A change-point estimator with the hazard ratio

The problem of estimation of parameters in hazard rate change models with a change-point is considered. A change-point estimator using the hazard ratio is suggested and compared with the previously developed change-point estimators. The proposed estimator is shown to be consistent. The performance of the proposed estimator is checked and compared with other change-point estimators via simulation.     

Estimating the case fatality rate using a constant cure-death hazard ratio

The case fatality rate is an important indicator of the severity of a disease, and unbiased and accurate estimates of it during an outbreak are important in the study of epidemic diseases, including severe acute respiratory syndrome (SARS). In this paper, estimation methods are developed using a constant cure-death hazard ratio. A semiparametric model is presented, in which the cure-death hazard ratio is a parameter of interest, and a profile likelihood-based technique is proposed for estimating the case fatality rate. An extensive simulation was carried out to investigate the performance of this technique for small and medium sample sizes, using both summary and individual data. The results show that the performance depends on the model validity but is not heavily dependent on the sample size. The method was applied to summary SARS data obtained from Hong Kong and Singapore.     

On a class exhibiting trend change in average failure rate

Abstract

Weak convergence and moment convergence issues are investigated for the New Better than Average Failure Rate (NBAFR) family (introduced by Loh (1984 Loh, W. Y. 1984. A new generalization of the class of NBU distributions. IEEE Transactions on Reliability R-33 :97–113[Crossref], [Web of Science ®] , [Google Scholar])). We explore the validity of these results in the context of a more general ageing class that we introduce. We prove some new properties of this class and derive its interrelationships with other non-monotonic ageing families. Reliability and moment bounds are obtained and an interesting characterization of exponentiality is proved. Special cases of our results lead to new theorems for the NBAFR class. Finally weak convergence and related issues are established for this class.     

Improving the biomarker diagnostic capacity via functional transformations

The use of the area under the receiver-operating characteristic, ROC, curve (AUC) as an index of diagnostic accuracy is overwhelming in fields such as biomedical science and machine learning. It seems that a larger AUC value has become synonymous with a better performance. The functional transformation of the marker values has been proposed in the specialized literature as a procedure for increasing the AUC and therefore the diagnostic accuracy. However, the classification process is based on some regions (classification subsets) which support the decision made; one subject is classified as positive if its marker is within this region and classified as negative otherwise. In this paper we study the capacity of improving the classification performance of univariate biomarkers via functional transformations and the impact of this transformation on the final classification regions based on a real-world dataset. Particularly, we consider the problem of determining the gender of a subject based on the Mode frequency of his/her voice. The shape of the cumulative distribution function of this characteristic in both the male and the female groups makes the resulting classification problem useful for illustrating the differences between having useful diagnostic rules and obtaining an optimal AUC value. Our point is that improving the AUC by means of a functional transformation can produce classification regions with no practical interpretability. We propose to improve the classification accuracy by making the selection of the classification subsets more flexible while preserving their interpretability. Besides, we provide different graphical approximations which allow us a better understanding of the classification problem.     

A class of exponential-type ratio estimators of a general parameter

This paper addresses the problem of estimating a general parameter using information on an auxiliary variable X. We have suggested a class of exponential-type ratio estimators for the parameter and its properties are studied. It is identified that the estimators due to Upadhyaya et al. [Journal of Statistical Theory and Practice (2011), 5(2), 285–302] and Yadav and Kadilar [Revista Columbiana de Estadistica, (2013), 36(1), 145–152] are members of the proposed estimator. We have also shown that the suggested estimator is more efficient than the estimators of Upadhyaya et al. (2011 Upadhyaya, L.N., Singh, H.P., Chatterjee, S., Yadav, R. (2011). Improved ratio and product exponential type estimators. J. Stat. Theo. Pract. 5 (2): 285–302.[Taylor &; Francis Online] , [Google Scholar]) and Yadav and Kadilar (2013 Yadav, S.K., Kadilar, C. (2013). Improved exponential type ratio estimator of population variance. Revis. Colum. de Estadist. 36(1): 145–152. [Google Scholar]). Numerical illustration is provided in support of the present study.     

