Conditional Studentized Survival Tests for Randomly Censored Models

It is shown that in the case of heterogenous censoring distributions Studentized survival tests can be carried out as conditional permutation tests given the order statistics and their censoring status. The result is based on a conditional central limit theorem for permutation statistics. It holds for linear test statistics as well as for sup-statistics. The procedure works under one of the following general circumstances for the two-sample problem: the unbalanced sample size case, highly censored data, certain non-convergent weight functions or under alternatives. For instance, the two-sample log rank test can be carried out asymptotically as a conditional test if the relative amount of uncensored observations vanishes asymptotically as long as the number of uncensored observations becomes infinite. Similar results hold whenever the sample sizes and are unbalanced in the sense that and hold.     

Block permutation principles for the change analysis of dependent data

We study an AMOC model with an abrupt change in the mean and dependent errors that form a linear process. Different kinds of statistics are considered, such as maximum-type statistics (particularly different CUSUM procedures) or sum-type statistics. Approximations of the critical values for change-point tests are obtained through permutation methods. The theoretical results show that the original test statistics and their corresponding block permutation counterparts follow the same distributional asymptotics. The main step in the proof is to obtain limit theorems for the corresponding rank statistics and then use laws of large numbers to obtain the permutation asymptotics conditionally on the given data.     

A toolbox of permutation tests for structural change

The sup $LM$ test for structural change is embedded into a permutation test framework for a simple location model. The resulting conditional permutation distribution is compared to the usual (unconditional) asymptotic distribution, showing that the power of the test can be clearly improved in small samples. Furthermore, the permutation test is embedded into a general framework that encompasses tools for binary and multivariate dependent variables as well as model-based permutation testing for structural change. It is also demonstrated that the methods can not only be employed for analyzing structural changes in time series data but also for recursive partitioning of cross-section data. The procedures suggested are illustrated using both artificial data and empirical applications (number of youth homicides, employment discrimination data, carbon flux in tropical forests, stock returns, and demand for economics journals).     

Exact permutation inference for two sample repeated measures data

Experiments in which very few units are measured many times sometimes present particular difficulties. Interest often centers on simple location shifts between two treatment groups, but appropriate modeling of the error distribution can be challenging. For example, normality may be difficult to verify, or a single transformation stabilizing variance or improving normality for all units and all measurements may not exist. We propose an analysis of two sample repeated measures data based on the permutation distribution of units. This provides a distribution free alternative to standard analyses. The analysis includes testing, estimation and confidence intervals. By assuming a certain structure in the location shift model, the dimension of the problem is reduced by analyzing linear combinations of the marginal statistics. Recently proposed algorithms for computation of two sample permutation distributions, require only a few seconds for experiments having as many as 100 units and any number of repeated measures. The test has high asymptotic efficiency and good power with respect to tests based on the normal distribution. Since the computational burden is minimal, approximation of the permutation distribution is unnecessary.     

Weighted Logrank Permutation Tests for Randomly Right Censored Life Science Data

In biomedical research, weighted logrank tests are frequently applied to compare two samples of randomly right censored survival times. We address the question how to combine a number of weighted logrank statistics to achieve good power of the corresponding survival test for a whole linear space or cone of alternatives, which are given by hazard rates. This leads to a new class of semiparametric projection tests that are motivated by likelihood ratio tests for an asymptotic model. We show that these tests can be carried out as permutation tests and discuss their asymptotic properties. A simulation study together with the analysis of a classical data set illustrates the advantages.     

Permutation Tests for Linear Models

Several approximate permutation tests have been proposed for tests of partial regression coefficients in a linear model based on sample partial correlations. This paper begins with an explanation and notation for an exact test. It then compares the distributions of the test statistics under the various permutation methods proposed, and shows that the partial correlations under permutation are asymptotically jointly normal with means 0 and variances 1. The method of Freedman & Lane (1983) is found to have asymptotic correlation 1 with the exact test, and the other methods are found to have smaller correlations with this test. Under local alternatives the critical values of all the approximate permutation tests converge to the same constant, so they all have the same asymptotic power. Simulations demonstrate these theoretical results.     

A general procedure for deriving distributions

A general procedure for deriving the exact and asymptotic distributions of a certain class of test statistics in multivariate analysis is proposed. The method is based on an asymptotic expansion of gamma ratios in terms of generalized Bernoulli polynomials. The exact and asymptotic results are obtained and the method is illustrated in the problem of testing linear hypotheses in the multinomial case. In this problem the method yields Box's (1949) expansion as a special case.     

