Linear Rank Tests Research Articles

Effect Modification in a Randomized Trial under Non-Ignorable Non-Compliance: An Application to the Alpha-Tocopherol Beta-Carotene Study

Summary We propose an inference procedure that allows an estimation of the effects of treatment modifying factors on a survival end point in a placebo-controlled randomized trial with non-ignorable non-compliance. Against prior expectation the alpha-tocopherol beta-carotene (ATBC) study showed an increase in the incidence of lung cancer and total mortality following supplementation with beta-carotene. Any attempt to assess interactions between beta-carotene supplementation and base-line covariates needs to address the issue of non-compliance. In the ATBC study about 25% of the participants dropped out from the active follow-up prematurely for reasons other than death and they stopped receiving the study supplementation. Survival status was observed regardless of drop-out for all participants. Drop-out is related to base-line covariates, to the treatment arm and presumably also to the health status of the individuals through both measured and unmeasured factors. Non-ignorable non-compliance is induced by unmeasured covariates related both to drop-out from active participation in the study and to health status and death. Instead of comparing survival between the randomized groups we focus on an estimation of the effect of beta-carotene actually received and its interaction with base-line covariates such as age, smoking and alcohol consumption. We use an accelerated failure time model to link a potential treatment-free survival time to the actual observed survival time and to the treatment received and its interactions with base-line covariates. We refer to this as the structural model. The estimation of the parameters in the structural model is based on minimizing a score test statistic obtained from the partial likelihood specified for the treatment-free survival time under the structural model. We compare the method proposed with Robins's g-estimation based on a linear rank test statistic. If the structural model holds, then the method proposed can substantially increase efficiency. We propose diagnostic techniques for evaluating the adequacy of the structural model and discuss how a misspecification of the model may induce bias into the score equations.

A testing procedure for survival data with few responders

In the course of designing a clinical trial, investigators are often faced with the possibility that only a fraction of the patients will benefit from the experimental treatment. A proper clinical trial design requires prospective specification of the testing procedures to be used in the analysis. In the absence of reliable prognostic factors capable of identifying the appropriate subset of patients, there is a need for a test procedure that will be sensitive to a range of possible fractions of responders. Focusing on survival data, we propose guidelines for selecting a proper test procedure based on the anticipated proportion of responding patients. These guidelines suggest that the logrank test should be used when the fraction of responders is expected to be greater than 0.5, otherwise procedures based on weighted linear rank tests are preferable. Overall this approach provides good power properties when the treatment affects only a small proportion of patients while protecting against substantial loss of power when all patients are affected. Use of the procedure is illustrated with data from two published randomized studies.

Unmasking the Trend Sought by Jonckheere‐Type Tests for Right‐Censored Data

Medical and epidemiological studies often involve groups of subjects associated with increasing levels of exposure to a risk factor. Survival of the groups is expected to follow the same order as the level of exposure. Formal tests for this trend fall into the regression framework if one knows what function of exposure to use as a covariate. When unknown, a linear function of exposure level is often used. Jonckheere‐type tests for trend have generated continued interest largely because they do not require specification of a covariate. This paper shows that the Jonckheere‐type test statistics are special cases of a generalized linear rank statistic with time‐dependent covariates which unfortunately depend on the initial group sizes and censoring distributions. Using asymptotic relative efficiency calculations, the Jonckheere tests are compared to standard linear rank tests based on a linear covariate over a spectrum of shapes for the true trend.

Drawbacks to integer scoring for ordered categorical data.

Linear rank tests are widely used when testing for independence against stochastic order in a 2 x J contingency table with two treatments and J ordered outcome levels. For this purpose, numerical scores are assigned, possibly by default, to the J outcome levels. When the choice of scores is not apparent, integer (equally spaced) scores are often assigned. We show that this practice generally leads to unnecessarily conservative tests. The use of slightly perturbed scores will result in a less conservative and uniformly more powerful test.

Linear rank tests under general alternatives, with application to summary statistics computed from repeated measures data

Linear rank tests are used extensively for comparing two or more groups of continuous outcomes. Tests in this class retain proper test size with minimal assumptions and can have high efficiency towards an alternative of interest. In recent years, these tests have been increasingly used in settings where an individual's observation is itself a scalar summary of several outcome measures. Here, simple distributional structures on the outcome variables can lead to complex differences between the distributions of summary statistics of the comparison groups. The local asymptotic power of linear rank tests when the groups are assumed to differ by a location or scale alternative has been studied in detail. However, not much is known about their behavior for other types of alternatives. To address this, we derive the asymptotic distribution of linear rank tests under a general contiguous alternative and then investigate the implications for location–scale families and more general settings, including an example drawn from an AIDS clinical trial where the continuous outcome is a summary statistic computed from repeated measures of a biological marker.

THE POWER AND SIZE OF NONPARAMETRIC TESTS FOR COMMON DISTRIBUTIONAL CHARACTERISTICS

This paper considers the power and size properties of some well known nonparametric linear rank tests for location and scale as well as the Kolmogorov-Smirnov omnibus test and proposed alternatives to it. Independence between some classes of linear rank tests is established facilitating their joint application. Monte Carlo study confirms the asymptotic power properties of the linear rank tests but raises concerns about their application in more general and practically relevant circumstances. It also indicates that the new omnibus tests constitute viable alternatives with superior properties to the Kolmogorov-Smirnov test in certain circumstances.

