Hybrid pairwise‐likelihood estimation methods for incomplete longitudinal binary data

In longitudinal data, missing observations occur commonly with incomplete responses and covariates. Missing data can have a ‘missing not at random’ mechanism, a non‐monotone missing pattern, and moreover response and covariates can be missing not simultaneously. To avoid complexities in both modelling and computation, a two‐stage estimation method and a pairwise‐likelihood method are proposed. The two‐stage estimation method enjoys simplicities in computation, but incurs more severe efficiency loss. On the other hand, the pairwise approach leads to estimators with better efficiency, but can be cumbersome in computation. In this paper, we develop a compromise method using a hybrid pairwise‐likelihood framework. Our proposed approach has better efficiency than the two‐stage method, but its computational cost is still reasonable compared to the pairwise approach. The performance of the methods is evaluated empirically by means of simulation studies. Our methods are used to analyse longitudinal data obtained from the National Population Health Study.     

Reference‐based sensitivity analysis for time‐to‐event data

The analysis of time‐to‐event data typically makes the censoring at random assumption, ie, that—conditional on covariates in the model—the distribution of event times is the same, whether they are observed or unobserved (ie, right censored). When patients who remain in follow‐up stay on their assigned treatment, then analysis under this assumption broadly addresses the de jure, or “while on treatment strategy” estimand. In such cases, we may well wish to explore the robustness of our inference to more pragmatic, de facto or “treatment policy strategy,” assumptions about the behaviour of patients post‐censoring. This is particularly the case when censoring occurs because patients change, or revert, to the usual (ie, reference) standard of care. Recent work has shown how such questions can be addressed for trials with continuous outcome data and longitudinal follow‐up, using reference‐based multiple imputation. For example, patients in the active arm may have their missing data imputed assuming they reverted to the control (ie, reference) intervention on withdrawal. Reference‐based imputation has two advantages: (a) it avoids the user specifying numerous parameters describing the distribution of patients' postwithdrawal data and (b) it is, to a good approximation, information anchored, so that the proportion of information lost due to missing data under the primary analysis is held constant across the sensitivity analyses. In this article, we build on recent work in the survival context, proposing a class of reference‐based assumptions appropriate for time‐to‐event data. We report a simulation study exploring the extent to which the multiple imputation estimator (using Rubin's variance formula) is information anchored in this setting and then illustrate the approach by reanalysing data from a randomized trial, which compared medical therapy with angioplasty for patients presenting with angina.     

How to Make Model‐free Feature Screening Approaches for Full Data Applicable to the Case of Missing Response?

It is quite a challenge to develop model‐free feature screening approaches for missing response problems because the existing standard missing data analysis methods cannot be applied directly to high dimensional case. This paper develops some novel methods by borrowing information of missingness indicators such that any feature screening procedures for ultrahigh‐dimensional covariates with full data can be applied to missing response case. The first method is the so‐called missing indicator imputation screening, which is developed by proving that the set of the active predictors of interest for the response is a subset of the active predictors for the product of the response and missingness indicator under some mild conditions. As an alternative, another method called Venn diagram‐based approach is also developed. The sure screening property is proven for both methods. It is shown that the complete case analysis can also keep the sure screening property of any feature screening approach with sure screening property.     

Handling of missing data in long‐term clinical trials: a case study

Missing data in clinical trials is a well‐known problem, and the classical statistical methods used can be overly simple. This case study shows how well‐established missing data theory can be applied to efficacy data collected in a long‐term open‐label trial with a discontinuation rate of almost 50%. Satisfaction with treatment in chronically constipated patients was the efficacy measure assessed at baseline and every 3 months postbaseline. The improvement in treatment satisfaction from baseline was originally analyzed with a paired t‐test ignoring missing data and discarding the correlation structure of the longitudinal data. As the original analysis started from missing completely at random assumptions regarding the missing data process, the satisfaction data were re‐examined, and several missing at random (MAR) and missing not at random (MNAR) techniques resulted in adjusted estimate for the improvement in satisfaction over 12 months. Throughout the different sensitivity analyses, the effect sizes remained significant and clinically relevant. Thus, even for an open‐label trial design, sensitivity analysis, with different assumptions for the nature of dropouts (MAR or MNAR) and with different classes of models (selection, pattern‐mixture, or multiple imputation models), has been found useful and provides evidence towards the robustness of the original analyses; additional sensitivity analyses could be undertaken to further qualify robustness. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonlinear mixed‐effect models with nonignorably missing covariates

Nonlinear mixed‐effect models are often used in the analysis of longitudinal data. However, it sometimes happens that missing values for some of the model covariates are not purely random. Motivated by an application to HTV viral dynamics, where this situation occurs, the author considers likelihood inference for this type of problem. His approach involves a Monte Carlo EM algorithm, along with a Gibbs sampler and rejection/importance sampling methods. A concrete application is provided.     

