Frailty models of manufacturing effects

The median service lifetime of respirator safety devices produced by different manufacturers is determined using frailty models to account for unobserved differences in manufacturing processes and raw materials. The gamma and positive stable frailty distributions are used to obtain survival distribution estimates when the baseline hazard is assumed to be Weibull. Frailty distributions are compared using laboratory test data of the failure times for 104 respirator cartridges produced by 10 different manufacturers tested with three different challenge agents. Likelihood ratio tests indicate that both frailty models provide a significant improvement over a Weibull model assuming independence. Results are compared to fixed effects approaches for analysis of this data.     

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.     

Validation of A Heteroscedastic Hazards Regression Model

A Cox-type regression model accommodating heteroscedasticity, with a power factor of the baseline cumulative hazard, is investigated for analyzing data with crossing hazards behavior. Since the approach of partial likelihood cannot eliminate the baseline hazard, an overidentified estimating equation (OEE) approach is introduced in the estimation procedure. Its by-product, a model checking statistic, is presented to test for the overall adequacy of the heteroscedastic model. Further, under the heteroscedastic model setting, we propose two statistics to test the proportional hazards assumption. Implementation of this model is illustrated in a data analysis of a cancer clinical trial.     

Estimation in proportional hazard and log-linear models

Proportional hazard models and models where the dependent variables follow a linear model are shown to be equivalent if and only if the hazard rate is the product of a non-negative periodic function and a Weibull factor. Estimates based on rank statistics are proposed for the parameters in the proportional hazard model.     

Progressively Type-II censored competing risks data for exponential distributions based on sequential order statistics

We consider the progressively Type-II censored competing risks model based on sequential order statistics. It is assumed that the latent failure times are independent and the failure of each unit influences the lifetime distributions of the latent failure times of surviving units. We provide explicit expressions for the likelihood function of the available data under the conditional proportional hazard rate (CPHR) and the power trend conditional proportional hazard rate (PTCPHR) models. Under CPHR and PTCPHR models and assumption that the baseline distributions of the latent failure times are exponential, classical and Bayesian estimates of the unknown parameters are provided. Monte Carlo simulations are then performed for illustrative purposes. Finally, two datasets are analyzed.     

On Goodness-of-Fit Tests for Aalen's Additive Risk Model

Abstract.  In this paper we propose goodness-of-fit tests for Aalen's additive risk model. They are based on test statistics the asymptotic distributions of which are determined under both the null and alternative hypotheses. The results are derived using martingale techniques for counting processes. An important feature of these tests is that they can be adjusted to particular alternatives. One of the alternatives we consider is Cox's multiplicative risk model. It is perhaps remarkable that such a test needs no estimate of the baseline hazard in the Cox model. We present simulation studies which give an impression of the performance of the proposed tests. In addition, the tests are applied to real data sets.     

The two-sample problem with multivariate censored data: a new rank test family

A new rank test family is proposed to test the equality of two multivariate failure times distributions with censored observations. The tests are very simple: they are based on a transformation of the multivariate rank vectors to a univariate rank score and the resulting statistics belong to the familiar class of the weighted logrank test statistics. The new procedure is also applicable to multivariate observations in general, such as repeated measures, some of which may be missing. To investigate the performance of the proposed tests, a simulation study was conducted with bivariate exponential models for various censoring rates. The size and power of these tests against Lehmann alternatives were compared to the size and power of two other tests (Wei and Lachin, 1984 and Wei and Knuiman, 1987). In all simulations the new procedures provide a relatively good power and an accurate control over the size of the test. A real example from the National Cooperative Gallstone Study is given     

Rank tests for independence based on partially right-censored pairs

Linear rank procedures are developed for testing independence with right-censored matched pairs. It is assumed that censoring Is Independent of the random variables under study. The test statistics are derived as score statistics (Hajek and Sidak, 1967) based on the probability of the generalised rank vectors (Prentice, 1978). Applications to survival data analysis are also discussed.     

