Simultaneous Confidence Bands Research Articles

Empirical likelihood confidence band for the difference of survival functions under proportional hazards model

Under the stratified Cox model, an empirical likelihood based simultaneous confidence band for the difference of two individualized survival functions is proposed. Simulation studies demonstrate its superiority over the normal approximation based competitors.

On the restricted mean survival time curve in survival analysis.

For a study with an event time as the endpoint, its survival function contains all the information regarding the temporal, stochastic profile of this outcome variable. The survival probability at a specific time point, say t, however, does not transparently capture the temporal profile of this endpoint up to t. An alternative is to use the restricted mean survival time (RMST) at time t to summarize the profile. The RMST is the mean survival time of all subjects in the study population followed up to t, and is simply the area under the survival curve up to t. The advantages of using such a quantification over the survival rate have been discussed in the setting of a fixed-time analysis. In this article, we generalize this approach by considering a curve based on the RMST over time as an alternative summary to the survival function. Inference, for instance, based on simultaneous confidence bands for a single RMST curve and also the difference between two RMST curves are proposed. The latter is informative for evaluating two groups under an equivalence or noninferiority setting, and quantifies the difference of two groups in a time scale. The proposal is illustrated with the data from two clinical trials, one from oncology and the other from cardiology.

Nonparametric inference for time-dependent incremental cost-effectiveness ratios.

As the costs of medical care increase, more studies are evaluating cost in addition to effectiveness of treatments. Cost-effectiveness analyses in randomized clinical trials have typically been conducted only at the end of follow-up. However, cost-effectiveness may change over time. We therefore propose a nonparametric estimator to assess the incremental cost-effectiveness ratio over time. We also derive the asymptotic variance of our estimator and present formulation of Fieller-based simultaneous confidence bands. Simulation studies demonstrate the performance of our point estimators, variance estimators, and confidence bands. We also illustrate our methods using data from a randomized clinical trial, the second Multicenter Automatic Defibrillator Implantation Trial. This trial studied the effects of implantable cardioverter-defibrillators on patients at high risk for cardiac arrhythmia. Results show that our estimator performs well in large samples, indicating promising future directions in the field of cost-effectiveness. Copyright © 2015 John Wiley & Sons, Ltd.

Simultaneous Confidence Bands in Nonlinear Regression Models with Nonstationarity

We consider nonparametric estimation of the regression function g(*) in a nonlinear regression model Yt = g(Xt) o(Xt)et, where the regressor Xt is a nonstationary unit root process and the error et is s sequence of independent and identically distributed (i.i.d.) random variables. With proper centering and scaling, the maximum deviation of the local linear estimator of g is shown to be asymptotically Gumbel. Based on the latter result, we construct simultaneous confidence bands for g, which can be used to test patterns of the regression function g. Our results substantially extend existing ones which typically require independent or stationary weakly dependent regressors. Furthermore, we provide a simulation study to illustrate the finite sample behavior of the proposed approach.

On the Residual Plot in a Mixture Model

The residual plot is a graphical diagnostic that can tell whether a mixture model is necessary. We establish the approximate joint normality of the residual function nonparametrically. We propose an approach to constructing simultaneous asymptotic confidence bands. Our investigation is extended to the empirical log-ratio plot to obtain better visual effect sometimes. The type I error rate is under control to some degree in the proposed procedures with confidence bands based on nonparametric estimation. The proposed procedures are adopted in proteomic and ecological applications. It is found that the probabilities being detected on gels vary over proteins, and that there is no evidence for beetles to be removed with different probabilities.

Simultaneous confidence bands for Cox regression from semiparametric random censorship.

Cox regression is combined with semiparametric random censorship models to construct simultaneous confidence bands (SCBs) for subject-specific survival curves. Simulation results are presented to compare the performance of the proposed SCBs with the SCBs that are based only on standard Cox. The new SCBs provide correct empirical coverage and are more informative. The proposed SCBs are illustrated with two real examples. An extension to handle missing censoring indicators is also outlined.

Diagnosing Nonlinearity With Confidence Envelopes for a Semiparametric Approach to Modeling Bivariate Nonlinear Relations Among Latent Variables

During the early phases of research, semiparametric models (SPMs) have the advantage of recovering latent nonlinearity over parametric counterparts. Structural equation mixture models (Bauer, 2005) can be applied as SPMs to flexibly recover and describe the form of the unknown latent relationship with minimal distributional assumptions. This short report extends the work on this SPM (Bauer, 2005; Pek, Losardo & Bauer, 2011) by developing approximate simultaneous confidence bands or confidence envelopes (CEs) to evaluate potential nonlinearity of the unknown latent function. A line-finding algorithm to be used in conjunction with these CEs is also developed as an implementation of an informal test to diagnose nonlinearity. Coverage of the CEs and performance of the algorithm in terms of rates of detecting latent nonlinearity are evaluated by Monte Carlo. Recommendations for the use of these CEs and the algorithm for detecting nonlinearity are suggested.