Data Driven Rank Test for Two‐Sample Problem

Traditional linear rank tests are known to possess low power for large spectrum of alternatives. In this paper we introduce a new rank test possessing a considerably larger range of sensitivity than linear rank tests. The new test statistic is a sum of squares of some linear rank statistics while the number of summands is chosen via a data‐based selection rule. Simulations show that the new test possesses high and stable power in situations when linear rank tests completely break down, while simultaneously it has almost the same power under alternatives which can be detected by standard linear rank tests. Our approach is illustrated by some practical examples. Theoretical support is given by deriving asymptotic null distribution of the test statistic and proving consistency of the new test under essentially any alternative.

Asymptotic Behavior of Linear Permutation Tests under General Alternatives, with Application to Test Selection and Study Design

Tests based on the permutation of observations are a common and attractive method of comparing two groups of outcomes, in part because they retain proper test size with minimal assumptions and can have high efficiency toward specific alternatives of interest. In addition, permutation tests may be used with discrete or categorical outcomes, for which linear rank tests are not designed. Permutation tests are now increasingly used to analyze discrete or continuous responses that themselves are functions of several statistics. Examples of such summary statistics include the area under the curve generated by repeated measures of a laboratory marker or an overall composite score from a quality of life study. Here even simple structures for the joint distribution of the component statistics can lead to complex differences between the distributions of summary statistics of the comparison groups. Despite their attractive features, surprisingly little is known about the behavior of linear permutation tests when the two groups differ even in simple ways. This lack of knowledge limits an assessment of the relative efficiency of different tests or the planning of the size of a study based on a permutation test. To address these issues, we derive the asymptotic distribution of permutation tests under a general contiguous alternative, and then investigate the implications for test selection and study design for several diverse areas of application. For discrete outcomes, areas of application include permutation tests for ordinal responses and for count data. For continuous outcomes, we explore several applications, including general results for location-scale families, a comparison of different data transformations, and a comparison to linear rank tests.

Linear rank tests for censored survival data

We describe a class of rank test procedures for the two-sample problem with right censored survival data. The class of tests is directly generalized from the linear rank tests by assigning each observation a rank according to its corresponding Wilcoxon scores. It allows a flexible choice of score functions, in particular, those powerful against scale differences between the two survival distributions. Monte Carlo simulations have shown that some members of this class have great power in detecting crossing-curve alternatives (alternatives where underlying survival curves cross over). The class also contains tests essentially equivalent to the Gehan-Wilcoxon and the logrank tests.

A group sequential test for survival trials: an alternative to rank-based procedures.

A method of interim monitoring is described for survival trials in which the proportional hazards assumption may not hold. This method extends the test statistics based on the cumulative weighted difference in the Kaplan-Meier estimates (Pepe and Fleming, 1989, Biometrics 45, 497-507) to the sequential setting. Therefore, it provides a useful alternative to the group sequential linear rank tests. With an appropriate weight function, the test statistic itself provides an estimator for the cumulative weighted difference in survival probabilities, which is an interpretable measure for the treatment difference, especially when the proportional hazards model fails. The method is illustrated based on the design of a real trial. The operating characteristics are studied through a small simulation.

Convex Hull Test for Ordered Categorical Data

When testing for stochastic order in ordered 2 x J contingency tables, it is common to select the cutoff required to declare significance so as to ensure that the size of the test is exactly alpha conditionally on the margins. It is valid, however, to use the margins to select not only the cutoff but also the form of the test. Linear rank tests, which are locally most powerful and frequently used in practice, suffer from the drawback that they may have power as low as zero to detect some alternatives of interest when the margins satisfy certain conditions. The Smirnov and convex hull tests are shown, through exact conditional power calculations and simulations, to avoid this drawback. The convex hull test is also admissible and palindromic invariant and minimizes the required significance level to have limiting power of one as the alternative moves away from the null in any direction.

An Order-Directed Score Test for Trend in Ordered 2 x K Tables

This paper presents a new test procedure for detecting trend in ordered 2 X K tables. Using an order-directed score statistic, the procedure does not require a set of scores preassigned to the ordinal categories under consideration. Thus the problem of varying p-values of linear rank tests, due to choices of different scoring systems, is avoided. The proposed test procedure can be easily generalized to handle stratified analysis where data are represented by several 2 x K tables. Examples are given to illustrate the method.

On the Lagrange gamma distribution

Statistical inference on the Lagrange gamma distribution is considered in this paper. The method of moments, the maximum-likelihood estimation technique, and the estimation of the scale parameter by order statistics are discussed. A linear rank test statistic is proposed for the distribution. A numerical example on the application of the Lagrange gamma distribution is given. In the numerical example, the Lagrange gamma distribution provided a better fit than the gamma distribution.

