The weighted log-rank class of permutation tests: P-values and confidence intervals using saddlepoint methods
TL;DR: In this paper, the authors used saddlepoint methods to determine mid-p-values from the null permutation distributions of tests from the weighted log-rank class, which can be used to compare treatment with control when there is right censoring.
Abstract: Test statistics from the weighted log-rank class are commonly used to compare treatment with control when there is right censoring. This paper uses saddlepoint methods to determine mid-p-values from the null permutation distributions of tests from the weighted log-rank class. Analytical saddlepoint computations replace the permutation simulations and provide mid-p-values that are virtually exact for all practical purposes. The speed of these saddlepoint computations makes it practicable to invert the weighted log-rank tests to determine nominal 95% confidence intervals for the treatment effect with right-censored data. Such analytical inversions lead to permutation confidence intervals that are easily computed and virtually identical to the exact intervals that would normally require massive amounts of simulation.
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TL;DR: This paper reviews the method of nonparametric combination of dependent permutation tests and its main properties along with some new results in experimental and observational situations (robust testing, multi-sided alternatives and testing for survival functions).
Abstract: In recent years permutation testing methods have increased both in number of applications and in solving complex multivariate problems. When available permutation tests are essentially of an exact nonparametric nature in a conditional context, where conditioning is on the pooled observed data set which is often a set of sufficient statistics in the null hypothesis. Whereas, the reference null distribution of most parametric tests is only known asymptotically. Thus, for most sample sizes of practical interest, the possible lack of efficiency of permutation solutions may be compensated by the lack of approximation of parametric counterparts. There are many complex multivariate problems, quite common in empirical sciences, which are difficult to solve outside the conditional framework and in particular outside the method of nonparametric combination (NPC) of dependent permutation tests. In this paper we review such a method and its main properties along with some new results in experimental and observational situations (robust testing, multi-sided alternatives and testing for survival functions).
59 citations
TL;DR: In this article, a review of permutation testing methods is presented, along with a number of applications in different experimental and observational situations (e.g., zero-inflated data and testing for a stochastic ordering) and properties specific to this methodology, such as: for a given number of subjects, when the number of variables diverges and the noncentrality of the combined test diverges accordingly, then the power of combination-based permutation tests converges to one.
Abstract: In recent years permutation testing methods have increased both in number of applications and in solving complex multivariate problems. A large number of testing problems may also be usefully and effectively solved by traditional parametric or rank-based nonparametric methods, although in relatively mild conditions their permutation counterparts are generally asymptotically as good as the best ones. Permutation tests are essentially of an exact nonparametric nature in a conditional context, where conditioning is on the pooled observed data as a set of sufficient statistics in the null hypothesis. Instead, the reference null distribution of most parametric tests is only known asymptotically. Thus, for most sample sizes of practical interest, the possible lack of efficiency of permutation solutions may be compensated by the lack of approximation of parametric counterparts. There are many complex multivariate problems (quite common in biostatistics, clinical trials, engineering, the environment, epidemiology, experimental data, industrial statistics, pharmacology, psychology, social sciences, etc.) which are difficult to solve outside the conditional framework and outside the nonparametric combination (NPC) method for dependent permutation tests. In this paper we review this method along with a number of applications in different experimental and observational situations (e.g. multi-sided alternatives, zero-inflated data and testing for a stochastic ordering) and we present properties specific to this methodology, such as: for a given number of subjects, when the number of variables diverges and the noncentrality of the combined test diverges accordingly, then the power of combination-based permutation tests converges to one.
46 citations
Cites methods from "The weighted log-rank class of perm..."
...…statistics (Downer, 2002), principal component analysis (Dray, 2008), shape analysis (Brombin and Salmaso, 2009), gene expression data (Xu and Li, 2003; Jung, 2005), stochastic ordering problems (Finos et al., 2007; Basso and Salmaso, 2009), and survival analysis (Abd-Elfattah and Butler, 2007)....
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TL;DR: In this paper, the authors used saddlepoint methods to determine the mid-P-values for such permutation tests for any test statistic in the weighted log-rank class, which is almost always more accurate than normal approximations.
