Missing data in clinical trials: control‐based mean imputation and sensitivity analysis

In some randomized (drug versus placebo) clinical trials, the estimand of interest is the between‐treatment difference in population means of a clinical endpoint that is free from the confounding effects of “rescue” medication (e.g., HbA1c change from baseline at 24 weeks that would be observed without rescue medication regardless of whether or when the assigned treatment was discontinued). In such settings, a missing data problem arises if some patients prematurely discontinue from the trial or initiate rescue medication while in the trial, the latter necessitating the discarding of post‐rescue data. We caution that the commonly used mixed‐effects model repeated measures analysis with the embedded missing at random assumption can deliver an exaggerated estimate of the aforementioned estimand of interest. This happens, in part, due to implicit imputation of an overly optimistic mean for “dropouts” (i.e., patients with missing endpoint data of interest) in the drug arm. We propose an alternative approach in which the missing mean for the drug arm dropouts is explicitly replaced with either the estimated mean of the entire endpoint distribution under placebo (primary analysis) or a sequence of increasingly more conservative means within a tipping point framework (sensitivity analysis); patient‐level imputation is not required. A supplemental “dropout = failure” analysis is considered in which a common poor outcome is imputed for all dropouts followed by a between‐treatment comparison using quantile regression. All analyses address the same estimand and can adjust for baseline covariates. Three examples and simulation results are used to support our recommendations.     

Multiple imputation for ordinal longitudinal data with monotone missing data patterns

Missing data often complicate the analysis of scientific data. Multiple imputation is a general purpose technique for analysis of datasets with missing values. The approach is applicable to a variety of missing data patterns but often complicated by some restrictions like the type of variables to be imputed and the mechanism underlying the missing data. In this paper, the authors compare the performance of two multiple imputation methods, namely fully conditional specification and multivariate normal imputation in the presence of ordinal outcomes with monotone missing data patterns. Through a simulation study and an empirical example, the authors show that the two methods are indeed comparable meaning any of the two may be used when faced with scenarios, at least, as the ones presented here.     

Improving interim decisions in randomized trials by exploiting information on short‐term endpoints and prognostic baseline covariates

Conditional power calculations are frequently used to guide the decision whether or not to stop a trial for futility or to modify planned sample size. These ignore the information in short‐term endpoints and baseline covariates, and thereby do not make fully efficient use of the information in the data. We therefore propose an interim decision procedure based on the conditional power approach which exploits the information contained in baseline covariates and short‐term endpoints. We will realize this by considering the estimation of the treatment effect at the interim analysis as a missing data problem. This problem is addressed by employing specific prediction models for the long‐term endpoint which enable the incorporation of baseline covariates and multiple short‐term endpoints. We show that the proposed procedure leads to an efficiency gain and a reduced sample size, without compromising the Type I error rate of the procedure, even when the adopted prediction models are misspecified. In particular, implementing our proposal in the conditional power approach enables earlier decisions relative to standard approaches, whilst controlling the probability of an incorrect decision. This time gain results in a lower expected number of recruited patients in case of stopping for futility, such that fewer patients receive the futile regimen. We explain how these methods can be used in adaptive designs with unblinded sample size re‐assessment based on the inverse normal P‐value combination method to control Type I error. We support the proposal by Monte Carlo simulations based on data from a real clinical trial.     

A structured approach to choosing estimands and estimators in longitudinal clinical trials

An important evolution in the missing data arena has been the recognition of need for clarity in objectives. The objectives of primary focus in clinical trials can often be categorized as assessing efficacy or effectiveness. The present investigation illustrated a structured framework for choosing estimands and estimators when testing investigational drugs to treat the symptoms of chronic illnesses. Key issues were discussed and illustrated using a reanalysis of the confirmatory trials from a new drug application in depression. The primary analysis used a likelihood‐based approach to assess efficacy: mean change to the planned endpoint of the trial assuming patients stayed on drug. Secondarily, effectiveness was assessed using a multiple imputation approach. The imputation model—derived solely from the placebo group—was used to impute missing values for both the drug and placebo groups. Therefore, this so‐called placebo multiple imputation (a.k.a. controlled imputation) approach assumed patients had reduced benefit from the drug after discontinuing it. Results from the example data provided clear evidence of efficacy for the experimental drug and characterized its effectiveness. Data after discontinuation of study medication were not required for these analyses. Given the idiosyncratic nature of drug development, no estimand or approach is universally appropriate. However, the general practice of pairing efficacy and effectiveness estimands may often be useful in understanding the overall risks and benefits of a drug. Controlled imputation approaches, such as placebo multiple imputation, can be a flexible and transparent framework for formulating primary analyses of effectiveness estimands and sensitivity analyses for efficacy estimands. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of a long-term study of neurotic disorder, with insights into the process of non-response

