology. (2010).21(4):467-74

    2010

  43. [51]

    Practical Marginalized Multilevel Models.Stat

    Griswold ME, Swihart BJ, Caffo BS, Zeger SL. Practical Marginalized Multilevel Models.Stat. (2013).2(1)

    2013

  44. [52]

    A note on marginalization of regression parameters from mixed models of binary outcomes.Biometrics.(2018)74(1):354-361

    Hedeker D, du Toit SHC, Demirtas H, Gibbons RD. A note on marginalization of regression parameters from mixed models of binary outcomes.Biometrics.(2018)74(1):354-361

    2018

  45. [53]

    Ma J, Raina P, Beyene J, Thabane L. Comparison of population-averaged and cluster-specific models for the analysis of cluster randomized trials with missing binary outcomes: a simulation study.BMC Med Res Methodol.201313:9

  46. [54]

    Methods for longitudinal data: a generalized estimating equation approach.Biometrics1988;44:1049-60

    Zeger SL, Liang K-Y, Albert PS. Methods for longitudinal data: a generalized estimating equation approach.Biometrics1988;44:1049-60

  47. [55]

    Generalized Linear Models

    McCullagh P, Nelder JA. Generalized Linear Models. Chapman and Hall (London and Harp- enden) (1983). This work was completed before the start of the COVID-19 pandemic in the USA, and the references have not been updated after February, 2020. 31

    1983

Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.