Determination of the functional form of the relationship of covariates to the log hazard ratio in a Cox model

In this paper, we review available methods for determination of the functional form of the relation between a covariate and the log hazard ratio for a Cox model. We pay special attention to the detection of influential observations to the extent that they influence the estimated functional form of the relation between a covariate and the log hazard ratio. Our paper is motivated by a data set from a cohort study of lung cancer and silica exposure, where the nonlinear shape of the estimated log hazard ratio for silica exposure plotted against cumulative exposure and hereafter referred to as the exposure–response curve was greatly affected by whether or not two individuals with the highest exposures were included in the analysis. Formal influence diagnostics did not identify these two individuals but did identify the three highest exposed cases. Removal of these three cases resulted in a biologically plausible exposure–response curve.     

Estimation of the average causal effect via multiple propensity score stratification

Suppose we are interested in estimating the average causal effect (ACE) for the population mean from observational study. Because of simplicity and ease of interpretation, stratification by a propensity score (PS) is widely used to adjust for influence of confounding factors in estimation of the ACE. Appropriateness of the estimation by the PS stratification relies on correct specification of the PS. We propose an estimator based on stratification with multiple PS models by clustering techniques instead of model selection. If one of them correctly specifies, the proposed estimator removes bias and thus is more robust than the standard PS stratification.     

Non-existence of an optimum estimator in a class of ratio estimators

A number of estimators formulated in the field of the ratio method of estimation has been presented. A class of estimators encompassing these estimators is constructed. It is noted that an optimum estimator does not exist uniformly in this class. The “Optimum” so obtained reduces to the usual regression estimator.     

Design and analysis of a 3‐arm noninferiority trial with a prespecified margin for the hazard ratio

A 3‐arm trial design that includes an experimental treatment, an active reference treatment, and a placebo is useful for assessing the noninferiority of an experimental treatment. The inclusion of a placebo arm enables the assessment of assay sensitivity and internal validation, in addition to the testing of the noninferiority of the experimental treatment compared with the reference treatment. In 3‐arm noninferiority trials, various statistical test procedures have been considered to evaluate the following 3 hypotheses: (i) superiority of the experimental treatment over the placebo, (ii) superiority of the reference treatment over the placebo, and (iii) noninferiority of the experimental treatment compared with the reference treatment. However, hypothesis (ii) can be insufficient and may not accurately assess the assay sensitivity for the noninferiority of the experimental treatment compared with the reference treatment. Thus, demonstrating that the superiority of the reference treatment over the placebo is greater than the noninferiority margin (the nonsuperiority of the reference treatment compared with the placebo) can be necessary. Here, we propose log‐rank statistical procedures for evaluating data obtained from 3‐arm noninferiority trials to assess assay sensitivity with a prespecified margin Δ. In addition, we derive the approximate sample size and optimal allocation required to minimize the total sample size and that of the placebo treatment sample size, hierarchically.     

On the goodness-of-fit test based on the ratio of cumulative hazard functions with the Type II censored data

For comparing two cumulative hazard functions, we consider an extension of the Kullback–Leibler information to the cumulative hazard function, which is concerning the ratio of cumulative hazard functions. Then we consider its estimate as a goodness-of-fit test with the Type II censored data. For an exponential null distribution, the proposed test statistic is shown to outperform other test statistics based on the empirical distribution function in the heavy censoring case against the increasing hazard alternatives.     

Regression analysis of competing risks data via semi-parametric additive hazard model

When the subjects in a study possess different demographic and disease characteristics and are exposed to more than one types of failure, a practical problem is to assess the covariate effects on each type of failure as well as on all-cause failure. The most widely used method is to employ the Cox models on each cause-specific hazard and the all-cause hazard. It has been pointed out that this method causes the problem of internal inconsistency. To solve such a problem, the additive hazard models have been advocated. In this paper, we model each cause-specific hazard with the additive hazard model that includes both constant and time-varying covariate effects. We illustrate that the covariate effect on all-cause failure can be estimated by the sum of the effects on all competing risks. Using data from a longitudinal study on breast cancer patients, we show that the proposed method gives simple interpretation of the final results, when the primary covariate effect is constant in the additive manner on each cause-specific hazard. Based on the given additive models on the cause-specific hazards, we derive the inferences for the adjusted survival and cumulative incidence functions.     