Summary.  The paper studies the non-response process in a long-term study of neurotic dis-order by comparing the analysis based on the responses that were collected by the established practice of interviewing the subjects, at dates arranged in advance (appointments), with the analysis of the nearly complete set of responses that were collected by an extensive effort that involved attempts to interview without seeking a prior agreement. The method of multiple imputation is applied, and its properties are explored in a setting that is not perfectly suited for its application: a relatively small sample size, ordinal score outcomes and the likelihood that the outcomes are missing not at random.     

Coping with missing data in phase III pivotal registration trials: Tolvaptan in subjects with kidney disease,a case study

Missing data cause challenging issues, particularly in phase III registration trials, as highlighted by the European Medicines Agency (EMA) and the US National Research Council. We explore, as a case study, how the issues from missing data were tackled in a double‐blind phase III trial in subjects with autosomal dominant polycystic kidney disease. A total of 1445 subjects were randomized in a 2:1 ratio to receive active treatment (tolvaptan), or placebo. The primary outcome, the rate of change in total kidney volume, favored tolvaptan (P < .0001). The key secondary efficacy endpoints of clinical progression of disease and rate of decline in kidney function also favored tolvaptan. However, as highlighted by Food and Drug Administration and EMA, the interpretation of results was hampered by a high number of unevenly distributed dropouts, particularly early dropouts. In this paper, we outline the analyses undertaken to address the issue of missing data thoroughly. “Tipping point analyses” were performed to explore how extreme and detrimental outcomes among subjects with missing data must be to overturn the positive treatment effect attained in those subjects who had complete data. Nonparametric rank‐based analyses were also performed accounting for missing data. In conclusion, straightforward and transparent analyses directly taking into account missing data convincingly support the robustness of the preplanned analyses on the primary and secondary endpoints. Tolvaptan was confirmed to be effective in slowing total kidney volume growth, which is considered an efficacy endpoint by EMA, and in lessening the decline in renal function in patients with autosomal dominant polycystic kidney disease.     

Validity and efficiency in analyzing ordinal responses with missing observations

This article addresses issues in creating public-use data files in the presence of missing ordinal responses and subsequent statistical analyses of the dataset by users. The authors propose a fully efficient fractional imputation (FI) procedure for ordinal responses with missing observations. The proposed imputation strategy retrieves the missing values through the full conditional distribution of the response given the covariates and results in a single imputed data file that can be analyzed by different data users with different scientific objectives. Two most critical aspects of statistical analyses based on the imputed data set,  validity  and  efficiency, are examined through regression analysis involving the ordinal response and a selected set of covariates. It is shown through both theoretical development and simulation studies that, when the ordinal responses are missing at random, the proposed FI procedure leads to valid and highly efficient inferences as compared to existing methods. Variance estimation using the fractionally imputed data set is also discussed. The Canadian Journal of Statistics 48: 138–151; 2020 © 2019 Statistical Society of Canada     

Recurrent time-to-event models with ordinal outcomes

A model to accommodate time-to-event ordinal outcomes was proposed by Berridge and Whitehead. Very few studies have adopted this approach, despite its appeal in incorporating several ordered categories of event outcome. More recently, there has been increased interest in utilizing recurrent events to analyze practical endpoints in the study of disease history and to help quantify the changing pattern of disease over time. For example, in studies of heart failure, the analysis of a single fatal event no longer provides sufficient clinical information to manage the disease. Similarly, the grade/frequency/severity of adverse events may be more important than simply prolonged survival in studies of toxic therapies in oncology. We propose an extension of the ordinal time-to-event model to allow for multiple/recurrent events in the case of marginal models (where all subjects are at risk for each recurrence, irrespective of whether they have experienced previous recurrences) and conditional models (subjects are at risk of a recurrence only if they have experienced a previous recurrence). These models rely on marginal and conditional estimates of the instantaneous baseline hazard and provide estimates of the probabilities of an event of each severity for each recurrence over time. We outline how confidence intervals for these probabilities can be constructed and illustrate how to fit these models and provide examples of the methods, together with an interpretation of the results.     

