Use of empirical likelihood to calibrate auxiliary information in partly linear monotone regression models

In statistical analysis, a regression model is needed if one is interested in finding the relationship between a response variable and covariates. When the response depends on the covariate, then it may also depend on the function of this covariate. If one has no knowledge of this functional form but expect for monotonic increasing or decreasing, then the isotonic regression model is preferable. Estimation of parameters for isotonic regression models is based on the pool‐adjacent‐violators algorithm (PAVA), where the monotonicity constraints are built in. With missing data, people often employ the augmented estimating method to improve estimation efficiency by incorporating auxiliary information through a working regression model. However, under the framework of the isotonic regression model, the PAVA does not work as the monotonicity constraints are violated. In this paper, we develop an empirical likelihood‐based method for isotonic regression model to incorporate the auxiliary information. Because the monotonicity constraints still hold, the PAVA can be used for parameter estimation. Simulation studies demonstrate that the proposed method can yield more efficient estimates, and in some situations, the efficiency improvement is substantial. We apply this method to a dementia study. Copyright © 2013 John Wiley & Sons, Ltd.     

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study. Copyright © 2013 John Wiley & Sons, Ltd.     

Statistical analysis of longitudinal quality of life data with missing measurements

The statistical analysis of longitudinal quality of life data in the presence of missing data is discussed. In cancer trials missing data are generated due to the fact that patients die, drop out, or are censored. These missing data are problematic in the monitoring of the quality of life during the trial. However, by means of assuming that the cause of the missing data lies in the observed history of the patients and not in their unobserved future, the missing data are ignorable. Consequently, all available data can be used to estimate quality of life change patterns with time. The computations that are required are illustrated with real quality of life data and three commonly used computer packages for statistical analysis.Paper read at meeting of the EORTC Quality of Life Study Group, November 1991, Leicester, UK.This research was supported by a grant from the Dutch Science Foundation (NWO).     

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 generalized partially linear mean‐covariance regression model for longitudinal proportional data,with applications to the analysis of quality of life data from cancer clinical trials

Motivated by the analysis of quality of life data from a clinical trial on early breast cancer, we propose in this paper a generalized partially linear mean‐covariance regression model for longitudinal proportional data, which are bounded in a closed interval. Cholesky decomposition of the covariance matrix for within‐subject responses and generalized estimation equations are used to estimate unknown parameters and the nonlinear function in the model. Simulation studies are performed to evaluate the performance of the proposed estimation procedures. Our new model is also applied to analyze the data from the cancer clinical trial that motivated this research. In comparison with available models in the literature, the proposed model does not require specific parametric assumptions on the density function of the longitudinal responses and the probability function of the boundary values and can capture dynamic changes of time or other interested variables on both mean and covariance of the correlated proportional responses. Copyright © 2017 John Wiley & Sons, Ltd.     

A bootstrap semiparametric homogeneity test for the distributions of multigroup proportional data,with applications to analysis of quality of life outcomes in clinical trials

This article is concerned about the test for the difference in the distributions of multigroup proportional data, which is motivated by the problem of comparing the distributions of quality of life (QoL) outcomes among different treatment groups in clinical trials. The proportional data, such as QoL outcomes assessed by answers to questions on a questionnaire, are bounded in a closed interval such as [0,1] with continuous observations in (0,1) and, in addition, excess observations taking the boundary values 0 and/or 1. Common statistical procedures used in practice, such as t- and rank-based tests, may not be very powerful since they ignore the specific feature of the proportional data. In this article, we propose a three-component mixture model for the proportional data and a density ratio model for the distributions of continuous observations in (0,1). A semiparametric test statistic for the homogeneity of distributions of multigroup proportional data is derived based on the empirical likelihood ratio principle and shown to be asymptotically distributed as a chi-squared random variable under null hypothesis. A nonparametric bootstrap procedure is proposed to further improve the performance of the semiparametric test. Simulation studies are performed to evaluate the empirical type I error and power of the proposed test procedure and compare it with likelihood ratio tests (LRTs) under parametric distribution assumptions, rank-based Kruskal-Wallis test, and Wald-type test. The proposed test procedure is also applied to the analysis of QoL outcomes from a clinical trial on colorectal cancer that motivated our study.     