A smooth simultaneous confidence band for conditional variance function

A smooth simultaneous confidence band (SCB) is obtained for heteroscedastic variance function in nonparametric regression by applying spline regression to the conditional mean function followed by Nadaraya–Waston estimation using the squared residuals. The variance estimator is uniformly oracally efficient, that is, it is as efficient as, up to order less than \(n^{-1/2}\), the infeasible kernel estimator when the conditional mean function is known, uniformly over the data range. Simulation experiments provide strong evidence that confirms the asymptotic theory while the computing is extremely fast. The proposed SCB has been applied to test for heteroscedasticity in the well-known motorcycle data and Old Faithful geyser data with different conclusions.

Oracally efficient estimation for single-index link function with simultaneous confidence band

Over the last twenty-five years, various $\sqrt{n}$-consistent estimators have been devised for the coefficient vector in the popular semiparametric single-index model. In this paper, we prove under general assumptions that the kernel estimator of the link function by a univariate regression on the index variable is oracally efficient, namely, the estimator with the true single-index coefficient vector is asymptotically indistinguishable from that with any $\sqrt{n}$-consistent coefficient vector estimator. As a mathematical byproduct of the oracle efficiency, a simultaneous confidence band is constructed for the link function based on the oracally efficient kernel estimator. Simulation experiments corroborate the theoretical results. The proposed simultaneous confidence band is applied to analyze and test hypothesis about the Boston housing data.

Asymptotic Confidence Bands for Copulas Based on the Local Linear Kernel Estimator

In this paper, we establish asymptotically optimal simultaneous confidence bands for the copula function based on the local linear kernel estimator proposed by Chen and Huang [1]. For this, we prove under smoothness conditions on the derivatives of the copula a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We also show that the bias term converges uniformly to zero with a precise rate. The performance of these bands is illustrated by a simulation study. An application based on pseudo-panel data is also provided for modeling the dependence structure of Senegalese households’ expense data in 2001 and 2006.

Functional Nonlinear Mixed Effects Models for Longitudinal Image Data.

Motivated by studying large-scale longitudinal image data, we propose a novel functional nonlinear mixed effects modeling (FNMEM) framework to model the nonlinear spatial-temporal growth patterns of brain structure and function and their association with covariates of interest (e.g., time or diagnostic status). Our FNMEM explicitly quantifies a random nonlinear association map of individual trajectories. We develop an efficient estimation method to estimate the nonlinear growth function and the covariance operator of the spatial-temporal process. We propose a global test and a simultaneous confidence band for some specific growth patterns. We conduct Monte Carlo simulation to examine the finite-sample performance of the proposed procedures. We apply FNMEM to investigate the spatial-temporal dynamics of white-matter fiber skeletons in a national database for autism research. Our FNMEM may provide a valuable tool for charting the developmental trajectories of various neuropsychiatric and neurodegenerative disorders.

Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems

We develop uniformly valid confidence regions for regression coefficients in a high-dimensional sparse median regression model with homoscedastic errors. Our methods are based on a moment equation that is immunized against nonregular estimation of the nuisance part of the median regression function by using Neyman’s orthogonalization. We establish that the resulting instrumental median regression estimator of a target regression coefficient is asymptotically normally distributed uniformly with respect to the underlying sparse model and is semiparametrically efficient. We also generalize our method to a general nonsmooth Z-estimation framework where the number of target parameters is possibly much larger than the sample size. We extend Huber's results on asymptotic normality to this setting, demonstrating uniform asymptotic normality of the proposed estimators over rectangles, constructing simultaneous confidence bands on all of the target parameters, and establishing asymptotic validity of the bands uniformly over underlying approximately sparse models.

Optimal Simultaneous Confidence Bands in Multiple Linear Regression with Predictor Variables Constrained in an Ellipsoidal Region

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model function, just as a confidence interval gives the plausible range of an unknown parameter. For a multiple linear regression model, confidence bands of different shapes, such as the hyperbolic band and the constant width band, can be constructed and the predictor variable region over which a confidence band is constructed can take various forms. One interesting but unsolved problem is to find the optimal (shape) confidence band over an ellipsoidal region χE under the Minimum Volume Confidence Set (MVCS) criterion of Liu and Hayter (2007) and Liu et al. (2009). This problem is challenging as it involves optimization over an unknown function that determines the shape of the confidence band over χE. As a step towards solving this difficult problem, in this paper, we introduce a family of confidence bands over χE, called the inner-hyperbolic bands, which includes the hyperbolic and constant-width bands as special cases. We then search for the optimal confidence band within this family under the MVCS criterion. The conclusion from this study is that the hyperbolic band is not optimal even within this family of inner-hyperbolic bands and so cannot be the overall optimal band. On the other hand, the constant width band can be optimal within the family of inner-hyperbolic bands when the region χE is small and so might be the overall optimal band.

