Analyzing multiple endpoints in a confirmatory randomized clinical trial—an approach that addresses stratification,missing values,baseline imbalance and multiplicity for strictly ordinal outcomes

Confirmatory randomized clinical trials with a stratified design may have ordinal response outcomes, ie, either ordered categories or continuous determinations that are not compatible with an interval scale. Also, multiple endpoints are often collected when 1 single endpoint does not represent the overall efficacy of the treatment. In addition, random baseline imbalances and missing values can add another layer of difficulty in the analysis plan. Therefore, the development of an approach that provides a consolidated strategy to all issues collectively is essential. For a real case example that is from a clinical trial comparing a test treatment and a control for the pain management for patients with osteoarthritis, which has all aforementioned issues, multivariate Mann‐Whitney estimators with stratification adjustment are applicable to the strictly ordinal responses with stratified design. Randomization based nonparametric analysis of covariance is applied to account for the possible baseline imbalances. Several approaches that handle missing values are provided. A global test followed by a closed testing procedure controls the family wise error rate in the strong sense for the analysis of multiple endpoints. Four outcomes indicating joint pain, stiffness, and functional status were analyzed collectively and also individually through the procedures. Treatment efficacy was observed in the combined endpoint as well as in the individual endpoints. The proposed approach is effective in addressing the aforementioned problems simultaneously and straightforward to implement.     

A joint marginalized multilevel model for longitudinal outcomes

The shared-parameter model and its so-called hierarchical or random-effects extension are widely used joint modeling approaches for a combination of longitudinal continuous, binary, count, missing, and survival outcomes that naturally occurs in many clinical and other studies. A random effect is introduced and shared or allowed to differ between two or more repeated measures or longitudinal outcomes, thereby acting as a vehicle to capture association between the outcomes in these joint models. It is generally known that parameter estimates in a linear mixed model (LMM) for continuous repeated measures or longitudinal outcomes allow for a marginal interpretation, even though a hierarchical formulation is employed. This is not the case for the generalized linear mixed model (GLMM), that is, for non-Gaussian outcomes. The aforementioned joint models formulated for continuous and binary or two longitudinal binomial outcomes, using the LMM and GLMM, will naturally have marginal interpretation for parameters associated with the continuous outcome but a subject-specific interpretation for the fixed effects parameters relating covariates to binary outcomes. To derive marginally meaningful parameters for the binary models in a joint model, we adopt the marginal multilevel model (MMM) due to Heagerty [13] and Heagerty and Zeger [14] and formulate a joint MMM for two longitudinal responses. This enables to (1) capture association between the two responses and (2) obtain parameter estimates that have a population-averaged interpretation for both outcomes. The model is applied to two sets of data. The results are compared with those obtained from the existing approaches such as generalized estimating equations, GLMM, and the model of Heagerty [13]. Estimates were found to be very close to those from single analysis per outcome but the joint model yields higher precision and allows for quantifying the association between outcomes. Parameters were estimated using maximum likelihood. The model is easy to fit using available tools such as the SAS NLMIXED procedure.     

Modeling longitudinal binomial responses: implications from two dueling paradigms

The generalized estimating equations (GEEs) and generalized linear mixed-effects model (GLMM) are the two most popular paradigms to extend models for cross-sectional data to a longitudinal setting. Although the two approaches yield well-interpreted models for continuous outcomes, it is quite a different story when applied to binomial responses. We discuss major modeling differences between the GEE- and GLMM-derived models by presenting new results regarding the model-driven differences. Our results show that GLMM induces some artifacts in the marginal models at assessment times, making it inappropriate when applied to such responses from real study data. The different interpretations of parameters resulting from the conceptual difference between the two modeling approaches also carry quite significant implications and ramifications with respect to data and power analyses. Although a special case involving a scale difference in parameters between GEE and GLMM has been noted in the literature, its implications in real data analysis has not been thoroughly addressed. Further, this special case has a very limited covariate structure and does not apply to most real studies, especially multi-center clinical trials. The new results presented fill a substantial gap in the literature regarding the model-driven differences between the two dueling paradigms.     