Goodness‐of‐fit tests for linear regression models with missing response data

The authors show how to test the goodness‐of‐fit of a linear regression model when there are missing data in the response variable. Their statistics are based on the L₂ distance between nonparametric estimators of the regression function and a ‐consistent estimator of the same function under the parametric model. They obtain the limit distribution of the statistics and check the validity of their bootstrap version. Finally, a simulation study allows them to examine the behaviour of their tests, whether the samples are complete or not.     

Evaluations of Bayesian and maximum likelihood methods in PK models with below‐quantification‐limit data

Pharmacokinetic (PK) data often contain concentration measurements below the quantification limit (BQL). While specific values cannot be assigned to these observations, nevertheless these observed BQL data are informative and generally known to be lower than the lower limit of quantification (LLQ). Setting BQLs as missing data violates the usual missing at random (MAR) assumption applied to the statistical methods, and therefore leads to biased or less precise parameter estimation. By definition, these data lie within the interval [0, LLQ], and can be considered as censored observations. Statistical methods that handle censored data, such as maximum likelihood and Bayesian methods, are thus useful in the modelling of such data sets. The main aim of this work was to investigate the impact of the amount of BQL observations on the bias and precision of parameter estimates in population PK models (non‐linear mixed effects models in general) under maximum likelihood method as implemented in SAS and NONMEM, and a Bayesian approach using Markov chain Monte Carlo (MCMC) as applied in WinBUGS. A second aim was to compare these different methods in dealing with BQL or censored data in a practical situation. The evaluation was illustrated by simulation based on a simple PK model, where a number of data sets were simulated from a one‐compartment first‐order elimination PK model. Several quantification limits were applied to each of the simulated data to generate data sets with certain amounts of BQL data. The average percentage of BQL ranged from 25% to 75%. Their influence on the bias and precision of all population PK model parameters such as clearance and volume distribution under each estimation approach was explored and compared. Copyright © 2009 John Wiley & Sons, Ltd.     

GRAPHICAL SENSITIVITY ANALYSIS WITH DIFFERENT METHODS OF IMPUTATION FOR A TRIAL WITH PROBABLE NON‐IGNORABLE MISSING DATA

Graphical sensitivity analyses have recently been recommended for clinical trials with non‐ignorable missing outcome. We demonstrate an adaptation of this methodology for a continuous outcome of a trial of three cognitive‐behavioural therapies for mild depression in primary care, in which one arm had unexpectedly high levels of missing data. Fixed‐value and multiple imputations from a normal distribution (assuming either varying mean and fixed standard deviation, or fixed mean and varying standard deviation) were used to obtain contour plots of the contrast estimates with their P‐values superimposed, their confidence intervals, and the root mean square errors. Imputation was based either on the outcome value alone, or on change from baseline. The plots showed fixed‐value imputation to be more sensitive than imputing from a normal distribution, but the normally distributed imputations were subject to sampling noise. The contours of the sensitivity plots were close to linear in appearance, with the slope approximately equal to the ratio of the proportions of subjects with missing data in each trial arm.     

Multiple imputation methods for recurrent event data with missing event category

Frequently in clinical and epidemiologic studies, the event of interest is recurrent (i.e., can occur more than once per subject). When the events are not of the same type, an analysis which accounts for the fact that events fall into different categories will often be more informative. Often, however, although event times may always be known, information through which events are categorized may potentially be missing. Complete‐case methods (whose application may require, for example, that events be censored when their category cannot be determined) are valid only when event categories are missing completely at random. This assumption is rather restrictive. The authors propose two multiple imputation methods for analyzing multiple‐category recurrent event data under the proportional means/rates model. The use of a proper or improper imputation technique distinguishes the two approaches. Both methods lead to consistent estimation of regression parameters even when the missingness of event categories depends on covariates. The authors derive the asymptotic properties of the estimators and examine their behaviour in finite samples through simulation. They illustrate their approach using data from an international study on dialysis.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.     