Estimations in homoscedastic linear regression models with censored data: an empirical process approach

Pertaining to the estimating equations proposed by Tsiatis, based on the linear rank test, we show the existence of local confounding between the baseline hazard function and the covariates. Due to the local confounding, an estimating equation in Tsiatis' family with a larger time point of truncation could contain less information about the regression parameter than the estimating equation with a smaller time point of truncation. This phenomenon further indicates significant loss of efficiency of Tsiatis' estimating equations as well as the power loss of log-rank type tests when the baseline hazard function goes up and down along the time scale. To take care of this local confounding without using nonparametric estimates of the derivative of the baseline hazard function, we propose the empirical process approach (EPA) based on an empirical process constructed from Tsiatis' log-rank estimating equation by varying its truncating time point. The EPA will provide very tractable estimations of the regression parameters as well as Pearson's chi-squared statistics for testing the model's assumptions. Specifically, the performance of the EPA estimator is shown to be very close to the best estimator in Tsiatis' family.

On the asymptotic normality of rank tests for independence

We show that the commonly used linear rank tests for independence have certain continuity properties with respect to the score functions which are applied to the ranks. Using these properties, we derive the asymptotic normality of the test statistics under general conditions. These conditions are closely related to those under which simple linear rank statistics are known to be asymptotically normal.

Empirical process approach in a two-sample location-scale model with censored data

To compare two samples of possibly right-censored failure times, a location-scale model, without assuming the distribution form, is considered for the log-transformed data. This new accelerated failure time model is introduced to accommodate the possible heterogeneity between within-treatment variations of the two groups considered. One distinct feature of this model is that, with the presence of heterogeneity, statistical inferences on both location and scale parameters based on log-rank tests or linear rank tests will become inappropriate. In this paper we propose the empirical process approach to construct a regression setup derived from the theory of strong approximation of the Kaplan-Meier product-limit empirical quantile process. A generalized least squares (GLS) estimator is obtained and shown to be semiparametric efficient. Also it is shown that this estimation is adaptive for a special case. At the end, results on the two-sample setting are applied to the K-sample problem.

Rank Invariant Tests for Interval Censored Data under the Grouped Continuous Model

This paper creates rank invariant score tests for grouped or interval censored data. This generalizes Finkelstein (1986, Biometrics 42, 845-854), who derived score tests for interval censored data assuming proportional hazards. We frame the problem as a linear rank test of a shift in location with a known error distribution. We discuss adjustments to the test that may be required when the number of observation times is large. We offer a graphical test of the assumption of the location shift model and discuss an alternative interpretation of the test using the logistic error when the location shift assumption does not hold.

Test of Significance Based on Wavelet Thresholding and Neyman's Truncation

Abstract Traditional nonparametric tests, such as the Kolmogorov—Smirnov test and the Cramer—Von Mises test, are based on the empirical distribution functions. Although these tests possess root-n consistency, they effectively use only information contained in the low frequencies. This leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high-frequency alternations. The drawback can be repaired via smoothing-based test statistics. In this article we propose two such kind of test statistics based on the wavelet thresholding and the Neyman truncation. We provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations, while maintaining the same capability in detecting smooth alternative densities as the traditional tests. Similar conclusions can be made for two-sample nonparametric tests of distribution functions. In that case, the traditional linear rank tests such as th...

Rank-Based Analysis of the Heteroscedastic Linear Model

Abstract Heteroscedasticity often causes problems in the analysis of linear models. Frequently for such cases, scale is a function of the response. Here, for such models, a methodology is presented based on well-known rank-based procedures. The procedure is iterative. Using the residuals from an initial R estimate of the regression coefficients, scale is estimated by inverting a linear rank test for scale. This in turn leads to a weighted R estimate of the regression coefficients and then to a final estimate of scale. Asymptotic linearity results for these estimates are derived, from which their asymptotic distribution is obtained. The weighted R estimate has the same asymptotic distribution as the optimal (known scale) R estimate; hence it is efficiently robust. Consistent estimates of the standard errors of the R estimates of scale and the regression coefficients are determined. Based on these results, a complete inference for the regression coefficients and the scale parameter is realized, including a ...

Rank-Based Analysis of the Heteroscedastic Linear Model

Abstract Heteroscedasticity often causes problems in the analysis of linear models. Frequently for such cases, scale is a function of the response. Here, for such models, a methodology is presented based on well-known rank-based procedures. The procedure is iterative. Using the residuals from an initial R estimate of the regression coefficients, scale is estimated by inverting a linear rank test for scale. This in turn leads to a weighted R estimate of the regression coefficients and then to a final estimate of scale. Asymptotic linearity results for these estimates are derived, from which their asymptotic distribution is obtained. The weighted R estimate has the same asymptotic distribution as the optimal (known scale) R estimate; hence it is efficiently robust. Consistent estimates of the standard errors of the R estimates of scale and the regression coefficients are determined. Based on these results, a complete inference for the regression coefficients and the scale parameter is realized, including a test for homogeneity. The procedure is illustrated by the analysis of a dose—response data set drawn from pharmaceutical science. Results of a simulation study that support the asymptotic theory are reported.