Latent class based multiple imputation approach for missing categorical data

In this paper we propose a latent class based multiple imputation approach for analyzing missing categorical covariate data in a highly stratified data model. In this approach, we impute the missing data assuming a latent class imputation model and we use likelihood methods to analyze the imputed data. Via extensive simulations, we study its statistical properties and make comparisons with complete case analysis, multiple imputation, saturated log-linear multiple imputation and the Expectation–Maximization approach under seven missing data mechanisms (including missing completely at random, missing at random and not missing at random). These methods are compared with respect to bias, asymptotic standard error, type I error, and 95% coverage probabilities of parameter estimates. Simulations show that, under many missingness scenarios, latent class multiple imputation performs favorably when jointly considering these criteria. A data example from a matched case–control study of the association between multiple myeloma and polymorphisms of the Inter-Leukin 6 genes is considered.     

On the recoding of continuous and bounded indexes to a binomial form: an application to quality-of-life scores

The statistical analysis of patient-reported outcomes (PROs) as endpoints has shown to be of great practical relevance. The resulting scores or indexes from the questionnaires used to measure PROs could be treated as continuous or ordinal. The goal of this study is to propose and evaluate a recoding process of the scores, so that they can be treated as binomial outcomes and, therefore, analyzed using logistic regression with random effects. The general methodology of recoding is based on the observable values of the scores. In order to obtain an optimal recoding, the evaluation of the recoding method is tested for different values of the parameters of the binomial distribution and different probability distributions of the random effects. We illustrate, evaluate and validate the proposed method of recoding with the Short Form-36 (SF-36) Survey and real data. The optimal recoding approach is very useful and flexible. Moreover, it has a natural interpretation, not only for ordinal scores, but also for questionnaires with many dimensions and different profiles, where a common method of analysis is desired, such as the SF-36.     

Latent variable model for mixed correlated power series and ordinal longitudinal responses with non ignorable missing values

We propose a joint model based on a latent variable for analyzing mixed power series and ordinal longitudinal data with and without missing values. A bivariate probit regression model is used for the missing mechanisms. Random effects are used to take into account the correlation between longitudinal responses. A full likelihood-based approach is used to yield maximum-likelihood estimates of the model parameters. Our model is applied to a medical data set, obtained from an observational study on women where the correlated responses are the ordinal response of osteoporosis of the spine and the power series response of the number of joint damages. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms and overdispersion of the model on likelihood displacement.     

General location multivariate latent variable models for mixed correlated bounded continuous,ordinal, and nominal responses with non-ignorable missing data

Using a multivariate latent variable approach, this article proposes some new general models to analyze the correlated bounded continuous and categorical (nominal or/and ordinal) responses with and without non-ignorable missing values. First, we discuss regression methods for jointly analyzing continuous, nominal, and ordinal responses that we motivated by analyzing data from studies of toxicity development. Second, using the beta and Dirichlet distributions, we extend the models so that some bounded continuous responses are replaced for continuous responses. The joint distribution of the bounded continuous, nominal and ordinal variables is decomposed into a marginal multinomial distribution for the nominal variable and a conditional multivariate joint distribution for the bounded continuous and ordinal variables given the nominal variable. We estimate the regression parameters under the new general location models using the maximum-likelihood method. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms of the model on the maximal normal curvature. The proposed models are applied to two data sets: BMI, Steatosis and Osteoporosis data and Tehran household expenditure budgets.     

