robustness and power of analysis of covariance applied to ordinal scaled data as arising in randomized controlled trials in clinical trials comparing two treatments ordinal scales of three four or five points are often used to assess severity both prior to and after treatment analysis of covariance is an attractive technique however the data clearly violate the normality assumption and in the presence of small samples and large sample theory may not apply the robustness and power of various versions of parametric analysis of covariance applied to small samples of ordinal scaled data are investigated through computer simulation subjects are randomized to one of two competing treatments and the pre treatment or baseline assessment is used as the covariate we compare two parametric analysis of covariance tests that vary according to the treatment of the homogeneity of regressions slopes and the two independent samples t test on difference scores under the null hypothesis of no difference in adjusted treatment means we estimated actual significance levels by comparing observed test statistics to appropriate critical values from the f and t distributions for nominal significance levels of 0 10 0 05 0 02 and 0 01 we estimated power by similar comparisons under various alternative hypotheses the model which assumes homogeneous slopes and the t test on difference scores were robust in the presence of three four and five point ordinal scales the hierarchical approach which first tests for homogeneity of regression slopes and then fits separate slopes if there is significant non homogeneity produced significance levels that exceeded the nominal levels especially when the sample sizes were small the model which assumes homogeneous regression slopes produced the highest power among competing tests for all of the configurations investigated the t test on difference scores also produced good power in the presence of small samples

maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

robustness and power of analysis of covariance applied to data distorted from normality by floor effects homogeneous regression slopes we investigate through computer simulations the robustness and power of two group analysis of covariance test applied to small samples distorted normality by floor effects when the regression slopes are homogeneous we consider four parametric analysis of covariance tests that vary according to the treatment of the homogeneity of regression slopes and two t tests on unadjusted means and on difference scores under the null hypothesis of no difference in means we estimated actual significance levels by comparing observed test statistics to appropriate values from the f and t distributions for nominal significance levels of 0 10 0 05 0 02 and 0 01 we estimated power by similar comparisons under various alternative hypothesis the hierarchical approach that adjusts for non homogeneous slopes if found significant the test that assumes homogeneous regression slopes and the test that estimates separate regression slopes in each treatment were robust in general each test produced power at least equal to that expected from normal theory the textbook approach which does not test for mean differences when there is significant non homogeneity was conservative but also had good power the t tests were robust but had poorer power properties than the above procedures

statistical properties of a nonuniformly hyperbolic map of the interval we prove a power law upper bound for the decay of the correlations for holder observables in the case of a nonuniformly hyperbolic map of the interval introduced by gaspard and wang as a piecewise linear approximation of the intermittent map of manneville pomeau the result is then applied to compute the central limit theorem for the same class of observables

determining the appropriate sample size for nonparametric tests for location shift for the case in which two independent samples arc to be compared using a nonparametric test for location shift we propose a bootstrap technique for estimating the sample sizes required to achieve a specified power the estimator called boot uses information from a small pilot experiment for the special case of the wilcoxon test a simulation study is conducted to compare boot to two other sample size estimators one method called anpv is based on the assumption that the underlying distribution is normal with a variance estimated from the pilot data the other method called noether adapts the sample size formula of noether for use with a location shift alternative the boot and noether sample size estimators are particularly appropriate for this nonparametric setting because they do not require assumptions about the shape of the underlying continuous probability distribution the simulation study shows that a sample size estimates can have large uncertainty b boot is at least as accurate as an

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

power and sample size calculations for discrete bounded outcome scores we consider power and sample size calculations for randomized trials with a bounded outcome score bos as primary response adjusted for a priori chosen covariates we define bos to be a random variable restricted to a finite interval typically a bos has a j or u shaped distribution hindering traditional parametric methods of analysis when no adjustment for covariates is needed a non parametric test could be chosen however there is still a problem with calculating the power since the common location shift alternative does not hold in general for a bos in this paper we consider a parametric approach and assume that the observed bos is a coarsened version of a true bos which has a logit normal distribution in each treatment group allowing correction for covariates a two step procedure is used to calculate the power firstly the power function is defined conditionally on the covariate values secondly the marginal power is obtained by averaging the conditional power with respect to an assumed distribution for the covariates using monte carlo integration a simulation study evaluates the performance of our method which is also applied to the ecass 1 stroke study

