A class of robust stepwise alternatives to Hotelling's T 2 tests

Hotelling's T 2 test is known to be optimal under multivariate normality and is reasonably validity-robust when the assumption fails. However, some recently introduced robust test procedures have superior power properties and reasonable type I error control with non-normal populations. These, including the tests due to Tiku & Singh (1982), Tiku & Balakrishnan (1988) and Mudholkar & Srivastava (1999b, c), are asymptotically valid but are useful with moderate size samples only if the population dimension is small. A class of B-optimal modifications of the stepwise alternatives to Hotellings T 2 introduced by Mudholkar & Subbaiah (1980) are simple to implement and essentially equivalent to the T 2 test even with small samples. In this paper we construct and study the robust versions of these modified stepwise tests using trimmed means instead of sample means. We use the robust one- and two-sample trimmed- t procedures as in Mudholkar et al. (1991) and propose statistics based on combining them. The results of an extensive Monte Carlo experiment show that the robust alternatives provide excellent type I error control and a substantial gain in power.     

Likelihood ratio test for testing equality of location parameters of two exponential distributions from doubly censored samples

The Likelihood Ratio (LR) test for testing equality of two exponential distributions with common unknown scale parameter is obtained. Samples are assumed to be drawn under a type II doubly censored sampling scheme. Effects of left and right censoring on the power of the test are studied. Further, the performance of the LR test is compared with the Tiku(1981) test.     

Testing the equality of variance-covariance matrices the robust way

The robust statistic T² _Dproposed by Tiku and Singh (1982) for testing the equality of mean vectors of two mu1 t ivariate populations is modified to test the equality of variance-covariance matrices.     

Testing equality of location parameters of two exponential distributions from censored samples

A modification to Tiku's (1981) test, which may be seriously biased, is proposed. The modified test is only marginally biased if at all and is substantially more powerful. A ratio test based on Tiku’s (1967) modified likelihood function is also proposed, and shown to have power comparable to the power of the ratio test based on the likelihood function. The proposed ratio test is, however, much easier from a computational viewpoint.     

Generalization of the robust bivariate t2 statistic to multivariate populations

The robust bivariate Hotelling–type T² statistics proposed by Tiku and Balakrishnan (1988) is extendend to p–variate (p ≧ 3) populations.     

Robust tests for equality of two population means under the normal model

A robust procedure is developed for testing the equality of means in the two sample normal model. This is based on the weighted likelihood estimators of Basu et al. (1993). When the normal model is true the tests proposed have the same asymptotic power as the two sample Student's t-statistic in the equal variance case. However, when the normality assumptions are only approximately true the proposed tests can be substantially more powerful than the classical tests. In a Monte Carlo study for the equal variance case under various outlier models the proposed test using Hellinger distance based weighted likelihood estimator compared favorably with the classical test as well as the robust test proposed by Tiku (1980).     

Using Auxiliary Time-Dependent Covariates to Recover Information in Nonparametric Testing with Censored Data

Murrayand Tsiatis (1996) described a weighted survival estimate thatincorporates prognostic time-dependent covariate informationto increase the efficiency of estimation. We propose a test statisticbased on the statistic of Pepe and Fleming (1989, 1991) thatincorporates these weighted survival estimates. As in Pepe andFleming, the test is an integrated weighted difference of twoestimated survival curves. This test has been shown to be effectiveat detecting survival differences in crossing hazards settingswhere the logrank test performs poorly. This method uses stratifiedlongitudinal covariate information to get more precise estimatesof the underlying survival curves when there is censored informationand this leads to more powerful tests. Another important featureof the test is that it remains valid when informative censoringis captured by the incorporated covariate. In this case, thePepe-Fleming statistic is known to be biased and should not beused. These methods could be useful in clinical trials with heavycensoring that include collection over time of covariates, suchas laboratory measurements, that are prognostic of subsequentsurvival or capture information related to censoring.     

A robust procedure for testing the equality of mean vectors of two bivariate populations with unequal covariance matrices

A robust test is developed for testing equality of the mean vectors of two bivariate (multivariate) populations when the variance-covariance matrices are not necessarily equal. The test is an extension of the univariate robust test given by Tiku and Singh (1981).     

