In precision medicine, there is much interest in estimating the expected-to-benefit (EB) subset, i.e. the subset of patients who are expected to benefit from a new treatment based on a collection of baseline characteristics. There are many statistical methods for estimating the EB subset, most of which produce a \'point estimate\' without a confidence statement to address uncertainty. Confidence intervals for the EB subset have been defined only recently, and their construction is a new area for methodological research. This article proposes a pseudo-response approach to EB subset estimation and confidence interval construction. Compared to existing methods, the pseudo-response approach allows us to focus on modelling a conditional treatment effect function (as opposed to the conditional mean outcome given treatment and baseline covariates) and is able to incorporate information from baseline covariates that are not involved in defining the EB subset. Simulation results show that incorporating such covariates can improve estimation efficiency and reduce the size of the confidence interval for the EB subset. The methodology is applied to a randomized clinical trial comparing two drugs for treating HIV infection.

The mean residual life (MRL) function is one of the basic parameters of interest in survival analysis. In this paper, we develop three procedures based on modified versions of empirical likelihood (EL) to construct confidence intervals of the MRL function with length-biased data. The asymptotic results corresponding to the procedures have been established. The proposed methods exhibit better finite sample performance over other existing procedures, especially in small sample sizes. Simulations are conducted to compare coverage probabilities and the mean lengths of confidence intervals under different scenarios for the proposed methods and some existing methods. Two real data applications are provided to illustrate the methods of constructing confidence intervals.

In many problems, a sensible estimator of a possibly multivariate monotone function may fail to be monotone. We study the correction of such an estimator obtained via projection onto the space of functions monotone over a finite grid in the domain. We demonstrate that this corrected estimator has no worse supremal estimation error than the initial estimator, and that analogously corrected confidence bands contain the true function whenever the initial bands do, at no loss to band width. Additionally, we demonstrate that the corrected estimator is asymptotically equivalent to the initial estimator if the initial estimator satisfies a stochastic equicontinuity condition and the true function is Lipschitz and strictly monotone. We provide simple sufficient conditions in the special case that the initial estimator is asymptotically linear, and illustrate the use of these results for estimation of a G-computed distribution function. Our stochastic equicontinuity condition is weaker than standard uniform stochastic equicontinuity, which has been required for alternative correction procedures. This allows us to apply our results to the bivariate correction of the local linear estimator of a conditional distribution function known to be monotone in its conditioning argument. Our experiments suggest that the projection step can yield significant practical improvements.

Classical simultaneous confidence bands for survival functions (i.e., Hall-Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson-Aalen or Kaplan-Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.

The cause of failure in cohort studies that involve competing risks is frequently incompletely observed. To address this, several methods have been proposed for the semiparametric proportional cause-specific hazards model under a missing at random assumption. However, these proposals provide inference for the regression coefficients only, and do not consider the infinite dimensional parameters, such as the covariate-specific cumulative incidence function. Nevertheless, the latter quantity is essential for risk prediction in modern medicine. In this paper we propose a unified framework for inference about both the regression coefficients of the proportional cause-specific hazards model and the covariate-specific cumulative incidence functions under missing at random cause of failure. Our approach is based on a novel computationally efficient maximum pseudo-partial-likelihood estimation method for the semiparametric proportional cause-specific hazards model. Using modern empirical process theory we derive the asymptotic properties of the proposed estimators for the regression coefficients and the covariate-specific cumulative incidence functions, and provide methodology for constructing simultaneous confidence bands for the latter. Simulation studies show that our estimators perform well even in the presence of a large fraction of missing cause of failures, and that the regression coefficient estimator can be substantially more efficient compared to the previously proposed augmented inverse probability weighting estimator. The method is applied using data from an HIV cohort study and a bladder cancer clinical trial.

We consider the optimal design problem for a comparison of two regression curves, which is used to establish the similarity between the dose response relationships of two groups. An optimal pair of designs minimizes the width of the confidence band for the difference between the two regression functions. Optimal design theory (equivalence theorems, efficiency bounds) is developed for this non standard design problem and for some commonly used dose response models optimal designs are found explicitly. The results are illustrated in several examples modeling dose response relationships. It is demonstrated that the optimal pair of designs for the comparison of the regression curves is not the pair of the optimal designs for the individual models. In particular it is shown that the use of the optimal designs proposed in this paper instead of commonly used \"non-optimal\" designs yields a reduction of the width of the confidence band by more than 50%.

In this paper, we present a class of graphical tests of the proportional hazards hypothesis for two-sample censored survival data. The proposed tests are improvements over some existing tests based on asymptotic confidence bands of certain functions of the estimated cumulative hazard functions. The new methods are based on the comparison of unrestricted estimates of the said functions and their restricted versions under the hypothesis. They combine the rigour of analytical tests with the descriptive value of plots. Monte Carlo simulations suggest that the proposed asymptotic procedures have reasonable small sample properties. The power is much higher than existing graphical tests and comparable with existing analytical tests. The method is then illustrated through the analysis of a data set on bone marrow transplantation for Leukemia patients.

In longitudinal data analysis, there is great interest in assessing the impact of predictors on the time-varying trajectory in a response variable. In such settings, an important issue is to account for heterogeneity in the shape of the trajectory among subjects, while allowing the impact of the predictors to vary across subjects. We propose a flexible semiparametric Bayes approach for addressing this issue relying on a local partition process prior, which allows flexible local borrowing of information across subjects. Local hypothesis testing and credible bands are developed for the identification of time windows across which a predictor has a significant impact, while adjusting for multiple comparisons. Posterior computation proceeds via an efficient MCMC algorithm using the exact block Gibbs sampler. The methods are assessed using simulation studies and applied to a yeast cell-cycle gene expression data set.

There is substantial inter-subject variability in intervertebral range of motion (ROM) in the cervical spine. This makes it difficult to define \"normal\" ROM, and to assess the effects of age, injury, and surgical procedures on spine kinematics. The objective of this study was to define normal intervertebral kinematics in the cervical spine during dynamic functional loading. Twenty-nine participants performed dynamic flexion\\extension, axial rotation, and lateral bending while biplane radiographs were collected at 30 images/s. Vertebral motion was tracked with sub-millimeter accuracy using a validated volumetric model-based tracking process that matched subject-specific CT-based bone models to the radiographs. Gaussian point-by-point and bootstrap techniques were used to determine 90% prediction bands for the intervertebral kinematic curves at 1% intervals of each movement cycle. Cross validation was performed to estimate the true achieved coverage for each method. For a targeted coverage of 90%, the estimated true coverage using bootstrap prediction bands averaged 86±5%, while the estimated true coverage using Gaussian point-by-point intervals averaged 56±10% over all movements and all motion segments. Bootstrap prediction bands are recommended as the standard for evaluating full ROM cervical spine kinematic curves. The data presented here can be used to identify abnormal motion in patients presenting with neck pain, to drive computational models, and to assess the biofidelity of in vitro loading paradigms.

The case-cohort design facilitates economical investigation of risk factors in a large survival study, with covariate data collected only from the cases and a simple random subset of the full cohort. Methods that accommodate the design have been developed for various semiparametric models, but most inference procedures are based on asymptotic distribution theory. Such inference can be cumbersome to derive and implement, and does not permit confidence band construction. While bootstrap is an obvious alternative, how to resample is unclear because of complications from the two-stage sampling design. We establish an equivalent sampling scheme, and propose a novel and versatile nonparametric bootstrap for robust inference with an appealingly simple single-stage resampling. Theoretical justification and numerical assessment are provided for a number of procedures under the proportional hazards model.