estimating the power of the two sample wilcoxon test for location shift summary traditional methods for calculating the power of a statistical test for location shift require knowledge of the shape of the underlying probability distribution the distribution shape however may be unknown this paper describes a bootstrap method for using observed data or pilot data to approximate the power no assumptions need be made about the shape of the underlying continuous probability distribution simulation evidence shows that when applied to the wilcoxon two sample test for location shift the suggested method is reliable the evidence also shows that it is more accurate than a benchmark traditional approach the bootstrap method is applied to a real data example the analysis demonstrates how the method can be used to determine sample sizes and how to choose the more powerful of two alternative tests for location shift

robustness of the two independent samples t test when applied to ordinal scaled data one may encounter the application of the two independent samples t test to ordinal scaled data for example data that assume only the values 0 1 2 3 from small samples this situation clearly violates the underlying normality assumption for the t test and one cannot appeal to large sample theory for validity in this paper we report the results of an investigation of the t test s robustness when applied to data of this form for samples of sizes 5 to 20 our approach consists of complete enumeration of the sampling distributions and comparison of actual levels of significance with the significance level expected if the data followed a normal distribution we demonstrate under general conditions the robustness of the t test in that the maximum actual level of significance is close to the declared level

calculation of power and sample size with bounded outcome scores the two sample wilcoxon rank sum test is the most popular non parametric test for the comparison of two samples when the underlying distributions are not normal although the underlying distributions need not be known in detail to calculate the null distribution of the test statistic parametric assumptions are often made to determine the power of the test or the sample size we encountered difficulties with this approach in the planning of a recent clinical trial in stroke patients it is shown that for power and sample size estimation it can be dangerous to apply the classical formulae routinely especially with outcome scores having a u shaped or a j shaped distribution as an example we have taken the barthel index a quality of life outcome measure in stroke patients further we have investigated alternative methods by means of monte carlo simulation the distributional characteristics of the estimated powers were compared our findings suggest more appropriate computer software is necessary for the calculation of power and sample size when efficacy is measured by a non parametric method

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

on minimal surfaces in a kahler manifold of constant holomorphic sectional curvature this paper studies minimal surfaces in kshler manifolds of constant holomorphic sectional curvature using the technique of the moving frame in particular we provide a classification of the minimal two spheres in cpst complex projective sf space equipped with the fubini study metric this classification can be described as follows to each holomorphic curve in c pt1 classically there is associated a particular framing of c l l called the frenet frame each element of the frenet frame induces a minimal surface in cp the classification theorem states that all minimal surfaces of topological type of the two sphere occur in this manner the theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in kshler manifolds of constant holomorphic sectional curvature together with the riemann roch theorem

sample size calculations for ordered categorical data many clinical trials yield data on an ordered categorical scale such as very good good moderate poor under the assumption of proportional odds such data can be analysed using techniques of logistic regression in simple comparisons of two treatments this approach becomes equivalent to the mann whitney test in this paper sample size formulae consistent with an eventual logistic regression analysis are derived the influence on efficiency of the number and breadth of categories will be examined effects of misclassification and of stratification are discussed and examples of the calculations are given

the use of bootstrap methods for estimating sample size and analysing health related quality of life outcomes health related quality of life hrqol measures are increasingly used in trials as primary outcome measures investigators are now asking statisticians for advice on how to plan and analyse studies using such outcomes hrqol outcomes like the sf 36 are usual measured on an ordinal scale although most investigators assume that there exists an underlying continuous latent variable and that the actual measured outcomes the ordered categories reflect contiguous intervals along this continuum the ordinal scaling of hrqol measures means they tend to generate data that have discrete bounded and skewed distributions thus standard methods of analysis that assume normality and constant variance may not be appropriate for this reason conventional statistical advice would suggest non parametric methods be used to analyse hrqol data the bootstrap is one such computer intensive non parametric method for estimating sample sizes and analysing data we describe three methods of estimating sample sizes for two group cross sectional comparisons of hrqol outcomes we then compared the power of the three methods for a two group cross sectional study design using bootstrap simulation the results showed that under the location shift alternative hypothesis conventional methods of sample size estimation performed well particularly whitehead s method whitehead s method is recommended if the hrqol outcome has a limited number of discrete values 7 and or the expected proportion of cases at either of the bounds is high if a pilot data set is readily available then bootstrap simulation will provide a more accurate and reliable estimate than conventional methods finally we used the bootstrap for hypothesis testing and the estimation of standard errors and confidence intervals for parameters in an example data set we then compared and contrasted the bootstrap with standard methods of analysing hrqol outcomes in the data set studied with the sf 36 outcome the use of the bootstrap for estimating sample sizes and analysing hrqol data produces results similar to conventional statistical methods these results suggest that bootstrap methods are not more appropriate for analysing hrqol outcome data than standard methods

