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.     

Hypothesis Testing of Hazard Ratio Parameters in Marginal Models for Multivariate Failure Time Data

Marginal hazard models for multivariate failure time data have been studied extensively in recent literature. However, standard hypothesis test statistics based on the likelihood method are not exactly appropriate for this kind of model. In this paper, extensions of the three commonly used likelihood hypothesis test statistics are discussed. Generalized Wald, generalized score and generalized likelihood ratio tests for hazard ratio parameters in a marginal hazard model for multivariate failure time data are proposed and their asymptotic distributions examined. The finite sample properties of these statistics are studied through simulations. The proposed method is applied to data from Busselton Population Health Surveys.     

Alternative Methods of Estimating the Likelihood Ratio in Classification of Multivariate Normal Observations

An observation is to be classified into one of two multivariate normal distributions with equal covariance matrices. When the parameters are unknown, four methods of estimating the likelihood ratio, that is, the plug-in method, the test procedure, the Bayesian approach, and the best invariant estimate method, are reviewed. The assumptions, interpretations, and consequences of the four approaches are given. It is shown that the last three methods yield the same classification procedure.     

Testing the Homogeneity of Two Survival Functions Against a Mixture Alternative Based on Censored Data

When the survival distribution in a treatment group is a mixture of two distributions of the same family, traditional parametric methods that ignore the existence of mixture components or the nonparametric methods may not be very powerful. We develop a modified likelihood ratio test (MLRT) for testing homogeneity in a two sample problem with censored data and compare the actual type I error and power of the MLRT with that nonparametric log-rank test and parametric test through Monte-Carlo simulations. The proposed test is also applied to analyze data from a clinical trial on early breast cancer.     

Comparison Between Two Partial Likelihood Approaches for the Competing Risks Model with Missing Cause of Failure

In many clinical studies where time to failure is of primary interest, patients may fail or die from one of many causes where failure time can be right censored. In some circumstances, it might also be the case that patients are known to die but the cause of death information is not available for some patients. Under the assumption that cause of death is missing at random, we compare the Goetghebeur and Ryan (1995, Biometrika, 82, 821–833) partial likelihood approach with the Dewanji (1992, Biometrika, 79, 855–857)partial likelihood approach. We show that the estimator for the regression coefficients based on the Dewanji partial likelihood is not only consistent and asymptotically normal, but also semiparametric efficient. While the Goetghebeur and Ryan estimator is more robust than the Dewanji partial likelihood estimator against misspecification of proportional baseline hazards, the Dewanji partial likelihood estimator allows the probability of missing cause of failure to depend on covariate information without the need to model the missingness mechanism. Tests for proportional baseline hazards are also suggested and a robust variance estimator is derived.     

Generalized Supremum Tests for the Equality of Cause Specific Hazard Rates

In this paper we propose two new classes of asymptotically distribution-free Renyi-type tests for testing the equality of two risks in a competing risk model with possible censoring. This work extends the work of Aly, Kochar and McKeague [1994, Journal of American Statistical Association, 89, 994–999] and many of the existing tests for this problem belong to these newly proposed classes. The asymptotic properties of the proposed tests are investigated. Simulation studies are done to compare the performance with existing tests. A competing risks data set is analyzed to demonstrate the usefulness of the procedure.     

A Class of Non-parametric Tests in the Competing Risks Model for Comparing Two Samples

We consider a model when a process involving the production of elements is under inspection. The elements have possible failures due to competing risks. We assume the availability of a data set of failure times, D₁, obtained when the process is under control. Our main goal is to test if the failure rates in D₁ are equal to or less than the failure rates in another data set D₂, against undesirable neighbouring alternatives. A class of tests based on a two-dimensional vector statistic is obtained. Linear test statistics with weight functions giving optimal local asymptotic power are derived. Martingale techniques are used. An example illustrates the derivation of reasonable tests     

A class of tests of proportional hazards assumption for left-truncated and right-censored data

In this paper, we proposed a class of tests of proportional hazards assumption for left-truncated and right-censored data based on a pair of estimators of the hazard ratio constant. Using counting process and martingale theory, the asymptotically normal distribution of the test statistic is derived and a family of consistent estimators of variance are also provided. Extensive simulation studies were conducted to evaluate the performance of the proposed test statistics under finite sample situations. Two real data sets are analyzed to illustrate our method.     

