Missing binary outcomes under covariate dependent missingness in cluster randomised trials
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Abstract:
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.Keywords:
Imputation (statistics)
Gee
Estimating equations
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In cluster randomized trials, clusters of subjects are randomized rather than subjects themselves, and missing outcomes are a concern as in individual randomized trials. We assessed strategies for handling missing data when analysing cluster randomized trials with a binary outcome; strategies included complete case, adjusted complete case, and simple and multiple imputation approaches. We performed a simulation study to assess bias and coverage rate of the population-averaged intervention-effect estimate. Both multiple imputation with a random-effects logistic regression model or classical logistic regression provided unbiased estimates of the intervention effect. Both strategies also showed good coverage properties, even slightly better for multiple imputation with a random-effects logistic regression approach. Finally, this latter approach led to a slightly negatively biased intracluster correlation coefficient estimate but less than that with a classical logistic regression model strategy. We applied these strategies to a real trial randomizing households and comparing ivermectin and malathion to treat head lice.
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Although missing outcome data are an important problem in randomized trials and observational studies, methods to address this issue can be difficult to apply. Using simulated data, the authors compared 3 methods to handle missing outcome data: 1) complete case analysis; 2) single imputation; and 3) multiple imputation (all 3 with and without covariate adjustment). Simulated scenarios focused on continuous or dichotomous missing outcome data from randomized trials or observational studies. When outcomes were missing at random, single and multiple imputations yielded unbiased estimates after covariate adjustment. Estimates obtained by complete case analysis with covariate adjustment were unbiased as well, with coverage close to 95%. When outcome data were missing not at random, all methods gave biased estimates, but handling missing outcome data by means of 1 of the 3 methods reduced bias compared with a complete case analysis without covariate adjustment. Complete case analysis with covariate adjustment and multiple imputation yield similar estimates in the event of missing outcome data, as long as the same predictors of missingness are included. Hence, complete case analysis with covariate adjustment can and should be used as the analysis of choice more often. Multiple imputation, in addition, can accommodate the missing-not-at-random scenario more flexibly, making it especially suited for sensitivity analyses.
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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 performed a simulation study to compare the performance of complete case (CC) analysis, the last observation carried forward (LOCF), the missingness pattern approach (MPA), multiple imputation (MI) and inverse-probability-of-missingness weighting (IPMW). We considered three mechanisms for non-monotone missing data which are common in observational studies using electronic health record data. Whereas CC analysis lead to biased estimates of the intervention effect in almost all scenarios, the performance of the other approaches varied across scenarios. The LOCF approach led to unbiased estimates only under a specific non-random mechanism in which confounder values were missing when their values remained unchanged since the previous measurement. In this scenario, MI, the MPA and IPMW were biased. MI and IPMW led to the estimation of unbiased effects when data were missing at random, given the covariates or the treatment but only MI was unbiased when the outcome was a predictor of missingness. Furthermore, IPMW generally lead to very large standard errors. Lastly, regardless of the missingness mechanism, the MPA led to unbiased estimates only when the failure to record a confounder at a given time-point modified the subsequent relationships between the partially observed covariate and the outcome.
Marginal structural model
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Evaluating the impact of imputations for missing participant outcome data in a network meta-analysis
Background In a meta-analysis of trials with missing outcome data, a parameter known as informative missing odds ratio (IMOR) can be used to quantify the relationship between informative missingness and a binary outcome. IMORs also account for the increased uncertainty due to missingness in the meta-analysis results. Purpose To extend the idea of IMOR into a network meta-analysis (NMA) setting in order to explore the impact of missing outcome data on the inferences about the relative effectiveness of several competing treatments in psychiatric trials. Methods IMORs were estimated in two datasets comparing anti-manic treatments and antidepressants. The outcome was response to treatments. In the original meta-analyses, missing participants were assumed to have failed regardless the treatment they were allocated to. To evaluate the robustness of this assumption in each dataset, several imputations of the missing outcomes were studied by an IMOR parameter in the NMA model. By comparing the odds ratios for efficacy under the initial analysis and under several assumptions about the missingness, we assessed the consistency of the conclusions. The missing data mechanism was studied by comparing the prior with the posterior IMOR distribution. Models were fitted using Markov chain Monte Carlo (MCMC) in WinBUGS. Results In both datasets, the relative effectiveness of the treatments seems to be affected only by the two extreme imputation scenarios of worst- and best-case analyses. Moreover, heterogeneity increases in both datasets under these two extreme scenarios. Overall, there is a non-significant change on the ranking of the anti-manic and antidepressant treatments. The posterior and prior IMOR distributions are very similar showing that the data do not provide any information about the true outcome in missing participants. There is a very weak indication that missing participants tend to fail in placebo and paroxetine, while the opposite occurs for sertraline, fluoxetine, and fluvoxamine. Limitations Investigation of informative missingness was limited two classes of treatments and for dichotomous outcome measures. The proportion of missing outcomes was very low overall, and hence, the power of detecting any differences in effectiveness estimated under the various imputation methods is small. Conclusions Sensitivity analysis to account for missing outcome data and their uncertainty in the NMA can be undertaken by extending the idea of IMOR. In two case examples, we found no differences between the various models due to low missing data rate. In line with previous observations, data carry little information about the reason of missingness.
