Meta‐regression with partial information on summary trial or patient characteristics

We present a model for meta‐regression in the presence of missing information on some of the study level covariates, obtaining inferences using Bayesian methods. In practice, when confronted with missing covariate data in a meta‐regression, it is common to carry out a complete case or available case analysis. We propose to use the full observed data, modelling the joint density as a factorization of a meta‐regression model and a conditional factorization of the density for the covariates. With the inclusion of several covariates, inter‐relations between these covariates are modelled. Under this joint likelihood‐based approach, it is shown that the lesser assumption of the covariates being Missing At Random is imposed, instead of the more usual Missing Completely At Random (MCAR) assumption. The model is easily programmable in WinBUGS, and we examine, through the analysis of two real data sets, sensitivity and robustness of results to the MCAR assumption. Copyright © 2010 John Wiley & Sons, Ltd.     

Bias and efficiency of multiple imputation compared with complete‐case analysis for missing covariate values

When missing data occur in one or more covariates in a regression model, multiple imputation (MI) is widely advocated as an improvement over complete‐case analysis (CC). We use theoretical arguments and simulation studies to compare these methods with MI implemented under a missing at random assumption. When data are missing completely at random, both methods have negligible bias, and MI is more efficient than CC across a wide range of scenarios. For other missing data mechanisms, bias arises in one or both methods. In our simulation setting, CC is biased towards the null when data are missing at random. However, when missingness is independent of the outcome given the covariates, CC has negligible bias and MI is biased away from the null. With more general missing data mechanisms, bias tends to be smaller for MI than for CC. Since MI is not always better than CC for missing covariate problems, the choice of method should take into account what is known about the missing data mechanism in a particular substantive application. Importantly, the choice of method should not be based on comparison of standard errors. We propose new ways to understand empirical differences between MI and CC, which may provide insights into the appropriateness of the assumptions underlying each method, and we propose a new index for assessing the likely gain in precision from MI: the fraction of incomplete cases among the observed values of a covariate (FICO). Copyright © 2010 John Wiley & Sons, Ltd.     

Reference‐based multiple imputation for missing data sensitivity analyses in trial‐based cost‐effectiveness analysis

Missing data are a common issue in cost‐effectiveness analysis (CEA) alongside randomised trials and are often addressed assuming the data are ‘missing at random’. However, this assumption is often questionable, and sensitivity analyses are required to assess the implications of departures from missing at random. Reference‐based multiple imputation provides an attractive approach for conducting such sensitivity analyses, because missing data assumptions are framed in an intuitive way by making reference to other trial arms. For example, a plausible not at random mechanism in a placebo‐controlled trial would be to assume that participants in the experimental arm who dropped out stop taking their treatment and have similar outcomes to those in the placebo arm. Drawing on the increasing use of this approach in other areas, this paper aims to extend and illustrate the reference‐based multiple imputation approach in CEA. It introduces the principles of reference‐based imputation and proposes an extension to the CEA context. The method is illustrated in the CEA of the CoBalT trial evaluating cognitive behavioural therapy for treatment‐resistant depression. Stata code is provided. We find that reference‐based multiple imputation provides a relevant and accessible framework for assessing the robustness of CEA conclusions to different missing data assumptions.     

Avoiding zero between‐study variance estimates in random‐effects meta‐analysis

Fixed‐effects meta‐analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty. Random‐effects meta‐analysis and meta‐regression are therefore typically used to accommodate explained and unexplained between‐study variability. However, it is not unusual to obtain a boundary estimate of zero for the (residual) between‐study standard deviation, resulting in fixed‐effects estimates of the other parameters and their standard errors. To avoid such boundary estimates, we suggest using Bayes modal (BM) estimation with a gamma prior on the between‐study standard deviation. When no prior information is available regarding the magnitude of the between‐study standard deviation, a weakly informative default prior can be used (with shape parameter 2 and rate parameter close to 0) that produces positive estimates but does not overrule the data, leading to only a small decrease in the log likelihood from its maximum. We review the most commonly used estimation methods for meta‐analysis and meta‐regression including classical and Bayesian methods and apply these methods, as well as our BM estimator, to real datasets. We then perform simulations to compare BM estimation with the other methods and find that BM estimation performs well by (i) avoiding boundary estimates; (ii) having smaller root mean squared error for the between‐study standard deviation; and (iii) better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity. Copyright © 2013 John Wiley & Sons, Ltd.     

