Additive Risk Model Using Piecewise Constant Hazard Function

Goodness-of-fit tests for Cox's proportional hazards model and Aalen's additive risk model, in which each model is compared on an equal footing with the best fitting fully nonparametric model, are developed. The goodness-of-fit statistics are based on differences each model, with a fully nonparametric estimator of d recently introduced by the authors. Here A( , z) denotes the conditional hazard function of the survival time of an individual with covariate vector z. Comparison of the results of the tests makes it possible to decide whether Cox's proportional hazards or Aalen's additive risk model gives a better fit to the data. In addition, a goodness-of-fit test for Cox's model within the family of all proportional hazards models A(t, z) = AO(t)r(z), where AO is a baseline hazard function and r is a general relative risk function, is developed. Additive hazards and proportional hazards regression models used in the analysis of censored survival data can give substantially different results. For instance, in connection with a study of cancer mortality among Japanese atomic bomb survivors, Muirhead & Darby (1987) have noted that the two models give substantially different estimates of the age-spe- cific probability that an individual will develop radiation induced cancer. Muirhead & Darby (see also Aranda-Ordaz, 1983) introduced a generalized parametric model which contains parametric additive hazards and proportional hazards models as special cases. The goodness- of-fit of each model is then obtained by comparing with the best fitting model within the generalized family, allowing the two special models to be treated on an equal footing. Beyond the parametric setting, much effort has been devoted to the development of goodness-of-fit tests for Cox's (1972) proportional hazards model

Estimating the survival functions for right-censored and interval-censored data with piecewise constant hazard functions

As a useful alternative to Cox's proportional hazard model, the additive risk model assumes that the hazard function is the sum of the baseline hazard function and the regression function of covariates. This article is concerned with estimation and prediction for the additive risk models with right censored survival data, especially when the dimension of the covariates is comparable to or larger than the sample size. Principal component regression is proposed to give unique and numerically stable estimators. Asymptotic properties of the proposed estimators, component selection based on the weighted bootstrap, and model evaluation techniques are discussed. This approach is illustrated with analysis of the primary biliary cirrhosis clinical data and the diffuse large B-cell lymphoma genomic data. It is shown that this methodology is numerically stable and effective in dimension reduction, while still being able to provide satisfactory prediction and classification results.

SUMMARY The additive risk model assumes that the hazard function associated with a set of covariates is the sum of the baseline hazard function and the regression function of covariates. We propose two different test procedures for checking the adequacy of two-sample additive risk models for randomly censored observations. One is based on the martingale residuals and the other on the difference between weighted estimators of the excess risk. The test statistics are shown to be asymptotically normal under appropriate regularity conditions and consistent under any model misspecifications. Finally, two real examples are provided, along with results of a simulation study.

In this article, we consider modeling arbitrarily censored survival data with spatio‐temporal covariates. We demonstrate that under the piecewise constant hazard function, the likelihood for uncensored or right‐censored subjects is proportional to the likelihood of multiple conditionally independent Poisson random variables. To address left‐ or interval‐censored subjects, we propose to impute the exact event times and convert them into uncensored subjects, enabling the application of the integrated nested Laplace approximation to update model parameters using the imputed data. We introduce an iterative algorithm that alternates between imputing event times for left‐ and interval‐censored subjects and re‐estimating model parameters. The proposed method is assessed through a simulation study and applied to analyze a spatio‐temporal survival dataset in a wildlife disease study investigating bovine tuberculosis in white‐tailed deer in Michigan.

Additive excess risk model for epidemiologic interaction in retrospective studies

One problem of interest is to relate genes to survival outcomes of patients for the purpose of building regression models to predict future patientspsila survival based on their gene expression data. Applying semiparametric additive risk model of survival analysis, we propose a new approach to conduct the analysis of gene expression data with the focus on modelpsilas predictive ability. The method modifies the correlation principal component regression to handle the censoring problem of survival data. In addition, we employ the time dependent AUC and RMSEP to assess how well the model predicts the survival time. Furthermore, the proposed method is able to identify significant genes that are related to the disease. Finally, this proposed approach is illustrated by the diffuse large B-cell lymphoma (DLBCL) data set. The results show that the model fits the data set very well.