A Simultaneous Confidence Band for Dense Longitudinal Regression

We present a method of using local linear smoothing to construct simultaneous confidence bands for the mean function of densely spaced functional data. Our approach works well under mild conditions. In addition, the local linear estimator and its accompanying confidence band enjoy semiparametric efficiency in the sense that they are asymptotically equivalent to the counterparts obtained from the random trajectories entirely observed without errors. We illustrate the performance of the proposed confidence band through a simulation study. Furthermore, an application in food science is presented.

Quantifying and comparing dynamic predictive accuracy of joint models for longitudinal marker and time-to-event in presence of censoring and competing risks.

Thanks to the growing interest in personalized medicine, joint modeling of longitudinal marker and time-to-event data has recently started to be used to derive dynamic individual risk predictions. Individual predictions are called dynamic because they are updated when information on the subject's health profile grows with time. We focus in this work on statistical methods for quantifying and comparing dynamic predictive accuracy of this kind of prognostic models, accounting for right censoring and possibly competing events. Dynamic area under the ROC curve (AUC) and Brier Score (BS) are used to quantify predictive accuracy. Nonparametric inverse probability of censoring weighting is used to estimate dynamic curves of AUC and BS as functions of the time at which predictions are made. Asymptotic results are established and both pointwise confidence intervals and simultaneous confidence bands are derived. Tests are also proposed to compare the dynamic prediction accuracy curves of two prognostic models. The finite sample behavior of the inference procedures is assessed via simulations. We apply the proposed methodology to compare various prediction models using repeated measures of two psychometric tests to predict dementia in the elderly, accounting for the competing risk of death. Models are estimated on the French Paquid cohort and predictive accuracies are evaluated and compared on the French Three-City cohort.

Estimation and inference in functional single-index models

We propose a functional single-index model (FSiM) to study the link between a scalar response variable and multiple functional predictors, in which the mean of the response is related to the linear predictors via an unknown link function. The FSiM serves as a good tool for dimension reduction in regression with multiple predictors and it is more flexible than functional linear models. Assuming that the functional predictors are observed at discrete points, we use B-spline basis functions to estimate the slope functions and the link function based on the least-squares criterion, and propose an iterative estimating procedure. Moreover, we provide uniform convergence rates of the proposed spline estimators in the FSiM, and construct asymptotic simultaneous confidence bands for the slope functions for inference. Our proposed method is illustrated by simulation studies and by an analysis of a diffusion tensor imaging data application.

A martingale approach to estimating confidence band with censored data

Abstract This paper develops some non-parametric simultaneous confidence bands for survival function when data are randomly censored on the right. To construct the confidence bands, a computer-assisted method is utilized and this approach requires no distributional assumptions, so the confidence bands can be easily estimated. The procedures are based on the integrated martingale whose distribution is approximated by a Gaussian process. The supremum distribution of the Gaussian process generated by simulation techniques leads to the construction of the confidence bands. To improve the estimation procedures for the finite sample sizes, the log-minus-log transformation is employed. The proposed confidence bands are assessed using numerical simulations and applied to a real-world data set regarding leukemia.

Inference for Local Autocorrelations in Locally Stationary Models

For nonstationary processes, the time-varying correlation structure provides useful insights into the underlying model dynamics. We study estimation and inferences for local autocorrelation process in locally stationary time series. Our constructed simultaneous confidence band can be used to address important hypothesis testing problems, such as whether the local autocorrelation process is indeed time-varying and whether the local autocorrelation is zero. In particular, our result provides an important generalization of the R function acf()to locally stationary Gaussian processes. Simulation studies and two empirical applications are developed. For the global temperature series, we find that the local autocorrelations are time-varying and have a “V” shape during 1910–1960. For the S&P 500 index, we conclude that the returns satisfy the efficient-market hypothesis whereas the magnitudes of returns show significant local autocorrelations.

Simultaneous confidence bands for a percentile line in linear regression

Construction of simultaneous confidence bands for a percentile line in linear regression has been considered by several authors. But only conservative symmetric bands, which use critical constants over the whole covariate range (−∞,∞), are available in the literature. New methods allow the construction of exact symmetric bands for a percentile line over a finite interval of the covariate x. The exact symmetric bands can be substantially narrower than the corresponding conservative symmetric bands. Several exact symmetric confidence bands are compared under the average band width criterion. Furthermore, new asymmetric confidence bands for a percentile line are proposed. They are uniformly and can be very substantially narrower than the corresponding exact symmetric bands. Therefore, asymmetric bands should always be used under the average band width criterion. The proposed methods are illustrated with a real example.

Simultaneous Small Sample Inference for Linear Combinations of Generalized Linear Model Parameters

A method is proposed to construct simultaneous confidence intervals for multiple linear combinations of generalized linear model parameters, that uses a multivariate normal- or t-distribution together with the signed likelihood root statistic. In an application to a case study simultaneous confidence bands for logistic regression are calculated. A simulation study based on the example evaluation suggests superior performance compared to the common Wald-type approaches. The proposed methods are readily implemented in the R extension package mcprofile.