Functional inference in semiparametric models using the piggyback bootstrap

Abstract. This paper studies the representation and large‐sample consistency for non‐parametric maximum likelihood estimators (NPMLEs) of an unknown baseline continuous cumulative‐hazard‐type function and parameter of group survival difference, based on right‐censored two‐sample survival data with marginal survival function assumed to follow a transformation model, a slight generalization of the class of frailty survival regression models. The paper's main theoretical results are existence and unique a.s. limit, characterized variationally, for large data samples of the NPMLE of baseline nuisance function in an appropriately defined neighbourhood of the true function when the group difference parameter is fixed, leading to consistency of the NPMLE when the difference parameter is fixed at a consistent estimator of its true value. The joint NPMLE is also shown to be consistent. An algorithm for computing it numerically, based directly on likelihood equations in place of the expectation‐maximization (EM) algorithm, is illustrated with real data.

AIDS Research and Human RetrovirusesVol. 30, No. S1 Product Acceptability and AdherenceParticipation of Individuals from Fishing Communities in a Simulated Vaccine Efficacy Trial in Preparation for Future HIV Prevention WorkGershim Asiki, Andrew Abaasa, Ubaldo Bahemuka, Jerry Mulondo, Freddie Kibengo, Eugene Ruzagira, Anatoli Kamali, and Pat FastGershim AsikiMedical Research Council/Uganda Virus Research Institute, Entebbe, UgandaSearch for more papers by this author, Andrew AbaasaMedical Research Council/Uganda Virus Research Institute, Entebbe, UgandaSearch for more papers by this author, Ubaldo BahemukaMedical Research Council/Uganda Virus Research Institute, Entebbe, UgandaSearch for more papers by this author, Jerry MulondoMedical Research Council/Uganda Virus Research Institute, Entebbe, UgandaSearch for more papers by this author, Freddie KibengoMedical Research Council/Uganda Virus Research Institute, Entebbe, UgandaSearch for more papers by this author, Eugene RuzagiraMedical Research Council/Uganda Virus Research Institute, Entebbe, UgandaSearch for more papers by this author, Anatoli KamaliMedical Research Council/Uganda Virus Research Institute, Entebbe, UgandaSearch for more papers by this author, and Pat FastInternational AIDS Vaccine Initiative, New York, NY, United StatesSearch for more papers by this authorPublished Online:30 Oct 2014https://doi.org/10.1089/aid.2014.5338.abstractAboutSectionsView articleView Full TextPDF/EPUB ToolsPermissionsDownload CitationsTrack CitationsAdd to favorites Back To Publication ShareShare onFacebookTwitterLinked InRedditEmail View article"Participation of Individuals from Fishing Communities in a Simulated Vaccine Efficacy Trial in Preparation for Future HIV Prevention Work." AIDS Research and Human Retroviruses, 30(S1), p. A163FiguresReferencesRelatedDetails Volume 30Issue S1Oct 2014 InformationCopyright 2014, Mary Ann Liebert, Inc.To cite this article:Gershim Asiki, Andrew Abaasa, Ubaldo Bahemuka, Jerry Mulondo, Freddie Kibengo, Eugene Ruzagira, Anatoli Kamali, and Pat Fast.Participation of Individuals from Fishing Communities in a Simulated Vaccine Efficacy Trial in Preparation for Future HIV Prevention Work.AIDS Research and Human Retroviruses.Oct 2014.A163-A163.http://doi.org/10.1089/aid.2014.5338.abstractPublished in Volume: 30 Issue S1: October 30, 2014PDF download

