A linear rank test for use when the main interest is in differences in cure rates

For diseases with a positive probability of being cured, a family of alternatives to the null hypothesis of equality of survival distributions is introduced, which is designed to focus power against alternatives with differences in cure rates. The optimal linear rank test for this alternative is derived, and found to be substantially more efficient than the log-rank test for this alternative when cure rates are less than 50%, while there is little difference between the tests if the cure rates are 50% or greater. The simple test based on the difference of Kaplan-Meier estimates of the proportion cured is also examined, and found to be fully efficient for this alternative with no censoring, while its efficiency rapidly drops as censoring is increased. The new test is not a pure test of equality of cure rates when the data are censored, but rather is a test of equality of survival distributions that focuses power against late differences in the survival curves.     

Generalized Rank Transformations for Tests of Survival

The asymptotic equivalence of nonparametric tests and parametric tests based on rank-transformed data (CONOVER and IMAN , 1981) can be extended to the case of censoring. This paper presents generalized rank transformations for analyses of censored data, of interval-censored data and of survival data with uncertain causes of death. A Monte Carlo study and an analysis of leukemia remission times demonstrate excellent agreement of suggested procedures with GEHAN 'S (1965) and PRENTICE 'S (1978) tests.     

Linear Rank Statistics for the Simple Tree Alternatives

The problem of comparing several treatments with a control is considered. It is formulated as a test of homogeneity with one-sided alternatives that at least one treatment is better than the control, while no treatment is worse than the control. A class of linear rank tests for the simple tree alternatives is proposed, assuming a location model. The most efficient test of the proposed class for detecting a specified pattern is derived. Two optimal tests for use in the case of vaguely specified tree alternatives are derived. In a simulation study the power of these two tests are compared with the power of Kruskal-Wallis test. Recommendations about the use of the proposed tests are made.     

Survival Tests for r Groups

In this paper we give the generalization of the score tests covering the case of ties and we give examples where the expressions in matrix form are completely specified for the weighted tests and the score tests for the case of r groups. It is worth mentioning that although the score tests are not generally included in the commercial software, these tests should be used if it can be assumed that the censoring mechanism is equal in the r groups or if there is no censoring (Lawless , 1982). We establish the equivalence between “numerators” of these families of tests. As result of this equivalence we define four new tests that complete the classification of score and weighted tests. The Kruskal‐Wallis test (1952) appears as a particular case of the score tests for the case of non‐censoring. A simulation study has been done in order to compare the performance of the tests described in this paper. An example is included to make the understanding of the paper easier.     

Exact,Distribution Free Confidence Intervals for Late Effects in Censored Matched Pairs

When comparing censored survival times for matched treated and control subjects, a late effect on survival is one that does not begin to appear until some time has passed. In a study of provider specialty in the treatment of ovarian cancer, a late divergence in the Kaplan–Meier survival curves hinted at superior survival among patients of gynecological oncologists, who employ chemotherapy less intensively, when compared to patients of medical oncologists, who employ chemotherapy more intensively; we ask whether this late divergence should be taken seriously. Specifically, we develop exact, permutation tests, and exact confidence intervals formed by inverting the tests, for late effects in matched pairs subject to random but heterogeneous censoring. Unlike other exact confidence intervals with censored data, the proposed intervals do not require knowledge of censoring times for patients who die. Exact distributions are consequences of two results about signs, signed ranks, and their conditional independence properties. One test, the late effects sign test, has the binomial distribution; the other, the late effects signed rank test, uses nonstandard ranks but nonetheless has the same exact distribution as Wilcoxon's signed rank test. A simulation shows that the late effects signed rank test has substantially more power to detect late effects than do conventional tests. The confidence statement provides information about both the timing and magnitude of late effects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rank tests for survival analysis: a comparison by analogy with games

Several commonly used two-sample linear rank tests for censored data have been shown to be closely linked in mathematical structure. This pedagogical paper elucidates the relationships among the tests by introducing a videogame played between a boys' team and a girls' team with different rules of payment by the consecutive losers considered. Depending on the rule of payment, the total amount of money the girls win as a team becomes the Wilcoxon statistic, the Savage statistic, or their generalizations under censoring.     

