Confidence Bands for the Difference of Two Survival Curves Under Proportional Hazards Model

A common approach to testing for differences between the survival rates of two therapies is to use a proportional hazards regression model which allows for an adjustment of the two survival functions for any imbalance in prognostic factors in the comparison. When the relative risk of one treatment to the other is not constant over time the question of which therapy has a survival advantage is difficult to determine from the Cox model. An alternative approach to this problem is to plot the difference between the two predicted survival functions with a confidence band that provides information about when these two treatments differ. Such a band will depend on the covariate values of a given patient. In this paper we show how to construct a confidence band for the difference of two survival functions based on the proportional hazards model. A simulation approach is used to generate the bands. This approach is used to compare the survival probabilities of chemotherapy and allogeneic bone marrow transplants for chronic leukemia.     

Disentangling the sources of variation in the survival of the European dipper

The population growth rate of the European dipper has been shown to decrease with winter temperature and population size. We examine here the demographic mechanism for this effect by analysing how these factors affect the survival rate. Using more than 20 years of capture-mark-recapture data (1974-1997) based on more than 4000 marked individuals, we perform analyses using open capture-mark-recapture models. This allowed us to estimate the annual apparent survival rates (probability of surviving and staying on the study site from one year to the next one) and the recapture probabilities. We partitioned the variance of the apparent survival rates into sampling variance and process variance using random effects models, and investigated which variables best accounted for temporal process variation. Adult males and females had similar apparent survival rates, with an average of 0.52 and a coefficient of variation of 40%. Chick apparent survival was lower, averaging 0.06 with a coefficient of variation of 42%. Eighty percent of the variance in apparent survival rates was explained by winter temperature and population size for adults and 48% by winter temperature for chicks. The process variance outweighed the sampling variance both for chick and adult survival rates, which explained that shrunk estimates obtained under random effects models were close to MLE estimates. A large proportion of the annual variation in the apparent survival rate of chicks appears to be explained by inter-year differences in dispersal rates.     

Disentangling the sources of variation in the survival of the European dipper

The population growth rate of the European dipper has been shown to decrease with winter temperature and population size. We examine here the demographic mechanism for this effect by analysing how these factors affect the survival rate. Using more than 20 years of capture-mark-recapture data (1974-1997) based on more than 4000 marked individuals, we perform analyses using open capture-mark-recapture models. This allowed us to estimate the annual apparent survival rates (probability of surviving and staying on the study site from one year to the next one) and the recapture probabilities. We partitioned the variance of the apparent survival rates into sampling variance and process variance using random effects models, and investigated which variables best accounted for temporal process variation. Adult males and females had similar apparent survival rates, with an average of 0.52 and a coefficient of variation of 40%. Chick apparent survival was lower, averaging 0.06 with a coefficient of variation of 42%. Eighty percent of the variance in apparent survival rates was explained by winter temperature and population size for adults and 48% by winter temperature for chicks. The process variance outweighed the sampling variance both for chick and adult survival rates, which explained that shrunk estimates obtained under random effects models were close to MLE estimates. A large proportion of the annual variation in the apparent survival rate of chicks appears to be explained by inter-year differences in dispersal rates.     

Analysis of Variance by Randomization when Variances are Unequal

If there are significant factor and interaction effects with analysis of variance using ran-domization inference, they can be detected by tests that compare the F -statistics for the real data with the distributions of these statistics obtained by randomly allocating either the original observations or the residuals to the various factor combinations. Such tests involve the assumption that the effect of factors or interactions is to shift the observations for a factor combination by a fixed amount, without changing the amount of variation at that combination. In reality the expected amount of variation at each factor combination, as measured by the variance, may not be constant, which may upset the properties of the tests for the effects of factors and interactions. This paper discusses several possible methods for adjusting the randomization procedure to allow for this type of problem, including generalizations of methods that have been proposed for comparing the means of several samples when there is unequal variance but no factor structure. A simulation study shows that the best of the methods examined is one for which the randomized sets of data are designed to approximate the distributions of F -statistics when unequal variance is present.     

An index of local sensitivity to non-ignorability for parametric survival models with potential non-random missing covariate: an application to the SEER cancer registry data

Several survival regression models have been developed to assess the effects of covariates on failure times. In various settings, including surveys, clinical trials and epidemiological studies, missing data may often occur due to incomplete covariate data. Most existing methods for lifetime data are based on the assumption of missing at random (MAR) covariates. However, in many substantive applications, it is important to assess the sensitivity of key model inferences to the MAR assumption. The index of sensitivity to non-ignorability (ISNI) is a local sensitivity tool to measure the potential sensitivity of key model parameters to small departures from the ignorability assumption, needless of estimating a complicated non-ignorable model. We extend this sensitivity index to evaluate the impact of a covariate that is potentially missing, not at random in survival analysis, using parametric survival models. The approach will be applied to investigate the impact of missing tumor grade on post-surgical mortality outcomes in individuals with pancreas-head cancer in the Surveillance, Epidemiology, and End Results data set. For patients suffering from cancer, tumor grade is an important risk factor. Many individuals in these data with pancreas-head cancer have missing tumor grade information. Our ISNI analysis shows that the magnitude of effect for most covariates (with significant effect on the survival time distribution), specifically surgery and tumor grade as some important risk factors in cancer studies, highly depends on the missing mechanism assumption of the tumor grade. Also a simulation study is conducted to evaluate the performance of the proposed index in detecting sensitivity of key model parameters.     