Full and conditional likelihood approaches for hazard change-point estimation with truncated and censored data

Efficiency and robustness are two essential concerns on statistical estimation. Unfortunately, it was widely accepted that there existed a contradiction between achieving efficiency and robustness simultaneously. For parametric models with complete data, the minimum Hellinger distance estimation introduced by Beran (Ann Stat 5:445–463, 1977) has been shown that it can reconcile this contradiction. Because data in biostatistics, actuarial science or economics are often subject to censoring and even involve a fraction of long-term survivors, our study aims to extend the minimum Hellinger distance estimation to a two-sample semiparametric cure rate model with right-censored survival data. The asymptotic properties such as consistency, efficiency, normality, and robustness of the proposed estimator have been considered and its performances are examined via simulation studies in comparison with those of the maximum semiparametric conditional likelihood estimator introduced by Shen et al. (J Am Stat Assoc 102:1235–1244, 2007). Finally, our method is illustrated by analyzing a real data set: Bone Marrow Transplant Data.

The distribution-free method of conformal prediction has gained considerable attention in computer science, machine learning, and statistics. Candès etal. extended this method to right-censored survival data, addressing right-censoring complexity by creating a covariate shift setting, extracting a subcohort of subjects with censoring times exceeding a fixed threshold. Their approach only estimates the lower prediction bound for type I censoring, where all subjects have available censoring times regardless of their failure status. In medical applications, we often encounter more general right-censored data, observing only the minimum of failure time and censoring time. Subjects with observed failure times have unavailable censoring times. To address this, we propose a bootstrap method to construct 1- as well as 2-sided conformal predictive intervals for general right-censored survival data under different working regression models. Through simulations, our method demonstrates excellent average coverage for the lower bound and good coverage for the 2-sided predictive interval, regardless of working model is correctly specified or not, particularly under moderate censoring. We further extend the proposed method to several directions in medical applications. We apply this method to predict breast cancer patients' future survival times based on tumor characteristics and treatment.

ObjectivesWhen extrapolating time-to-event data the Bagust and Beale (B&B) approach uses the Kaplan-Meier survival function until a manually chosen time point, after which a constant hazard is assumed. This study demonstrates an objective statistical approach to estimate this time point. MethodsWe estimate piecewise exponential models (PEMs), whereby the hazard function is partitioned into segments each with constant hazards. The boundaries of these segments are known as change points. Our approach determines the location and number of change points in PEMs from which the hazard in the final segment is used to model long-term survival. We reviewed previous applications of the B&B approach in National Institute for Health and Care Excellence Technology Appraisals (TAs) completed between July 2011 and June 2017. The time points after which constant hazards were assumed were compared between PEMs and the B&B approaches. When further survival data were published following the original TA, we compared these updated estimates to predicted survival from the PEM and other parametric models adjusted for general population mortality. ResultsSix of the 59 TAs in this review considered the B&B approach. There was general agreement between the location of time points identified through the PEM and the B&B approaches. In 2 of the identified TAs the best fitting model to the data was a no-change-point model. Of the 3 TAs for which further survival data became available, PEM provided the closest prediction for survival outcomes in 2 TAs. ConclusionsPEMs are useful for survival extrapolation when a long-term constant hazard trend for the disease is clinically plausible.

Three models of risk were examined for 277 African American children from single‐mother‐headed homes: cumulative, additive, and indirect models. Risk factors were mother‐reported community risks, inadequate income, maternal depressive symptoms, and inadequate parenting. Child‐reported internalizing and externalizing difficulties served as dependent measures of adjustment. Whereas the cumulative risk model identified a subset of children within this group as at‐risk for adjustment difficulties, only more proximal family variables (maternal depressive symptoms and inadequate parenting) accounted for unique variance in child outcomes (additive risk model). However, the more distal risk factors—community risk and inadequate income—were linked to both internalizing and externalizing difficulties through the proximal family variables (indirect effects model).

