Weighted Logrank Tests Research Articles

Exact and Asymptotic Weighted Logrank Tests for Interval Censored Data: TheintervalRPackage

For right-censored data perhaps the most commonly used tests are weighted logrank tests, such as the logrank and Wilcoxon-type tests. In this paper we review several generalizations of those weighted logrank tests to interval-censored data and present an R package, interval, to implement many of them. The interval package depends on the perm package, also presented here, which performs exact and asymptotic linear permutation tests. The perm package performs many of the tests included in the already available coin package, and provides an independent validation of coin. We review analysis methods for interval-censored data, and we describe and show how to use the interval and perm packages.

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).

Testing the equality of two survival functions with right truncated data

To compare the survival functions based on right-truncated data, Lagakos et al. proposed a weighted logrank test based on a reverse time scale. This is in contrast to Bilker and Wang, who suggested a semi-parametric version of the Mann-Whitney test by assuming that the distribution of truncation times is known or can be estimated parametrically. The approach of Lagakos et al. is simple and elegant, but the weight function in their method depends on the underlying cumulative hazard functions even under proportional hazards models. On the other hand, a semi-parametric test may have better efficiency, but it may be sensitive to misspecification of the distribution of truncation times. Therefore, this paper proposes a non-parametric test statistic based on the integrated weighted difference between two estimated survival functions in forward time. The comparative results from a simulation study are presented and the implementation of these methods to a real data set is demonstrated.

A partial score test for difference among heterogeneous populations

In event time data analysis, comparisons between distributions are made by the logrank test. When the data appear to contain crossing hazards phenomena, nonparametric weighted logrank statistics are usually suggested to accommodate different-weighted functions to increase the power. However, the gain in power by imposing different weights has its limits since differences before and after the crossing point may balance each other out. In contrast to the weighted logrank tests, we propose a score-type statistic based on the semiparametric-, heteroscedastic-hazards regression model of Hsieh [2001. On heteroscedastic hazards regression models: theory and application. J. Roy. Statist. Soc. Ser. B 63, 63–79.], by which the nonproportionality is explicitly modeled. Our score test is based on estimating functions derived from partial likelihood under the heteroscedastic model considered herein. Simulation results show the benefit of modeling the heteroscedasticity and power of the proposed test to two classes of weighted logrank tests (including Fleming–Harrington's test and Moreau's locally most powerful test), a Renyi-type test, and the Breslow's test for acceleration. We also demonstrate the application of this test by analyzing actual data in clinical trials.

A note on testing in Aalen's additive hazards regression models

Aalen's additive hazards regression model is a useful alternative to the proportional hazards model for censored data regression. When used to compare treatments this approach leads to weighted comparisons of the crude estimate of the hazard rate of each group as compared to a baseline group. This is contrasted to the weighted log rank test from the proportional hazards model which compares each treatment's rate to the pooled rate. We show in this brief note that Aalen's suggestion for weights in this test leads to inconsistent tests in the sense that the test statistic depends on which group we pick for a baseline group. We show that 'consistent' tests are obtained by using common weight functions for all comparisons and we make some suggestions.

Multiple testing procedures based on weighted Kaplan–Meier statistics for right-censored survival data

In clinical trials or drug development studies, researchers are often interested in identifying which treatments or dosages are more effective than the standard one. Recently, several multiple testing procedures based on weighted logrank tests have been proposed to compare several treatments with a control in a one-way layout where survival data are subject to random right-censorship. However, weighted logrank tests are based on ranks, and these tests might not be sensitive to the magnitude of the difference in survival times against a specific alternative. Therefore, it is desirable to develop a more robust and powerful multiple testing procedure. This paper proposes multiple testing procedures based on two-sample weighted Kaplan-Meier statistics, each comparing an individual treatment with the control, to determine which treatments are more effective than the control. The comparative results from a simulation study are presented and the implementation of these methods to the prostate cancer clinical trial and the renal carcinoma tumour study are presented.