A unified approach to rank tests for mixed models

The nonparametric version of the classical mixed model is considered and the common hypotheses of (parametric) main effects and interactions are reformulated in a nonparametric setup. To test these nonparametric hypotheses, the asymptotic distributions of quadratic forms of rank statistics are derived in a general framework which enables the derivation of the statistics for the nonparametric hypotheses of the fixed treatment effects and interactions in an arbitrary mixed model. The procedures given here are not restricted to semiparametric models or models with additive effects. Moreover, they are robust to outliers since only the ranks of the observations are needed. They are also applicable to pure ordinal data and since no continuity of the distribution functions is assumed, they can also be applied to data with ties. Some approximations for small sample sizes are suggested and analyzed in a simulation study. The application of the statistics and the interpretation of the results is demonstrated in several worked-out examples where some data sets given in the literature are re-analyzed.     

A score test for monotonic trend in hazard rates for grouped survival data in stratified populations

Models for monotone trends in hazard rates for grouped survival data in stratified populations are introduced, and simple closed form score statistics for testing the significance of these trends are presented. The test statistics for some of the models understudy are shown to be independent of the assumed form of the function which relates the hazard rates to the sets of monotone scores assigned to the time intervals. The procedure is applied to test monotone trends in the recovery rates of erythematous response among skin cancer patients and controls that have been irradiated with a ultraviolent challenge.     

Union-intersection rank tests for ordered alternatives in some simple linear models

The theory of locally most powerful rank tests and the union intersection principle are incorporated in the formulation of some distribution-free rank tests for ordered alternatives in some simple linear models (including the classical one-way layout as a special case). Some approximations for the null-hypothesis distributions of the test statistics ( for finite sample sizes) are considered, and the theory is supplemented by some numerical and simulation results.     

The role of the constant and linear terms in cointegration analysis of nonstationary variables

The autoregressive model for cointegrated variables is analyzed with respect to the role of the constant and linear terms. Various models for 1(1) variables defined by restrictions on the deterministic terms are discussed, and it is shown that statistical inference can be performed by reduced rank regression. The asymptotic distributions of the test statistics and estimators are found. A similar analysis is given for models for 1(2) variables with a constant term.     

The role of the constant and linear terms in cointegration analysis of nonstationary variables

The autoregressive model for cointegrated variables is analyzed with respect to the role of the constant and linear terms. Various models for 1(1) variables defined by restrictions on the deterministic terms are discussed, and it is shown that statistical inference can be performed by reduced rank regression. The asymptotic distributions of the test statistics and estimators are found. A similar analysis is given for models for 1(2) variables with a constant term.     

Wavelet Estimation of an Unknown Function Observed with Correlated Noise

Tests based on rank statistics are introduced to test for systematic changes in a sequence of independent observations. Proposed tests include a rank test analogous to the parametric likelihood ratio test and others analogous to parametric Bayes tests. The tests are usable with either one- or two-sided alternative hypotheses, and their asymptotic distributions are studied. The results of the general model are applied to two special cases, and their asymptotic distributions are also investigated. A Monte Carlo study verifies the applicability of asymptotic critical points in samples of moderate size, and other simulation studies compare power of the competing tests and their special-case versions. Finally, these tests are applied to a data set of traffic fatalities.     

A New Exponential GOF Test for Data Subject to Multiply Type II Censoring

This article presents a new goodness-of-fit (GOF) test statistic for multiply Type II censored Exponential data. The new test also applies to ordinary Type II censored samples and complete samples, since those cases are special cases of multiply Type II censoring. This test statistic is based on a ratio of linear functions of order statistics. Empirical power studies confirm that this ratio test compares favorably to currently available GOF tests for ordinary Type II censored data. Three data analysis examples are provided that demonstrate the usefulness of this new test statistic.     