Lianne Sheppard (University of Washington, Seattle) As both the papers by Gelman et al. and Wakefield and Salway emphasize, a deeper understanding of ecologic studies is vital to their application. There is a fundamental misalignment in many ecologic studies because the level of analysis, the group, does not match the level of inference, the individual. In their stimulating papers, Gelman and his colleagues and Wakefield and Salway address this misalignment by concentrating on features of ecologic regressions under fundamentally different frameworks. Gelman and his colleagues restrict their attention to binary or categorical covariates and cast their models in terms that are familiar in the political science literature. Within this structure they suggest some workable approaches to identifying violations of model assumptions. They rightly highlight the fundamental inability to evaluate all the assumptions that are necessary in an ecologic regression with the data at hand. Wakefield and Salway focus on the more general challenge of coping with continuous covariates in a group level setting. They present their framework for ecologic studies in the context of environmental epidemiologic applications. In all applications, group level studies, unlike multilevel studies, cannot separately estimate covariate effects that operate differently through groups and individuals. Since the additional effects of group membership (often called contextual effects (Firebaugh, 1978)) are much more likely to derive from the social rather than the physical environment, the intended application is crucial to the potential for valid inference from group level studies. It is natural to assume that environmental epidemiologic applications will require fewer untestable assumptions than ecologic studies in the social and behavioural sciences. Wakefield and Salway emphasize the importance of the within-area joint distribution of the exposure variables. Of particular value, they clarify the variance of an aggregate outcome (conditional on the exposure) induced under a range of assumptions about the exposure. They restrict their detailed treatment to cases where the within-area outcomes are independent. In many applications this assumption will be too strong and consequently there will be additional dependences in the variance models. A second feature of ignoring the within-area exposure distribution is its influence on the bias of the relative risk parameter estimates for all but the additive risk model. For instance, in their epidemiologic example, Gelman and his colleagues substitute average covariate measurements for individual values in a non-linear model. This is common practice in epidemiology (see for example Guthrie and Sheppard's (2000) discussion of Greenland and Robins's (1994) treatment of a similar radon model). The fitted ecologic model is clearly misspecified when inference is directed at the individual. The implications of this misspecification depend on the full within-area exposure distribution, not just the mean, features which are not included in the ecologic model. Gelman and his colleagues describe ramifications that are specific to radon exposures, whereas Wakefield and Salway give a more general perspective. These papers, as well as others in this issue, give fresh insights into many important considerations for ecologic studies.

This study aimed at developing a reliable instrument to identify reading, writing and numeracy problems at school entry in Bahrain. Three curricular areas, reading, writing and numeracy were targeted for building this indtrument. The reading section consisted of items derived from the research literature relating to phonological awareness and concept of print. The writing section consisted of items requiring the child to write their own name, draw a human figure, write a letter to a favourite person and copy geometric shapes. The numeracy section composed of addition items derived from the Maths Recovery programme (Wright, Martland, & Stafford, 2006). Item analyses and analyses of reliability and validity were carried out. A sample consisting of 145 Bahraini children from preschool (N=47, 14 boys and 30 girls, mean age 72.77 months, range 63 months to 95 months) and grade one (N=98, 47 boys and 51 girls, mean age 79.63 months, range 71 months to 115 months) were used for validation. All of the preschool children had teacher-identified learning difficulties and one group of the grade one pupils (N=48, 22 boys and 26 girls, mean age 82.33 months, range 76 months to 115 months) had learning difficulties while a second group (N=50, 25 boys and 25 girls, mean age 76.92 months, range 71 months to 84 months) were typically-developing. The results revealed satisfactory levels of internal consistency reliability for the final selection of items (Cronbach alpha was 0.73) and concurrent validity based upon teacher evaluation was in the range 0.81-0.90. The instrument also distinguished children with difficulties from their typically-developing peers (insert some details). Analyses of sensitivity and specificity based upon an additive risk model (reference) yielded sensitivity of 0.93 and specificity of 0.67, in both cases based upon problems in two or more of the three curricular areas of the test. The instrument reliably identified over 90% of the children with difficulties and can serve as a diagnostic tool to assist teachers in assessing problems in literacy and numeracy. However, in the light of the lower levels of specificity, use of the instrument as a screening test is not recommended.

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.