Bayesian point null hypothesis testing via the posterior likelihood ratio

This paper gives an exposition of the use of the posterior likelihood ratio for testing point null hypotheses in a fully Bayesian framework. Connections between the frequentist P-value and the posterior distribution of the likelihood ratio are used to interpret and calibrate P-values in a Bayesian context, and examples are given to show the use of simple posterior simulation methods to provide Bayesian tests of common hypotheses.     

Estimation and simulation of conditional hazard function in the quasi-associated framework when the observations are linked via a functional single-index structure

We consider the estimation of the conditional hazard function of a scalar response variable Y given a Hilbertian random variable X when the observations are linked via a single-index structure in the quasi-associated framework. We establish the pointwise almost complete convergence and the uniform almost complete convergence (with the rate) of the estimate of this model. A simulation is given to illustrate the good behavior in the practice of our methodology.     

Bayesian estimation of the Cox model under different hazard rate shape assumptions via slice sampling

In this paper, we provide a full Bayesian analysis for Cox's proportional hazards model under different hazard rate shape assumptions. To this end, we select the modified Weibull distribution family to model failure rates. A novel Markov chain Monte Carlo method allows one to tackle both exact and right-censored failure time data. Both simulated and real data are used to illustrate the methods.     

Generalizing the standardized hazard ratio to multivariate proportional hazards regression,with an application to clinical~genomic studies

The standardized hazard ratio for univariate proportional hazards regression is generalized as a scalar to multivariate proportional hazards regression. Estimators of the standardized log hazard ratio are developed, with corrections for bias and for regression to the mean in high-dimensional analyses. Tests of point and interval null hypotheses and confidence intervals are constructed. Cohort sampling study designs, commonly used in prospective–retrospective clinical genomic studies, are accommodated.     

Comparison of the restricted mean survival time with the hazard ratio in superiority trials with a time‐to‐event end point

With the emergence of novel therapies exhibiting distinct mechanisms of action compared to traditional treatments, departure from the proportional hazard (PH) assumption in clinical trials with a time‐to‐event end point is increasingly common. In these situations, the hazard ratio may not be a valid statistical measurement of treatment effect, and the log‐rank test may no longer be the most powerful statistical test. The restricted mean survival time (RMST) is an alternative robust and clinically interpretable summary measure that does not rely on the PH assumption. We conduct extensive simulations to evaluate the performance and operating characteristics of the RMST‐based inference and against the hazard ratio–based inference, under various scenarios and design parameter setups. The log‐rank test is generally a powerful test when there is evident separation favoring 1 treatment arm at most of the time points across the Kaplan‐Meier survival curves, but the performance of the RMST test is similar. Under non‐PH scenarios where late separation of survival curves is observed, the RMST‐based test has better performance than the log‐rank test when the truncation time is reasonably close to the tail of the observed curves. Furthermore, when flat survival tail (or low event rate) in the experimental arm is expected, selecting the minimum of the maximum observed event time as the truncation timepoint for the RMST is not recommended. In addition, we recommend the inclusion of analysis based on the RMST curve over the truncation time in clinical settings where there is suspicion of substantial departure from the PH assumption.     

Empirical likelihood ratio confidence intervals in terms of cumulative hazard function for left-truncated and right-censored data

Abstract

Based on the approach of Pan and Zhou, we demonstrate that empirical likelihood ratios in terms of cumulative hazard function for left-truncated and right-censored (LTRC) data can be used to form confidence intervals for the parameters that are linear functionals of the cumulative hazard function. Simulation studies indicate that the empirical likelihood ratio based confidence intervals work well in finite samples.