A class of partially linear single‐index survival models

The authors define a class of “partially linear single‐index” survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single‐index form. It is then possible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single‐index component. The partially linear hazards model and the single‐index hazards model are special cases of the proposed model. The authors develop a likelihood‐based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite‐sample behaviour through simulation, and use a set of real data to illustrate their approach.     

Local influence for generalized linear mixed models

The authors describe a method for assessing model inadequacy in maximum likelihood estimation of a generalized linear mixed model. They treat the latent random effects in the model as missing data and develop the influence analysis on the basis of a Q‐function which is associated with the conditional expectation of the complete‐data log‐likelihood function in the EM algorithm. They propose a procedure to detect influential observations in six model perturbation schemes. They also illustrate their methodology in a hypothetical situation and in two real cases.     

Type I error rates from likelihood‐based repeated measures analyses of incomplete longitudinal data

The last observation carried forward (LOCF) approach is commonly utilized to handle missing values in the primary analysis of clinical trials. However, recent evidence suggests that likelihood‐based analyses developed under the missing at random (MAR) framework are sensible alternatives. The objective of this study was to assess the Type I error rates from a likelihood‐based MAR approach – mixed‐model repeated measures (MMRM) – compared with LOCF when estimating treatment contrasts for mean change from baseline to endpoint (Δ). Data emulating neuropsychiatric clinical trials were simulated in a 4 × 4 factorial arrangement of scenarios, using four patterns of mean changes over time and four strategies for deleting data to generate subject dropout via an MAR mechanism. In data with no dropout, estimates of Δ and SE_Δ from MMRM and LOCF were identical. In data with dropout, the Type I error rates (averaged across all scenarios) for MMRM and LOCF were 5.49% and 16.76%, respectively. In 11 of the 16 scenarios, the Type I error rate from MMRM was at least 1.00% closer to the expected rate of 5.00% than the corresponding rate from LOCF. In no scenario did LOCF yield a Type I error rate that was at least 1.00% closer to the expected rate than the corresponding rate from MMRM. The average estimate of SE_Δ from MMRM was greater in data with dropout than in complete data, whereas the average estimate of SE_Δ from LOCF was smaller in data with dropout than in complete data, suggesting that standard errors from MMRM better reflected the uncertainty in the data. The results from this investigation support those from previous studies, which found that MMRM provided reasonable control of Type I error even in the presence of MNAR missingness. No universally best approach to analysis of longitudinal data exists. However, likelihood‐based MAR approaches have been shown to perform well in a variety of situations and are a sensible alternative to the LOCF approach. MNAR methods can be used within a sensitivity analysis framework to test the potential presence and impact of MNAR data, thereby assessing robustness of results from an MAR method. Copyright © 2004 John Wiley & Sons, Ltd.     

Semi‐parametric small‐area estimation by combining time‐series and cross‐sectional data methods

In survey sampling, policymaking regarding the allocation of resources to subgroups (called small areas) or the determination of subgroups with specific properties in a population should be based on reliable estimates. Information, however, is often collected at a different scale than that of these subgroups; hence, the estimation can only be obtained on finer scale data. Parametric mixed models are commonly used in small‐area estimation. The relationship between predictors and response, however, may not be linear in some real situations. Recently, small‐area estimation using a generalised linear mixed model (GLMM) with a penalised spline (P‐spline) regression model, for the fixed part of the model, has been proposed to analyse cross‐sectional responses, both normal and non‐normal. However, there are many situations in which the responses in small areas are serially dependent over time. Such a situation is exemplified by a data set on the annual number of visits to physicians by patients seeking treatment for asthma, in different areas of Manitoba, Canada. In cases where covariates that can possibly predict physician visits by asthma patients (e.g. age and genetic and environmental factors) may not have a linear relationship with the response, new models for analysing such data sets are required. In the current work, using both time‐series and cross‐sectional data methods, we propose P‐spline regression models for small‐area estimation under GLMMs. Our proposed model covers both normal and non‐normal responses. In particular, the empirical best predictors of small‐area parameters and their corresponding prediction intervals are studied with the maximum likelihood estimation approach being used to estimate the model parameters. The performance of the proposed approach is evaluated using some simulations and also by analysing two real data sets (precipitation and asthma).     