A weighted estimating equation for linear regression with missing covariate data

Linear regression is one of the most popular statistical techniques. In linear regression analysis, missing covariate data occur often. A recent approach to analyse such data is a weighted estimating equation. With weighted estimating equations, the contribution to the estimating equation from a complete observation is weighted by the inverse 'probability of being observed'. In this paper, we propose a weighted estimating equation in which we wrongly assume that the missing covariates are multivariate normal, but still produces consistent estimates as long as the probability of being observed is correctly modelled. In simulations, these weighted estimating equations appear to be highly efficient when compared to the most efficient weighted estimating equation as proposed by Robins et al. and Lipsitz et al. However, these weighted estimating equations, in which we wrongly assume that the missing covariates are multivariate normal, are much less computationally intensive than the weighted estimating equations given by Lipsitz et al. We compare the weighted estimating equations proposed in this paper to the efficient weighted estimating equations via an example and a simulation study. We only consider missing data which are missing at random; non-ignorably missing data are not addressed in this paper.     

Systematically missing confounders in individual participant data meta‐analysis of observational cohort studies

One difficulty in performing meta‐analyses of observational cohort studies is that the availability of confounders may vary between cohorts, so that some cohorts provide fully adjusted analyses while others only provide partially adjusted analyses. Commonly, analyses of the association between an exposure and disease either are restricted to cohorts with full confounder information, or use all cohorts but do not fully adjust for confounding. We propose using a bivariate random‐effects meta‐analysis model to use information from all available cohorts while still adjusting for all the potential confounders. Our method uses both the fully adjusted and the partially adjusted estimated effects in the cohorts with full confounder information, together with an estimate of their within‐cohort correlation. The method is applied to estimate the association between fibrinogen level and coronary heart disease incidence using data from 154 012 participants in 31 cohorts.^? One hundred and ninety‐nine participants from the original 154 211 withdrew their consent and have been removed from this analysis. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiple imputation for IPD meta‐analysis: allowing for heterogeneity and studies with missing covariates

Recently, multiple imputation has been proposed as a tool for individual patient data meta‐analysis with sporadically missing observations, and it has been suggested that within‐study imputation is usually preferable. However, such within study imputation cannot handle variables that are completely missing within studies. Further, if some of the contributing studies are relatively small, it may be appropriate to share information across studies when imputing. In this paper, we develop and evaluate a joint modelling approach to multiple imputation of individual patient data in meta‐analysis, with an across‐study probability distribution for the study specific covariance matrices. This retains the flexibility to allow for between‐study heterogeneity when imputing while allowing (i) sharing information on the covariance matrix across studies when this is appropriate, and (ii) imputing variables that are wholly missing from studies. Simulation results show both equivalent performance to the within‐study imputation approach where this is valid, and good results in more general, practically relevant, scenarios with studies of very different sizes, non‐negligible between‐study heterogeneity and wholly missing variables. We illustrate our approach using data from an individual patient data meta‐analysis of hypertension trials. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

A latent‐variable marginal method for multi‐level incomplete binary data

Incomplete multi‐level data arise commonly in many clinical trials and observational studies. Because of multi‐level variations in this type of data, appropriate data analysis should take these variations into account. A random effects model can allow for the multi‐level variations by assuming random effects at each level, but the computation is intensive because high‐dimensional integrations are often involved in fitting models. Marginal methods such as the inverse probability weighted generalized estimating equations can involve simple estimation computation, but it is hard to specify the working correlation matrix for multi‐level data. In this paper, we introduce a latent variable method to deal with incomplete multi‐level data when the missing mechanism is missing at random, which fills the gap between the random effects model and marginal models. Latent variable models are built for both the response and missing data processes to incorporate the variations that arise at each level. Simulation studies demonstrate that this method performs well in various situations. We apply the proposed method to an Alzheimer's disease study. Copyright © 2012 John Wiley & Sons, Ltd.     

