Missing outcomes are a commonly occurring problem for cluster randomised trials, which can lead to biased and inefficient inference if ignored or handled inappropriately. Two approaches for analysing such trials are cluster-level analysis and individual-level analysis. In this study, we assessed the performance of unadjusted cluster-level analysis, baseline covariate adjusted cluster-level analysis, random effects logistic regression (RELR) and generalised estimating equations (GEE) when binary outcomes are missing under a baseline covariate dependent missingness mechanism. Missing outcomes were handled using complete records analysis (CRA) and multilevel multiple imputation (MMI). We analytically show that cluster-level analyses for estimating risk ratio (RR) using complete records are valid if the true data generating model has log link and the intervention groups have the same missingness mechanism and the same covariate effect in the outcome model. We performed a simulation study considering four different scenarios, depending on whether the missingness mechanisms are the same or different between the intervention groups and whether there is an interaction between intervention group and baseline covariate in the outcome model. Based on the simulation study and analytical results, we give guidance on the conditions under which each approach is valid.
Attrition is a common occurrence in cluster randomised trials (CRTs) which leads to missing outcome data. Two approaches for analysing such trials are cluster-level analysis and individual-level analysis. This paper compares the performance of unadjusted cluster-level analysis, baseline covariate adjusted cluster-level analysis and linear mixed model (LMM) analysis, under baseline covariate dependent missingness (CDM) in continuous outcomes, in terms of bias, average estimated standard error and coverage probability. The methods of complete case analysis (CCA) and multiple imputation (MI) are …
Covariate adjustment has the potential to increase power in the analysis of randomised trials, but mis-specification of the adjustment model could cause error. We explore what error is possible when the adjustment model omits a covariate by randomised treatment interaction, in a setting where the covariate is perfectly balanced between randomised treatments. We use mathematical arguments and analyses of single hypothetical data sets. We show that analysis by a generalised linear model with the canonical link function leads to no error under the null -- that is, if treatment effect is truly zer…
Analyses of randomised trials are often based on regression models which adjust for baseline covariates, in addition to randomised group. Based on such models, one can obtain estimates of the marginal mean outcome for the population under assignment to each treatment, by averaging the model based predictions across the empirical distribution of the baseline covariates in the trial. We identify under what conditions such estimates are consistent, and in particular show that for canonical generalised linear models, the resulting estimates are always consistent. We show that a recently proposed v…
Background: It has long been advised to account for baseline covariates in the analysis of confirmatory randomised trials, with the main statistical justifications being that this increases power and, when a randomisation scheme balanced covariates, permits a valid estimate of experimental error. There are various methods available to account for covariates but it is not clear how to choose among them. Methods: Taking the perspective of writing a statistical analysis plan, we consider how to choose between the three most promising broad approaches: direct adjustment, standardisation and invers…
Complete randomization allows for consistent estimation of the average treatment effect based on the difference in means of the outcomes without strong modeling assumptions on the outcome-generating process. Appropriate use of the pretreatment covariates can further improve the estimation efficiency. However, missingness in covariates is common in experiments and raises an important question: should we adjust for covariates subject to missingness, and if so, how? The unadjusted difference in means is always unbiased. The complete-covariate analysis adjusts for all completely observed covariate…
We derive the closed-form restricted maximum likelihood (REML) estimator and Kenward-Roger's variance estimator for fixed effects in the mixed effects model for repeated measures (MMRM) when the missing data pattern is monotone. As an important application of the analytic result, we present the formula for calculating the power of treatment comparison using the Wald t test with the Kenward-Roger adjusted variance estimate in MMRM. It allows adjustment for baseline covariates without the need to specify the covariate distribution in randomized trials. A simple two-step procedure is proposed to …
Randomised trials with continuous outcomes are often analysed using ANCOVA, with adjustment for prognostic baseline covariates. In an article published recently, Wang \etal proved that in this setting the model based standard error estimator for the treamtent effect is consistent under outcome model misspecification, provided the probability of randomisation to each treatment is 1/2. In this article, we extend their results allowing for unequal randomisation. These demonstrate that the model based standard error is in general inconsistent when the randomisation probability differs from 1/2. In…
In this paper, we propose an empirical likelihood-based weighted estimator of regression parameter in quantile regression model with nonignorable missing covariates. The proposed estimator is computationally simple and achieves semiparametric efficiency if the probability of missingness on the fully observed variables is correctly specified. The efficiency gain of the proposed estimator over the complete-case-analysis estimator is quantified theoretically and illustrated via simulation and a real data application.