Simplex Mixed‐Effects Models for Longitudinal Proportional Data

Abstract. Continuous proportional outcomes are collected from many practical studies, where responses are confined within the unit interval (0,1). Utilizing Barndorff‐Nielsen and Jørgensen's simplex distribution, we propose a new type of generalized linear mixed‐effects model for longitudinal proportional data, where the expected value of proportion is directly modelled through a logit function of fixed and random effects. We establish statistical inference along the lines of Breslow and Clayton's penalized quasi‐likelihood (PQL) and restricted maximum likelihood (REML) in the proposed model. We derive the PQL/REML using the high‐order multivariate Laplace approximation, which gives satisfactory estimation of the model parameters. The proposed model and inference are illustrated by simulation studies and a data example. The simulation studies conclude that the fourth order approximate PQL/REML performs satisfactorily. The data example shows that Aitchison's technique of the normal linear mixed model for logit‐transformed proportional outcomes is not robust against outliers.     

Analysis of longitudinal data with non-ignorable non-monotone missing values

A full likelihood method is proposed to analyse continuous longitudinal data with non-ignorable (informative) missing values and non-monotone patterns. The problem arose in a breast cancer clinical trial where repeated assessments of quality of life were collected: patients rated their coping ability during and after treatment. We allow the missingness probabilities to depend on unobserved responses, and we use a multivariate normal model for the outcomes. A first-order Markov dependence structure for the responses is a natural choice and facilitates the construction of the likelihood; estimates are obtained via the Nelder–Mead simplex algorithm. Computations are difficult and become intractable with more than three or four assessments. Applying the method to the quality-of-life data results in easily interpretable estimates, confirms the suspicion that the data are non-ignorably missing and highlights the likely bias of standard methods. Although treatment comparisons are not affected here, the methods are useful for obtaining unbiased means and estimating trends over time.     

Efficient estimation of regression parameters from multistage studies with validation of outcome and covariates

Often the variables in a regression model are difficult or expensive to obtain so auxiliary variables are collected in a preliminary step of a study and the model variables are measured at later stages on only a subsample of the study participants called the validation sample. We consider a study in which at the first stage some variables, throughout called auxiliaries, are collected; at the second stage the true outcome is measured on a subsample of the first-stage sample, and at the third stage the true covariates are collected on a subset of the second-stage sample. In order to increase efficiency, the probabilities of selection into the second and third-stage samples are allowed to depend on the data observed at the previous stages. In this paper we describe a class of inverse-probability-of-selection-weighted semiparametric estimators for the parameters of the model for the conditional mean of the outcomes given the covariates. We assume that a subject's probability of being sampled at subsequent stages is bounded away from zero and depends only on the subject's data collected at the previous sampling stages. We show that the asymptotic variance of the optimal estimator in our class is equal to the semiparametric variance bound for the model. Since the optimal estimator depends on unknown population parameters it is not available for data analysis. We therefore propose an adaptive estimation procedure for locally efficient inferences. A simulation study is carried out to study the finite sample properties of the proposed estimators.     

A hierarchical modelling approach to analysing longitudinal data with drop-out and non-compliance, with application to an equivalence trial in paediatric acquired immune deficiency syndrome

Longitudinal clinical trials with long follow-up periods almost invariably suffer from a loss to follow-up and non-compliance with the assigned therapy. An example is protocol 128 of the AIDS Clinical Trials Group, a 5-year equivalency trial comparing reduced dose zidovudine with the standard dose for treatment of paediatric acquired immune deficiency syndrome patients. This study compared responses to treatment by using both clinical and cognitive outcomes. The cognitive outcomes are of particular interest because the effects of human immunodeficiency virus infection of the central nervous system can be more acute in children than in adults. We formulate and apply a Bayesian hierarchical model to estimate both the intent-to-treat effect and the average causal effect of reducing the prescribed dose of zidovudine by 50%. The intent-to-treat effect quantifies the causal effect of assigning the lower dose, whereas the average causal effect represents the causal effect of actually taking the lower dose. We adopt a potential outcomes framework where, for each individual, we assume the existence of a different potential outcomes process at each level of time spent on treatment. The joint distribution of the potential outcomes and the time spent on assigned treatment is formulated using a hierarchical model: the potential outcomes distribution is given at the first level, and dependence between the outcomes and time on treatment is specified at the second level by linking the time on treatment to subject-specific effects that characterize the potential outcomes processes. Several distributional and structural assumptions are used to identify the model from observed data, and these are described in detail. A detailed analysis of AIDS Clinical Trials Group protocol 128 is given; inference about both the intent-to-treat effect and average causal effect indicate a high probability of dose equivalence with respect to cognitive functioning.     