choice of test for comparing two groups with particular application to skewed outcomes we consider a clinical trial where a skewed outcome variable is to be compared between two groups while comparison of sample means may lack power we show that power also depends on the nature of the anticipated treatment effect for any given distribution in the control arm there is a family of true distributions in the intervention arm for which the most powerful test is a comparison of arithmetic means similar results hold for a comparison of geometric means and approximately for the wilcoxon rank sum test and a comparison of medians we discuss how these methods could be used in planning the analysis of a clinical trial in which the intervention effect alters the shape of the distribution these ideas are illustrated by a trial in community psychiatry where the primary outcome days in hospital was highly skewed but the intervention was mainly expected to reduce the frequency of values in the tail we show that a comparison of sample means is a reasonable choice in this case despite the skewness

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

decay of correlations for certain quadratic maps we prove exponential decay of correlations for f wheref belongs in a positive measure set of quadratic maps of the interval and is its absolutely continuous invariant measure these results generalize to other interval maps

a bootstrap semiparametric homogeneity test for the distributions of multigroup proportional data with applications to analysis of quality of life outcomes in clinical trials this article is concerned about the test for the difference in the distributions of multigroup proportional data which is motivated by the problem of comparing the distributions of quality of life qol outcomes among different treatment groups in clinical trials the proportional data such as qol outcomes assessed by answers to questions on a questionnaire are bounded in a closed interval such as 0 1 with continuous observations in 0 1 and in addition excess observations taking the boundary values 0 and or 1 common statistical procedures used in practice such as t and rank based tests may not be very powerful since they ignore the specific feature of the proportional data in this article we propose a three component mixture model for the proportional data and a density ratio model for the distributions of continuous observations in 0 1 a semiparametric test statistic for the homogeneity of distributions of multigroup proportional data is derived based on the empirical likelihood ratio principle and shown to be asymptotically distributed as a chi squared random variable under null hypothesis a nonparametric bootstrap procedure is proposed to further improve the performance of the semiparametric test simulation studies are performed to evaluate the empirical type i error and power of the proposed test procedure and compare it with likelihood ratio tests lrts under parametric distribution assumptions rank based kruskal wallis test and wald type test the proposed test procedure is also applied to the analysis of qol outcomes from a clinical trial on colorectal cancer that motivated our study

approximating the power of wilcoxon s rank sum test against shift alternatives three methods of approximating the power of wilcoxon s rank sum test against shift alternatives are studied they are obtained by using a gaussian assumption edgeworth expansion or bootstrap it is assumed that a historical data set is available to use in estimating the shape of the distribution the methods are compared through simulation across several different distributional types the results indicate that the bootstrap generally gives the most reliable approximation however the edgeworth expansion has the practical advantage that a lower bound on the power can be roughly approximated the methods are illustrated on muscle strength data from patients with osteogenesis imperfecta published in 1999 by john wiley sons ltd this is us government work and is in the public domain in the united states

sample size determination for some common nonparametric tests abstract the article discusses the problem of determining the number of observations required by some common nonparametric tests so that the tests have power at least 1 against alternatives that differ sufficiently from the hypothesis being tested it is shown that the number of observations depends on certain simple probabilities a method is suggested for fixing the value of the appropriate probability when determining sample size

alternative estimation procedures for pr x y in categorized data summary consider two independent random variables x and y the functional r pr x y or x pr x y pr y x is of practical importance in many situations including clinical trials genetics and reliability in this paper several approaches to estimation of x when x and y are presented in discretized categorical form are analyzed and compared asymptotic formulas for the variances of the estimators are derived use of the bootstrap to estimate variances is also discussed computer simulations indicate that the choice of the best estimator depends on the value of x the underlying distribution and the sparseness of the data it is shown that the bootstrap provides a robust estimate of variance several examples are treated