accurate and efficient power calculations for 2 x m tables in unmatched case control designs the aim of this article was to derive an approximation of the distribution of pearson s statistic for 2 x m contingency tables to calculate power for case control studies accurately and efficiently in terms of computer time we first prove that rather than a non central chi square distribution pearson s statistic is asymptotically equivalent with the sum of squares of m independent normal random variables whose variances are typically different under the alternative hypothesis based on this asymptotically equivalent ae we derive a cost effective approximation ce numerical results show that ce was almost as precise as ae but computationally more efficient although somewhat less costly in terms of cpu time we show that a commonly used approximation using the non central chi square distribution can be very inaccurate and overestimate power in some scenarios and underestimate it in others simulations and exact distributions on the other hand are accurate but computationally very intensive compared to our ce the ce reached its lowest accuracy under the null hypothesis where it has a chi square distribution with m 1 degree of freedom this suggests that ce can be used to calculate power for tables where researchers would feel comfortable using pearson s chi square test

the spectral gap for some spin chains with discrete symmetry breaking we prove that for any finite set of generalized valence bond solid gvbs states of a quantum spin chain there exists a translation invariant finite range hamiltonian for which this set is the set of ground states this result implies that there are gvbs models with arbitrary broken discrete symmetries that are described as combinations of lattice translations lattice reflections and local unitary or anti unitary transformations we also show that all gvbs models that satisfy some natural conditions have a spectral gap the existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics this general scheme is interesting in its own right and threfore although the basic idea is not new we present it in a system independent setting the results are illustrated with a number of examples

the foundations of finite sample estimation in stochastic processes generally in the literature on stochastic processes estimation is investigated in terms of asymptotic properties in this paper we establish some finite sample optimal estimation let y1 y2 be a discrete stochastic process since we will be discussing results for samples of finite size n we restrict the process to r in the following the term parameter is used in the same broad sense as by godambe thompson 1984 let y be a class of probability distributions f on r and 0 0 f f e jz be a real parameter let hi be a real function of y 1 yi and 0 be such that

allocation of subjects to test null relative risks smaller than one allocating a proportion k 1 1 radicalr 0 of subjects to an intervention is a practical approach to approximately maximize power for testing whether an intervention reduces relative risk of disease below a null ratio r 0 or r 0 0 33 2 3 for 0 33 or r 0 0 17 and 3 4 for 0 17 or r 0 or 0 10 both k and k s are easily calculated and invariant to alterations in disease rate estimates under null and alternative hypotheses when r 0 remains constant in examples that we studied allocating k or k s subjects to intervention achieved close to the minimum possible sample size given test size and power equivalently maximum power given test size and sample size for likelihood score tests compared to equal allocation k and k s reduced sample sizes by amounts ranging from approximately 5 5 per cent for r 0 0 50 to approximately 24 per cent for r 0 0 10 these sample size savings may be particularly important for large studies of prophylactic interventions such as vaccines while k was derived from variance minimization for an arcsine transformation we do not recommend the arcsine test since its true size exceeded the nominal value in contrast the true size for the uncorrected score test was less than the nominal size a skewness correction made the size of the score test very close to the nominal level and slightly increased power we recommend using the score test or the skewness corrected score test for planing studies designed to show a ratio of proportions is less than a prespecified null ratio r 0 1

subordination in the wide sense for levy processes given t t 0 a transient convolution semigroup over rd we characterize the transient semi groups which are subordinate in the wide sense to the latter via some quotient relation between the two levy generators this is also equivalent to an inclusion between excessive cones thanks to a theorem of hirsch this allows us in big dimensions to give a probabilistic description of some measures satisfying the classical maximum principle

right inverses of nonsymmetric levy processes we analyze the existence and properties of right inverses k for nonsymmetric levy processes x extending recent work of evans 7 in the symmetric setting first both x and x have right inverses if and only if x is recurrent and has a nontrivial gaussian component our main result is then a description of the excursion measure n z of the strong markov process z x l reflected process where lt inf x 0 kx t specifically n z is essentially the restriction of n x to the excursions starting negative second when only asking for right inverses of x a certain strength of asymmetry is needed millar s 9 notion of creeping turns out necessary but not sufficient for the existence of right inverses we analyze this both in the bounded and unbounded variation case with a particular emphasis on results in terms of the levy khintchine characteristics 1 introduction consider a real valued levy process that is a continuous time process with stationary independent increments and cadlag paths right inverses of levy processes have first been studied in the symmetric setting by evans 7 he defines an increasing process k to be a right inverse of a levy process x if it satisfies x kx x for all x 0 the minimal such k turns out to be a subordinator i e an increasing levy process specifically evans characterized the symmetric levy processes that have right inverses he introduced lt inf x 0 kx t and the reflected process z x l and showed that z is strong markov with local time l at zero he gave some fluctuation type identities for z and l and proved formulae for the entrance laws to the excursion measure n z of z evans also showed that k is distributed like a linear time change of the inverse local time of x at zero in this work we basically answer three questions left open by evans first what happens when symmetry fails the only result of evans that is clearly based on symmetry is the coincidence of the laws of k and the inverse local time of x in zero also the class of processes possessing inverses is not restricted to those having a positive gaussian component in the symmetric setting this is evans s characterization result but those form again an important class we provide a further characterization in the bounded variation case and we also point