Evaluation of an analysis approach used to account for extra-variation in clustered categorical responses

This article presents the results of a simulation study investigating the performance of an approach developed by Miller and Landis (1991) for the analysis of clustered categorical responses. Evaluation of this “two-step” approach, which utilizes the method of moments to estimate the extra-variation pardmeters and subsequently incorporates these parameters into estimating equations for modelling the marginal expectations, is carried out in an experimental setting involving a comparison between two groups of observations. We assume that data for both groups are collected from each cluster and responses are measured on a three-point ordinal scale. The performance of the estimators used in both “steps” of the analysisis investigated and comparisons are made to an alternative analysismethod that ignores the clustering. The results indicate that in the chosen setting the test for a difference between groups generally operatbs at the nominal α=0.05 for 10 or more clusters and hasincreasing power with both an increasing number of clusters and an inrreasing treatment effect. These results provide a striking contrasc to those obtained from an improper analysis that ignores clustering.     

Joint modeling of hierarchically clustered and overdispersed non‐gaussian continuous outcomes for comet assay data

Multivariate longitudinal or clustered data are commonly encountered in clinical trials and toxicological studies. Typically, there is no single standard endpoint to assess the toxicity or efficacy of the compound of interest, but co‐primary endpoints are available to assess the toxic effects or the working of the compound. Modeling the responses jointly is thus appealing to draw overall inferences using all responses and to capture the association among the responses. Non‐Gaussian outcomes are often modeled univariately using exponential family models. To accommodate both the overdispersion and hierarchical structure in the data, Molenberghs et al. A family of generalized linear models for repeated measures with normal and conjugate random effects. Statistical Science 2010; 25:325–347 proposed using two separate sets of random effects. This papers considers a model for multivariate data with hierarchically clustered and overdispersed non‐Gaussian data. Gamma random effect for the over‐dispersion and normal random effects for the clustering in the data are being used. The two outcomes are jointly analyzed by assuming that the normal random effects for both endpoints are correlated. The association structure between the response is analytically derived. The fit of the joint model to data from a so‐called comet assay are compared with the univariate analysis of the two outcomes. Copyright © 2012 John Wiley & Sons, Ltd.     

Consultants' forum: should post hoc sample size calculations be done?

Pre‐study sample size calculations for clinical trial research protocols are now mandatory. When an investigator is designing a study to compare the outcomes of an intervention, an essential step is the calculation of sample sizes that will allow a reasonable chance (power) of detecting a pre‐determined difference (effect size) in the outcome variable, at a given level of statistical significance. Frequently studies will recruit fewer patients than the initial pre‐study sample size calculation suggested. Investigators are faced with the fact that their study may be inadequately powered to detect the pre‐specified treatment effect and the statistical analysis of the collected outcome data may or may not report a statistically significant result. If the data produces a “non‐statistically significant result” then investigators are frequently tempted to ask the question “Given the actual final study size, what is the power of the study, now, to detect a treatment effect or difference?” The aim of this article is to debate whether or not it is desirable to answer this question and to undertake a power calculation, after the data have been collected and analysed. Copyright © 2008 John Wiley & Sons, Ltd.     

A PRESS statistic for working correlation structure selection in generalized estimating equations

Generalized estimating equations (GEE) is one of the most commonly used methods for regression analysis of longitudinal data, especially with discrete outcomes. The GEE method accounts for the association among the responses of a subject through a working correlation matrix and its correct specification ensures efficient estimation of the regression parameters in the marginal mean regression model. This study proposes a predicted residual sum of squares (PRESS) statistic as a working correlation selection criterion in GEE. A simulation study is designed to assess the performance of the proposed GEE PRESS criterion and to compare its performance with its counterpart criteria in the literature. The results show that the GEE PRESS criterion has better performance than the weighted error sum of squares SC criterion in all cases but is surpassed in performance by the Gaussian pseudo-likelihood criterion. Lastly, the working correlation selection criteria are illustrated with data from the Coronary Artery Risk Development in Young Adults study.     