Testing for Covariate Effect in the Cox Proportional Hazards Regression Model

This article presents methods for testing covariate effect in the Cox proportional hazards model based on Kullback–Leibler divergence and Renyi's information measure. Renyi's measure is referred to as the information divergence of order γ (γ ≠ 1) between two distributions. In the limiting case γ → 1, Renyi's measure becomes Kullback–Leibler divergence. In our case, the distributions correspond to the baseline and one possibly due to a covariate effect. Our proposed statistics are simple transformations of the parameter vector in the Cox proportional hazards model, and are compared with the Wald, likelihood ratio and score tests that are widely used in practice. Finally, the methods are illustrated using two real-life data sets.     

Testing for Imperfect Debugging in Software Reliability

This paper continues the study of the software reliability model of Fakhre-Zakeri & Slud (1995), an "exponential order statistic model" in the sense of Miller (1986) with general mixing distribution, imperfect debugging and large-sample asymptotics reflecting increase of the initial number of bugs with software size. The parameters of the model are θ (proportional to the initial number of bugs in the software), G (·, μ) (the mixing df, with finite dimensional unknown parameter μ, for the rates λ _i with which the bugs in the software cause observable system failures), and p (the probability with which a detected bug is instantaneously replaced with another bug instead of being removed). Maximum likelihood estimation theory for (θ, p , μ) is applied to construct a likelihood-based score test for large sample data of the hypothesis of "perfect debugging" ( p = 0) vs "imperfect" ( p > 0) within the models studied. There are important models (including the Jelinski–Moranda) under which the score statistics with 1/√ n normalization are asymptotically degenerate. These statistics, illustrated on a software reliability data of Musa (1980), can serve nevertheless as important diagnostics for inadequacy of simple models     

Exact Likelihood Ratio Testing for Homogeneity of the Exponential Distribution

The aim of this article is to discuss homogeneity testing of the exponential distribution. We introduce the exact likelihood ratio test of homogeneity in the subpopulation model, ELR, and the exact likelihood ratio test of homogeneity against the two-components subpopulation alternative, ELR2. The ELR test is asymptotically optimal in the Bahadur sense when the alternative consists of sampling from a fixed number of components. Thus, in some setups the ELR is superior to frequently used tests for exponential homogeneity which are based on the EM algorithm (like the MLRT, ADDS, and D-tests). One important example of superiority of ELR and ELR2 tests is the case of lower contamination. We demonstrate this fact by both theoretical comparisons and simulations.     

Estimating the Basic Reproductive Number in the General Epidemic Model with an Unknown Initial Number of Susceptible Individuals

Abstract. In any epidemic, there may exist an unidentified subpopulation which might be naturally immune or isolated and who will not be involved in the transmission of the disease. Estimation of key parameters, for example, the basic reproductive number, without accounting for this possibility would underestimate the severity of the epidemics. Here, we propose a procedure to estimate the basic reproductive number ( R ₀ ) in an epidemic model with an unknown initial number of susceptibles. The infection process is usually not completely observed, but is reconstructed by a kernel‐smoothing method under a counting process framework. Simulation is used to evaluate the performance of the estimators for major epidemics. We illustrate the procedure using the Abakaliki smallpox data.     

Testing for the Equality of Two Autoregressive Functions Using Quasi-Residuals

This article discusses the problem of testing the equality of two nonparametric autoregressive functions against one-sided alternatives. The heteroscedastic errors and stationary densities of the two independent strong mixing strictly stationary time series can be possibly different. The article adapts the idea of using sum of quasi-residuals to construct the test and derives its asymptotic null distributions. The article also shows that the test is consistent for general alternatives and obtains its limiting distributions under a sequence of local alternatives. Then a Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. We also compare the test to an existing lag matched test theoretically and by Monte Carlo experiments.     

