Order test for high-dimensional two-sample means

We propose a new method to test the order between two high-dimensional mean curves. The new statistic extends the approach of Follmann (1996) to high-dimensional data by adapting the strategy of Bai and Saranadasa (1996). The proposed procedure is an alternative to the non-negative basis matrix factorization (NBMF) based test of Lee et al. (2008) for the same hypothesis, but it is much easier to implement. We derive the asymptotic mean and variance of the proposed test statistic under the null hypothesis of equal mean curves. Based on theoretical results, we put forward a permutation procedure to approximate the null distribution of the new test statistic. We compare the power of the proposed test with that of the NBMF-based test via simulations. We illustrate the approach by an application to tidal volume traces.     

The new robust two-sample test for randomly right-censored data

Using a minimum p-value principle, a new two-sample test MIN3 is proposed in the paper. The cumulative distribution function of the MIN3 test statistic is studied and approximated by the Beta distribution of the third kind. Lower percentage points of the distribution of the new test statistic under the null hypothesis are computed. Also the test power for a lot of types of alternative hypotheses (with 0, 1 and 2 point(-s) of the intersection(-s) of survival functions) is studied and we found that the usage of the MIN3 test is a preferred strategy by the Wald and Savage decision-making criteria under risk and uncertainty. The results of application of the MIN3 test are shown for two examples from lifetime data analysis.     

A nonparametric test for diagnosis of the proportionality assumption

We propose a nonparametric test for diagnosis of the proportionality assumption between hazard functions based on a functional equation. Because of involvement of censoring distribution, we consider the test procedure in an asymptotic manner and obtain the asymptotic normality for the proposed test statistic. Then we discuss the rationale of use of the functional equation for the initial effect model. Finally we compare our test with others using an example.     

Moments of generalized quadratic forms and distributions of certain multivariate test statistics

Summary Moments and distributions of quadratic forms or quadratic expressions in normal variables are available in literature. Such quadratic expressions are shown to be equivalent to a linear function of independent central or noncentral chi-square variables. Some results on linear functions of generalized quadratic forms are also available in literature. Here we consider an arbitrary linear function of matrix-variate gamma variables. Moments of the determinant of such a linear function are evaluated when the matrix-variate gammas are independently distributed. By using these results, arbitrary non-null moments as well as the non-null distribution of the likelihood ratio criterion for testing the hypothesis of equality of covariance matrices in independent multivariate normal populations are derived. As a related result, the distribution of a linear function of independent matrix-variate gamma random variables, which includes linear functions of independent Wishart matrices, is also obtained. Some properties of generalized special functions of several matrix arguments are used in deriving these results.     

The limit test statistic distribution of the maximum value test for right-censored data

ABSTRACT

In this paper, the maximum value test is proposed and considered for two-sample problem solving with lifetime data. This test is a distribution-free test under non-censoring and is a not distribution-free test under censoring. The formula of the limit distribution of the proposed maximal value test is represented in the general case. The distribution of the test statistic has been studied experimentally. Also, we propose the estimate of a p-value calculation of the maximum value test instead of the Monte-Carlo simulation. This test is useful and applicable in case of choosing among the logrank test, the Cox–Mantel test, the Q test and Generalized Wilcoxon tests, for instance, the Gehan's Generalized Wilcoxon test and the Peto and Peto's Generalized Wilcoxon test.     

A new two-sample test for choosing between log-rank and Wilcoxon tests with right-censored data

When differences of survival functions are located in early time, a Wilcoxon test is the best test, but when differences of survival functions are located in late time, using a log-rank test is better. Therefore, a researcher needs a stable test in these situations. In this paper, a new two-sample test is proposed and considered. This test is distribution-free. This test is useful for choosing between log-rank and Wilcoxon tests. Its power is roughly the maximal power of the log-rank test and Wilcoxon test.     

The exact distribution function of the ratio of two dependent quadratic forms

A closed-form representation of the distribution function of the ratio of two linear combinations of Chi-squared variables is derived. The ratio is of the following form R = (X + aY)/(bY + Z), where X, Y, Z are independent Chi-square variables and a, b > 0. Two methods of obtaining the distribution function of this ratio are used. The exact density function of such a ratio is then obtained by differentiation. Two numerical examples are provided.     

A modified chi-square test for Bertholon model with censored data

In this work, we propose the construction of a chi-squared goodness-of-fit test in censored data case, for Bertholon model which can analyse various competing risks of failure or death. This test is based on a modification of the Nikulin-Rao-Robson (NRR) statistic proposed by Bagdonavicius and Nikulin (2011a Bagdonavicius, V., Nikulin, M. (2011a). Chi-squared tests for general composite hypotheses from censored samples. Comptes Rendus Mathématiques: Series I 349(3–4):219–223. [Google Scholar], 2011b Bagdonavicius, V., Nikulin, M. (2011b). Chi-squared goodness-of-fit test for right censored data. International Journal of Applied Mathematics and Statistics 24:30–50. [Google Scholar]) for censored data. We applied this test to numerical examples from simulated samples and real data.     

A nonparametric test for Type I censored data: r samples

This paper is concerned with testing that r random samples are all from the same population when the data are left (or right) censored. A statistic is developed which has elements of a goodness-of-fit statistic and of a modified Kruskal-Wallis statistic. The efficiency of this statistic relative to some other commonly used statistics is calculated. Some Monte Carlo comparisons are given.     