BackgroundFemale sex workers (FSWs) at substantial risk of HIV are potentially a suitable group for HIV prevention trials including vaccine trials. Few HIV vaccine preparatory studies have been conducted among FSWs in Sub-Saharan Africa (SSA); data are therefore limited on acceptability of vaccine trial procedures. We determined vaccination completion and one-year retention among FSWs in Kampala, Uganda.MethodsWe conducted a prospective study that simulated a vaccine efficacy trial among HIV negative FSWs (18–49 years). Hepatitis B vaccine (Engerix B) was used to mimic an HIV vaccine product. Volunteers received 1 ml intramuscular injection at 0, 1 and 6 months, and made additional visits (3 days post-vaccination and months 3, 9 and 12). They were censored at that visit if diagnosed as HIV positive or pregnant. We collected socio-demographic, behavioral and clinical data at baseline, 6 and 12 months and fitted Poisson regression models with robust standard error to find factors associated with vaccination completion and retention.ResultsWe enrolled 290 volunteers (median age 27 years) of whom 230 reached a study end-point as follows: 7 became HIV infected, 11 became pregnant and 212 completed both the vaccination schedule and 12-month visit giving a retention of 77.9% (212/272). Vaccination completion was 82.4%.Non-retention at 1 year was more likely among those reporting symptoms of genital ulcer disease (GUD) in the past 3 months (IRR 1.90; 95% CI 1.09–3.32) and those < 35 years; (IRR 6.59; 95% CI 2.11–20.57). Non-completion of the vaccination schedule was associated with being < 35 years (IRR 13.10; 95% CI 1.89–90.92, reporting GUD symptoms (IRR 3.02; 95% CI 1.71–5.33) and reporting consistent condom use with new sexual partners (IRR 2.57; 95% CI 1.10–6.07).ConclusionsFSWs are at substantial risk of HIV infection and yet willing to participate in HIV vaccine and prevention research; young FSWs should be empowered, and those reporting GUD symptoms need close follow up to improve participation in future HIV vaccine trials.

Case-cohort analysis with semiparametric transformation models

Middle censoring introduced by Jammalamadaka and Mangalam (2003), refers to data arising in situations where the exact lifetime becomes unobservable if it falls within a random censoring interval, otherwise it is observable. In the present paper we propose a semi-parametric regression model for such lifetime data, arising from an unknown population and subject to middle censoring. We provide an algorithm to find the nonparametric maximum likelihood estimator (NPMLE) for regression parameters and the survival function. The consistency of the estimators are established. We report simulation studies to assess the finite sample properties of the estimators. We then analyze a real life data on survival times for diabetic patients studied by Lee et al. (1988).

The frailty model is a generalization of Cox's proportional hazard model, where a shared unobserved quantity in the intensity induces a positive correlation among the survival times. Murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator (NPMLE) for the shared gamma-frailty model without covariates. In this paper we extend this result to the correlated gamma-frailty model, and we allow for covariates. We discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator, which goes back to Wald. Our proof of the consistency for the NPMLE is essentially the same as the classical proof for the maximum likelihood estimator. A new central limit theorem for processes of bounded variation is given. Furthermore, we prove that a consistent estimator for the asymptotic variance of the NPMLE is given by the inverse of a discrete observed information matrix.

The cumulative incidence is the probability of failure from the cause of interest over a certain time period in the presence of other risks. A semiparametric regression model proposed by Fine and Gray (1999) has become the method of choice for formulating the effects of covariates on the cumulative incidence. Its estimation, however, requires modeling of the censoring distribution and is not statistically efficient. In this paper, we present a broad class of semiparametric transformation models which extends the Fine and Gray model, and we allow for unknown causes of failure. We derive the nonparametric maximum likelihood estimators (NPMLEs) and develop simple and fast numerical algorithms using the profile likelihood. We establish the consistency, asymptotic normality, and semiparametric efficiency of the NPMLEs. In addition, we construct graphical and numerical procedures to evaluate and select models. Finally, we demonstrate the advantages of the proposed methods over the existing ones through extensive simulation studies and an application to a major study on bone marrow transplantation.

Causal mediation modeling has become a popular approach for studying the effect of an exposure on an outcome through mediators. Currently, the literature on mediation analyses with survival outcomes largely focused on settings with a single mediator and quantified the mediation effects on the hazard, log hazard and log survival time (Lange and Hansen 2011; VanderWeele 2011). In this article, we propose a multi-mediator model for survival data by employing a flexible semiparametric probit model. We characterize path-specific effects (PSEs) of the exposure on the outcome mediated through specific mediators. We derive closed form expressions for PSEs on a transformed survival time and the survival probabilities. Statistical inference on the PSEs is developed using a nonparametric maximum likelihood estimator under the semiparametric probit model and the functional Delta method. Results from simulation studies suggest that our proposed methods perform well in finite sample. We illustrate the utility of our method in a genomic study of glioblastoma multiforme survival.

Although there had been few textbooks dealing with frailty models in survival analysis in the past, Chapman and Hall/CRC Press published two books in 2011 devoted to this topic. I reviewed the book...