A latent variable model for analyzing mixed longitudinal (k,l)-inflated count and ordinal responses

A random effects model for analyzing mixed longitudinal count and ordinal data is presented where the count response is inflated in two points (k and l) and an (k,l)-Inflated Power series distribution is used as its distribution. A full likelihood-based approach is used to obtain maximum likelihood estimates of parameters of the model. For data with non-ignorable missing values models with probit model for missing mechanism are used.The dependence between longitudinal sequences of responses and inflation parameters are investigated using a random effects approach. Also, to investigate the correlation between mixed ordinal and count responses of each individuals at each time, a shared random effect is used. In order to assess the performance of the model, a simulation study is performed for a case that the count response has (k,l)-Inflated Binomial distribution. Performance comparisons of count-ordinal random effect model, Zero-Inflated ordinal random effects model and (k,l)-Inflated ordinal random effects model are also given. The model is applied to a real social data set from the first two waves of the national longitudinal study of adolescent to adult health (Add Health study). In this data set, the joint responses are the number of days in a month that each individual smoked as the count response and the general health condition of each individual as the ordinal response. For the count response there is incidence of excess values of 0 and 30.     

Identifying treatment effects using trimmed means when data are missing not at random

Patients often discontinue from a clinical trial because their health condition is not improving or they cannot tolerate the assigned treatment. Consequently, the observed clinical outcomes in the trial are likely better on average than if every patient had completed the trial. If these differences between trial completers and non-completers cannot be explained by the observed data, then the study outcomes are missing not at random (MNAR). One way to overcome this problem—the trimmed means approach for missing data due to study discontinuation—sets missing values as the worst observed outcome and then trims away a fraction of the distribution from each treatment arm before calculating differences in treatment efficacy (Permutt T, Li F. Trimmed means for symptom trials with dropouts. Pharm Stat. 2017;16(1):20–28). In this paper, we derive sufficient and necessary conditions for when this approach can identify the average population treatment effect. Simulation studies show the trimmed means approach's ability to effectively estimate treatment efficacy when data are MNAR and missingness due to study discontinuation is strongly associated with an unfavorable outcome, but trimmed means fail when data are missing at random. If the reasons for study discontinuation in a clinical trial are known, analysts can improve estimates with a combination of multiple imputation and the trimmed means approach when the assumptions of each hold. We compare the methodology to existing approaches using data from a clinical trial for chronic pain. An R package trim implements the method. When the assumptions are justifiable, using trimmed means can help identify treatment effects notwithstanding MNAR data.     

How to avoid concerns with the interpretation of two primary endpoints if significant superiority in one is sufficient for formal proof of efficacy

Formal proof of efficacy of a drug requires that in a prospective experiment, superiority over placebo, or either superiority or at least non-inferiority to an established standard, is demonstrated. Traditionally one primary endpoint is specified, but various diseases exist where treatment success needs to be based on the assessment of two primary endpoints. With co-primary endpoints, both need to be “significant” as a prerequisite to claim study success. Here, no adjustment of the study-wise type-1-error is needed, but sample size is often increased to maintain the pre-defined power. Studies that use an at-least-one concept have been proposed where study success is claimed if superiority for at least one of the endpoints is demonstrated. This is sometimes also called the dual primary endpoint concept, and an appropriate adjustment of the study-wise type-1-error is required. This concept is not covered in the European Guideline on multiplicity because study success can be claimed if one endpoint shows significant superiority, despite a possible deterioration in the other. In line with Röhmel's strategy, we discuss an alternative approach including non-inferiority hypotheses testing that avoids obvious contradictions to proper decision-making. This approach leads back to the co-primary endpoint assessment, and has the advantage that minimum requirements for endpoints can be modeled flexibly for several practical needs. Our simulations show that, if planning assumptions are correct, the proposed additional requirements improve interpretation with only a limited impact on power, that is, on sample size.     

Assessment of the information theory approach to evaluating time-to-event surrogate and true endpoints in a meta-analytic setting

In many disease areas, commonly used long-term clinical endpoints are becoming increasingly difficult to implement due to long follow-up times and/or increased costs. Shorter-term surrogate endpoints are urgently needed to expedite drug development, the evaluation of which requires robust and reliable statistical methodology to drive meaningful clinical conclusions about the strength of relationship with the true long-term endpoint. This paper uses a simulation study to explore one such previously proposed method, based on information theory, for evaluation of time to event surrogate and long-term endpoints, including the first examination within a meta-analytic setting of multiple clinical trials with such endpoints. The performance of the information theory method is examined for various scenarios including different dependence structures, surrogate endpoints, censoring mechanisms, treatment effects, trial and sample sizes, and for surrogate and true endpoints with a natural time-ordering. Results allow us to conclude that, contrary to some findings in the literature, the approach provides estimates of surrogacy that may be substantially lower than the true relationship between surrogate and true endpoints, and rarely reach a level that would enable confidence in the strength of a given surrogate endpoint. As a result, care is needed in the assessment of time to event surrogate and true endpoints based only on this methodology.     

