A class of partially linear single‐index survival models

The authors define a class of “partially linear single‐index” survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single‐index form. It is then possible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single‐index component. The partially linear hazards model and the single‐index hazards model are special cases of the proposed model. The authors develop a likelihood‐based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite‐sample behaviour through simulation, and use a set of real data to illustrate their approach.     

Effects of neckbands on survival and fidelity of white-fronted and Canada geese captured as non-breeding adults

We conducted an experiment to examine the effect of neckbands, controlling for differences in sex, species and year of study (1991-1997), on probabilities of capture, survival, reporting, and fidelity in non-breeding small Canada ( Branta canadensis hutchinsi ) and white-fronted ( Anser albifrons frontalis ) geese. In Canada's central arctic, we systematically double-marked about half of the individuals from each species with neckbands and legbands, and we marked the other half only with legbands. We considered 48 a priori models that included combinations of sex, species, year, and neckband effects on the four population parameters produced by Burnham's (1993) model, using AIC for model selection. The four best approximating models each included a negative effect of neckbands on survival, and effect size varied among years. True survival probability of neckbanded birds annually ranged from 0.006 to 0.23 and 0.039 to 0.22 (Canada and white-fronted geese, respectively) lower than for conspecifics without neckbands. Changes in estimates of survival probability in neckbanded birds appeared to attenuate more recently, particularly in Canada Geese, a result that we suspect was related to lower retention rates of neckbands. We urge extreme caution in use of neckbands for estimation of certain population parameters, and discourage their use for estimation of unbiased survival probability in these two species.     

A nonparametric test of conditional autoregressive heteroscedasticity for threshold autoregressive models

Threshold autoregressive models are widely used in time‐series applications. When building or using such a model, it is important to know whether conditional heteroscedasticity exists. The authors propose a nonparametric test of this hypothesis. They develop the large‐sample theory of a test of nonlinear conditional heteroscedasticity adapted to nonlinear autoregressive models and study its finite‐sample properties through simulations. They also provide percentage points for carrying out this test, which is found to have very good power overall.     

Bayesian and maximin optimal designs for heteroscedastic regression models

The authors consider the problem of constructing standardized maximin D‐optimal designs for weighted polynomial regression models. In particular they show that by following the approach to the construction of maximin designs introduced recently by Dette, Haines & Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian _q‐optimal designs. They further demonstrate that the results are more broadly applicable to certain families of nonlinear models. The authors examine two specific weighted polynomial models in some detail and illustrate their results by means of a weighted quadratic regression model and the Bleasdale–Nelder model. They also present a capstone example involving a generalized exponential growth model.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Closed‐form likelihoods for stochastic differential equation growth models

The authors derive closed‐form expressions for the full, profile, conditional and modified profile likelihood functions for a class of random growth parameter models they develop as well as Garcia's additive model. These expressions facilitate the determination of parameter estimates for both types of models. The profile, conditional and modified profile likelihood functions are maximized over few parameters to yield a complete set of parameter estimates. In the development of their random growth parameter models the authors specify the drift and diffusion coefficients of the growth parameter process in a natural way which gives interpretive meaning to these coefficients while yielding highly tractable models. They fit several of their random growth parameter models and Garcia's additive model to stock market data, and discuss the results. The Canadian Journal of Statistics 38: 474–487; 2010 © 2010 Statistical Society of Canada     

Exact and approximate sum representations for the Dirichlet process

The Dirichlet process can be regarded as a random probability measure for which the authors examine various sum representations. They consider in particular the gamma process construction of Ferguson (1973) and the “stick‐breaking” construction of Sethuraman (1994). They propose a Dirichlet finite sum representation that strongly approximates the Dirichlet process. They assess the accuracy of this approximation and characterize the posterior that this new prior leads to in the context of Bayesian nonpara‐metric hierarchical models.     