Distribution theory of quadratic forms for matrix multivariate elliptical distribution

This paper proposes the density and characteristic functions of a general matrix quadratic form X^(?)AX${\mathbf{X}}^{(?)}\mathbf{AX}$, when A=A^(?)$\mathbf{A}={\mathbf{A}}^{(?)}$ is a positive semidefinite matrix, X$\mathbf{X}$ has a matrix multivariate elliptical distribution and X^(?)${\mathbf{X}}^{(?)}$ denotes the usual conjugate transpose of X$\mathbf{X}$. These results are obtained for real normed division algebras. With particular cases we obtained the density and characteristic functions of matrix quadratic forms for matrix multivariate normal, Pearson type VII, t and Cauchy distributions.     

A two-sample test for mean functions with increasing number of projections

We propose a test for equality of two means when data are functions and obtain the asymptotic properties of the test statistic as data dimension increases with the sample size. We also derive the asymptotic power of the test under some local alternatives and show that the test statistic is root-n consistent. A simulation study is conducted to evaluate the performance of the test numerically and to compare the proposed test with other existing four popular tests.     

Symbolic computation for approximating distributions of some families of one and two-sample nonparametric test statistics

A conditional saddlepoint approximation was provided by Gatto and Jammalamadaka (1999) for computing the distribution function of many test statistics based on dependent quantities like multinomial frequencies, spacing frequencies, etc. The considerable complexity of the formulas involved can be bypassed by symbolic computation. This article illustrates the effectiveness of symbolic computation to evaluate the saddlepoint approximation for the likelihood ratio, the exponential score, and the Wald-Wolfowitz test statistics. The case of composite hypotheses is also discussed.     

The rank von neumann test as a test for autocorrelation in regression models

The rank Von Neumann test, which performs extremely well as a test for serial correlation in raw data, is here compared with the Durbin-Watson and Geary tests as a test for autocorrelation in regression residuals. The test convincingly outperforms the Geary test but it is less robust than the Durbin-Watson test     

An extensive power evaluation of a novel two-sample density-based empirical likelihood ratio test for paired data with an application to a treatment study of attention-deficit/hyperactivity disorder and severe mood dysregulation

In many case-control studies, it is common to utilize paired data when treatments are being evaluated. In this article, we propose and examine an efficient distribution-free test to compare two independent samples, where each is based on paired observations. We extend and modify the density-based empirical likelihood ratio test presented by Gurevich and Vexler [7] to formulate an appropriate parametric likelihood ratio test statistic corresponding to the hypothesis of our interest and then to approximate the test statistic nonparametrically. We conduct an extensive Monte Carlo study to evaluate the proposed test. The results of the performed simulation study demonstrate the robustness of the proposed test with respect to values of test parameters. Furthermore, an extensive power analysis via Monte Carlo simulations confirms that the proposed method outperforms the classical and general procedures in most cases related to a wide class of alternatives. An application to a real paired data study illustrates that the proposed test can be efficiently implemented in practice.     

Confidence intervals for secondary parameters following a sequential test

In sequential studies, formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest. Valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study. However, the completed data set may warrant a more detailed analysis, involving the estimation of parameters corresponding to effects that were not used to determine when to stop, and yet correlated with those that were. This paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used. The method assumes that information accumulates on the primary and secondary parameters at proportional rates. This requirement will be valid in many potential applications, but only in limited situations in survival analysis.     

Mood's test for dispersion as a counting of triplets

The well-known equivalence of Wilcoxon and Mann-Whitney location statistics is herein extended to dispersion tests. Mood (1954) statistic is related to a statistic based on “triplets”. The triplet version of Mood statistic is useful for proving the asymptotic normality (under alternatives) of the test.     

An exact density-based empirical likelihood ratio test for paired data

The Wilcoxon rank-sum test and its variants are historically well-known to be very powerful nonparametric decision rules for testing no location difference between two groups given paired data versus a shift alternative. In this title, we propose a new alternative empirical likelihood (EL) ratio approach for testing the equality of marginal distributions given that sampling is from a continuous bivariate population. We show that in various shift alternative scenarios the proposed exact test is superior to the classic nonparametric procedures, which may break down completely or are frequently inferior to the density-based EL ratio test. This is particularly true in the cases where there is a nonconstant shift under the alternative or the data distributions are skewed. An extensive Monte Carlo study shows that the proposed test has excellent operating characteristics. We apply the density-based EL ratio test to analyze real data from two medical studies.     

Evaluation of a sequential global test of improved recovery following stroke as applied to the ICTUS trial of citicoline

The International Citicoline Trial in acUte Stroke is a sequential phase III study of the use of the drug citicoline in the treatment of acute ischaemic stroke, which was initiated in 2006 in 56 treatment centres. The primary objective of the trial is to demonstrate improved recovery of patients randomized to citicoline relative to those randomized to placebo after 12 weeks of follow-up. The primary analysis will take the form of a global test combining the dichotomized results of assessments on three well-established scales: the Barthel Index, the modified Rankin scale and the National Institutes of Health Stroke Scale. This approach was previously used in the analysis of the influential National Institute of Neurological Disorders and Stroke trial of recombinant tissue plasminogen activator in stroke.The purpose of this paper is to describe how this trial was designed, and in particular how the simultaneous objectives of taking into account three assessment scales, performing a series of interim analyses and conducting treatment allocation and adjusting the analyses to account for prognostic factors, including more than 50 treatment centres, were addressed.