On local odds and hazard rate models in survival analysis

The Fundamentals of Survival and Event History Analysis Introduction: What Is Survival and Event History Analysis? Key Concepts and Terminology Censoring and Truncation Mathematical Expression and Relation of Basic Statistical Functions How Do the Survivor, Density and Hazard Function Relate? Why Use Survival and Event History Analysis? Overview of Survival and Event History Models Exercises Using R and Other Computer Programs for Survival and Event History Analysis Introduction: Computer Programs for Survival and Event History Analysis Conducting Serious Data Analysis: Life Lessons Why Use R? Downloading R on Your Personal Computer Add-On Packages Running R Determining and Setting your Working Directory Help and Documentation Importing Data Into R Working With Data: Opening and Accessing Variables from a Data Frame Saving Output as File, Workspace and History and Quitting R Exercises Your First Session: Using the Survival Package and Exploring Data Via Descriptive Statistics and Graphs Your First Session Using the 'Survival' Package In F Loading and Examining the Survival Package and Rcmdrplugin.Survival Plug-In Opening and Examining Data The Surv Object: Packaging the 'Survival Variable' Basic Descriptive Statistics Descriptive Data Exploration with Graphs Exercises Data and Data Reconstruction Introduction: Why Discuss Data and Data Preparation? Sources of Event History Data Single-Episode Data for Single Transition Analyses Multi-Episode Data for Recurrent Event and Frailty Analyses Subject-(Person)-Period Data for Discrete-Time Hazard Models The Counting Process and Episode Splitting A Note on Dates Exercises Non-Parametric Methods: Estimating and Comparing Survival Curves Using the Kaplan-Meier Estimator Introduction The Kaplan-Meier Estimator Producing Kaplan-Meier Estimates Plotting the Kaplan-Meier Survival Curve Testing Differences Between Two Groups Using Survdiff Stratifying the Analysis by a Covariate Exercises The Cox Proportional-Hazards Regression Introduction: Why is The Cox Model So Popular? The Cox Regression Model Estimating and Interpreting The Cox Model with Fixed Covariates The Cox Regression Model with Time-Varying Covariates Exercises Parametric Models Introduction: What are Parametric Models and Why Use Them? Proportional Hazards (Ph) Versus Accelerated Failure Time (Aft) Models The Path to Choosing a Model Estimating and Interpreting Parametric Survival Models Exponential and Piecewise Constant Exponential Model Weibull Model Log-Logistic and Log-Normal Models Additional Parametric Models Finding the Best Fitting Parametric Model Exercises Model Building and Diagnostics Introduction Model Building and Selection of Covariates Assessing the Overall Goodness of Fit of Your Model What is Residual Analysis? Testing Overall Model Adequacy: Cox-Snell Residuals Testing the Proportional Hazards Assumption: Schoenfeld Residuals Checking For Influential Observations: Score Residuals (Dfbeta Statistics) Assessing Nonlinearity: Martingale Residual and Component-Plus-Residual Plots Exercises Correlated and Discrete-Time Survival Data: Frailty, Recurrent Events and Discrete-Time Models Introduction Shared Frailty: Modeling Recurrent Events and Clustering In Groups Other Frailty Models: Unshared, Nested, Joint and Additive Models Estimating Frailty Models in R Example of Frailty Model Estimation and Interpretation Discrete-Time and Count Models Exercises Multiple Events and Entire Histories: Competing Risk, Multistate Models and Sequence Analysis Introduction Competing Risk Models Multistate Models Sequence Analysis: Modeling Entire Histories Exercises Appendix : Datasets Used in this Book