sample size determination for logistic regression revisited there is no consensus on the approach to compute the power and sample size with logistic regression some authors use the likelihood ratio test some use the test on proportions some suggest various approximations to handle the multivariate case we advocate the use of the wald test since the z score is routinely used for statistical significance testing of regression coefficients the null variance formula became popular from early studies which contradicts modern software which utilizes the method of maximum likelihood estimation mle when the variance of the mle is estimated at the mle not at the null we derive general wald based power and sample size formulas for logistic regression and then apply them to binary exposure and confounder to obtain a closed form expression these formulas are applied to minimize the total sample size in a case control study to achieve a given power by optimizing the ratio of controls to cases approximately the optimal number of controls to cases is equal to the square root of the alternative odds ratio our sample size and power calculations can be carried out online at www dartmouth edu approximately eugened

power for studies with random group sizes in any study it is essential to select the sample size carefully to ensure adequate power for many studies this is simple recruit a desired number of subjects within each group conduct measurements and perform the statistical test in some studies e g observational studies however the group membership is unknown at recruitment in this paper we examine the effect of random group sizes on power additionally we consider the situation when the group proportions are unknown and specified by a prior distribution the problem that initially motivated this research is presented power for a 2 by 2 table as are examples using continuous outcomes we find that standard estimates of power using expected group sizes can over or underestimate power sas macros are available at http www phs wfubmc edu public wambrosi randompower

embedding processes in brownian motion in r n we give a potential theoretic characterization of the right continuous processes which can be embedded in brownian motion in r by means of an increasing family of standard stopping times in general it is necessary to use a brownian motion process whose filtration is richer than the natural one

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

formulae and tables for the determination of sample sizes and power in clinical trials for testing differences in proportions for the two sample design a review this paper is a compendium of exact and asymptotic formulae and tables for estimating the sample size in a clinical trial with two treatment groups and a dichotomous outcome the paper provides separate formulae for equal and unequal treatment group sizes formulae for the calculation of power given the sample size and complete references for all formulae and tables cited

challenge of multiple co primary endpoints a new approach there are many disorders where regulatory agencies have required a new treatment to demonstrate efficacy on multiple co primary endpoints all significant at the one sided 2 5 per cent level before accepting the treatment s effect for the disorder this requirement rooted in the intersection union iu test has led many researchers to increase the study sample size to make up for the reduction in the statistical power at the study level unfortunately the increase in sample size could be substantial when the endpoints are minimally correlated and the treatment effects on the multiple endpoints are comparable in this paper we demonstrate that the frequentist concept of controlling the maximum false positive rate even when applied to a restricted null space has only limited success in keeping the sample size increase at a reasonable level we therefore propose an approach that is based on the notion of controlling an average type i error rate by employing an upper bound for the average type i error rate the new approach provides an adjustment to the significance level that depends only on the correlation among the endpoints for the most common case of two or three co primary endpoints the adjusted significance level is at most 5 per cent one sided when the endpoints are moderately correlated we show how sample size could be calculated under the proposed approach and contrast the needed sample size with that required under the iu test we provide additional comments and discuss why the new approach is consistent with the principle requiring evidence of significance in the drug development and approval process

logistic regression for dependent binary observations summary the likelihood of a set of binary dependent outcomes with or without explanatory variables is expressed as a product of conditional probabilities each of which is assumed to be logistic the models are called regressive logistic models they provide a simple but relatively unknown parametrization of the multivariate distribution they have the theoretical and practical advantage that they can be analyzed and fitted as in logistic regression for independent outcomes and with the same computer programs the paper is largely expository and is intended to motivate the development and usage of the regressive logistic models the discussion includes serially dependent outcomes equally predictive outcomes more specialized patterns of dependence multidimensional tables and three examples this paper seeks to promote the development and usage of a simple but relatively unknown approach to the specification of models for dependent binary outcomes that are useful in situations where it is apt to describe the dependence in terms of conditional probabilities the joint probability is decomposed into a product of successive conditional probabilities each of which is assumed to be univariate logistic since the probability of an observation is conditioned on all preceding observations the models will be referred to as regressive logistic models two broad situations are distinguished in this paper the first concerning sequences of binary outcomes possibly with measures on associated explanatory variables can usually be supported by simple patterns of dependence these will be discussed in sections 2 5 the second broad situation concerns dependent binary outcomes crossclassified into multidimensional tables that admit a fuller parametrization this is discussed in section 6 for all the regressive logistic models discussed here the parametrization is such that jointly dependent binary observations with or without explanatory variables can be fitted with existing computer programs for the standard logistic regression for independent observations some of the possibilities of the approach are illustrated with real data in section 8

global cross ratio models for bivariate discrete ordered responses summary a family of statistical models is presented for bivariate discrete response to a regressor when both components of the response have ordered categories association between components is expressed in terms of global cross ratios cross product ratios of quadrant probabilities for each double dichotomy of the response table of probabilities into quadrants pearson and heron 1913 biometrika 9 159 315 these models are extensions to the work of plackett 1965 journal of the american statistical association 60 516 522 and mantel and brown 1973 biometrics 29 649 665 the marginal cumulative probabilities may satisfy linear logistic or other generalized linear models mccullagh 1980 journal of the royal statistical society series b 42 109 142 an analysis of patients postoperative pain level and medication frequency illustrates these methods