Effect of design specifications in dose‐finding trials for combination therapies in oncology

Model‐based dose‐finding methods for a combination therapy involving two agents in phase I oncology trials typically include four design aspects namely, size of the patient cohort, three‐parameter dose‐toxicity model, choice of start‐up rule, and whether or not to include a restriction on dose‐level skipping. The effect of each design aspect on the operating characteristics of the dose‐finding method has not been adequately studied. However, some studies compared the performance of rival dose‐finding methods using design aspects outlined by the original studies. In this study, we featured the well‐known four design aspects and evaluated the impact of each independent effect on the operating characteristics of the dose‐finding method including these aspects. We performed simulation studies to examine the effect of these design aspects on the determination of the true maximum tolerated dose combinations as well as exposure to unacceptable toxic dose combinations. The results demonstrated that the selection rates of maximum tolerated dose combinations and UTDCs vary depending on the patient cohort size and restrictions on dose‐level skipping However, the three‐parameter dose‐toxicity models and start‐up rules did not affect these parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

A Quantile Survival Model for Censored Data

In this paper we propose a quantile survival model to analyze censored data. This approach provides a very effective way to construct a proper model for the survival time conditional on some covariates. Once a quantile survival model for the censored data is established, the survival density, survival or hazard functions of the survival time can be obtained easily. For illustration purposes, we focus on a model that is based on the generalized lambda distribution (GLD). The GLD and many other quantile function models are defined only through their quantile functions, no closed‐form expressions are available for other equivalent functions. We also develop a Bayesian Markov Chain Monte Carlo (MCMC) method for parameter estimation. Extensive simulation studies have been conducted. Both simulation study and application results show that the proposed quantile survival models can be very useful in practice.     

Hierarchical-Likelihood Approach for Mixed Linear Models with Censored Data

Mixed linear models describe the dependence via random effects in multivariate normal survival data. Recently they have received considerable attention in the biomedical literature. They model the conditional survival times, whereas the alternative frailty model uses the conditional hazard rate. We develop an inferential method for the mixed linear model via Lee and Nelder's (1996) hierarchical-likelihood (h-likelihood). Simulation and a practical example are presented to illustrate the new method.     

Choice of Estimators Based on Different Observations: Modified AIC and LCV Criteria

Abstract. It is quite common in epidemiology that we wish to assess the quality of estimators on a particular set of information, whereas the estimators may use a larger set of information. Two examples are studied: the first occurs when we construct a model for an event which happens if a continuous variable is above a certain threshold. We can compare estimators based on the observation of only the event or on the whole continuous variable. The other example is that of predicting the survival based only on survival information or using in addition information on a disease. We develop modified Akaike information criterion (AIC) and Likelihood cross‐validation (LCV) criteria to compare estimators in this non‐standard situation. We show that a normalized difference of AIC has a bias equal to o ( n ^{? 1} ) if the estimators are based on well‐specified models; a normalized difference of LCV always has a bias equal to o ( n ^{? 1} ). A simulation study shows that both criteria work well, although the normalized difference of LCV tends to be better and is more robust. Moreover in the case of well‐specified models the difference of risks boils down to the difference of statistical risks which can be rather precisely estimated. For ‘compatible’ models the difference of risks is often the main term but there can also be a difference of mis‐specification risks.     

Two‐stage phase II survival trial design

Recently, molecularly targeted agents and immunotherapy have been advanced for the treatment of relapse or refractory cancer patients, where disease progression‐free survival or event‐free survival is often a primary endpoint for the trial design. However, methods to evaluate two‐stage single‐arm phase II trials with a time‐to‐event endpoint are currently processed under an exponential distribution, which limits application of real trial designs. In this paper, we developed an optimal two‐stage design, which is applied to the four commonly used parametric survival distributions. The proposed method has advantages compared with existing methods in that the choice of underlying survival model is more flexible and the power of the study is more adequately addressed. Therefore, the proposed two‐stage design can be routinely used for single‐arm phase II trial designs with a time‐to‐event endpoint as a complement to the commonly used Simon's two‐stage design for the binary outcome.     

A new class of multivariate skew distributions with applications to bayesian regression models

Abstract: The authors develop a new class of distributions by introducing skewness in multivariate elliptically symmetric distributions. The class, which is obtained by using transformation and conditioning, contains many standard families including the multivariate skew‐normal and t distributions. The authors obtain analytical forms of the densities and study distributional properties. They give practical applications in Bayesian regression models and results on the existence of the posterior distributions and moments under improper priors for the regression coefficients. They illustrate their methods using practical examples.     

Penalized likelihood regression: General formulation and efficient approximation

The authors consider a formulation of penalized likelihood regression that is sufficiently general to cover canonical and noncanonical links for exponential families as well as accelerated life models with censored survival data. They present an asymptotic analysis of convergence rates to justify a simple approach to the lower‐dimensional approximation of the estimates. Such an approximation allows for much faster numerical calculation, paving the way to the development of algorithms that scale well with large data sets.     