Delayed treatment effects,treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology

In the analysis of survival times, the logrank test and the Cox model have been established as key tools, which do not require specific distributional assumptions. Under the assumption of proportional hazards, they are efficient and their results can be interpreted unambiguously. However, delayed treatment effects, disease progression, treatment switchers or the presence of subgroups with differential treatment effects may challenge the assumption of proportional hazards. In practice, weighted logrank tests emphasizing either early, intermediate or late event times via an appropriate weighting function may be used to accommodate for an expected pattern of non-proportionality. We model these sources of non-proportional hazards via a mixture of survival functions with piecewise constant hazard. The model is then applied to study the power of unweighted and weighted log-rank tests, as well as maximum tests allowing different time dependent weights. Simulation results suggest a robust performance of maximum tests across different scenarios, with little loss in power compared to the most powerful among the considered weighting schemes and huge power gain compared to unfavorable weights. The actual sources of non-proportional hazards are not obvious from resulting populationwise survival functions, highlighting the importance of detailed simulations in the planning phase of a trial when assuming non-proportional hazards.We provide the required tools in a software package, allowing to model data generating processes under complex non-proportional hazard scenarios, to simulate data from these models and to perform the weighted logrank tests.     

Semivarying coefficient least-squares support vector regression for analyzing high-dimensional gene-environmental data

In the context of genetics and genomic medicine, gene-environment (G×E) interactions have a great impact on the risk of human diseases. Some existing methods for identifying G×E interactions are considered to be limited, since they analyze one or a few number of G factors at a time, assume linear effects of E factors, and use inefficient selection methods. In this paper, we propose a new method to identify significant main effects and G×E interactions. This is based on a semivarying coefficient least-squares support vector regression (LS-SVR) technique, which is devised by utilizing flexible semiparametric LS-SVR approach for censored survival data. This semivarying coefficient model is used to deal with the nonlinear effects of E factors. We also derive a generalized cross validation (GCV) function for determining the optimal values of hyperparameters of the proposed method. This GCV function is also used to identify significant main effects and G×E interactions. The proposed method is evaluated through numerical studies.     

Random effects and shrinkage estimation in capture-recapture models

We discuss the analysis of random effects in capture-recapture models, and outline Bayesian and frequentists approaches to their analysis. Under a normal model, random effects estimators derived from Bayesian or frequentist considerations have a common form as shrinkage estimators. We discuss some of the difficulties of analysing random effects using traditional methods, and argue that a Bayesian formulation provides a rigorous framework for dealing with these difficulties. In capture-recapture models, random effects may provide a parsimonious compromise between constant and completely time-dependent models for the parameters (e.g. survival probability). We consider application of random effects to band-recovery models, although the principles apply to more general situations, such as Cormack-Jolly-Seber models. We illustrate these ideas using a commonly analysed band recovery data set.     

Random effects and shrinkage estimation in capture-recapture models

We discuss the analysis of random effects in capture-recapture models, and outline Bayesian and frequentists approaches to their analysis. Under a normal model, random effects estimators derived from Bayesian or frequentist considerations have a common form as shrinkage estimators. We discuss some of the difficulties of analysing random effects using traditional methods, and argue that a Bayesian formulation provides a rigorous framework for dealing with these difficulties. In capture-recapture models, random effects may provide a parsimonious compromise between constant and completely time-dependent models for the parameters (e.g. survival probability). We consider application of random effects to band-recovery models, although the principles apply to more general situations, such as Cormack-Jolly-Seber models. We illustrate these ideas using a commonly analysed band recovery data set.     

A shared component model for detecting joint and selective clustering of two diseases

The study of spatial variations in disease rates is a common epidemiological approach used to describe the geographical clustering of diseases and to generate hypotheses about the possible 'causes' which could explain apparent differences in risk. Recent statistical and computational developments have led to the use of realistically complex models to account for overdispersion and spatial correlation. However, these developments have focused almost exclusively on spatial modelling of a single disease. Many diseases share common risk factors (smoking being an obvious example) and, if similar patterns of geographical variation of related diseases can be identified, this may provide more convincing evidence of real clustering in the underlying risk surface. We propose a shared component model for the joint spatial analysis of two diseases. The key idea is to separate the underlying risk surface for each disease into a shared and a disease-specific component. The various components of this formulation are modelled simultaneously by using spatial cluster models implemented via reversible jump Markov chain Monte Carlo methods. We illustrate the methodology through an analysis of oral and oesophageal cancer mortality in the 544 districts of Germany, 1986–1990.     