A residuals-based transition model for longitudinal analysis with estimation in the presence of missing data

We propose a transition model for analysing data from complex longitudinal studies. Because missing values are practically unavoidable in large longitudinal studies, we also present a two-stage imputation method for handling general patterns of missing values on both the outcome and the covariates by combining multiple imputation with stochastic regression imputation. Our model is a time-varying auto-regression on the past innovations (residuals), and it can be used in cases where general dynamics must be taken into account, and where the model selection is important. The entire estimation process was carried out using available procedures in statistical packages such as SAS and S-PLUS. To illustrate the viability of the proposed model and the two-stage imputation method, we analyse data collected in an epidemiological study that focused on various factors relating to childhood growth. Finally, we present a simulation study to investigate the behaviour of our two-stage imputation procedure.     

Allowing for missing outcome data and incomplete uptake of randomised interventions, with application to an Internet-based alcohol trial

Missing outcome data and incomplete uptake of randomised interventions are common problems, which complicate the analysis and interpretation of randomised controlled trials, and are rarely addressed well in practice. To promote the implementation of recent methodological developments, we describe sequences of randomisation-based analyses that can be used to explore both issues. We illustrate these in an Internet-based trial evaluating the use of a new interactive website for those seeking help to reduce their alcohol consumption, in which the primary outcome was available for less than half of the participants and uptake of the intervention was limited. For missing outcome data, we first employ data on intermediate outcomes and intervention use to make a missing at random assumption more plausible, with analyses based on general estimating equations, mixed models and multiple imputation. We then use data on the ease of obtaining outcome data and sensitivity analyses to explore departures from the missing at random assumption. For incomplete uptake of randomised interventions, we estimate structural mean models by using instrumental variable methods. In the alcohol trial, there is no evidence of benefit unless rather extreme assumptions are made about the missing data nor an important benefit in more extensive users of the intervention. These findings considerably aid the interpretation of the trial's results. More generally, the analyses proposed are applicable to many trials with missing outcome data or incomplete intervention uptake. To facilitate use by others, Stata code is provided for all methods.     

Modelling the rate of change in a longitudinal study with missing data, adjusting for contact attempts

The Collaborative Ankle Support Trial (CAST) is a longitudinal trial of treatments for severe ankle sprains in which interest lies in the rate of improvement, the effectiveness of reminders and potentially informative missingness. A model is proposed for continuous longitudinal data with non-ignorable or informative missingness, taking into account the nature of attempts made to contact initial non-responders. The model combines a non-linear mixed model for the outcome model with logistic regression models for the reminder processes. A sensitivity analysis is used to contrast this model with the traditional selection model, where we adjust for missingness by modelling the missingness process. The conclusions that recovery is slower, and less satisfactory with age and more rapid with below knee cast than with a tubular bandage do not alter materially across all models investigated. The results also suggest that phone calls are most effective in retrieving questionnaires.     

A comparison of two methods for the estimation of precision with incomplete longitudinal data, jointly modelled with a time-to-event outcome

Several methods for the estimation and comparison of rates of change in longitudinal studies with staggered entry and informative drop-outs have been recently proposed. For multivariate normal linear models, REML estimation is used. There are various approaches to maximizing the corresponding log-likelihood; in this paper we use a restricted iterative generalized least squares method (RIGLS) combined with a nested EM algorithm. An important statistical problem in such approaches is the estimation of the standard errors adjusted for the missing data (observed data information matrix). Louis has provided a general technique for computing the observed data information in terms of completed data quantities within the EM framework. The multiple imputation (MI) method for obtaining variances can be regarded as an alternative to this. The aim of this paper is to develop, apply and compare the Louis and a modified MI method in the setting of longitudinal studies where the source of missing data is either death or disease progression (informative) or end of the study (assumed non-informative). Longitudinal data are simultaneously modelled with the missingness process. The methods are illustrated by modelling CD4 count data from an HIV-1 clinical trial and evaluated through simulation studies. Both methods, Louis and MI, are used with Monte Carlo simulations of the missing data using the appropriate conditional distributions, the former with 100 simulations, the latter with 5 and 10. It is seen that naive SEs based on the completed data likelihood can be seriously biased. This bias was largely corrected by Louis and modified MI methods, which gave broadly similar estimates. Given the relative simplicity of the modified MI method, it may be preferable.     

