Restricted tests for testing independence of time to failure and cause of failure in a competing-risks model

We consider the competing-risks problem without making any assumption concerning the independence of the risks. Maximum-likelihood estimates of the cause-specific hazard rates are obtained under the condition that their ratio is monotone. We also consider the likelihood-ratio test for testing the proportionality of two cause-specific hazard rates against the alternative that the ratio of these two hazard rates is monotonic. This testing problem is equivalent to testing independence against likelihood-ratio dependence of the time to failure and the cause of failure in the competing-risks setup. We allow for random censoring on the right. The asymptotic null distribution of the test statistic is obtained and is found to be of the chi-bar-square type. The problem is extended to the case of more than two risks. A numerical example is given to illustrate the procedure.     

Two tests for monotone failure rate with incomplete data

Two statistics are proposed for testing for the exponential distribution against monotone failure rate alternatives when ran-domly right censored data are available. One of them is a general-ization of the Billmann, Antle and Bain test based on the MLE of the shape parameter of the Weibull distribution. The second has the advantage of being given in closed form. For this test the asymptotic null distribution is given. Consistency of the two tests is proved starting from an expected value inequality characterizing monotone failure rate.     

The Kumaraswamy modified Weibull distribution: theory and applications

A five-parameter extension of the Weibull distribution capable of modelling a bathtub-shaped hazard rate function is introduced and studied. The beauty and importance of the new distribution lies in its ability to model both monotone and non-monotone failure rates that are quite common in lifetime problems and reliability. The proposed distribution has a number of well-known lifetime distributions as special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull (MW) distributions, among others. We obtain quantile and generating functions, mean deviations, Bonferroni and Lorenz curves and reliability. We provide explicit expressions for the density function of the order statistics and their moments. For the first time, we define the log-Kumaraswamy MW regression model to analyse censored data. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is determined. Two applications illustrate the potentiality of the proposed distribution.     

A log-location-scale lifetime distribution with censored data: Regression modeling,residuals, and global influence

In survival analysis applications, the presence of failure rate functions with non monotone shapes is common. Therefore, models that can accommodate such different shapes are needed. In this article, we present a location regression model based on the complementary exponentiated exponential geometric distribution as an alternative to the usual bathtub, increasing, and decreasing failure rates in lifetime data. Assuming censored data, we consider the maximum likelihood inference for analysis, graphical verification for residuals, and test statistics for influential points.     

Exponentiated Weibull regression for time-to-event data

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.     

Tests for Comparing Mark-Specific Hazards and Cumulative Incidence Functions

It is of interest in some applications to determine whether there is a relationship between a hazard rate function (or a cumulative incidence function) and a mark variable which is only observed at uncensored failure times. We develop nonparametric tests for this problem when the mark variable is continuous. Tests are developed for the null hypothesis that the mark-specific hazard rate is independent of the mark versus ordered and two-sided alternatives expressed in terms of mark-specific hazard functions and mark-specific cumulative incidence functions. The test statistics are based on functionals of a bivariate test process equal to a weighted average of differences between a Nelson-Aalen-type estimator of the mark-specific cumulative hazard function and a nonparametric estimator of this function under the null hypothesis. The weight function in the test process can be chosen so that the test statistics are asymptotically distribution-free. Asymptotically correct critical values are obtained through a simple simulation procedure. The testing procedures are shown to perform well in numerical studies, and are illustrated with an AIDS clinical trial example. Specifically, the tests are used to assess if the instantaneous or absolute risk of treatment failure depends on the amount of accumulation of drug resistance mutations in a subject's HIV virus. This assessment helps guide development of anti-HIV therapies that surmount the problem of drug resistance.     

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.     

The exponentiated generalized gamma distribution with application to lifetime data

A four-parameter extension of the generalized gamma distribution capable of modelling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma and generalized Rayleigh, among others. We derive two infinite sum representations for its moments. We calculate the density of the order statistics and two expansions for their moments. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is obtained. Finally, a real data set from the medical area is analysed.     

Comparison of goodness of fit tests for the cox proportional hazards model

The proportional hazards regression model of Cox(1972) is widely used in analyzing survival data. We examine several goodness of fit tests for checking the proportionality of hazards in the Cox model with two-sample censored data, and compare the performance of these tests by a simulation study. The strengths and weaknesses of the tests are pointed out. The effects of the extent of random censoring on the size and power are also examined. Results of a simulation study demonstrate that Gill and Schumacher's test is most powerful against a broad range of monotone departures from the proportional hazards assumption, but it may not perform as well fail for alternatives of nonmonotone hazard ratio. For the latter kind of alternatives, Andersen's test may detect patterns of irregular changes in hazards.     

Partitioning chi-squape in contingency tables: A teaching approach

A representation of sums and differences of the form 2n log n, the lnn function, is introduced to express likelihood-ratio chi-square test statistics in contingency table analysis. This is a concise explicit form to display when partitioning chi-square statistics in accordance with hierarchical models. The lnn representation gives students insights into the construction of test statistics, and assists in relating identical forms under differing model sets. Hierarchies are presented for independence and equi-probability in two-way tables, for symmetry in correlated square tables, for independence-and-homogeneity of two-way responses across levels of a factor, and for mutual independence in three-way tables, along with relevant partitions of chi-square.     