Model selection with missing covariates for policy considerations in fox enclosures

Foxhound training enclosures are facilities where wild-trapped foxes are placed into large fenced areas for dog training purposes. Although the purpose of these facilities is to train dogs without harming foxes, dog-related mortality has been reported to be an issue in some enclosures. Using data from a fox enclosure in Virginia, we investigate factors that influence fox survival in these dog training facilities and propose a set of policies to improve fox survival. In particular, a Bayesian hierarchical model is formulated to compute fox survival probabilities based on a fox's time in the enclosure and the number of dogs allowed in the enclosure at one time. These calculations are complicated by missing information on the number of dogs in the enclosure for many days during the study. We elicit expert knowledge for a prior on the number of dogs to account for the uncertainty in the missing data. Reversible jump Markov Chain Monte Carlo is used for model selection in the presence of missing covariates. We then use our model to examine possible changes to foxhound training enclosure policy and what effect those changes may have on fox survival.     

Optimal crossover designs when carryover effects are proportional to direct effects

Crossover designs are used for a variety of different applications. While these designs have a number of attractive features, they also induce a number of special problems and concerns. One of these is the possible presence of carryover effects. Even with the use of washout periods, which are for many applications widely accepted as an indispensable component, the effect of a treatment from a previous period may not be completely eliminated. A model that has recently received renewed attention in the literature is the model in which first-order carryover effects are assumed to be proportional to direct treatment effects. Under this model, assuming that the constant of proportionality is known, we identify optimal and efficient designs for the direct effects for different values of the constant of proportionality. We also consider the implication of these results for the case that the constant of proportionality is not known.     

A Smooth Test in Proportional Hazard Survival Models Using Local Partial Likelihood Fitting

Proportional hazard models for survival data, even though popular and numerically handy, suffer from the restrictive assumption that covariate effects are constant over survival time. A number of tests have been proposed to check this assumption. This paper contributes to this area by employing local estimates allowing to fit hazard models in which covariate effects are smoothly varying with time. A formal test is derived to check for proportional hazards against smooth hazards as alternative. The test proves to possess omnibus power in that it is powerful against arbitrary but smooth alternatives. Comparative simulations and two data examples accompany the presentation. Extensions are provided to multiple covariate settings, where the focus of interest is to decide which of the covariate effects vary with time.     

Survival analysis using inverse probability of treatment weighted methods based on the generalized propensity score

In survival analysis, treatment effects are commonly evaluated based on survival curves and hazard ratios as causal treatment effects. In observational studies, these estimates may be biased due to confounding factors. The inverse probability of treatment weighted (IPTW) method based on the propensity score is one of the approaches utilized to adjust for confounding factors between binary treatment groups. As a generalization of this methodology, we developed an exact formula for an IPTW log‐rank test based on the generalized propensity score for survival data. This makes it possible to compare the group differences of IPTW Kaplan–Meier estimators of survival curves using an IPTW log‐rank test for multi‐valued treatments. As causal treatment effects, the hazard ratio can be estimated using the IPTW approach. If the treatments correspond to ordered levels of a treatment, the proposed method can be easily extended to the analysis of treatment effect patterns with contrast statistics. In this paper, the proposed method is illustrated with data from the Kyushu Lipid Intervention Study (KLIS), which investigated the primary preventive effects of pravastatin on coronary heart disease (CHD). The results of the proposed method suggested that pravastatin treatment reduces the risk of CHD and that compliance to pravastatin treatment is important for the prevention of CHD. Copyright © 2009 John Wiley & Sons, Ltd.     

Job Durations With Worker- and Firm-Specific Effects: MCMC Estimation With Longitudinal Employer–Employee Data

We study job durations using a multivariate hazard model allowing for worker-specific and firm-specific unobserved determinants. The latter are captured by unobserved heterogeneity terms or random effects, one at the firm level and another at the worker level. This enables us to decompose the variation in job durations into the relative contribution of the worker and the firm. We also allow the unobserved terms to be correlated in a model that is primarily relevant for markets with small firms. For the empirical analysis, we use a Portuguese longitudinal matched employer–employee dataset. The model is estimated with a Bayesian Markov chain Monte Carlo (MCMC) estimation method. The results imply that unobserved firm characteristics explain almost 40% of the systematic variation in log job durations. In addition, we find a positive correlation between unobserved worker and firm characteristics.     