In longitudinal data a response variable is measured over time, or under different conditions, for a cohort of individuals. In many situations all intended measurements are not available which results in missing values. If the missing value is never followed by an observed measurement, this leads to dropout pattern. The missing values could be in the response variable, the covariates or in both. The missingness mechanism is termed non-random when the probability of missingness depends on the missing value and may be on the observed values. In this case the missing values should be considered i…
Handling incomplete outcomes and covariates in cluster-randomized trials: doubly-robust estimation, efficiency considerations, and sensitivity analysis
Semi-parametric methods are often used for the estimation of intervention effects on correlated outcomes in cluster-randomized trials (CRTs). When outcomes are missing at random (MAR), Inverse Probability Weighted (IPW) methods incorporating baseline covariates can be used to deal with informative missingness. Also, augmented generalized estimating equations (AUG) correct for imbalance in baseline covariates but need to be extended for MAR outcomes. However, in the presence of interactions between treatment and baseline covariates, neither method alone produces consistent estimates for the mar…
Non-adherence to assigned treatment is a common issue in cluster randomised trials (CRTs). In these settings, the efficacy estimand may be also of interest. Many methodological contributions in recent years have advocated using instrumental variables to identify and estimate the local average treatment effect (LATE). However, the clustered nature of randomisation in CRTs adds to the complexity of such analyses. In this paper, we show that under certain assumptions, the LATE can be estimated via two-stage least squares (TSLS) using cluster-level summaries of outcomes and treatment received. Imp…
Marginal structural models (MSMs) are commonly used to estimate causal intervention effects in longitudinal non-randomised studies. A common issue when analysing data from observational studies is the presence of incomplete confounder data, which might lead to bias in the intervention effect estimates if they are not handled properly in the statistical analysis. However, there is currently no recommendation on how to address missing data on covariates in MSMs under a variety of missingness mechanisms encountered in practice. We reviewed existing methods to handling missing data in MSMs and per…
We consider missing data in the context of hidden Markov models with a focus on situations where data is missing not at random (MNAR) and missingness depends on the identity of the hidden states. In simulations, we show that including a submodel for state-dependent missingness reduces bias when data is MNAR and state-dependent, whilst not reducing accuracy when data is missing at random (MAR). When missingness depends on time but not the hidden states, a model which only allows for state-dependent missingness is biased, whilst a model that allows for both state- and time-dependent missingness …
Missing data is a common problem in clinical data collection, which causes difficulty in the statistical analysis of such data. In this article, we consider the problem under a framework of a semiparametric partially linear model when observations are subject to missingness with complex patterns. If the correct model structure of the additive partially linear model is available, we propose to use a new imputation method called Partial Replacement IMputation Estimation (PRIME), which can overcome problems caused by incomplete data in the partially linear model. Also, we use PRIME in conjunction…
The missing data issue often complicates the task of estimating generalized linear models (GLMs). We describe why the pseudo-marginal Metropolis-Hastings algorithm, used in this setting, is an effective strategy for parameter estimation. This approach requires fewer assumptions, it provides joint inferences on the parameters in the likelihood, the covariate model, and the parameters of the missingness-mechanism, and there is no logical inconsistency of assuming that there are multiple posterior distributions. Moreover, this approach is asymptotically exact, just like most other Markov chain Mo…
We propose a general Bayesian nonparametric (BNP) approach to causal inference in the point treatment setting. The joint distribution of the observed data (outcome, treatment, and confounders) is modeled using an enriched Dirichlet process. The combination of the observed data model and causal assumptions allows us to identify any type of causal effect - differences, ratios, or quantile effects, either marginally or for subpopulations of interest. The proposed BNP model is well-suited for causal inference problems, as it does not require parametric assumptions about the distribution of confoun…
