Linear mixed models are commonly used in analyzing stepped-wedge cluster randomized trials. A key consideration for analyzing a stepped-wedge cluster randomized trial is accounting for the potentially complex correlation structure, which can be achieved by specifying random-effects. The simplest random effects structure is random intercept but more complex structures such as random cluster-by-period, discrete-time decay, and more recently, the random intervention structure, have been proposed. Specifying appropriate random effects in practice can be challenging: assuming more complex correlation structures may be reasonable but they are vulnerable to computational challenges. To circumvent these challenges, robust variance estimators may be applied to linear mixed models to provide consistent estimators of standard errors of fixed effect parameters in the presence of random-effects misspecification. However, there has been no empirical investigation of robust variance estimators for stepped-wedge cluster randomized trials. In this article, we review six robust variance estimators (both standard and small-sample bias-corrected robust variance estimators) that are available for linear mixed models in R, and then describe a comprehensive simulation study to examine the performance of these robust variance estimators for stepped-wedge cluster randomized trials with a continuous outcome under different data generators. For each data generator, we investigate whether the use of a robust variance estimator with either the random intercept model or the random cluster-by-period model is sufficient to provide valid statistical inference for fixed effect parameters, when these working models are subject to random-effect misspecification. Our results indicate that the random intercept and random cluster-by-period models with robust variance estimators performed adequately. The CR3 robust variance estimator (approximate jackknife) estimator, coupled with the number of clusters minus two degrees of freedom correction, consistently gave the best coverage results, but could be slightly conservative when the number of clusters was below 16. We summarize the implications of our results for the linear mixed model analysis of stepped-wedge cluster randomized trials and offer some practical recommendations on the choice of the analytic model.

Individually randomized group treatment (IRGT) trials, in which the clustering of outcome is induced by group-based treatment delivery, are increasingly popular in public health research. IRGT trials frequently incorporate longitudinal measurements, of which the proper sample size calculations should account for correlation structures reflecting both the treatment-induced clustering and repeated outcome measurements. Given the relatively sparse literature on designing longitudinal IRGT trials, we propose sample size procedures for continuous and binary outcomes based on the generalized estimating equations approach, employing the block exchangeable correlation structures with different correlation parameters for the treatment arm and for the control arm, and surveying five marginal mean models with different assumptions of time effect: no-time constant treatment effect, linear-time constant treatment effect, categorical-time constant treatment effect, linear time by treatment interaction, and categorical time by treatment interaction. Closed-form sample size formulas are derived for continuous outcomes, which depends on the eigenvalues of the correlation matrices; detailed numerical sample size procedures are proposed for binary outcomes. Through simulations, we demonstrate that the empirical power agrees well with the predicted power, for as few as eight groups formed in the treatment arm, when data are analyzed using the matrix-adjusted estimating equations for the correlation parameters with a bias-corrected sandwich variance estimator.

Multivariate interval-censored data arise when there are multiple types of events or clusters of study subjects, such that the event times are potentially correlated and when each event is only known to occur over a particular time interval. We formulate the effects of potentially time-varying covariates on the multivariate event times through marginal proportional hazards models while leaving the dependence structures of the related event times unspecified. We construct the nonparametric pseudolikelihood under the working assumption that all event times are independent, and we provide a simple and stable EM-type algorithm. The resulting nonparametric maximum pseudolikelihood estimators for the regression parameters are shown to be consistent and asymptotically normal, with a limiting covariance matrix that can be consistently estimated by a sandwich estimator under arbitrary dependence structures for the related event times. We evaluate the performance of the proposed methods through extensive simulation studies and present an application to data from the Atherosclerosis Risk in Communities Study.