Distance Testing for Selecting the Best Population

Consider testing the null hypothesis that a given population has location parameter greater than or equal to the largest location parameter of k competing populations. This paper generalizes tests proposed by Gupta and Bartholomew by considering tests based on p -distances from the parameter estimate to the null parameter space. It is shown that all tests are equivalent when k →∞ for a class of distributions that includes the normal and the uniform. The paper proposes the use of adaptive quantiles. Under suitable assumptions the resulting tests are asymptotically equivalent to the uniformly most powerful test for the case that the location parameters of all but one of the populations are known. The increase in power obtained by using adaptive tests is confirmed by a simulation study.     

A test for equality of variances with censored samples

A variance homogeneity test for type II right-censored samples is proposed. The test is based on Bartlett's statistic. The asymptotic distribution of the statistic is investigated. The limiting distribution is that of a linear combination of i.i.d. chi-square variables with 1 degree of freedom. By using simulation, the critical values of the null distribution of the modified Bartlett's statistic for testing the homogeneity of variances of two normal populations are obtained when the sample sizes and censoring levels are not equal. Also, we investigate the properties of the proposed test (size, power and robustness). Results show that the distribution of the test statistic depends on the censoring level. An example of the use of the new methodology in animal science involving reproduction in ewes is provided.     

Most powerful tests for use in matched pair experiments when data may be censored

Four distribution-free tests are developed for use in matched pair experiments when data may be censored: a bootstrap based on estimates of the median difference, and three rerandomization tests. The latter include a globally almost most powerful (GAMP) test which uses the original data and two modified Gilbert-Gehan tests which use the ranks. Computation time is reduced by using a binary count to generate subsamples and by restricting subsampling to the uncensored pairs. In Monte Carlo simulations against normal alternatives, mixed normal alternatives, and exponential alternatives, the GAMP test is most powerful with light censoring, the rank test is most powerful with heavy censoring. The bootstrap degenerates to the sign test and is least powerful.     

Empirical Likelihood for Censored Linear Regression

In this paper we investigate the empirical likelihood method in a linear regression model when the observations are subject to random censoring. An empirical likelihood ratio for the slope parameter vector is defined and it is shown that its limiting distribution is a weighted sum of independent chi-square distributions. This reduces to the empirical likelihood to the linear regression model first studied by Owen (1991) if there is no censoring present. Some simulation studies are presented to compare the empirical likelihood method with the normal approximation based method proposed in Lai et al. (1995). It was found that the empirical likelihood method performs much better than the normal approximation method.     

A TEST OF HOMOGENEITY AGAINST SCALE ALTERNATIVES USING SUBSAMPLE EXTREMA1

In this paper a new class of non-parametric tests for testing homogeneity of several populations against scale alternatives is proposed. For this, independent samples of fixed sizes are drawn from each population and from these samples, all possible sub-samples of the same size are drawn and their maxima and minima are computed. Using these extreme the class of tests is obtained. Tests of this type have been offered for the two-sample slippage problem by Kochar (1978). Under certain conditions, this class of tests is shown to be consistent against ‘difference in scale’ alternatives. The test has been compared with Bhapkar's V-test (1961), Deshpande's D-test (1965), Sugiura's D_rs-test (1965) and with a classical test given by Lehmann (1959, pp. 273–275). It is shown that some members of this proposed class of tests are more efficient than the first three tests in the case of uniform, Laplace and normal distributions, when the number of populations compared is small.     

C55. Quick and accurate rule of thumb for determining the critical value of the coefficient of correlation

Life table analysis techniques in epidemiology depend upon the asymptotic properties of the statistical test methods employed. In some instances, the statistical procedures indicate highly significant results which are, in reality, unjustified. The phenomenon may occur when the asymptotic methods are applied in situations where the cases of interest are few in number. This situation is illustrated by the 20 multiple myeloma deaths observed in the RERF Life Span Study cohort. A permutation test is applied to the life table data, although the test requires the false assumption that the censoring distribution is independent of the radiation dose. A simulation test is developed which does not require equal censoring, which has the same asymptotics as the usual test methods, and which is less likely to overestimate significance in small samples. It is found that both of these small-sample tests provide reasonable numerical solutions. In addition, the simulation test is recommended in general for analyzing life table data with unequal censoring. Finally, by using the small-sample tests, the frequency of death from multiple myeloma is shown to be positively associated with radiation dose (P<0.01).     