PARAMETRIC FRACTIONAL IMPUTATION FOR NON‐IGNORABLE CATEGORICAL MISSING DATA WITH FOLLOW‐UP

Incomplete data subject to non‐ignorable non‐response are often encountered in practice and have a non‐identifiability problem. A follow‐up sample is randomly selected from the set of non‐respondents to avoid the non‐identifiability problem and get complete responses. Glynn, Laird, & Rubin analyzed non‐ignorable missing data with a follow‐up sample under a pattern mixture model. In this article, maximum likelihood estimation of parameters of the categorical missing data is considered with a follow‐up sample under a selection model. To estimate the parameters with non‐ignorable missing data, the EM algorithm with weighting, proposed by Ibrahim, is used. That is, in the E‐step, the weighted mean is calculated using the fractional weights for imputed data. Variances are estimated using the approximated jacknife method. Simulation results are presented to compare the proposed method with previously presented methods.     

Control‐based imputation for sensitivity analyses in informative censoring for recurrent event data

In clinical trials, missing data commonly arise through nonadherence to the randomized treatment or to study procedure. For trials in which recurrent event endpoints are of interests, conventional analyses using the proportional intensity model or the count model assume that the data are missing at random, which cannot be tested using the observed data alone. Thus, sensitivity analyses are recommended. We implement the control‐based multiple imputation as sensitivity analyses for the recurrent event data. We model the recurrent event using a piecewise exponential proportional intensity model with frailty and sample the parameters from the posterior distribution. We impute the number of events after dropped out and correct the variance estimation using a bootstrap procedure. We apply the method to an application of sitagliptin study.     

On quasi‐likelihood inference in generalized linear mixed models with two components of dispersion

The authors propose a quasi‐likelihood approach analogous to two‐way analysis of variance for the estimation of the parameters of generalized linear mixed models with two components of dispersion. They discuss both the asymptotic and small‐sample behaviour of their estimators, and illustrate their use with salamander mating data.     

Addressing potential prior‐data conflict when using informative priors in proof‐of‐concept studies

Bayesian methods are increasingly used in proof‐of‐concept studies. An important benefit of these methods is the potential to use informative priors, thereby reducing sample size. This is particularly relevant for treatment arms where there is a substantial amount of historical information such as placebo and active comparators. One issue with using an informative prior is the possibility of a mismatch between the informative prior and the observed data, referred to as prior‐data conflict. We focus on two methods for dealing with this: a testing approach and a mixture prior approach. The testing approach assesses prior‐data conflict by comparing the observed data to the prior predictive distribution and resorting to a non‐informative prior if prior‐data conflict is declared. The mixture prior approach uses a prior with a precise and diffuse component. We assess these approaches for the normal case via simulation and show they have some attractive features as compared with the standard one‐component informative prior. For example, when the discrepancy between the prior and the data is sufficiently marked, and intuitively, one feels less certain about the results, both the testing and mixture approaches typically yield wider posterior‐credible intervals than when there is no discrepancy. In contrast, when there is no discrepancy, the results of these approaches are typically similar to the standard approach. Whilst for any specific study, the operating characteristics of any selected approach should be assessed and agreed at the design stage; we believe these two approaches are each worthy of consideration. Copyright © 2015 John Wiley & Sons, Ltd.     

Likelihood‐based Inference with Missing Data Under Missing‐at‐Random

Likelihood‐based inference with missing data is challenging because the observed log likelihood is often an (intractable) integration over the missing data distribution, which also depends on the unknown parameter. Approximating the integral by Monte Carlo sampling does not necessarily lead to a valid likelihood over the entire parameter space because the Monte Carlo samples are generated from a distribution with a fixed parameter value. We consider approximating the observed log likelihood based on importance sampling. In the proposed method, the dependency of the integral on the parameter is properly reflected through fractional weights. We discuss constructing a confidence interval using the profile likelihood ratio test. A Newton–Raphson algorithm is employed to find the interval end points. Two limited simulation studies show the advantage of the Wilks inference over the Wald inference in terms of power, parameter space conformity and computational efficiency. A real data example on salamander mating shows that our method also works well with high‐dimensional missing data.