Monotonicity of Bayes sequential tests for multidimensional and censored observations

For statistical models admitting a sufficient, transitive sequence of one-dimensional statistics, Brown, Cohen and Strawderman proved in 1979 that, under simple conditions verified in many examples, all Bayes sequential tests are monotone. We extend the definition of monotone test to higher dimension in a suitable way, and show that the same result holds for multidimensional statistics. Our work was partly motivated by the observation that deterministically censored observations from a statistical model admitting a sufficient, transitive sequence of m-dimensional statistics can be viewed as observations from another statistical model, still admitting a sufficient, transitive sequence of statistics, but with values in a higher-dimensional space. A typical application to a reliability problem is also discussed.     

Weighted Logrank Permutation Tests for Randomly Right Censored Life Science Data

In biomedical research, weighted logrank tests are frequently applied to compare two samples of randomly right censored survival times. We address the question how to combine a number of weighted logrank statistics to achieve good power of the corresponding survival test for a whole linear space or cone of alternatives, which are given by hazard rates. This leads to a new class of semiparametric projection tests that are motivated by likelihood ratio tests for an asymptotic model. We show that these tests can be carried out as permutation tests and discuss their asymptotic properties. A simulation study together with the analysis of a classical data set illustrates the advantages.     

Transformation of the bathtub failure rate data in reliability for using Weibull-model analysis

All statistical methods involve basic model assumptions, which if violated render results of the analysis dubious. A solution to such a contingency is to seek an appropriate model or to modify the customary model by introducing additional parameters. Both of these approaches are in general cumbersome and demand uncommon expertise. An alternative is to transform the data to achieve compatibility with a well understood and convenient customary model with readily available software. The well-known example is the Box–Cox data transformation developed in order to make the normal theory linear model usable even when the assumptions of normality and homoscedasticity are not met.In reliability analysis the model appropriateness is determined by the nature of the hazard function. The well-known Weibull distribution is the most commonly employed model for this purpose. However, this model, which allows only a small spectrum of monotone hazard rates, is especially inappropriate if the data indicate bathtub-shaped hazard rates.In this paper, a new model based on the use of data transformation is presented for modeling bathtub-shaped hazard rates. Parameter estimation methods are studied for this new (transformation) approach. Examples and results of comparisons between the new model and other bathtub-shaped models are shown to illustrate the applicability of this new model.     

Bayesian analysis for monotone hazard ratio

We propose a Bayesian approach for estimating the hazard functions under the constraint of a monotone hazard ratio. We construct a model for the monotone hazard ratio utilizing the Cox’s proportional hazards model with a monotone time-dependent coefficient. To reduce computational complexity, we use a signed gamma process prior for the time-dependent coefficient and the Bayesian bootstrap prior for the baseline hazard function. We develope an efficient MCMC algorithm and illustrate the proposed method on simulated and real data sets.     

Some asymptotic results for hypothesis testing in multivariate failure time data

This paper presents a class of generalized Wald, generalized score and generalized likelihood ratio statistics for hypothesis testing and model selection for multivariate failure time data. These statistics are based on a marginal hazard model with a common baseline hazard function. The large sample distributions of these statistics are examined. It is shown that the proposed test statistics follow asymptotically a weighted sum of independent χ₁² distributions.     

Adaptive Test for Testing the Difference in Survival Distributions

An adaptive test is proposed for the problem of testing the difference in survival distributions when the shape of the hazard ratio is unknown, hence the efficient test is unknown. The proposed adaptive test selects a test statistic from a finite set of the weighted logrank statistics T on the basis of the estimates of the efficiencies of the tests in T for given data. The efficiency estimator uses the length of the test based nonparametric confidence interval for the shift in a time transformed shift model. The suggested adaptive test is shown to be asymptotically efficient among the tests in T under the time transformed shift model and conditions commonly used in survival analysis. Simulations demonstrate that the adaptive test enjoys good small sample properties and in most situations is more powerful than the test using the maximum of the tests in T.     

Goodness‐of‐Fit Test for Monotone Functions

Abstract. In this article, we develop a test for the null hypothesis that a real‐valued function belongs to a given parametric set against the non‐parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right‐censoring model with monotone hazard rate. The criterion for testing is an ‐distance between a Grenander‐type non‐parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

Small sample power of multivariate nonparametric tests for monotone trend

Bhattacharyya and Kioiz (1966) propose two multivariate nonparametric tests for monotone trend, one involving coordinate-wise Mann statistics and the other, coordinate-wise Spearman statistics. Dietz and Killeen (1981) propose a different test statistic based on coordinate-wise Mann statistics. The Pitman asymptotic relative efficiency of all three tests with respect to a normal theory competitor equals the cube root of the efficiency of a multivariate signed rank test with respect to Hotelling's T2. In this article, the small sample power of the nonparametric tests, the normal theory test, and a Bonferroni approach involving coordinate-wise univariate Mann or Spearman tests is examined in a simulation study. The Mann statistic of Dietz and Killeen and the Spearman statistic of Bhattacharyya and Klotz are found to perform well under both null and alternative hypotheses