Haplotype analysis in the presence of informatively missing genotype data

It is common to have missing genotypes in practical genetic studies, but the exact underlying missing data mechanism is generally unknown to the investigators. Although some statistical methods can handle missing data, they usually assume that genotypes are missing at random, that is, at a given marker, different genotypes and different alleles are missing with the same probability. These include those methods on haplotype frequency estimation and haplotype association analysis. However, it is likely that this simple assumption does not hold in practice, yet few studies to date have examined the magnitude of the effects when this simplifying assumption is violated. In this study, we demonstrate that the violation of this assumption may lead to serious bias in haplotype frequency estimates, and haplotype association analysis based on this assumption can induce both false-positive and false-negative evidence of association. To address this limitation in the current methods, we propose a general missing data model to characterize missing data patterns across a set of two or more markers simultaneously. We prove that haplotype frequencies and missing data probabilities are identifiable if and only if there is linkage disequilibrium between these markers under our general missing data model. Simulation studies on the analysis of haplotypes consisting of two single nucleotide polymorphisms illustrate that our proposed model can reduce the bias both for haplotype frequency estimates and association analysis due to incorrect assumption on the missing data mechanism. Finally, we illustrate the utilities of our method through its application to a real data set.     

On assessing model fit for distribution‐free longitudinal models under missing data

The generalized estimating equation (GEE), a distribution‐free, or semi‐parametric, approach for modeling longitudinal data, is used in a wide range of behavioral, psychotherapy, pharmaceutical drug safety, and healthcare‐related research studies. Most popular methods for assessing model fit are based on the likelihood function for parametric models, rendering them inappropriate for distribution‐free GEE. One rare exception is a score statistic initially proposed by Tsiatis for logistic regression (1980) and later extended by Barnhart and Willamson to GEE (1998). Because GEE only provides valid inference under the missing completely at random assumption and missing values arising in most longitudinal studies do not follow such a restricted mechanism, this GEE‐based score test has very limited applications in practice. We propose extensions of this goodness‐of‐fit test to address missing data under the missing at random assumption, a more realistic model that applies to most studies in practice. We examine the performance of the proposed tests using simulated data and demonstrate the utilities of such tests with data from a real study on geriatric depression and associated medical comorbidities. Copyright © 2013 John Wiley & Sons, Ltd.     

Statistical methodology for estimating the mean difference in a meta‐analysis without study‐specific variance information

Statistical inference for analyzing the results from several independent studies on the same quantity of interest has been investigated frequently in recent decades. Typically, any meta‐analytic inference requires that the quantity of interest is available from each study together with an estimate of its variability. The current work is motivated by a meta‐analysis on comparing two treatments (thoracoscopic and open) of congenital lung malformations in young children. Quantities of interest include continuous end‐points such as length of operation or number of chest tube days. As studies only report mean values (and no standard errors or confidence intervals), the question arises how meta‐analytic inference can be developed. We suggest two methods to estimate study‐specific variances in such a meta‐analysis, where only sample means and sample sizes are available in the treatment arms. A general likelihood ratio test is derived for testing equality of variances in two groups. By means of simulation studies, the bias and estimated standard error of the overall mean difference from both methodologies are evaluated and compared with two existing approaches: complete study analysis only and partial variance information. The performance of the test is evaluated in terms of type I error. Additionally, we illustrate these methods in the meta‐analysis on comparing thoracoscopic and open surgery for congenital lung malformations and in a meta‐analysis on the change in renal function after kidney donation. Copyright © 2017 John Wiley & Sons, Ltd.     