OBJECTIVE: Marginal models with generalized estimating equations (GEE) are usually recommended for analyzing correlated ordinal outcomes which are commonly seen in a longitudinal study or clustered randomized trial (CRT). Within-cluster association is often of interest in longitudinal studies or CRTs, and can be estimated with paired estimating equations. However, the estimators for within-cluster association parameters and variances may be subject to finite-sample biases when the number of clusters is small. The objective of this article is to introduce a newly developed R package ORTH.Ord for analyzing correlated ordinal outcomes using GEE models with finite-sample bias corrections.
METHODS: The R package ORTH.Ord implements a modified version of alternating logistic regressions with estimation based on orthogonalized residuals (ORTH), which use paired estimating equations to jointly estimate parameters in marginal mean and association models. The within-cluster association between ordinal responses is modeled by global pairwise odds ratios (POR). The R package also provides a finite-sample bias correction to POR parameter estimates based on matrix multiplicative adjusted orthogonalized residuals (MMORTH) for correcting estimating equations, and bias-corrected sandwich estimators with different options for covariance estimation.
RESULTS: A simulation study shows that MMORTH provides less biased global POR estimates and coverage of their 95% confidence intervals closer to the nominal level than uncorrected ORTH. An analysis of patient-reported outcomes from an orthognathic surgery clinical trial illustrates features of ORTH.Ord.
CONCLUSIONS: This article provides an overview of the ORTH method with bias-correction on both estimating equations and sandwich estimators for analyzing correlated ordinal data, describes the features of the ORTH.Ord R package, evaluates the performance of the package using a simulation study, and finally illustrates its application in an analysis of a clinical trial.

Cluster-randomized trials (CRTs) involve randomizing entire groups of participants-called clusters-to treatment arms but are often comprised of a limited or fixed number of available clusters. While covariate adjustment can account for chance imbalances between treatment arms and increase statistical efficiency in individually randomized trials, analytical methods for individual-level covariate adjustment in small CRTs have received little attention to date. In this paper, we systematically investigate, through extensive simulations, the operating characteristics of propensity score weighting and multivariable regression as two individual-level covariate adjustment strategies for estimating the participant-average causal effect in small CRTs with a rare binary outcome and identify scenarios where each adjustment strategy has a relative efficiency advantage over the other to make practical recommendations. We also examine the finite-sample performance of the bias-corrected sandwich variance estimators associated with propensity score weighting and multivariable regression for quantifying the uncertainty in estimating the participant-average treatment effect. To illustrate the methods for individual-level covariate adjustment, we reanalyze a recent CRT testing a sedation protocol in 31 pediatric intensive care units.

Although logistic regression is the most popular for modelling regression relationships with binary responses, many find relative risk (RR), or risk ratio, easier to interpret and prefer to use this measure of risk in regression analysis. Indeed, since Zou published his modified Poisson regression approach for modelling RR for cross-sectional data, his paper has been cited over 7 000 times, demonstrating the popularity of this alternative measure of risk in regression analysis involving binary responses. As longitudinal studies have become increasingly popular in clinical trials and observational studies, it is imperative to extend Zou\'s approach for longitudinal data. The two most popular approaches for longitudinal data analysis are the generalised linear mixed-effects model (GLMM) and generalised estimating equations (GEE). However, the parametric GLMM cannot be used for the extension within the current context, because Zou\'s approach treats the binary response as a Poisson variable, which is at odds with the Bernoulli distribution for the binary response. On the other hand, as it imposes no mathematical model on data distributions, the semiparametric GEE is coherent with Zou\'s modified Poisson regression. In this paper, we develop a GEE-based longitudinal model for binary responses to provide inference about RR.

While statistical methods for analyzing cluster randomized trials with continuous and binary outcomes have been extensively studied and compared, little comparative evidence has been provided for analyzing cluster randomized trials with survival outcomes in the presence of competing risks. Motivated by the Strategies to Reduce Injuries and Develop Confidence in Elders trial, we carried out a simulation study to compare the operating characteristics of several existing population-averaged survival models, including the marginal Cox, marginal Fine and Gray, and marginal multi-state models. For each model, we found that adjusting for the intraclass correlations through the sandwich variance estimator effectively maintained the type I error rate when the number of clusters is large. With no more than 30 clusters, however, the sandwich variance estimator can exhibit notable negative bias, and a permutation test provides better control of type I error inflation. Under the alternative, the power for each model is differentially affected by two types of intraclass correlations-the within-individual and between-individual correlations. Furthermore, the marginal Fine and Gray model occasionally leads to higher power than the marginal Cox model or the marginal multi-state model, especially when the competing event rate is high. Finally, we provide an illustrative analysis of Strategies to Reduce Injuries and Develop Confidence in Elders trial using each analytical strategy considered.