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In analysis of covariance (ANCOVA), as a result of covariate adjustment, the estimated mean difference between the two comparative treatment groups may have a better precision than the unadjusted estimate. The extent of improvement of precision depends on the correlation between the outcome variable and the covariate selected for adjustment. Therefore, for this purpose, it is desirable to apply a proper transformation to this covariate so that the transformed covariate has a stronger correlation with the outcome variable. The best predictor from the covariate for the outcome variable is the conditional expectation of the outcome variable given the covariate. Thus, a viable strategy is using regression modeling approach to search for a statistical model to well approximate the conditional expectation based on external and/or current trial data. We propose an adaptive strategy to achieve this goal if the current data are needed to help the search.
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Missing outcomes are a commonly occurring problem in cluster randomised trials, which
can lead to biased and inefficient inference if ignored or handled inappropriately. Handling
missing data in CRTs is complicated due to the hierarchical structure of the
data. Two approaches for analysing such trials are cluster-level analysis and individual level
analysis. An assumption regarding missing outcomes in CRTs that is sometimes
plausible is that missingness depends on baseline covariates, but conditioning on these
baseline covariates, not on the outcome itself, which is known as a covariate dependent
missingness (CDM) mechanism. The aim of my thesis was to investigate the validity
of the approaches to the analysis of CRTs for the three most common outcome types:
continuous, binary and time-to-event, when outcomes are missing under the assumption
of CDM. Missing outcomes were handled using complete records analysis (CRA) and
multilevel multiple imputation (MMI).
We investigated analytically, and through simulations, the validity of the different combinations
of the analysis model and missing data handling approach for each of the three
outcome types. Simulations studies were performed considering scenarios depending on
whether the missingness mechanism is the same between the intervention groups and
whether the covariate effect is the same between the intervention groups in the outcome
model. Based on our analytical and simulations results, we give recommendations for
which methods to use when the CDM assumption is thought to be plausible for missing
outcomes. The key findings of this thesis are as follows.
Continuous outcomes:
Cluster-level analyses using CRA are in general biased unless the intervention
groups have the same missingness mechanism and the same covariate effects on
outcome in the data generating model.
In the case of individual-level analysis, the linear mixed model (LMM) using CRA
adjusted for covariates such that the CDM assumption holds gives unbiased estimates
of intervention effect regardless of whether the missingness mechanism is
the same or different between the intervention groups, and whether there is an
interaction between intervention and baseline covariates in the data generating
model for outcome, provided that such interaction is included in the model when
required.
There is no gain in terms of bias or efficiency of the estimates using MMI over
CRA as long as both approaches use the same functional form of the same set of
baseline covariates.
Binary outcomes:
The adjusted cluster-level estimator for estimating risk ratio (RR) using full data
is consistent if the true data generating model is a log link model, the functional
form of the baseline covariates is the same between the intervention groups, and
the random effects distribution is the same between the intervention groups.