Meta‐analysis of single‐arm survival studies: a distribution‐free approach for estimating summary survival curves with random effects

In epidemiologic studies and clinical trials with time‐dependent outcome (for instance death or disease progression), survival curves are used to describe the risk of the event over time. In meta‐analyses of studies reporting a survival curve, the most informative finding is a summary survival curve. In this paper, we propose a method to obtain a distribution‐free summary survival curve by expanding the product‐limit estimator of survival for aggregated survival data. The extension of DerSimonian and Laird's methodology for multiple outcomes is applied to account for the between‐study heterogeneity. Statistics I² and H² are used to quantify the impact of the heterogeneity in the published survival curves. A statistical test for between‐strata comparison is proposed, with the aim to explore study‐level factors potentially associated with survival. The performance of the proposed approach is evaluated in a simulation study. Our approach is also applied to synthesize the survival of untreated patients with hepatocellular carcinoma from aggregate data of 27 studies and synthesize the graft survival of kidney transplant recipients from individual data from six hospitals. Copyright © 2014 John Wiley & Sons, Ltd.     

Meta‐analysis of aggregate data on medical events

Meta‐analyses of clinical trials often treat the number of patients experiencing a medical event as binomially distributed when individual patient data for fitting standard time‐to‐event models are unavailable. Assuming identical drop‐out time distributions across arms, random censorship, and low proportions of patients with an event, a binomial approach results in a valid test of the null hypothesis of no treatment effect with minimal loss in efficiency compared with time‐to‐event methods. To deal with differences in follow‐up—at the cost of assuming specific distributions for event and drop‐out times—we propose a hierarchical multivariate meta‐analysis model using the aggregate data likelihood based on the number of cases, fatal cases, and discontinuations in each group, as well as the planned trial duration and groups sizes. Such a model also enables exchangeability assumptions about parameters of survival distributions, for which they are more appropriate than for the expected proportion of patients with an event across trials of substantially different length. Borrowing information from other trials within a meta‐analysis or from historical data is particularly useful for rare events data. Prior information or exchangeability assumptions also avoid the parameter identifiability problems that arise when using more flexible event and drop‐out time distributions than the exponential one. We discuss the derivation of robust historical priors and illustrate the discussed methods using an example. We also compare the proposed approach against other aggregate data meta‐analysis methods in a simulation study. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple imputation for handling systematically missing confounders in meta‐analysis of individual participant data

A variable is ‘systematically missing’ if it is missing for all individuals within particular studies in an individual participant data meta‐analysis. When a systematically missing variable is a potential confounder in observational epidemiology, standard methods either fail to adjust the exposure–disease association for the potential confounder or exclude studies where it is missing. We propose a new approach to adjust for systematically missing confounders based on multiple imputation by chained equations. Systematically missing data are imputed via multilevel regression models that allow for heterogeneity between studies. A simulation study compares various choices of imputation model. An illustration is given using data from eight studies estimating the association between carotid intima media thickness and subsequent risk of cardiovascular events. Results are compared with standard methods and also with an extension of a published method that exploits the relationship between fully adjusted and partially adjusted estimated effects through a multivariate random effects meta‐analysis model. We conclude that multiple imputation provides a practicable approach that can handle arbitrary patterns of systematic missingness. Bias is reduced by including sufficient between‐study random effects in the imputation model. Copyright © 2013 John Wiley & Sons, Ltd.     

Bias due to missing exposure data using complete-case analysis in the proportional hazards regression model

We studied bias due to missing exposure data in the proportional hazards regression model when using complete-case analysis (CCA). Eleven missing data scenarios were considered: one with missing completely at random (MCAR), four missing at random (MAR), and six non-ignorable missingness scenarios, with a variety of hazard ratios, censoring fractions, missingness fractions and sample sizes. When missingness was MCAR or dependent only on the exposure, there was negligible bias (2-3 per cent) that was similar to the difference between the estimate in the full data set with no missing data and the true parameter. In contrast, substantial bias occurred when missingness was dependent on outcome or both outcome and exposure. For models with hazard ratio of 3.5, a sample size of 400, 20 per cent censoring and 40 per cent missing data, the relative bias for the hazard ratio ranged between 7 per cent and 64 per cent. We observed important differences in the direction and magnitude of biases under the various missing data mechanisms. For example, in scenarios where missingness was associated with longer or shorter follow-up, the biases were notably different, although both mechanisms are MAR. The hazard ratio was underestimated (with larger bias) when missingness was associated with longer follow-up and overestimated (with smaller bias) when associated with shorter follow-up. If it is known that missingness is associated with a less frequently observed outcome or with both the outcome and exposure, CCA may result in an invalid inference and other methods for handling missing data should be considered.     