The marginal Fine-Gray proportional subdistribution hazards model is a popular approach to directly study the association between covariates and the cumulative incidence function with clustered competing risks data, which often arise in multicenter randomized trials or multilevel observational studies. To account for the within-cluster correlations between failure times, the uncertainty of the regression parameters estimators is quantified by the robust sandwich variance estimator, which may have unsatisfactory performance with a limited number of clusters. To overcome this limitation, we propose four bias-corrected variance estimators to reduce the negative bias of the usual sandwich variance estimator, extending the bias-correction techniques from generalized estimating equations with noncensored exponential family outcomes to clustered competing risks outcomes. We further compare their finite-sample operating characteristics through simulations and two real data examples. In particular, we found the Mancl and DeRouen (MD) type sandwich variance estimator generally has the smallest bias. Furthermore, with a small number of clusters, the Wald t -confidence interval with the MD sandwich variance estimator carries close to nominal coverage for the cluster-level effect parameter. The t -confidence intervals based on the sandwich variance estimator with any one of the three types of multiplicative bias correction or the z -confidence interval with the Morel, Bokossa and Neerchal (MBN) type sandwich variance estimator have close to nominal coverage for the individual-level effect parameter. Finally, we develop a user-friendly R package crrcbcv implementing the proposed sandwich variance estimators to assist practical applications.

As our understanding of the microbiome has expanded, so has the recognition of its critical role in human health and disease, thereby emphasizing the importance of testing whether microbes are associated with environmental factors or clinical outcomes. However, many of the fundamental challenges that concern microbiome surveys arise from statistical and experimental design issues, such as the sparse and overdispersed nature of microbiome count data and the complex correlation structure among samples. For example, in the human microbiome project (HMP) dataset, the repeated observations across time points (level 1) are nested within body sites (level 2), which are further nested within subjects (level 3). Therefore, there is a great need for the development of specialized and sophisticated statistical tests. In this paper, we propose multilevel zero-inflated negative-binomial models for association analysis in microbiome surveys. We develop a variational approximation method for maximum likelihood estimation and inference. It uses optimization, rather than sampling, to approximate the log-likelihood and compute parameter estimates, provides a robust estimate of the covariance of parameter estimates and constructs a Wald-type test statistic for association testing. We evaluate and demonstrate the performance of our method using extensive simulation studies and an application to the HMP dataset. We have developed an R package MZINBVA to implement the proposed method, which is available from the GitHub repository https://github.com/liudoubletian/MZINBVA.

Inverse probability weighted Cox models can be used to estimate marginal hazard ratios under different point treatments in observational studies. To obtain variance estimates, the robust sandwich variance estimator is often recommended to account for the induced correlation among weighted observations. However, this estimator does not incorporate the uncertainty in estimating the weights and tends to overestimate the variance, leading to inefficient inference. Here we propose a new variance estimator that combines the estimation procedures for the hazard ratio and weights using stacked estimating equations, with additional adjustments for the sum of terms that are not independently and identically distributed in a Cox partial likelihood score equation. We prove analytically that the robust sandwich variance estimator is conservative and establish the asymptotic equivalence between the proposed variance estimator and one obtained through linearization by Hajage et al. in 2018. In addition, we extend our proposed variance estimator to accommodate clustered data. We compare the finite sample performance of the proposed method with alternative methods through simulation studies. We illustrate these different variance methods in both independent and clustered data settings, using a bariatric surgery dataset and a multiple readmission dataset, respectively. To facilitate implementation of the proposed method, we have developed an R package ipwCoxCSV.