A Note on Optimality of Certain Rank Tests for Ordered Alternatives in Randomized Blocks

Nonparametric tests for ordered alternatives in randomised block designs based on within block rankings have been proposed by many authors. This note is concerned with the optimality of the choice of so-called regression constants usually considered in such rank tests. Some examples are discussed.     

Statistical Inference Methods for Two Crossing Survival Curves: A Comparison of Methods

A common problem that is encountered in medical applications is the overall homogeneity of survival distributions when two survival curves cross each other. A survey demonstrated that under this condition, which was an obvious violation of the assumption of proportional hazard rates, the log-rank test was still used in 70% of studies. Several statistical methods have been proposed to solve this problem. However, in many applications, it is difficult to specify the types of survival differences and choose an appropriate method prior to analysis. Thus, we conducted an extensive series of Monte Carlo simulations to investigate the power and type I error rate of these procedures under various patterns of crossing survival curves with different censoring rates and distribution parameters. Our objective was to evaluate the strengths and weaknesses of tests in different situations and for various censoring rates and to recommend an appropriate test that will not fail for a wide range of applications. Simulation studies demonstrated that adaptive Neyman’s smooth tests and the two-stage procedure offer higher power and greater stability than other methods when the survival distributions cross at early, middle or late times. Even for proportional hazards, both methods maintain acceptable power compared with the log-rank test. In terms of the type I error rate, Renyi and Cramér—von Mises tests are relatively conservative, whereas the statistics of the Lin-Xu test exhibit apparent inflation as the censoring rate increases. Other tests produce results close to the nominal 0.05 level. In conclusion, adaptive Neyman’s smooth tests and the two-stage procedure are found to be the most stable and feasible approaches for a variety of situations and censoring rates. Therefore, they are applicable to a wider spectrum of alternatives compared with other tests.     

Nonparametric Tests for Interval‐Censored Failure Time Data

This paper discusses two sample nonparametric comparison of survival functions when only interval‐censored failure time data are available. The problem considered often occurs in, for example, biological and medical studies such as medical follow‐up studies and clinical trials. For the problem, we present and study several nonparametric test procedures that include methods based on both absolute and squared survival differences as well as simple survival differences. The presented tests provide alternatives to existing methods, most of which are rank‐based tests and not sensitive to nonproportional or nonmonotone alternatives. Simulation studies are performed to evaluate and compare the proposed methods with existing methods and suggest that the proposed tests work well for nonmonotone alternatives as well as monotone alternatives. An illustrative example is presented.     

Robust covariate-adjusted log-rank statistics and corresponding sample size formula for recurrent events data

Summary .   Recurrent events data are frequently encountered in clinical trials. This article develops robust covariate-adjusted log-rank statistics applied to recurrent events data with arbitrary numbers of events under independent censoring and the corresponding sample size formula. The proposed log-rank tests are robust with respect to different data-generating processes and are adjusted for predictive covariates. It reduces to the Kong and Slud (1997, Biometrika 84, 847–862) setting in the case of a single event. The sample size formula is derived based on the asymptotic normality of the covariate-adjusted log-rank statistics under certain local alternatives and a working model for baseline covariates in the recurrent event data context. When the effect size is small and the baseline covariates do not contain significant information about event times, it reduces to the same form as that of Schoenfeld (1983, Biometrics 39, 499–503) for cases of a single event or independent event times within a subject. We carry out simulations to study the control of type I error and the comparison of powers between several methods in finite samples. The proposed sample size formula is illustrated using data from an rhDNase study.     