BAYESIAN HIDDEN MARKOV MODELS FOR LONGITUDINAL COUNTS

This paper describes a Bayesian approach to modelling carcinogenity in animal studies where the data consist of counts of the number of tumours present over time. It compares two autoregressive hidden Markov models. One of them models the transitions between three latent states: an inactive transient state, a multiplying state for increasing counts and a reducing state for decreasing counts. The second model introduces a fourth tied state to describe non‐zero observations that are neither increasing nor decreasing. Both these models can model the length of stay upon entry of a state. A discrete constant hazards waiting time distribution is used to model the time to onset of tumour growth. Our models describe between‐animal‐variability by a single hierarchy of random effects and the within‐animal variation by first‐order serial dependence. They can be extended to higher‐order serial dependence and multi‐level hierarchies. Analysis of data from animal experiments comparing the influence of two genes leads to conclusions that differ from those of Dunson (2000). The observed data likelihood defines an information criterion to assess the predictive properties of the three‐ and four‐state models. The deviance information criterion is appropriately defined for discrete parameters.     

Analysis of band‐recovery data in a multistate capture‐recapture framework

Dead recoveries of marked animals are commonly used to estimate survival probabilities. Band‐recovery models can be parameterized either by r (the probability of recovering a band conditional on death of the animal) or by f (the probability that an animal will be killed, retrieved, and have its band reported). The T parametrization can be implemented in a capture‐recapture framework with two states (alive and newly dead), mortality being the transition probability between the two states. The authors show here that the f parametrization can also be implemented in a multistate framework by imposing simple constraints on some parameters. They illustrate it using data on the mallard and the snow goose. However, they mention that because it does not entirely separate the individual survival and encounter processes, the f parametrization must be used with care on reduced models, or in the presence of estimates at the boundary of the parameter space. As they show, a multistate framework allows the use of powerful software for model fitting or testing the goodness‐of‐fit of models; it also affords the implementation of complex models such as those based on mixture of information or uncertain states     

Bayesian Semiparametric Modelling in Quantile Regression

Abstract. We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non‐parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non‐parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.     

Improvement over bayes prediction in small samples in the presence of model uncertainty

In an online prediction context, the authors introduce a new class of mongrel criteria that allow for the weighing of candidate models and the combination of their predictions based both on model‐based and empirical measures of their performance. They present simulation results which show that model averaging using the mongrel‐derived weights leads, in small samples, to predictions that are more accurate than that obtained by Bayesian weight updating, provided that none of the candidate models is too distant from the data generator.     

Bayesian Spatial Analysis for the Evaluation of Breast Cancer Detection Methods

This study investigated the impact of spatial location on the effectiveness of population‐based breast screening in reducing breast cancer mortality compared to other detection methods among Queensland women. The analysis was based on linked population‐based datasets from BreastScreen Queensland and the Queensland Cancer Registry for the period of 1997–2008 for women aged less than 90 years at diagnosis. A Bayesian hierarchical regression modelling approach was adopted and posterior estimation was performed using Markov Chain Monte Carlo techniques. This approach accommodated sparse data resulting from rare outcomes in small geographic areas, while allowing for spatial correlation and demographic influences to be included. A relative survival model was chosen to evaluate the relative excess risk for each breast cancer related factor. Several models were fitted to examine the influence of demographic information, cancer stage, geographic information and detection method on women's relative survival. Overall, the study demonstrated that including the detection method and geographic information when assessing the relative survival of breast cancer patients helped capture unexplained and spatial variability. The study also found evidence of better survival among women with breast cancer diagnosed in a screening program than those detected otherwise, as well as lower risk for those residing in a more urban or socio‐economically advantaged region, even after adjusting for tumour stage, environmental factors and demographics. However, no evidence of dependency between method of detection and geographic location was found. This project provides a sophisticated approach to examining the benefit of a screening program while considering the influence of geographic factors.     

Semiparametric Proportional Odds Models for Spatially Correlated Survival Data

The last decade has witnessed major developments in Geographical Information Systems (GIS) technology resulting in the need for statisticians to develop models that account for spatial clustering and variation. In public health settings, epidemiologists and health-care professionals are interested in discerning spatial patterns in survival data that might exist among the counties. This paper develops a Bayesian hierarchical model for capturing spatial heterogeneity within the framework of proportional odds. This is deemed more appropriate when a substantial percentage of subjects enjoy prolonged survival. We discuss the implementation issues of our models, perform comparisons among competing models and illustrate with data from the SEER (Surveillance Epidemiology and End Results) database of the National Cancer Institute, paying particular attention to the underlying spatial story.