SOME VERSATILE TESTS BASED ON THE SIMULTANEOUS USE OF WEIGHTED LOGRANK AND WEIGHTED KAPLAN-MEIER STATISTICS

It is well known that the logrank test is locally most powerful under proportional hazards model for right censored data when two groups have same censoring distributions. For a given data set, however, one seldom knows what the exact alternative is. Therefore, Lee [1] constructed versatile tests by combining four weighted logrank (WLR) tests with various weight functions detecting proportional, early, late and middle hazards differences, respectively. However, Pepe and Fleming [2] had pointed out that WLR tests are based on ranks, and these tests might not be sensitive to the magnitude of the difference in survival times against a specific alternative. Hence, they constructed weighted Kaplan-Meier (WKM) tests that are more sensitive than the logrank test under various alternatives. Therefore, another type of versatile tests based on the simultaneous use of WLR and WKM tests is developed in this paper and the performances of the proposed tests and Lee's tests are compared by simulation. None of the tests investigated in this paper is uniformly better than the others. Lee's maximum test is more robust for detecting various alternatives; however, for computation simplicity, the linear combination of WLR and WKM tests is recommended to apply in practice and its overall performance is better than Lee's linear combination test.

Weighted Kaplan-Meier Tests for Umbrella Alternatives

The problem of testing for umbrella alternatives in a one-way layout with right-censored survival data is considered. Testing procedures based on the two-sample weighted Kaplan-Meier statistics suggested by Pepe and Fleming (1989, Biometrics, 45, 497–507; 1991, J. Roy. Statist. Soc. Ser. B, 53, 341–352) are suggested for both cases when the peak of the umbrella is known or unknown. The asymptotic relative efficiency of the weighted Kaplan-Meier test and the weighted logrank test proposed by Chen and Wolfe (2000, Statist. Sinica, 10, 595–612) is computed for the umbrella peak-known setting where the piecewise exponential survival distributions have the proportional or crossing hazards, or the related hazards differ at early or late times. Moreover, the results of a Monte Carlo study are presented to investigate the level and power performances of the umbrella tests. Finally, application of the proposed procedures to an appropriated data set is illustrated.

Weighted Logrank Testing Procedures for the Simple Tree Alternatives

Weighted logrank testing procedures for comparing r treatments with a control when some of the data are randomly censored are discussed. Four kinds of test statistics for the simple tree alternatives are considered. The weighted logrank statistics based on pairwise ranking scheme is proposed and the covariances of the test statistics are explicitly obtained. This class of test statistics can be viewed as the general statistics of constructing the test procedures for various order restricted alternatives by modifying weights. Four kinds of weighted logrank tests are illustrated with an example. Simulation studies are performed to compare the sizes and the powers of the considered tests with the other.

Unobserved covariates in the two‐sample comparison of survival times: a maximin efficiency robust test

In analysing a clinical trial with the logrank test, the hazards between the two groups are usually assumed to be proportional. Nevertheless, this hypothesis is no longer valid with unobserved covariates. As a consequence, there is a loss of power of the logrank test for testing the null hypothesis H0 of no treatment effect. We propose a test suited for taking into account unobserved covariates. The proposed approach is based on a proportional hazard frailty model whereby the omitted covariates are considered as an unobserved frailty variable. The procedure is as follows. In a first step, the weighted logrank test optimal for testing H0 against a general proportional hazard frailty model is obtained and its specialization for a gamma frailty variable is derived. In a second step, the proposed test is obtained by combining the maximin efficiency robustness principle and the gamma frailty distribution properties. Simulation studies investigate the power properties of the test for different frailty distributions. A breast cancer clinical trial is analysed as an example. The proposed test might be recommended rather than the logrank for practical situations in which one expects heterogeneity related to omitted covariates. Copyright © 1999 John Wiley & Sons, Ltd.

Frailty models and rank tests.