Multiple imputation of censored survival data in the presence of missing covariates using restricted mean survival time

Missing covariates data with censored outcomes put a challenge in the analysis of clinical data especially in small sample settings. Multiple imputation (MI) techniques are popularly used to impute missing covariates and the data are then analyzed through methods that can handle censoring. However, techniques based on MI are available to impute censored data also but they are not much in practice. In the present study, we applied a method based on multiple imputation by chained equations to impute missing values of covariates and also to impute censored outcomes using restricted survival time in small sample settings. The complete data were then analyzed using linear regression models. Simulation studies and a real example of CHD data show that the present method produced better estimates and lower standard errors when applied on the data having missing covariate values and censored outcomes than the analysis of the data having censored outcome but excluding cases with missing covariates or the analysis when cases with missing covariate values and censored outcomes were excluded from the data (complete case analysis).     

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.     

Joint modeling of mixed skewed continuous and ordinal longitudinal responses: a Bayesian approach

In this paper, a joint model for analyzing multivariate mixed ordinal and continuous responses, where continuous outcomes may be skew, is presented. For modeling the discrete ordinal responses, a continuous latent variable approach is considered and for describing continuous responses, a skew-normal mixed effects model is used. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is adopted for parameter estimation. Some simulation studies are performed for illustration of the proposed approach. The results of the simulation studies show that the use of the separate models or the normal distributional assumption for shared random effects and within-subject errors of continuous and ordinal variables, instead of the joint modeling under a skew-normal distribution, leads to biased parameter estimates. The approach is used for analyzing a part of the British Household Panel Survey (BHPS) data set. Annual income and life satisfaction are considered as the continuous and the ordinal longitudinal responses, respectively. The annual income variable is severely skewed, therefore, the use of the normality assumption for the continuous response does not yield acceptable results. The results of data analysis show that gender, marital status, educational levels and the amount of money spent on leisure have a significant effect on annual income, while marital status has the highest impact on life satisfaction.     

Applying functional data analysis to assess tele-interpersonal psychotherapy's efficacy to reduce depression

The use of parametric linear mixed models and generalized linear mixed models to analyze longitudinal data collected during randomized control trials (RCT) is conventional. The application of these methods, however, is restricted due to various assumptions required by these models. When the number of observations per subject is sufficiently large, and individual trajectories are noisy, functional data analysis (FDA) methods serve as an alternative to parametric longitudinal data analysis techniques. However, the use of FDA in RCTs is rare. In this paper, the effectiveness of FDA and linear mixed models (LMMs) was compared by analyzing data from rural persons living with HIV and comorbid depression enrolled in a depression treatment randomized clinical trial. Interactive voice response systems were used for weekly administrations of the 10-item Self-Administered Depression Scale (SADS) over 41 weeks. Functional principal component analysis and functional regression analysis methods detected a statistically significant difference in SADS between telphone-administered interpersonal psychotherapy (tele-IPT) and controls but linear mixed effects model results did not. Additional simulation studies were conducted to compare FDA and LMMs under a different nonlinear trajectory assumption. In this clinical trial with sufficient per subject measured outcomes and individual trajectories that are noisy and nonlinear, we found FDA methods to be a better alternative to LMMs.     

Assessing Type I error and power of multistate Markov models for panel data—A simulation study

Ordinal outcomes collected at multiple follow-up visits are common in clinical trials. Sometimes, one visit is chosen for the primary analysis and the scale is dichotomized amounting to loss of information. Multistate Markov models describe how a process moves between states over time. Here, simulation studies are performed to investigate the Type I error and power characteristics of multistate Markov models for panel data with limited non-adjacent state transitions. The results suggest that the multistate Markov models preserve the Type I error and adequate power is achieved with modest sample sizes for panel data with limited non-adjacent state transitions.     

