Generalized Wilcoxon Statistics Research Articles

A moment generating function of a combination of linear rank tests and its asymptotic efficiency

When testing hypotheses in two-sample problems, the Lepage test has often been used to jointly test the location and scale parameters, and has been discussed by many authors over the years. The Lepage test was a combination of the Wilcoxon statistic and the Ansari–Bradley statistic. Various Lepage-type tests were proposed with discussions of an asymptotic relative efficiency (Duran et al., Biometrika 63:173–176, 1976; Goria, Stat Neerl 36:3–13, 1982), a robustness and a power comparison (Neuhauser, Commun Stat Theory Methods 29:67–78, 2000; Buning, J Appl Stat 29:907–924, 2002) and an adaptive test (Buning and Thadewald, J Stat Comput Sim 65:287–310, 2000). We derive an expression for the moment generating function of a linear combination of two linear rank statistics. As a suggested Lepage-type test, we use a combination of the generalized Wilcoxon statistic and the generalized Mood statistic. Deriving the exact critical value of the statistic can be difficult when the sample sizes are increasing. In this situation, an approximation method to the distribution function of the test statistic can be useful with a higher order moment. We use a moment-based approximation with an adjusted gamma polynomial to evaluate the upper tail probability of a Lepage-type test for a finite sample size. We determine the asymptotic efficiencies of the Lepage and Lepage-type tests for various distributions.

Nontransitivity in a class of weighted logrank statistics under nonproportional hazards

Transitivity is an important property of any statistic applied in the setting of multi-arm clinical trials and non-inferiority trials where active-controls are used. The G ρ , γ class of weighted logrank statistics for right-censored survival data as proposed by Fleming and Harrington [1991. Counting Processes and Survival Analysis. Wiley, New York] is often used to improve efficiency in the setting of nonproportional hazards. These statistics utilize a weighting scheme based upon the combined Kaplan–Meier estimate of survival for all comparison groups. Members of this class include the usual logrank statistic as well as the generalized Wilcoxon statistic. It is demonstrated that all useful members of this class exhibit nontransitivity. We propose a general modification of the G ρ , γ statistic which asymptotically achieves transitivity.

Testing for Crossover of Two Hazard Functions Using Gail and Simon's Method

Crossover of two hazard functions is sometimes called qualitative nonproportionality where the hazard ratio could be >1 in some time intervals but <1 in some other intervals. Investigators often wish to know whether a beneficial treatment effect exists over a long period of time (i.e., no crossover). This information is important for the management of safety and efficacy of a new treatment in long-term use. Also, if crossover occurs, the commonly used statistical methods, such as Cox proportional hazards model or linear rank tests, may not be appropriate. Graphical display may be used to visually examine whether there are crossovers in the observed hazard functions. A relevant question is whether the observed crossover of two hazard functions is due to chance variation. In this article, we propose a class of tests for crossover of two hazard functions. The study follow-up period is divided into nonoverlapping time intervals, and the weighted linear rank statistic, including logrank and generalized Wilcoxon statistics, can be calculated from each interval. These statistics are asymptotically independent and have normal distributions. Treating each interval as a “patient subset,” qualitative tests of interactions between treatment and patient subsets can naturally apply. For our purpose, we consider the likelihood ratio test proposed by Gail and Simon. Two examples are used to illustrate this approach. The proposed test procedures are also studied through simulations.

A note on the gehan-wilcoxon statistic

The Wilcoxon statistic is often introduced from different points of view, and has been considered mainly as a linear rank statistic or as a U-statistic. For survival data analysis, it has also been considered as a weighted Mantel-Haenszel statistic. As a result, different approaches to censoring have led to different versions of generalized Wilcoxon statistics. The Gehan-Wilcoxon statistic has been widely used in practice and is available in many statistical software packages, even though it has been criticized as inappropriate for censored data and as inferior to Peto-Peto's generalization of the Wilcoxon statistic. A new and intuitive interpretation of Gehan’s and Peto-Peto's generalized Wilcoxon statistic is presented here to illustrate their fundamental differences, and to show the possible hidden pitfalls of Gehan's generalization in the application of clinical trial data monitoring.

The appropriateness of the Wilcoxon test in ordinal data.

In two-sample studies with ordinal responses, the Wilcoxon rank-sum test is generally chosen to test equality of the distributions, in spite of it being a specific test of location shift. I compared the power of the exact tests based on the Wilcoxon statistic, O'Brien's generalized Wilcoxon statistic, and the omnibus Smirnov statistic in the presence of location shift and scale alternatives. All three tests can gain power as a function of the scale parameter, depending on its magnitude relative to the shift parameter. As the relative influence of the scale parameter grows, the O'Brien test is the most powerful. I also compared the power of the asymptotic Wilcoxon test with its exact version, and found that it either inflates or deflates power if there are scale changes present. I present results of a simulation study.

Symmetry of the Comparative Two‐sample Tests for Censored Reliability Data

Summarizes part of the work carried out to date on the two‐sample problem with censored data in the context of reliability and maintenance optimization applications. For this purpose, non‐parametric two‐sample tests for censored data are discussed and their regular usage in common engineering problems is reviewed. Calculation aspects of the generalized Wilcoxon statistics are detailed and a specific framework is set up for the hypothesis testing procedures involved.