A Test of Equality of Survival Distributions for Two Sample Censored Grouped Survival Data

A common testing problem for a life table or survival data is to test the equality of two survival distributions when the data is both grouped and censored. Several tests have been proposed in the literature which require various assumptions about the censoring distributions. It is shown that if these conditions are relaxed then the tests may no longer have the stated properties. The maximum likelihood test of equality when no assumptions are made about the censoring marginal distributions is derived. The properties of the test are found and it is compared to the existing tests. The fact that no assumptions are required about the censoring distributions make the test a useful initial testing procedure.     

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

Proposal of k Sample Tests for Bivariate Censored Data for Nondecreasing Ordered Alternatives

Tests are introduced which are designed to test for a nondecreasing ordered alternative among the survival functions of k populations consisting of multiple observations on each subject. Some of the observations could be right censored. A simulation study is conducted comparing the proposed tests on the basis of estimated power when the underlying distributions are multivariate normal. Equal sample sizes of 20 with 25% censoring, and 40 with both 25% and 50% censoring are considered for 3 and 4 populations. All of the tests hold their α‐values well. A recommendation is made as to the best overall test for the situations considered.     

A sample size formula for the supremum log-rank statistic

An advantage of the supremum log-rank over the standard log-rank statistic is an increased sensitivity to a wider variety of stochastic ordering alternatives. In this article, we develop a formula for sample size computation for studies utilizing the supremum log-rank statistic. The idea is to base power on the proportional hazards alternative, so that the supremum log rank will have the same power as the standard log rank in the setting where the standard log rank is optimal. This results in a slight increase in sample size over that required for the standard log rank. For example, a 5.733% increase occurs for a two-sided test having type I error 0.05 and power 0.80. This slight increase in sample size is offset by the significant gains in power the supremum log-rank test achieves for a wide range of nonproportional hazards alternatives. A small simulation study is used for illustration. These results should facilitate the wider use of the supremum log-rank statistic in clinical trials.     

Tail carriage in baboons (Papio cynocephalus): relationship to dominance rank and age.

Several metric and categorical variables were used to assess tail carriage in 717 photographs of 54 yellow baboons (Papio cynocephalus) of known identity, sex, dominance rank, and estimated or known age. Analyses of these data demonstrated that age was the primary factor, influencing tail carriage of baboons not engaged in social interaction. Specifically, the proximal segment of the tail was carried increasingly close to the vertical with increasing age; other changes in tail angles and segment curvature were also correlated with age. There was no relationship between dominance rank and neutral tail carriage in adult male or female baboons, nor was the neutral tail carriage of the first-ranking male or female either distinctive or characteristic. The morphological basis of ontogenetic changes in neutral tail carriage in baboons is also discussed.     

A frailty model for informative censoring

To account for the correlation between failure and censoring, we propose a new frailty model for clustered data. In this model, the risk to be censored is affected by the risk of failure. This model allows flexibility in the direction and degree of dependence between failure and censoring. It includes the traditional frailty model as a special case. It allows censoring by some causes to be analyzed as informative while treating censoring by other causes as noninformative. It can also analyze data for competing risks. To fit the model, the EM algorithm is used with Markov chain Monte Carlo simulations in the E-steps. Simulation studies and analysis of data for kidney disease patients are provided. Consequences of incorrectly assuming noninformative censoring are investigated.     

An omnibus consistent adaptive percentile modified Wilcoxon rank sum test with applications in gene expression studies

Summary We present an adaptive percentile modified Wilcoxon rank sum test for the two‐sample problem. The test is basically a Wilcoxon rank sum test applied on a fraction of the sample observations, and the fraction is adaptively determined by the sample observations. Most of the theory is developed under a location‐shift model, but we demonstrate that the test is also meaningful for testing against more general alternatives. The test may be particularly useful for the analysis of massive datasets in which quasi‐automatic hypothesis testing is required. We investigate the power characteristics of the new test in a simulation study, and we apply the test to a microarray experiment on colorectal cancer. These empirical studies demonstrate that the new test has good overall power and that it succeeds better in finding differentially expressed genes as compared to other popular tests. We conclude that the new nonparametric test is widely applicable and that its power is comparable to the power of the Baumgartner‐Weiß‐Schindler test.