Probabilistic Approach to Solution of the Neumann Problem for Some Nonlinear Equation

In this article we will consider the Neumann boundary-value problem for the nonlinear Helmholtz equation ? Δ?u + a?u = gexp?(u) + f₀. We will assume that there exists the solution to our problem and this permits us to construct an unbiased estimator on the trajectories of certain branching processes. To do so, we apply Green’s formula and an elliptic mean value theorem. This allows us to derive a special integral equation that gives the value of the function u(x) at the point x, with its integral over the domain D and on boundary of the domain ?D = G. The solution of the problem in the form of a mathematical expectation of some random variable is also obtained. In accordance with the probabilistic representation, a branching process is constructed and an unbiased estimator of the solution of the problem is built on its trajectories. The derived unbiased estimator has finite variance. The proposed branching process has a finite average number of branches, and easily simulated. We provide numerical results based on numerical experiments carried out with these algorithms.     

Estimation of Ratio of Two Finite-Population Means in the Presence of Non Response

It is well known that a ranked set sample under perfect ranking provides more information than an i.i.d. sample of the same size. Then it may be interesting to study how much information is lost due to imperfect ranking. In this article, we consider some ranking mechanisms and study the loss of the Fisher information according to the degree of imperfect ranking. Then we continue to discuss the optimal combination of the sample size and number of strata in terms of maximizing the Fisher information for the bivariate normal and exponential distributions.     

Testing for the Equality of Two Nonparametric Regression Curves with Long Memory Errors

This article discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives for uniform design on [0,1] with long memory moving average errors. The standard deviations and the long memory parameters are possibly different for the two errors. The article adapts the partial sum process idea used in the independent observations settings to construct the tests and derives their asymptotic null distributions. The article also shows that these tests are consistent for general alternatives and obtains their limiting distributions under a sequence of local alternatives. Since the limiting null distributions of these tests are unknown, we first conducted a Monte Carlo simulation study to obtain a few selected critical values of the proposed tests. Then based on these critical values, another Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. The article also contains a simulation study that assesses the effect of estimating the nonparametric regression function on an estimate of the long memory parameter of the errors. It is observed that the estimate based on direct observations is generally preferable over the one based on the estimated nonparametric residuals.     

Application of the Generalized Likelihood Ratio Test for Detecting Changes in the Mean of Multivariate GARCH Processes

We derive several multivariate control charts to monitor the mean vector of multi-variate GARCH processes under the presence of changes, by means of maximizing the generalized likelihood ratio. This presentation is rounded up by a comparative performance study based on extensive Monte Carlo simulations. An empirical illustration shows how the obtained results can be applied to real data.     

Non-parametric Estimation for the Location of a Change-point in an Otherwise Smooth Hazard Function under Random Censoring

A non-parametric wavelet based estimator is proposed for the location of a change-point in an otherwise smooth hazard function under non-informative random right censoring. The proposed estimator is based on wavelet coefficients differences via an appropriate parametrization of the time-frequency plane. The study of the estimator is facilitated by the strong representation theorem for the Kaplan–Meier estimator established by Lo and Singh (1986). The performance of the estimator is checked via simulations and two real examples conclude the paper.     

A Note on the Non-equivalence of the Neyman-Pearson and Generalized Likelihood Ratio Tests for Testing a Simple Null versus a Simple Alternative Hypothesis

Despite the great increase in the use of color in statistical graphics, we know very little about how color affects people's perception of the quantitative information on graphical displays. Perceptual psychologists have already demonstrated that color can cause optical illusions of various kinds. We ran a simple experiment to see if this can happen with a statistical map and found that an illusion did occur.     

On bates of convergence in the central limit theorem for parameter estimation in general autoregressive model

This paper explores the rate at which the estimates of the unknown parameters in an autoregressive process converge in distribution to the normal variate.