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

u 1 su 2 gauge invariance of non relativistic quantum mechanics and generalized hall effects we show that the non relativistic quantum mechanics of particles with spin coupled to an electromagnetic field has a naturalu 1 su 2 gauge invariance ward identities reflecting this gauge invariance combined with an assumption of incompressibility of a system of such particles in an appropriate external field and for suitable values of the particle density permit us to determine the form of the effective action of the system as a functional of small fluctuations in the electromagnetic field in the large distance adiabatic limit in this limit the action is found to have a universal form we present explicit results for two dimensional incompressible electron fluids and apply them to derive the equations of linear response theory describing a variety of generalized hall effects sum rules for the hall conductivities magnetic susceptibilities and other quantities of physical interest are found

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

adaptive group sequential design with population enrichment in phase 3 randomized controlled trials with two binary co primary endpoints the use of co primary endpoints in drug development allows investigators to capture an experimental intervention s multidimensional effect more comprehensively than a single primary endpoint we propose the theoretical basis and development of an adaptive population enrichment design with co primary endpoints provide stage wise boundary values for futility and efficacy and discuss power under different efficacy configurations subgroup prevalence and analysis times using a pre specified decision criterion we considered a two arm two stage parallel group design where population enrichment occurs at the interim analysis by dropping any non responsive subgroups a test for efficacy is conducted only in the enriched population two binary endpoints are evaluated as co primary endpoints our trial objective is to determine whether the experimental intervention is superior to the control intervention with superiority required in both endpoints we define the stopping boundary using alpha spending functions using a 0 025 significance level for each endpoint we obtain the stage i threshold boundary values for futility and efficacy as 0 1040 and 2 2761 respectively and the stage ii boundary value for futility and efficacy is 2 2419 we show that in the presence of substantial heterogeneity of treatment effect we gain more power to observe an effect in the subgroup where the benefits are greater by allowing the dropping of non responsive subgroups at an early stage our design reduces the likelihood of obtaining false negative results due to inclusion of the heterogeneous treatment effects of both subgroups which would dilute the responsive subgroup s results

sample size determination in group sequential clinical trials with two co primary endpoints we discuss sample size determination in group sequential designs with two endpoints as co primary we derive the power and sample size within two decision making frameworks one is to claim the test intervention s benefit relative to control when superiority is achieved for the two endpoints at the same interim timepoint of the trial the other is when superiority is achieved for the two endpoints at any interim timepoint not necessarily simultaneously we evaluate the behaviors of sample size and power with varying design elements and provide a real example to illustrate the proposed sample size methods in addition we discuss sample size recalculation based on observed data and evaluate the impact on the power and type i error rate copyright 2014 john wiley sons ltd

a simple analysis of a simple crossover trial with a dichotomous outcome measure for the two treatment two period crossover trial with a dichotomous outcome measure a simple and straightforward analysis is proposed compute the treatment difference for each of the treatment sequences ab and ba and then average over the sequence groups the mean difference is used in the statistical test and in the computation of the 95 per cent confidence interval the method is generalized to the case of a crossover trial including more than two treatments

planning and analysis of three arm non inferiority trials with binary endpoints three arm trials including an experimental treatment an active control and a placebo group are frequently preferred for the assessment of non inferiority in contrast to two arm non inferiority studies these designs allow a direct proof of efficacy of a new treatment by comparison with placebo as a further advantage the test problem for establishing non inferiority can be formulated in such a way that rejection of the null hypothesis assures that a pre defined portion of the unknown effect the reference shows versus placebo is preserved by the treatment under investigation we present statistical methods for this study design and the situation of a binary outcome variable asymptotic test procedures are given and their actual type i error rates are calculated approximate sample size formulae are derived and their accuracy is discussed furthermore the question of optimal allocation of the total sample size is considered power properties of the testing strategy including a pre test for assay sensitivity are presented the derived methods are illustrated by application to a clinical trial in depression

sample size determination in clinical trials with multiple co primary binary endpoints clinical trials often employ two or more primary efficacy endpoints one of the major problems in such trials is how to determine a sample size suitable for multiple co primary correlated endpoints we provide fundamental formulae for the calculation of power and sample size in order to achieve statistical significance for all the multiple primary endpoints given as binary variables on the basis of three association measures among primary endpoints we discuss five methods of power and sample size calculation the asymptotic normal method with and without continuity correction the arcsine method with and without continuity correction and fisher s exact method for all five methods the achieved sample size decreases as the value of association measure increases when the effect sizes among endpoints are approximately equal in particular a high positive association has a greater effect on the decrease in the sample size on the other hand such a relationship is not very strong when the effect sizes are different