Data depth‐based nonparametric scale tests

Liu and Singh (1993, 2006) introduced a depth‐based d‐variate extension of the nonparametric two sample scale test of Siegel and Tukey (1960). Liu and Singh (2006) generalized this depth‐based test for scale homogeneity of k ≥ 2 multivariate populations. Motivated by the work of Gastwirth (1965), we propose k sample percentile modifications of Liu and Singh's proposals. The test statistic is shown to be asymptotically normal when k = 2, and compares favorably with Liu and Singh (2006) if the underlying distributions are either symmetric with light tails or asymmetric. In the case of skewed distributions considered in this paper the power of the proposed tests can attain twice the power of the Liu‐Singh test for d ≥ 1. Finally, in the k‐sample case, it is shown that the asymptotic distribution of the proposed percentile modified Kruskal‐Wallis type test is χ² with k ? 1 degrees of freedom. Power properties of this k‐sample test are similar to those for the proposed two sample one. The Canadian Journal of Statistics 39: 356–369; 2011 © 2011 Statistical Society of Canada     

Testing equality of population variances the robust way

The purpose of this paper is to investigate the robustness (stability of Type I error to deviations from normality) and power properties of various tests for testing equality of population variances. It is shown that the tests based on Tiku’ s (1967, 1980, 1982) MML estimators have good robustness properties and are the most powerful overall.     

An analytic method for randomized trials with informative censoring: Part 1

Consider a randomized trial in which time to the occurrence of a particular disease, say pneumocystis pneumonia in an AIDS trial or breast cancer in a mammographic screening trial, is the failure time of primary interest. Suppose that time to disease is subject to informative censoring by the minimum of time to death, loss to and end of follow-up. In such a trial, the censoring time is observed for all study subjects, including failures. In the presence of informative censoring, it is not possible to consistently estimate the effect of treatment on time to disease without imposing additional non-identifiable assumptions. The goals of this paper are to specify two non-identifiable assumptions that allow one to test for and estimate an effect of treatment on time to disease in the presence of informative censoring. In a companion paper (Robins, 1995), we provide consistent and reasonably efficient semiparametric estimators for the treatment effect under these assumptions. In this paper we largely restrict attention to testing. We propose tests that, like standard weighted-log-rank tests, are asymptotically distribution-free -level tests under the null hypothesis of no causal effect of treatment on time to disease whenever the censoring and failure distributions are conditionally independent given treatment arm. However, our tests remain asymptotically distribution-free -level tests in the presence of informative censoring provided either of our assumptions are true. In contrast, a weighted log-rank test will be an -level test in the presence of informative censoring only if (1) one of our two non-identifiable assumptions hold, and (2) the distribution of time to censoring is the same in the two treatment arms. We also extend our methods to studies of the effect of a treatment on the evolution over time of the mean of a repeated measures outcome, such as CD-4 count.     

Gap-ratio goodness of fit tests for weibull or extreme value distribution assumptions with left or right censored data

With data collection in environmental science and bioassay, left censoring because of nondetects is a problem. Similarly in reliability and life data analysis right censoring frequently occurs. There is a need for goodness of fit tests that can adapt to left or right censored data and be used to check important distributional assumptions without becoming too difficult to regularly implement in practice. A new test statistic is derived from a plot of the standardized spacings between the order statistics versus their ranks. Any linear or curvilinear pattern is evidence against the null distribution. When testing the Weibull or extreme value null hypothesis this statistic has a null distribution that is approximately F for most combinations of sample size and censoring of practical interest. Our statistic is compared to the Mann-Scheuer-Fertig statistic which also uses the standardized spacings between the order statistics. The results of a simulation study show the two tests are competitive in terms of power. Although the Mann-Scheuer-Fertig statistic is somewhat easier to compute, our test enjoys advantages in the accuracy of the F approximation and the availability of a graphical diagnostic.     

A nonparametric test for current status data with unequal censoring

Nonparametric tests for the comparison of different treatments based on current status data are proposed. For this problem, most methods proposed in the literature require that observation times on all subjects follow the same distribution. In other words, censoring distributions are identical between the treatment groups. In this paper, we focus on the situation where the censoring distributions may be different for subjects in different treatment groups and the test that can take this unequal censoring into account is given. The asymptotic distribution of the test proposed is derived. The method proposed is applied to data arising from a tumorigenicity experiment.     

Testing for a finite mixture model with two components

Summary.  We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k =2 versus k 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.