On invariance and randomization in factorial designs with applications to D-optimal main effect designs of the symmetrical factorial

Under the setting of the columnwise orthogonal polynomial model in the context of the general factorial it is shown that (i) the determinant of the information matrix of a design relative to an admissible vector of effects is invariant under a permutation of levels; (ii) the unbiased estimation of a linear function of an admissible vector of effects can be obtained under equal probability randomization. These results extend the work on invariance and randomization carried out under the more restrictive assumption of the orthonormal polynomial model by Srivastava, Raktoe and Pesotan (1976) and Pesotan and Raktoe (1981). Moreover, the problem of the construction of D-optimal main effect designs in the symmetrical factorial is reduced to a study of a special class of (0,1)-matrices using the Helmat matrix model. Using this class of (0,1)-matrices and the determinant invariance result, some classes of D-optimal main effect designs of the s² and s³ factorial respectively are presented.     

Exact moments of the symmetric cubic assignment statistic

Hubert (1987Assignment Methods in Combinatorial Data Analysis) presented a class of permutation, or random assignment, techniques for assessing correspondence between general k-dimensional proximity measures on a set of “objects.” A major problem in higher-order assignment models is the prohibitive level of computation that is required. We present the first three exact moments of a test statistic for the symmetric cubic assignment model. Efficient computational formulas for the first three moments have been derived, thereby permitting approximation of the permutation distribution using well-known methods.     

Significance levels and confidence intervals for permutation tests

A computational algorithm is given which calculates exact significance levels of a wide class of permutation tests in the one and two sample problems. This class includes the permutation test based on the means, locally most powerful permutation tests and linear rank tests. When a shift model is assumed confidence intervals can also be obtained. Approximate methods, based on asymptotic expansions, are also presented.     

A general theory of hypothesis testing based on rankings

In nonparametric statistics, a hypothesis testing problem based on the ranks of the data gives rise to two separate permutation sets corresponding to the null and to the alternative hypothesis, respectively. A modification of Critchlow's unified approach to hypothesis testing is proposed. By defining the distance between permutation sets to be the average distance between pairs of permutations, one from each set, various test statistics are derived for the multi-sample location problem and the two-way layout. The asymptotic distributions of the test statistics are computed under both the null and alternative hypotheses. Some comparisons are made on the basis of the asymptotic relative efficiency.     

Wald,LM and LR test statistics of linear hypothese in a strutural equation model

For the linear hypothesis in a strucural equation model, the properties of test statistics based on the two stage least squares estimator (2SLSE) have been examined since these test statistics are easily derived in the instrumental variable estimation framework. Savin (1976) has shown that inequalities exist among the test statistics for the linear hypothesis, but it is well known that there is no systematic inequality among these statistics based on 2SLSE for the linear hypothesis in a structural equation model. Morimune and Oya (1994) derived the constrained limited information maximum likelihood estimator (LIMLE) subject to general linear constraints on the coefficients of the structural equation, as well as Wald, LM and Lr Test statistics for the adequacy of the linear constraints.

In this paper, we derive the inequalities among these three test statistics based on LIMLE and the local power functions based on Limle and 2SLSE to show that there is no test statistic which is uniformly most powerful, and the LR test statistic based on LIMLE is locally unbised and the other test statistics are not. Monte Carlo simulations are used to examine the actual sizes of these test statistics and some numerical examples of the power differences among these test statistics are given. It is found that the actual sizes of these test statistics are greater than the nominal sizes, the differences between the actual and nominal sizes of Wald test statistics are generally the greatest, those of LM test statistics are the smallest, and the power functions depend on the correlations between the endogenous explanatory variables and the error term of the structural equation, the asymptotic variance of estimator of coefficients of the structural equation and the number of restrictions imposed on the coefficients.     

Wald,LM and LR test statistics of linear hypothese in a strutural equation model

For the linear hypothesis in a strucural equation model, the properties of test statistics based on the two stage least squares estimator (2SLSE) have been examined since these test statistics are easily derived in the instrumental variable estimation framework. Savin (1976) has shown that inequalities exist among the test statistics for the linear hypothesis, but it is well known that there is no systematic inequality among these statistics based on 2SLSE for the linear hypothesis in a structural equation model. Morimune and Oya (1994) derived the constrained limited information maximum likelihood estimator (LIMLE) subject to general linear constraints on the coefficients of the structural equation, as well as Wald, LM and Lr Test statistics for the adequacy of the linear constraints.