Connections are established between the theories of weighted logrank tests and of frailty models. These connections arise because omission of a balanced covariate from a proportional hazards model generally leads to a model with non-proportional hazards, for which the simple logrank test is no longer optimal. The optimal weighting function and the asymptotic relative efficiencies of the simple logrank test and of the optimally weighted logrank test relative to the adjusted test that would be used if the covariate values were known, are expressible in terms of the Laplace transform of the hazard ratio for the distribution of the omitted covariate. For example if this hazard ratio has a gamma distribution, the optimal test is a member of the "G rho" class introduced by Harrington and Fleming (1982). We also consider positive stable, inverse Gaussian, displaced Poisson and two-point frailty distribution. Results are obtained for parametric and nonparametric tests and are extended to include random censoring. We show that the loss of efficiency from omitting a covariate is generally more important than the additional loss due to misspecification of the resulting non-proportional hazards model as a proportional hazards model. However two-point frailty distributions can provide exceptions to this rule. Censoring generally increases the efficiency of the simple logrank test to the adjusted logrank test.

Weighted Logrank Tests to Detect a Transient Improvement in Survivorship

In investigating whether an intervention has an effect on survival, we exploit the flexibility of weighted logrank tests for constructing optimal tests sensitive to specified patterns of changes in the ratios or differences of the hazards as a function of time. The tests presented seem appropriate if either the intervention on trial imposes changes in life style, such as a diet modification or an exercise regimen, or if the effect of therapy itself is limited in duration, such as in the case of drugs for AIDS that may induce drug resistant viruses. For such problems, we postulate no difference between the hazards of the two groups initially, increased discrepancies as time goes on, and an eventual leveling-off of the discrepancies in hazards perhaps due to compliance problems or to adoption of the behavior change under study by individuals in the control group. We suggest a quadratic weight function for such problems and show how to evaluate the power of the proposed tests as well as their efficiency relative to other logrank tests. Other uses of quadratic weight functions, for example for evaluating effects with lag times, and for some parametric procedures are also described. Additive and multiplicative models are presented and both discrete and continuous times are considered.

Sensitivity analysis of non-parametric tests for censored data

The use of non-parametric tests for the two sample problem such as say the weighted logrank tests for censored data is based on the hypothesis that the probability of belonging to a groupe is the same for each individual. Our problem is : what can be said about the significance level of such a test if this hypothesis does not hold ? Using a new formulation of the weighted logrank tests statistics, it is shown that, in presence of such an heterogeneity, bounds on the exact (and unknown) significance level may be given. This results allows to lead the sensitivity analysis of those test statistics which are shown to be asymptotically normal.

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

Three classes of censored data rank tests: Strengths and weaknesses under censoring

SUMMARY Nonparametric tests are useful when definite parametric alternatives are not available. The behaviour under censoring of extensions of the tests is used to delineate three distinct classes of censored data rank tests: asymptotically efficient, approximately distribution-free and approximately unbiased. Section 2 gives qualitative comparisons; ? 3 gives some representative numerical comparisons. The Pitman efficiencies approximated by Lee, Desu & Gehan (1975) from simulations with 50 observations in each group are shown to agree with asymptotic calculations. 1. WEIGHTED LOG RANK TESTS It is assumed that two samples of randomly censored observations are available, that is, that the true lifetimes Yij have cumulative distribution functions F(y I Oj) for j = 1, ..., n(i); i = 1,2, that the censoring times Cij have cumulative distribution functions H(y I Of), and that the observable quantities are Xij = min (Yij, C) and Dij = [Xij = Yij], where [A] denotes the indicator function of the set A. It is assumed that 01 = 0, that F(t 1 0) is continuous in 0 and that the null hypothesis is 61 = 02 = 0. The 'true' value of 02 will be denoted by 0 and the functions F(t I 0) and H(t I 0) will be abbreviated to F and H. Survival distributions will be denoted by a bar, so that F(t) = 1- F(t). The hazard rates of F and F(t 10) will be denoted by A and AO, respectively. The test statistics studied are functions of: Di(t), the number of deaths observed before time t in the ith sample; Ri(t), the number at risk at time t in the ith sample; and Aj, an estimator of the cumulative hazard function for the ith sample. These functions are defined formally by n(i) n(i)