Testing Goodness of Fit of Parametric AFT and PH Models with Residuals

In this article, we consider the problems of testing the goodness of fit of the parametric accelerated failure time model and the Cox proportional hazards model. We consider omnibus test statistics based on residuals. The statistical distributions of Kolmogorov, Cramer-von Mises–Smirnov, and Anderson–Darling statistics are all investigated by means of Monte Carlo simulations. Type-I, Type-II, and independent random censoring situations are all considered in this study. A Monte Carlo power study has also been carried out for these tests to distinguish between various baseline models, which reveals that the Anderson–Darling test performs better than the others.     

Heteroscedasticity in non-stationary time series,some Monte Carlo evidence

A frequent question raised by practitioners doing unit root tests is whether these tests are sensitive to the presence of heteroscedasticity. Theoretically this is not the case for a wide range of heteroscedastic models. However, for some limiting cases such as degenerate and integrated heteroscedastic processes it is not obvious whether this will have an effect. In this paper we report a Monte Carlo study analyzing the implications of various types of heteroscedasticity on three types of unit root tests: The usual Dickey-Fuller test, Phillips' (1987) semi-parametric test and finally a Dickey-Fuller type test using White's (1980) heteroscedasticity consistent standard errors. The sorts of heteroscedasticity we examine are the GARCH model of Bollerslev (1986) and the Exponential ARCH model of Nelson (1991). In particular, we call attention to situations where the conditional variances exhibit a high degree of persistence as is frequently observed for returns of financial time series, and the case where, in fact, the variance process for the first class of models becomes degenerate.     

WILCOXON-TYPE RANK-SUM PRECEDENCE TESTS

This paper introduces Wilcoxon‐type rank‐sum precedence tests for testing the hypothesis that two life‐time distribution functions are equal. They extend the precedence life‐test first proposed by Nelson in 1963. The paper proposes three Wilcoxon‐type rank‐sum precedence test statistics—the minimal, maximal and expected rank‐sum statistics—and derives their null distributions. Critical values are presented for some combinations of sample sizes, and the exact power function is derived under the Lehmann alternative. The paper examines the power properties of the Wilcoxon‐type rank‐sum precedence tests under a location‐shift alternative through Monte Carlo simulations, and it compares the power of the precedence test, the maximal precedence test and Wilcoxon rank‐sum test (based on complete samples). Two examples are presented for illustration.     

Nonparametric Tests for Stratum Effects in the Cox Model

Thispaper considers the stratified proportional hazards model witha focus on the assessment of stratum effects. The assessmentof such effects is often of interest, for example, in clinicaltrials. In this case, two relevant tests are the test of stratuminteraction with covariates and the test of stratum interactionwith baseline hazard functions. For the test of stratum interactionwith covariates, one can use the partial likelihood method (Kalbfleischand Prentice, 1980; Lin, 1994). For the test of stratum interactionwith baseline hazard functions, however, there seems to be noformal test available. We consider this problem and propose aclass of nonparametric tests. The asymptotic distributions ofthe tests are derived using the martingale theory. The proposedtests can also be used for survival comparisons which need tobe adjusted for covariate effects. The method is illustratedwith data from a lung cancer clinical trial.     

Exact distribution-free tests for equality of several linear models

Exact tests for the equality of several linear models are developed using permutation techniques. Two cases of the linear model, characterized by either stochastic or nonstochastic predictors, are considered: the linear regression model (LRM) and the general linear model (GLM). A general class of test statistics using the volume of simplexes as the basic unit of analysis is proposed for this problem. The resulting class of statistics is shown to be a natural generalization of the multi-response permutation procedure (MRPP) test statistics which have been shown to comprise many of the statistics used in both parametric and nonparametric analysis of the standard g—sample problem. In the LRM case, exact moments of all orders are derived for the permutation distribution of any test statistic in the general class. Moment-based approximation of significance levels is shown to be computationally feasible in the simple LRM.