In this paper, we derive the inequalities among these three test statistics based on LIMLE and the local power functions based on Limle and 2SLSE to show that there is no test statistic which is uniformly most powerful, and the LR test statistic based on LIMLE is locally unbised and the other test statistics are not. Monte Carlo simulations are used to examine the actual sizes of these test statistics and some numerical examples of the power differences among these test statistics are given. It is found that the actual sizes of these test statistics are greater than the nominal sizes, the differences between the actual and nominal sizes of Wald test statistics are generally the greatest, those of LM test statistics are the smallest, and the power functions depend on the correlations between the endogenous explanatory variables and the error term of the structural equation, the asymptotic variance of estimator of coefficients of the structural equation and the number of restrictions imposed on the coefficients.     

Saddlepoint p-values for a class of tests for comparing competing risks with censored data

One of the general problems in clinical trials and mortality rates is the comparison of competing risks. Most of the test statistics used for independent and dependent risks with censored data belong to the class of weighted linear rank tests in its multivariate version. In this paper, we introduce the saddlepoint approximations as accurate and fast approximations for the exact p-values of this class of tests instead of the asymptotic and permutation simulated calculations. Real data examples and extensive simulation studies showed the accuracy and stability performance of the saddlepoint approximations over different scenarios of lifetime distributions, sample sizes and censoring.     

Goodness of fit tests for the multiple logistic regression model

Several test statistics are proposed for the purpose of assessing the goodness of fit of the multiple logistic regression model. The test statistics are obtained by applying a chi-square test for a contingency table in which the expected frequencies are determined using two different grouping strategies and two different sets of distributional assumptions. The null distributions of these statistics are examined by applying the theory for chi-square tests of Moore Spruill (1975) and through computer simulations. All statistics are shown to have a chi-square distribution or a distribution which can be well approximated by a chi-square. The degrees of freedom are shown to depend on the particular statistic and the distributional assumptions.

The power of each of the proposed statistics is examined for the normal, linear, and exponential alternative models using computer simulations.     

p-Value for two-Sample Linear Rank Tests Using Permutation Simulations and the Normal Approximation Method

Test statistics from the class of two-sample linear rank tests are commonly used to compare a treatment group with a control group. Two independent random samples of sizes m and n are drawn from two populations. As a result, N = m + n observations in total are obtained. The aim is to test the null hypothesis of identical distributions. The alternative hypothesis is that the populations are of the same form but with a different measure of central tendency. This article examines mid p-values from the null permutation distributions of tests based on the class of two-sample linear rank statistics. The results obtained indicate that normal approximation-based computations are very close to the permutation simulations, and they provide p-values that are close to the exact mid p-values for all practical purposes.     

Optimal estimation of a finite population mean under generalized random permutation models

Employing certain generalized random permutation models and a general class of linear estimators of a finite population mean, it is shown that many of the conventional estimators are “optimal” in the sense of minimum average mean square error. Simple proofs are provided by using a well-known theorem on UMV estimation. The results also cover certain simple response error situations.     

Multivariate analysis with an autoregressive covariance model

This paper addresses the problem of testing the multivariate linear hypothesis when the errors follow an antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as a nonstationary autoregressive model of general order. Three test statistics are derived that provide analogs to three commonly used MANOVA statistics: Wilks' Lambda, the Lawley-Hotelling Trace, and Pillai's Trace. Formulas are given for each of these statistics that show how they can be obtained From any statistical computing package that calculates the usual MANOVA statistics. These antedependent statistics would be appropriate in analyzing certain multivariate data sets in which repeated measurements are taken on the same subjects over a period of time.     

Likelihood Ratio Tests for Fixed Model Terms using Residual Maximum Likelihood

Likelihood ratio tests for fixed model terms are proposed for the analysis of linear mixed models when using residual maximum likelihood estimation. Bartlett-type adjustments, using an approximate decomposition of the data, are developed for the test statistics. A simulation study is used to compare properties of the test statistics proposed, with or without adjustment, with a Wald test. A proposed test statistic constructed by dropping fixed terms from the full fixed model is shown to give a better approximation to the asymptotic χ²-distribution than the Wald test for small data sets. Bartlett adjustment is shown to improve the χ²-approximation for the proposed tests substantially.