In this paper we recast the problem of missing values in the covariates of a regression model as a latent Gaussian Markov random field (GMRF) model in a fully Bayesian framework. Our proposed approach is based on the definition of the covariate imputation sub-model as a latent effect with a GMRF structure. We show how this formulation works for continuous covariates and provide some insight on how this could be extended to categorical covariates. The resulting Bayesian hierarchical model naturally fits within the integrated nested Laplace approximation (INLA) framework, which we use for model …
Missing observations are common in cluster randomised trials. Approaches taken to handling such missing data include: complete case analysis, single-level multiple imputation that ignores the clustering, multiple imputation with a fixed effect for each cluster and multilevel multiple imputation. We conducted a simulation study to assess the performance of these approaches, in terms of confidence interval coverage and empirical bias in the estimated treatment effects. Missing-at-random clustered data scenarios were simulated following a full-factorial design. An Analysis of Variance was carried…
We study the problem of estimating a functional or a parameter in the context where outcome is subject to nonignorable missingness. We completely avoid modeling the regression relation, while allowing the propensity to be modeled by a semiparametric logistic relation where the dependence on covariates is unspecified. We discover a surprising phenomenon in that the estimation of the parameter in the propensity model as well as the functional estimation can be carried out without assessing the missingness dependence on covariates. This allows us to propose a general class of estimators for both …
When outcomes are missing for reasons beyond an investigator's control, there are two different ways to adjust a parameter estimate for covariates that may be related both to the outcome and to missingness. One approach is to model the relationships between the covariates and the outcome and use those relationships to predict the missing values. Another is to model the probabilities of missingness given the covariates and incorporate them into a weighted or stratified estimate. Doubly robust (DR) procedures apply both types of model simultaneously and produce a consistent estimate of the param…
Exploration of treatment effect heterogeneity (TEH) is an increasingly important aspect of modern statistical analysis for stratified medicine in randomised controlled trials (RCTs) as we start to gather more information on trial participants and wish to maximise the opportunities for learning from data. However, the analyst should refrain from including a large number of variables in a treatment interaction discovery stage. Because doing so can significantly dilute the power to detect any true outcome-predictive interactions between treatments and covariates. Current guidance is limited and m…
We propose a structural equation model, which reduces to a multidimensional latent class item response theory model, for the analysis of binary item responses with non-ignorable missingness. The missingness mechanism is driven by two sets of latent variables: one describing the propensity to respond and the other referred to the abilities measured by the test items. These latent variables are assumed to have a discrete distribution, so as to reduce the number of parametric assumptions regarding the latent structure of the model. Individual covariates may also be included through a multinomial …
Identifying a biomarker or treatment-dose threshold that marks a specified level of risk is an important problem, especially in clinical trials. This risk, viewed as a function of thresholds and possibly adjusted for covariates, we call the threshold-response function. Extending the work of Donovan, Hudgens and Gilbert (2019), we propose a nonparametric efficient estimator for the covariate-adjusted threshold-response function, which utilizes machine learning and Targeted Minimum-Loss Estimation (TMLE). We additionally propose a more general estimator, based on sequential regression, that also…
Identification of complier and noncomplier average causal effects in the presence of latent missing-at-random (LMAR) outcomes: a unifying view and choices of assumptions
Johns Hopkins Bloomberg School of Public Health（约翰斯·霍普金斯大学布隆伯格公共卫生学院）
Integrative analysis of datasets generated by multiple cohorts is a widely-used approach for increasing sample size, precision of population estimators, and generalizability of analysis results in epidemiological studies. However, often each individual cohort dataset does not have all variables of interest for an integrative analysis collected as a part of an original study. Such cohort-level missingness poses methodological challenges to the integrative analysis since missing variables have traditionally: (1) been removed from the data for complete case analysis; or (2) been completed by miss…