Recombinant human interleukin 1 receptor antagonist in the treatment of patients with sepsis syndrome. Results from a randomized, double-blind, placebo-controlled trial. Phase III rhIL-1ra Sepsis Syndrome Study Group

To further define the safety and efficacy of recombinant human interleukin 1 receptor antagonist (rhIL-1ra) in the treatment of sepsis syndrome. Randomized, double-blind, placebo-controlled, multicenter, multinational clinical trial. A total of 893 patients with sepsis syndrome received an intravenous loading dose of rhIL-1ra, 100 mg, or placebo followed by a continuous 72-hour intravenous infusion of rhIL-1ra (1.0 or 2.0 mg/kg per hour) or placebo. Twenty-eight-day all-cause mortality. There was not a significant increase in survival time for rhIL-1ra treatment compared with placebo among all patients who received the study medication (n = 893; generalized Wilcoxon statistic, P = .22) or among patients with shock at study entry (n = 713; generalized Wilcoxon statistic, P = .23), the two primary efficacy analyses specified a priori for this trial. Results from secondary analyses suggest an increase in survival time with rhIL-1ra treatment among patients with dysfunction of one or more organs (n = 563; linear dose-response, P = .009). Retrospective analysis demonstrated an increase in survival time with rhIL-1ra treatment among patients with a predicted risk of mortality of 24% or greater (n = 580; linear dose-response, P = .005) as well as among patients with both dysfunction of one or more organs and a predicted risk of mortality of 24% or greater (n = 411; linear dose-response, P = .002). There was not a statistically significant increase in survival time for rhIL-1ra treatment compared with placebo among all patients who received the study medication or among patients with shock at study entry. Secondary and retrospective analyses of efficacy suggest that treatment with rhIL-1ra results in a dose-related increase in survival time among patients with sepsis who have organ dysfunction and/or a predicted risk of mortality of 24% or greater.

Pathology and ploidy in the prognosis of gastric cancer with no extranodal metastasis.

The prognostic relevance of morphology in advanced gastric cancer is well known. Data on tumor cell DNA content are still inadequate and contradictory. Morphologic parameters and DNA ploidy were evaluated in 76 gastric cancer patients with no extranodal metastases (Stage I, 10 cases; Stage II, 20 cases; and Stage III, 46 cases), using formalin-fixed paraffin-embedded tissue. All cases were followed for at least 6 years after surgery or until death. Among the potential prognostic factors analyzed by Mantel-Cox and generalized Wilcoxon statistics, male sex (P = 0.02), cardiac location of neoplasia (P = 0.02), deeper infiltration of the gastric wall (P = 0.001), vascular neoplastic invasion (P = 0.006), metastatic lymph nodes (P = 0.001), pathologic stage (P = 0.0001), and aneuploidy (P = 0.01) were significantly associated with lower survival rate. Testing of all of the above-mentioned variables by the Cox stepwise multiple regression model disclosed that factors independently associated with survival were stage (P = 0.0001), ploidy (P = 0.0006), and vascular carcinomatous invasion (P = 0.01). In gastric cancer with no extranodal metastases, DNA ploidy was found to be the most significant prognostic parameter after pathologic stage.

Supremum Versions of the Log-Rank and Generalized Wilcoxon Statistics

Abstract Considerable progress has been made in generalizing to censored data hypothesis tests based on rank statistics. The most commonly used generalizations are the log-rank test statistic, which extends the Savage exponential scores statistic (Cox 1972; Mantel 1966) and the generalized Wilcoxon statistics (Gehan 1965; Peto and Peto 1972). Gill (1980) showed that in the two-sample problem the asymptotic efficiency properties of the Savage and Wilcoxon statistics are maintained in censored data by the log-rank and Peto and Peto Wilcoxon statistics, respectively, when censoring is the same in both samples. Heuristics and simulations have implied that some recently proposed supremum statistics, closely related to these two popular linear rank statistics, can be used to construct tests that provide good power against a much wider range of nonlocal alternatives (see, e.g., Fleming and Harrington 1981; Fleming, O'Brien, O'Fallon, and Harrington 1980; Gill 1980; Schumacher 1984). In this article we consider n...

A Monte Carlo Study of Some Two-Sample Rank Tests with Censored Data

Abstract The powers of several nonparametric tests for the two-sample problem with censored data are compared by simulation. The tests studied include Gehan's, Efron's, and Peto's, and Prentice's generalized Wilcoxon tests, along with the logrank test. The test with the greatest power changes with the sample sizes, censoring mechanism, and distribution of the random variables. The test that performed the best overall was the Peto-Prentice generalized Wilcoxon statistic with an asymptotic variance estimate.

A Monte Carlo Study of Some Two-Sample Rank Tests with Censored Data

Abstract The powers of several nonparametric tests for the two-sample problem with censored data are compared by simulation. The tests studied include Gehan's, Efron's, and Peto's, and Prentice's generalized Wilcoxon tests, along with the logrank test. The test with the greatest power changes with the sample sizes, censoring mechanism, and distribution of the random variables. The test that performed the best overall was the Peto-Prentice generalized Wilcoxon statistic with an asymptotic variance estimate.

Generalized Wilcoxon Statistics for the Two-Sample Problem with Censored Data

Generalized Wilcoxon Statistics for the Two-Sample Problem with Censored Data

Generalized Wilcoxon statistics for the two-sample problem with censored data

Generalized Wilcoxon statistics for the two-sample problem with censored data