Doubly robust estimation,optimally truncated inverse‐intensity weighting and increment‐based methods for the analysis of irregularly observed longitudinal data

Longitudinal data arising from routine follow‐up of patients will often have irregular measurement times. Existing methods for analysis include joint modelling of the outcome and measurement processes, and inverse‐intensity weighting (IIW). This work extends previously proposed analysis of increments to the case of irregular follow‐up, yielding a model for the increments that can be used as a stand‐alone method. Furthermore, we propose two ways of combining the increments and IIW estimators. First, we use the increment model to select the truncation point for the inverse‐intensity weights that minimises the mean squared error of the IIW estimator. Second, we use the increment model to augment the usual IIW estimating equations to form a doubly robust estimator. We evaluate the methods through simulation and apply these to a recent study of juvenile dermatomyositis. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal auxiliary‐covariate‐based two‐phase sampling design for semiparametric efficient estimation of a mean or mean difference,with application to clinical trials

To address the objective in a clinical trial to estimate the mean or mean difference of an expensive endpoint Y, one approach employs a two‐phase sampling design, wherein inexpensive auxiliary variables W predictive of Y are measured in everyone, Y is measured in a random sample, and the semiparametric efficient estimator is applied. This approach is made efficient by specifying the phase two selection probabilities as optimal functions of the auxiliary variables and measurement costs. While this approach is familiar to survey samplers, it apparently has seldom been used in clinical trials, and several novel results practicable for clinical trials are developed. We perform simulations to identify settings where the optimal approach significantly improves efficiency compared to approaches in current practice. We provide proofs and R code. The optimality results are developed to design an HIV vaccine trial, with objective to compare the mean ‘importance‐weighted’ breadth (Y) of the T‐cell response between randomized vaccine groups. The trial collects an auxiliary response (W) highly predictive of Y and measures Y in the optimal subset. We show that the optimal design‐estimation approach can confer anywhere between absent and large efficiency gain (up to 24 % in the examples) compared to the approach with the same efficient estimator but simple random sampling, where greater variability in the cost‐standardized conditional variance of Y given W yields greater efficiency gains. Accurate estimation of E[Y | W] is important for realizing the efficiency gain, which is aided by an ample phase two sample and by using a robust fitting method. Copyright © 2013 John Wiley & Sons, Ltd.     

Assessing potentially time‐dependent treatment effect from clinical trials and observational studies for survival data,with applications to the Women's Health Initiative combined hormone therapy trial

For risk and benefit assessment in clinical trials and observational studies with time‐to‐event data, the Cox model has usually been the model of choice. When the hazards are possibly non‐proportional, a piece‐wise Cox model over a partition of the time axis may be considered. Here, we propose to analyze clinical trials or observational studies with time‐to‐event data using a certain semiparametric model. The model allows for a time‐dependent treatment effect. It includes the important proportional hazards model as a sub‐model and can accommodate various patterns of time‐dependence of the hazard ratio. After estimation of the model parameters using a pseudo‐likelihood approach, simultaneous confidence intervals for the hazard ratio function are established using a Monte Carlo method to assess the time‐varying pattern of the treatment effect. To assess the overall treatment effect, estimated average hazard ratio and its confidence intervals are also obtained. The proposed methods are applied to data from the Women's Health Initiative. To compare the Women's Health Initiative clinical trial and observational study, we use the propensity score in building the regression model. Compared with the piece‐wise Cox model, the proposed model yields a better model fit and does not require partitioning of the time axis. Copyright © 2015 John Wiley & Sons, Ltd.