Abstract: Suppose p + 1 experimental groups correspond to increasing dose levels of a treatment and all groups are subject to right censoring. In such instances, permutation tests for trend can be performed based on statistics derived from the weighted log-rank class. This article uses saddlepoint methods to determine the mid-P-values for such permutation tests for any test statistic in the weighted log-rank class. Permutation sim- ulations are replaced by analytical saddlepoint computations which provide extremely accurate mid-P-values that are exact for most practical purposes and almost always more accurate than normal approximations. The speed of mid-P-value computation allows for the inversion of such tests to determine confidence intervals for the percentage increase in mean (or median) survival time per unit increase in dosage. The Canadian
17 citations
TL;DR: The R package FHtest is introduced, which implements the Fleming-Harrington class for right-censored and interval- censored survival data, and provides an integrated approach for performing two-sample, k-sample and trend tests based on either counting process theory, likelihood theory, or permutation distributions.
Abstract: The Fleming-Harrington class for right-censored data was first introduced by Harrington and Fleming (1982). This class is widely used in survival analysis studies and it is a subset of the so-called weighted logrank test statistics. Recently, Oller and Gomez (2012) proposed an extension of this class for interval-censored data. This paper introduces the R package FHtest, which implements the Fleming-Harrington class for right-censored and interval-censored survival data. It provides an integrated approach for performing two-sample, k-sample and trend tests based on either counting process theory, likelihood theory, or permutation distributions. In this paper, we summarize the main aspects of the theory framework and present several examples with R codes to illustrate the usage of the main functions of FHtest.
12 citations
Cites background from "The weighted log-rank class of perm..."
...As shown in Abd-Elfattah and Butler (2007), the statistic (7) can also be written in a linear form URC = n∑ i=1 zici, (9) where zi = ( Z1i, . . . , ZKi )> is a covariate vector of group indicators (Zji equals 1 or zero according to whether or not the ith individual is in the jth group) and ci is a…...
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TL;DR: This article considers the exact distribution of the weighted log-rank class of tests for censored data under the truncated binomial design and derives a double saddlepoint approximation for p-values of this class under this design.
Abstract: The randomization design used to collect the data provides basis for the exact distributions of the permutation tests. The truncated binomial design is one of the commonly used designs for forcing balance in clinical trials to eliminate experimental bias. In this article, we consider the exact distribution of the weighted log-rank class of tests for censored data under the truncated binomial design. A double saddlepoint approximation for p-values of this class is derived under the truncated binomial design. The speed and accuracy of the saddlepoint approximation over the normal asymptotic facilitate the inversion of the weighted log-rank tests to determine nominal 95% confidence intervals for treatment effect with right censored data.
11 citations
References
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9,803 citations
TL;DR: Efficient methods of analysis of randomized clinical trials in which the authors wish to compare the duration of survival among different groups of patients are described.
Abstract: Part I of this report appeared in the previous issue (Br. J. Cancer (1976) 34,585), and discussed the design of randomized clinical trials. Part II now describes efficient methods of analysis of randomized clinical trials in which we wish to compare the duration of survival (or the time until some other untoward event first occurs) among different groups of patients. It is intended to enable physicians without statistical training either to analyse such data themselves using life tables, the logrank test and retrospective stratification, or, when such analyses are presented, to appreciate them more critically, but the discussion may also be of interest to statisticians who have not yet specialized in clinical trial analyses.
8,334 citations
TL;DR: Some comparisons are made for five cases of varying degrees of censoring and tying between probabilities from the exact test and those from the proposed test and these suggest the test is appropriate under certain conditions when the sample size is five in each group.
Abstract: H2: F1(t) F2(t) (t F2(t) (t < T). The asymptotic efficiency of the test relative to the efficient parametric test when the distributions are exponential is at least 0 75 and increases with degree of censoring. When Ho is true, the test is not seriously affected by real differences in the percentage censored in the two groups. Some comparisons are made for five cases of varying degrees of censoring and tying between probabilities from the exact test and those from the proposed test and these suggest the test is appropriate under certain conditions when the sample size is five in each group. A worked example is presented and some discussion is given to further problems.
3,318 citations
"The weighted log-rank class of perm..." refers methods in this paper
...The most commonly used tests are the log-rank and generalized Wilcoxon, also known as Peto-Prentice, tests (Gehan, 1965; Peto & Peto, 1972; Prentice, 1978)....
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01 Mar 1972
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2,331 citations