A Note on Lancaster's Procedure for the Combination of Independent Events

Lancaster (1961) generalized Fisher's (1932) nonparametric procedure for combining independent p-values by transforming P_i from the i-th experiment to a chi-squared random variable with d_i degrees of freedom, with d_i not necessarily equal to 2. We explore the relationship between Lancaster's procedure and a weighted Lipták procedure (Koziol and Tuckwell, 1994) under which P_i is transformed to the standard normal scale. We investigate approximations to the null distribution of Lancaster's test procedure, chi-squared with d degrees of freedom. We find that the Cornish-Fisher (1960) expansions and the Lugannani-Rice (1980) saddlepoint approximations are quite accurate, for non-integral values of d, and for values of d as low as 20.     

Models of Independent Action for Combination Therapy

Two methods of applying Finney's concept of simple independent action are suggested for examining possible synergism in combination therapy. In one model the treatments act independently at one time to cure the disease. In the other model the treatments act independently continuously to prevent the disease from progressing to a terminal outcome. The implications of these models for the interaction terms of a proportional hazards model, a log-linear model on death rates, and a logistic model are discussed.     

Nonparametric Tests for the Gap Time Distributions of Serial Events Based on Censored Data

This article deals with the problem of comparing two populations with respect to the distribution of the gap time between two successive events when each subject can experience a series of events and when the event times are potentially right censored. Several families of nonparametric tests are developed, all of which allow arbitrary distributions and dependence structures for the serial events. The asymptotic and small-sample properties of the proposed tests are investigated. An illustration with data taken from a colon cancer study is provided. The related problem of testing the independence of two successive gap times is also studied.     

A Nonparametric Test for Differences in the Dispersion of Dependent Samples

A new statistic Δ to test the hypothesis of a difference in the dispersion of two dependent samples of ordinal data quality is proposed. It draws on the idea of rank assignment originally forwarded by Siegel and Tukey (1960). No exact probability levels can be given for this statistic for the time being, but it is shown that the statistic is linearly related to the so-called Hotelling-Pabst statistic D, and that one can use exact tables of the latter as a substitute in the statistical decision process with small samples. For larger samples, an approximation of Δ to the standard normal distribution is given. The problem of tied observations is not sufficiently solved yet. A conservative procedure of rank assignment is proposed as long as the exact distribution of Δ in the presence of ties is unknown.     

A nonparametric approach to the analysis of longitudinal data via a set of level crossing problems with application to the analysis of microarray time course experiments

Here we develop a completely nonparametric method for comparing two groups on a set of longitudinal measurements. No assumptions are made about the form of the mean response function, the covariance structure or the distributional form of disturbances around the mean response function. The solution proposed here is based on the realization that every longitudinal data set can also be thought of as a collection of survival data sets where the events of interest are level crossings. The method for testing for differences in the longitudinal measurements then is as follows: for an arbitrarily large set of levels, for each subject determine the first time the subject has an upcrossing and a downcrossing for each level. For each level one then computes the log rank statistic and uses the maximum in absolute value of all these statistics as the test statistic. By permuting group labels we obtain a permutation test of the hypothesis that the joint distribution of the measurements over time does not depend on group membership. Simulations are performed to investigate the power and it is applied to the area that motivated the method-the analysis of microarrays. In this area small sample sizes, few time points and far too many genes to consider genuine gene level longitudinal modeling have created a need for a simple, model free test to screen for interesting features in the data.     

A Bayesian Decision Procedure for Selecting Prognostic Variables Associated with Survival for Data in which Censoring is Prevalent

A Bayesian procedure is developed for the selection of concomitant variables in survival models. The variables are selected in a step-up procedure according to the criterion of maximum expected likelihood, where the expectation is over the prior parameter space. Prior knowledge of the influence of these covariates on patient prognosis is incorporated into the analysis. The step-up procedure is stopped when the Bayes factor in favor of omitting the variable selected in a particular step exceeds a specified value. The resulting model with the selected variables is fitted using Bayes estimates of the coefficients. This technique is applied to Hodgkin's disease data from a large Cooperative Clinical Trial Group and the results are compared to the results from the classical likelihood selection procedure.     

A Closed-form Jackknife Solution for the Behrens-Fisher Problem

A simple, closed-form jackknife estimate of the actual variance of the Mann-Whitney-Wilcoxon statistic, as opposed to the standard permutational variance under the test's null hypothesis has been derived which permits avoiding anticonservative performance in the presence of heterosce-dasticity. The formulation given allows modifications of the exponential scores test, of censored data tests by Gehan (1965), Peto & Peto (1977) and Prentice (1978), of tests for monotonic τ association by Kendall (1962) and for tests of ordered k-sample hypotheses. A Monte Carlo study supports recommendations for the jackknife procedures, but also shows their limited advantages in exponential scores and censored data versions. Thus, the paper extends results by Fligner & Policello (1981).     