Cluster-level analyses using CRA for estimating risk difference (RD) are in general
biased. For estimating RR, cluster-level analyses using CRA are valid if the…
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CRTS
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Abstract. In randomized controlled trials (RCTs), a common strategy to increase power to detect a treatment effect is adjustment for baseline covariates. However, adjustment with partly missing covariates, where complete cases are only used, is inefficient. We consider different alternatives in trials with discrete-time survival data, where subjects are measured in discrete-time intervals while they may experience an event at any point in time. The results of a Monte Carlo simulation study, as well as a case study of randomized trials in smokers with attention deficit hyperactivity disorder (ADHD), indicated that single and multiple imputation methods outperform the other methods and increase precision in estimating the treatment effect. Missing indicator method, which uses a dummy variable in the statistical model to indicate whether the value for that variable is missing and sets the same value to all missing values, is comparable to imputation methods. Nevertheless, the power level to detect the treatment effect based on missing indicator method is marginally lower than the imputation methods, particularly when the missingness depends on the outcome. In conclusion, it appears that imputation of partly missing (baseline) covariates should be preferred in the analysis of discrete-time survival data.
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Time point
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In longitudinal and multivariate settings incomplete data, due to missed visits, dropouts or non-return of
questionnaires are quite common.
A longitudinal trial in which potentially informative missingness occurs is the Collaborative Ankle Support
Trial (CAST). The aim of this study is to estimate the clinical effectiveness of four different methods of
mechanical support after severe ankle sprain. The clinical status of multiple subjects was measured at four
points in time via a questionnaire and, based on this, a continuous and bounded outcome score was calculated.
Motivated by this study, a model is proposed for continuous longitudinal data with non-ignorable or
informative missingness, taking into account the number of attempts made to contact initial non-responders.
The model combines a non-linear mixed model for the underlying response model with a logistic regression
model for the reminder process.
The outcome model enables us to analyze the rate of improvement including the dependence on explanatory
variables. The non-linear mixed model is derived under the assumption that the rate of improvement in a given
time interval is proportional to the current score and the still achievable score. Based on this assumption a
differential equation is solved in order to obtain the model of interest.
The response model relates the probability of response at each contact attempt and point in time to
covariates and to observed and missing outcomes.
Using this model the impact of missingness on the rate of improvement is evaluated for different missingness
processes.
Mixed model
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In randomized trials with missing or censored outcomes, standard maximum likelihood estimates of the effect of intervention on outcome are based on the assumption that the missing-data mechanism is ignorable. This assumption is violated if there is an unobserved baseline covariate that is informative, namely a baseline covariate associated with both outcome and the probability that the outcome is missing or censored. Incorporating informative covariates in the analysis has the desirable result of ameliorating the violation of this assumption. Although this idea of including informative covariates is recognized in the statistics literature, it is not appreciated in the literature on randomized trials. Moreover, to our knowledge, there has been no discussion on how to incorporate informative covariates into a general likelihood-based analysis with partially missing outcomes to estimate the quantities of interest. Our contribution is a simple likelihood-based approach for using informative covariates to estimate the effect of intervention on a partially missing outcome in a randomized trial. The first step is to create a propensity-to-be-missing score for each randomization group and divide the scores into a small number of strata based on quantiles. The second step is to compute stratum-specific estimates of outcome derived from a likelihood-analysis conditional on the informative covariates, so that the missing-data mechanism is ignorable. The third step is to average the stratum-specific estimates and compute the estimated effect of interventionon outcome. We discuss the computations for univariate, survival, and longitudinal outcomes, and present an application involving a randomized study of dual versus triple combinations of HIV-1 reverse transcriptase inhibitors.
Univariate
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Attrition is a common occurrence in cluster randomised trials 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 analysis, under baseline covariate dependent missingness in continuous outcomes, in terms of bias, average estimated standard error and coverage probability. The methods of complete records analysis and multiple imputation are used to handle the missing outcome data. We considered four scenarios, with the missingness mechanism and baseline covariate effect on outcome either the same or different between intervention groups. We show that both unadjusted cluster-level analysis and baseline covariate adjusted cluster-level analysis give unbiased estimates of the intervention effect only if both intervention groups have the same missingness mechanisms and there is no interaction between baseline covariate and intervention group. Linear mixed model and multiple imputation give unbiased estimates under all four considered scenarios, provided that an interaction of intervention and baseline covariate is included in the model when appropriate. Cluster mean imputation has been proposed as a valid approach for handling missing outcomes in cluster randomised trials. We show that cluster mean imputation only gives unbiased estimates when missingness mechanism is the same between the intervention groups and there is no interaction between baseline covariate and intervention group. Multiple imputation shows overcoverage for small number of clusters in each intervention group.
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