Analyzing multiple end points with a two-stage design in clinical trials

In many clinical trials, the assessment of the response to interventions can include a large variety of outcome variables which are generally correlated. The use of multiple significance tests is likely to increase the chance of detecting a difference in at least one of the outcomes between two treatments. Furthermore, univariate tests do not take into account the correlation structure. A new test is proposed that uses information from the interim analysis in a two-stage design to form the rejection region boundaries at the second stage. Initially, the test uses Hotelling’s T² at the end of the first stage allowing only, for early acceptance of the null hypothesis and an O’Brien ‘type’ procedure at the end of the second stage. This test allows one to ‘cheat’ and look at the data at the interim analysis to form rejection regions at the second stage, provided one uses the correct distribution of the final test statistic. This distribution is derived and the power of the new test is compared to the power of three common procedures for testing multiple outcomes: Bonferroni’s inequality, Hotelling’s T²and O’Brien’s test. O’Brien’s test has the best power to detect a difference when the outcomes are thought to be affected in exactly the same direction and the same magnitude or in exactly the same relative effects as those proposed prior to data collection. However, the statistic is not robust to deviations in the alternative parameters proposed a priori, especially for correlated outcomes. The proposed new statistic and the derivation of its distribution allows investigators to consider information from the first stage of a two-stage design and consequently base the final test on the direction observed at the first stage or modify the statistic if the direction differs significantly from what was expected a prior.     

A Bayesian shared parameter model for joint modeling of longitudinal continuous and binary outcomes

Joint modeling of associated mixed biomarkers in longitudinal studies leads to a better clinical decision by improving the efficiency of parameter estimates. In many clinical studies, the observed time for two biomarkers may not be equivalent and one of the longitudinal responses may have recorded in a longer time than the other one. In addition, the response variables may have different missing patterns. In this paper, we propose a new joint model of associated continuous and binary responses by accounting different missing patterns for two longitudinal outcomes. A conditional model for joint modeling of the two responses is used and two shared random effects models are considered for intermittent missingness of two responses. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is adopted for parameter estimation and model implementation. The validation and performance of the proposed model are investigated using some simulation studies. The proposed model is also applied for analyzing a real data set of bariatric surgery.     

On the efficiency of nonparametric variance estimation in sequential dose-finding

Dose-finding in clinical studies is typically formulated as a quantile estimation problem, for which a correct specification of the variance function of the outcomes is important. This is especially true for sequential study where the variance assumption directly involves in the generation of the design points and hence sensitivity analysis may not be performed after the data are collected. In this light, there is a strong reason for avoiding parametric assumptions on the variance function, although this may incur efficiency loss. In this paper, we investigate how much information one may retrieve by making additional parametric assumptions on the variance in the context of a sequential least squares recursion. By asymptotic comparison, we demonstrate that assuming homoscedasticity achieves only a modest efficiency gain when compared to nonparametric variance estimation: when homoscedasticity in truth holds, the latter is at worst 88% as efficient as the former in the limiting case, and often achieves well over 90% efficiency for most practical situations. Extensive simulation studies concur with this observation under a wide range of scenarios.     

Joint analysis of longitudinal data comprising repeated measures and times to events

In biomedical and public health research, both repeated measures of biomarkers Y as well as times T to key clinical events are often collected for a subject. The scientific question is how the distribution of the responses [ T , Y | X ] changes with covariates X . [ T | X ] may be the focus of the estimation where Y can be used as a surrogate for T . Alternatively, T may be the time to drop-out in a study in which [ Y | X ] is the target for estimation. Also, the focus of a study might be on the effects of covariates X on both T and Y or on some underlying latent variable which is thought to be manifested in the observable outcomes. In this paper, we present a general model for the joint analysis of [ T , Y | X ] and apply the model to estimate [ T | X ] and other related functionals by using the relevant information in both T and Y . We adopt a latent variable formulation like that of Fawcett and Thomas and use it to estimate several quantities of clinical relevance to determine the efficacy of a treatment in a clinical trial setting. We use a Markov chain Monte Carlo algorithm to estimate the model's parameters. We illustrate the methodology with an analysis of data from a clinical trial comparing risperidone with a placebo for the treatment of schizophrenia.