The events of interest in any survival analysis study are regularly subject to censoring. There are various censoring schemes, including right or left censoring, and interval censoring. The most frequent censoring scheme is the right censoring, where subjects might drop out of the study or simply because not all events of interest occur before the end of the study. Moreover, for each subject, additional information referred to as covariates is registered at the beginning or throughout the study, such as age, sex, undergoing treatment, etc. The classical model to study the distribution of the events of interest, while accounting for additional information, is the Cox model. The Cox model expresses the hazard function of a subject given a set of covariates in terms of a baseline hazard, for which all covariates are zero, and an exponential function of the covariates and corresponding regression parameters. The baseline hazard can be left completely unspecified while estimating the regression parameters. Nonetheless, in practice, there are numerous studies in which the baseline hazard appears to be monotone. Time to death or to the onset of a disease are observed to have a nondecreasing baseline hazard, while the survival or recovery time after a successful medical treatment usually exhibit a nonincreasing baseline hazard. The aim of this thesis is to study the behavior of nonparametric baseline hazard and baseline density estimators in the Cox model under monotonicity constraints. The event times are assumed to be right censored and the censoring mechanism is assumed to be independent of the event of interest and non-informative. The covariates are assumed to be time-independent, usually recorded at the beginning of the study. In addition to point estimates, interval estimates of a monotone baseline hazard will be provided, based on a likelihood ratio method, along with testing at a fixed point. Furthermore, kernel smoothed estimates of a monotone baseline hazard will be defined and their behavior will be investigated. In Chapter 2, we propose several nonparametric monotone estimators of a baseline hazard or a baseline density within the Cox model. We derive the nonparametric maximum likelihood estimator of a nondecreasing baseline hazard and we consider a Grenander-type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. The two estimators are then shown to be strongly consistent and asymptotically equivalent. Moreover, we derive their common limit distribution at a fixed point. The two equivalent estimators of a nonincreasing baseline hazard and their asymptotic properties are acquired similarly. Furthermore, we introduce a Grenander-type estimator of a nonincreasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function derived from the Breslow estimator. This estimator is proven to be strongly consistent and its asymptotic distribution at a fixed point is derived. Chapter 3 provides an asymptotic linear representation of the Breslow estimator of the baseline cumulative hazard function in the Cox model. This representation can be used to derive the asymptotic distribution of the Grenander type estimator of a monotone baseline hazard estimator. The representation consists of an average of independent random variables and a term involving the difference between the maximum partial likelihood estimator and the underlying regression parameter. The order of the remainder term is arbitrarily close to n^-1. Chapter 4 focuses on interval estimation and on testing whether a monotone baseline hazard function in the Cox model has a particular value at a fixed point, via a likelihood ratio method. Nonparametric maximum likelihood estimators under the null hypothesis are defined for both nondecreasing and nonincreasing baseline hazard functions. These characterizations, along with those of the monotone nonparametric maximum likelihood estimators provide the asymptotic distribution of the likelihood ratio test. This asymptotic distribution enables, via inversion, the construction of pointwise confidence intervals. This method of constructing confidence intervals avoids the issue of estimating the nuisance parameters, as in the case of confidence intervals based on the asymptotic distribution of the estimators. Simulations indicate that the two methods yield confidence intervals with comparable coverage probabilities. Nonetheless, the confidence intervals based on the likelihood ratio are smaller, on average. Finally, in chapter 5 we consider smooth baseline hazard estimators. The estimators are obtained by kernel smoothing the maximum likelihood and Grenander-type estimators of a monotone baseline hazard function. Three different estimators are proposed for a nondecreasing baseline hazard, which are provided by the interchange of the smoothing and isotonization step. With this respect, we define a smoothed maximum likelihood estimator (SMLE), as well as a smoothed Grenander type (SG) estimator and a Grenander type smoothed (GS) estimator. All estimators are shown to be strongly pointwise or uniformly consistent.

Assessing Goodness-of-Fit of Generalized Logit Models Based on Case-Control Data

SUMMARY We test the logistic regression assumption under a case-control sampling plan. After reparameterisation, the assumed logistic regression model is equivalent to a two-sample semiparametric model in which the log ratio of two density functions is linear in data. By identifying this model with a biased sampling model, we propose a Kolmogorov-Smirnovtype statistic to test the validity of the logistic link function. Moreover, we point out that this test statistic can also be used in mixture sampling. We present a bootstrap procedure along with some results on simulation and on analysis of two real datasets.

A semiparametric transformation model comprises a parametric component for covariate effects and a nonparametric component for the baseline hazard/intensity. The Breslow-type estimator has been proposed for estimating the nonparametric component in some inefficient estimation procedures. We show that introducing weights into this estimator leads to nonparametric maximum likelihood estimation, with the weights depending on the martingale residuals. The weighted Breslow-type estimator suggests an iterative reweighting algorithm for nonparametric maximum likelihood estimation, which can be implemented by a weighted variant of the existing algorithms for inefficient estimation, and can be computationally more efficient than an &lt;scp&gt;em&lt;/scp&gt;-type algorithm. The weighting idea is further extended to semiparametric transformation models with mismeasured covariates.