Missing data sensitivity analysis for recurrent event data using controlled imputation

Statistical analyses of recurrent event data have typically been based on the missing at random assumption. One implication of this is that, if data are collected only when patients are on their randomized treatment, the resulting de jure estimator of treatment effect corresponds to the situation in which the patients adhere to this regime throughout the study. For confirmatory analysis of clinical trials, sensitivity analyses are required to investigate alternative de facto estimands that depart from this assumption. Recent publications have described the use of multiple imputation methods based on pattern mixture models for continuous outcomes, where imputation for the missing data for one treatment arm (e.g. the active arm) is based on the statistical behaviour of outcomes in another arm (e.g. the placebo arm). This has been referred to as controlled imputation or reference‐based imputation. In this paper, we use the negative multinomial distribution to apply this approach to analyses of recurrent events and other similar outcomes. The methods are illustrated by a trial in severe asthma where the primary endpoint was rate of exacerbations and the primary analysis was based on the negative binomial model. Copyright © 2014 John Wiley & Sons, Ltd.     

Power and sample size for GEE analysis of incomplete paired outcomes in 2 × 2 crossover trials

The 2 × 2 crossover trial uses subjects as their own control to reduce the intersubject variability in the treatment comparison, and typically requires fewer subjects than a parallel design. The generalized estimating equations (GEE) methodology has been commonly used to analyze incomplete discrete outcomes from crossover trials. We propose a unified approach to the power and sample size determination for the Wald Z-test and t-test from GEE analysis of paired binary, ordinal and count outcomes in crossover trials. The proposed method allows misspecification of the variance and correlation of the outcomes, missing outcomes, and adjustment for the period effect. We demonstrate that misspecification of the working variance and correlation functions leads to no or minimal efficiency loss in GEE analysis of paired outcomes. In general, GEE requires the assumption of missing completely at random. For bivariate binary outcomes, we show by simulation that the GEE estimate is asymptotically unbiased or only minimally biased, and the proposed sample size method is suitable under missing at random (MAR) if the working correlation is correctly specified. The performance of the proposed method is illustrated with several numerical examples. Adaption of the method to other paired outcomes is discussed.     

Testing non-inferiority and superiority for two endpoints for several treatments with a control

Some multiple comparison procedures are described for multiple armed studies. The procedures are appropriate for testing all hypotheses for comparing two endpoints and multiple test arms to a single control group, for example three different fixed doses compared to a placebo. The procedure assumes that among the two endpoints, one is designated as a primary endpoint such that for a given treatment arm, no hypothesis for the secondary endpoint can be rejected unless the hypothesis for the primary endpoint was rejected. The procedures described control the family-wise error rate in the strong sense at a specified level α.     

A convenient formula for sample size calculations in clinical trials with multiple co-primary continuous endpoints

The clinical efficacy of a new treatment may often be better evaluated by two or more co-primary endpoints. Recently, in pharmaceutical drug development, there has been increasing discussion regarding establishing statistically significant favorable results on more than one endpoint in comparisons between treatments, which is referred to as a problem of multiple co-primary endpoints. Several methods have been proposed for calculating the sample size required to design a trial with multiple co-primary correlated endpoints. However, because these methods require users to have considerable mathematical sophistication and knowledge of programming techniques, their application and spread may be restricted in practice. To improve the convenience of these methods, in this paper, we provide a useful formula with accompanying numerical tables for sample size calculations to design clinical trials with two treatments, where the efficacy of a new treatment is demonstrated on continuous co-primary endpoints. In addition, we provide some examples to illustrate the sample size calculations made using the formula. Using the formula and the tables, which can be read according to the patterns of correlations and effect size ratios expected in multiple co-primary endpoints, makes it convenient to evaluate the required sample size promptly.