Exact Calculation of Permutational Distributions for Two Independent Samples

A simple Fortran Subroutine is given for the exact calculation of permutational distributions for two independent samples. Important special cases are the Fisher-Pitman randomization test, the Mann-Whitney U-test, the Mantel-Haenszel-test and the exact test for 2×k contingency tables. The algorithm needs polynomial time and is of use even for personal- or microcomputers.     

Application of Multiple Test Procedures to the Combination of Multivariate and Univariate Tests with Varying Variable Sets

The positive ascertainment of location differences in a multivariate comparison of two or more groups gives rise to the question for the contribution of the single variables or of subsets of variables to the multivariate difference. In this paper two methods are proposed to accomplish the original multivariate test by tests in variable subsets or in single variables using a closed test procedure and Holm's procedure, respectively. Both control the multiple level of the whole procedure.     

A Simple Numerical Procedure for Estimating the Prior Probability in the Biomathematical Proof of Paternity

This note gives a simple procedure for finding the maximum likelihood estimate of the prior probabilities in paternity cases. The procedure is based on a fixed point principle.     

A New Nonparametric Approach to the Comparison of K Independent Samples of Response Curves II: A K Sample Generalization of the FRIEDMAN Test

A K sample generalization of the FRIEDMAN test (1937) is introduced which can be used as a nonparametric procedure for testing the homogeneity of the profiles of K independent samples of response curves measured at T identical points of time. While a similar procedure in LEHMACHER & WALL (1978), section 3, is based on T combined tests, each of them at level a/T, here a finite and asymptotic test is presented which is based on a single test statistic. The application of the new multivariate test is illustrated by the same numerical example as in LEHMACHER & WALL (1978). The properties of this test are discussed and compared with the combined test mentioned above.     

A New Nonparametric Approach to the Comparison of K Independent Samples of Response Curves

Two simultaneous distribution free test procedures to test the homogeneity of K samples of response curves measured at T points of time each are proposed which react upon the shape of the response curve. One procedure is based on the comparison of K T-tupels of FRIEDMAN-rank sums, the other on the comparison of K T × T-incidence tables, which first were introduced by KANNEMANN (1976). The application of the two procedures is illustrated by a numerical example. In the discussion the tests are compared with each other and with concurring tests.     

A semiparametric approach for the nonparametric transformation survival model with multiple covariates

The nonparametric transformation model makes no parametric assumptions on the forms of the transformation function and the error distribution. This model is appealing in its flexibility for modeling censored survival data. Current approaches for estimation of the regression parameters involve maximizing discontinuous objective functions, which are numerically infeasible to implement with multiple covariates. Based on the partial rank (PR) estimator (Khan and Tamer, 2004), we propose a smoothed PR estimator which maximizes a smooth approximation of the PR objective function. The estimator is shown to be asymptotically equivalent to the PR estimator but is much easier to compute when there are multiple covariates. We further propose using the weighted bootstrap, which is more stable than the usual sandwich technique with smoothing parameters, for estimating the standard error. The estimator is evaluated via simulation studies and illustrated with the Veterans Administration lung cancer data set.     

A Nonparametrical Evaluation of Anova and MANOVA Designs Using Interaction Structure Analysis

The interaction structure analysis (ISA) is proposed as a nonparametric procedure for the evaluation of uni- and multivariate analysis of variance models. Main effects and interactions of (independent) treatment variables are replaced by interaction-types where types are defined as those treatment-response combinations which occur significantly more often than expected under a null hypothesis (H_o) of no treatment effects. The application of ISA and the typological interpretation of ISA results are illustrated for an ANOVA design from toxicology and for a MANOVA design from psychopharmacology.     

A Multivariate Permutation Test for the Analysis of Arbitrarily Censored Survival Data

The exact generalization of GEHAN's (1965) two-sample test for arbitrarily censored survival data has been overlooked by subsequent work on the multisample problem. We give this general covariance matrix and show how it may be used in test procedures. While this permutation test is less powerful than its competitors in cases where both apply, it may be used on types of data not previously discussed.     

A Procedure for the Common Estimation of Parameters Corresponding to Several Treatment Groups

In this article we give a procedure for the common estimation of parameters corresponding to several treatment groups. Thereby we assume that the distribution functions of the groups belong to the same family and differ only in the parameter values. The procedure allows the common estimation of some of these parameters. The parameters themselves will be estimated by the maximum likelihood method; the estimators will be calculated iteratively by the Newton-Raphson method. To prove if the common estimation is possible, we propose as a suitable test the maximum likelihood ratio test. Finally we show the application of our procedure in the case of the probit analysis.     

A set of highly discriminating microsatellite loci for the Galápagos marine iguana Amblyrhynchus cristatus

We describe here the cloning of 12 (7 dinucleotide, 1 trinucleotide and 4 tetranucleotide) microsatellite loci for the Galápagos marine iguana Amblyrhynchus cristatus. When tested for individuals from five different island populations on the Galápagos archipelago, high genetic diversities (9–20 alleles per locus) and heterozygosities (0.200–0.944) were observed. All loci showed no obvious deviations from Hardy–Weinberg equilibrium. The new set of microsatellite loci was able to assign individuals reliably to their island of origin, thus being able to discriminate between residents and migrants between islands.