Meta‐analysis methods and models with applications in evaluation of cholesterol‐lowering drugs

In this paper, we propose a class of multivariate random effects models allowing for the inclusion of study‐level covariates to carry out meta‐analyses. As existing algorithms for computing maximum likelihood estimates often converge poorly or may not converge at all when the random effects are multi‐dimensional, we develop an efficient expectation–maximization algorithm for fitting multi‐dimensional random effects regression models. In addition, we also develop a new methodology for carrying out variable selection with study‐level covariates. We examine the performance of the proposed methodology via a simulation study. We apply the proposed methodology to analyze metadata from 26 studies involving statins as a monotherapy and in combination with ezetimibe. In particular, we compare the low‐density lipoprotein cholesterol‐lowering efficacy of monotherapy and combination therapy on two patient populations (naïve and non‐naïve patients to statin monotherapy at baseline), controlling for aggregate covariates. The proposed methodology is quite general and can be applied in any meta‐analysis setting for a wide range of scientific applications and therefore offers new analytic methods of clinical importance. Copyright © 2012 John Wiley & Sons, Ltd.     

Semiparametric regression models for repeated measures of mortal cohorts with non‐monotone missing outcomes and time‐dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non‐response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time‐dependent covariates that are missing not at random with non‐monotone missingness patterns via inverse‐probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse‐probability‐weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age. Copyright © 2010 John Wiley & Sons, Ltd.     

A Bayesian framework to account for uncertainty due to missing binary outcome data in pairwise meta‐analysis

Missing outcome data are a common threat to the validity of the results from randomised controlled trials (RCTs), which, if not analysed appropriately, can lead to misleading treatment effect estimates. Studies with missing outcome data also threaten the validity of any meta‐analysis that includes them. A conceptually simple Bayesian framework is proposed, to account for uncertainty due to missing binary outcome data in meta‐analysis. A pattern‐mixture model is fitted, which allows the incorporation of prior information on a parameter describing the missingness mechanism. We describe several alternative parameterisations, with the simplest being a prior on the probability of an event in the missing individuals. We describe a series of structural assumptions that can be made concerning the missingness parameters. We use some artificial data scenarios to demonstrate the ability of the model to produce a bias‐adjusted estimate of treatment effect that accounts for uncertainty. A meta‐analysis of haloperidol versus placebo for schizophrenia is used to illustrate the model. We end with a discussion of elicitation of priors, issues with poor reporting and potential extensions of the framework. Our framework allows one to make the best use of evidence produced from RCTs with missing outcome data in a meta‐analysis, accounts for any uncertainty induced by missing data and fits easily into a wider evidence synthesis framework for medical decision making. © 2015 The Authors. Statistics in MedicinePublished by John Wiley & Sons Ltd.     

A Bayesian framework for health economic evaluation in studies with missing data

Health economics studies with missing data are increasingly using approaches such as multiple imputation that assume that the data are “missing at random.” This assumption is often questionable, as—even given the observed data—the probability that data are missing may reflect the true, unobserved outcomes, such as the patients' true health status. In these cases, methodological guidelines recommend sensitivity analyses to recognise data may be “missing not at random” (MNAR), and call for the development of practical, accessible approaches for exploring the robustness of conclusions to MNAR assumptions. Little attention has been paid to the problem that data may be MNAR in health economics in general and in cost‐effectiveness analyses (CEA) in particular. In this paper, we propose a Bayesian framework for CEA where outcome or cost data are missing. Our framework includes a practical, accessible approach to sensitivity analysis that allows the analyst to draw on expert opinion. We illustrate the framework in a CEA comparing an endovascular strategy with open repair for patients with ruptured abdominal aortic aneurysm, and provide software tools to implement this approach.