A spatial analysis of health and pharmaceutical firm survival

The presence of knowledge spillovers and shared human capital is at the heart of the Marhall–Arrow–Romer externalities hypothesis. Most of the earlier empirical contributions on knowledge externalities; however, considered data aggregated at a regional level so that conclusions are based on the arbitrary definition of jurisdictional spatial units: this is the essence of the so-called modifiable areal unit problem. A second limitation of these studies is constituted by the fact that, somewhat surprisingly, while concentrating on the effects of agglomeration on firm creation and growth, the literature has, conversely, largely ignored its effects on firm survival. The present paper aims at contributing to the existing literature by answering to some of the open methodological questions reconciling the literature of Cox proportional hazards model with that on point pattern and thus capturing the true nature of spatial information. We also present some empirical results based on Italian firm demography data collected and managed by the Italian National Institute of Statistics (ISTAT).     

Generalized Linear Mixed Models Based on Latent Markov Heterogeneity Structures

We describe a generalized linear mixed model in which all random effects may evolve over time. Random effects have a discrete support and follow a first‐order Markov chain. Constraints control the size of the parameter space and possibly yield blocks of time‐constant random effects. We illustrate with an application to the relationship between health education and depression in a panel of adolescents, where the random effects are highly dimensional and separately evolve over time.     

Nonparametric approach to rank global petroleum business opportunities

Crude oil continues to be one of the significant energy sources. Several countries do not have enough indigenous oil and gas resources. These countries resort to overseas business of Exploration and Production (E&P) of oil to secure a stable supply. Profitability, risk and growth guide overseas investment decisions. Selection of overseas investment opportunities are critical for a firm because of uncertainty in identifying and quantifying the attendant geological, commercial, social and political risks as well as return on investment. To secure overseas oil acreage, business entities intend to invest in overseas E&P destination having reasonable petroleum reserve, favorable contract terms (fiscal terms), well-developed infrastructure, sound legal system, minimum country risk (CR) (economic, social and political) and facilitate relative ease to do business in that country. The countries have varied mix of these parameters, and it leads to growing concern to screen and rank overseas investment opportunities. Methodologies to rank global opportunities should take into consideration the risk factors such as petroleum potential, infrastructure, geo-political scenario, contract terms, etc. We coin the term for the numerical rank as Globalization Index (GI), which is a function of the factors considered to affect the decision of a business entity in screening the global destinations for venturing in to E&P business of crude oil. This paper is an attempt to model these factors by invoking Alternating Conditional Expectation methodology to find GI.     

A proportional-hazards model for survival analysis and long-term survivors modeling: application to amyotrophic lateral sclerosis data

The majority of survival data are affected by explanatory variables. We develop a new regression model for survival data analysis. As an alternative to standard mixture models, another model is proposed to describe the eventual presence of a surviving fraction. The proposed models are based on the Marshall–Olkin extended generalized Gompertz distribution. A maximum-likelihood inference is presented in the presence of covariates and a censorship phenomenon. Explanatory variables are incorporated into the model through proportional-hazards to evaluate the effect of risk factors on overall survival under different assumptions. Parametric, semi-parametric, and non-parametric methods are applied to survival analysis of patients treated for amyotrophic lateral sclerosis. Interesting results about riluzole use and other treatment effects on patients'' survival have been obtained.     

Nonparametric Covariate-Adjusted Association Tests Based on the Generalized Kendall's Tau()

Identifying the risk factors for comorbidity is important in psychiatric research. Empirically, studies have shown that testing multiple, correlated traits simultaneously is more powerful than testing a single trait at a time in association analysis. Furthermore, for complex diseases, especially mental illnesses and behavioral disorders, the traits are often recorded in different scales such as dichotomous, ordinal and quantitative. In the absence of covariates, nonparametric association tests have been developed for multiple complex traits to study comorbidity. However, genetic studies generally contain measurements of some covariates that may affect the relationship between the risk factors of major interest (such as genes) and the outcomes. While it is relatively easy to adjust these covariates in a parametric model for quantitative traits, it is challenging for multiple complex traits with possibly different scales. In this article, we propose a nonparametric test for multiple complex traits that can adjust for covariate effects. The test aims to achieve an optimal scheme of adjustment by using a maximum statistic calculated from multiple adjusted test statistics. We derive the asymptotic null distribution of the maximum test statistic, and also propose a resampling approach, both of which can be used to assess the significance of our test. Simulations are conducted to compare the type I error and power of the nonparametric adjusted test to the unadjusted test and other existing adjusted tests. The empirical results suggest that our proposed test increases the power through adjustment for covariates when there exist environmental effects, and is more robust to model misspecifications than some existing parametric adjusted tests. We further demonstrate the advantage of our test by analyzing a data set on genetics of alcoholism.