on the o spektrum of regular operators and the spectrum of measures the banach algebra m g of all bounded regular borel measures on a locally compact group g is often represented on lz g by the mapping p t 2 t zf f fel2 g this representation has the advantage of bringing the hilbert space structure into play thus the involution is preserved and the spectrum of tu 2 is easy to compute if g is abelian or compact there is nevertheless an inconvenience the spectrum of in the banach algebra m g does not coincide with the spectrum of the operator tu z in general this could be overcome by considering li g instead of lz g another possibility which conserves the hilbert space structure makes use of the fact that the operators t 2 are regular operators on the banach lattice l2 g i e linear combinations of positive operators and indeed the representation of m g in the smaller banach algebra l2 g of all regular operators on l2 g behaves nicely if g is amenable it is an algebraic and lattice isomorphism of m g onto the full subalgebra and sublattice of those regular operators which commute with the right translations consequently the spectrum of in m g is the same as the spectrum of t 2 in the banach algebra r l2 g

an exact test for multiple inequality and equality constraints in the linear regression model abstract in this article we consider the linear regression model y x e where e is n 0 2i in this context we derive exact tests of the form h r r versus k rk for the case in which 2 is unknown we extend these results to consider hypothesis tests of the form h r1 r1 and r2 r2 versus k rk for each of these hypotheses tests we derive several equivalent forms of the test statistics using the duality theory of the quadratic programming for both tests we derive their exact distribution as a weighted sum of snedecor s f distributions normalized by the numerator degrees of freedom of each f distribution of the sum a methodology for computing critical values as well as probability values for the tests is discussed the relationship between this testing framework and the multivariate one sided hypothesis testing literature is also discussed in this context we show that for any size of the hypothesis test h 0 versus k rk the test statistic and critical value obtained a

on the descending central sequence of absolute galois groups let p be an odd prime number and f a field containing a primitive p th root of unity we prove a new restriction on the group theoretic structure of the absolute galois group g f of f namely the third subgroup g f 3 in the descending p central sequence of g f is the intersection of all open normal subgroups n such that g f n is 1 bbb z p 2 or the extra special group m p 3 of order p 3 and exponent p 2

the asymptotic properties of nonparametric tests for comparing survival distributions the asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary and possibly unequal right censoring the test statistics include equivalents to the log rank statistic the modified wilcoxon statistic and the class of rank invariant test procedures introduced by peto peto when there are equal censoring distributions and the hazard functions are proportional the sample size formula for the f test used to compare exponential samples is shown to be valid for the log rank test in certain situations the power of the log rank test falls as the amount of censoring decreases

polynomially riesz perturbations in this paper we investigate perturbations by holomorphically and polynomially riesz operators concerning the sets of upper and lower semi fredholm operators the sets of left and right fredholm operators as well as the sets of upper and lower semi browder operators and the sets of left and right browder operators we consider in particular perturbations of shifts by polynomially riesz operators 1 introduction and preliminaries let c denote the set of all complex numbers and let x be an infinite dimensional banach space we denote by b x the set of all linear bounded operators on x for a2 b x let r a and n a denote the range and the null space of a respectively and let a dimn a and a dimx r a codim r a the sets of upper and lower semi fredholm operators respectively are defined as x fa2 b x a 1 and r a is closedg and x fa 2 b x a 1g set x x x for the set of semi fredholm operators and x x x for the set of fredholm operators for a2 x the index is defined by i a a a the calkin algebra over x is the quotient algebra c x b x k x and let b x c x denote the natural homomorphism it is well known that a2 x if and only if a is invertible in c x 5 theorem 3 2 8 an operator a2 b x is left fredholm if a 1 and r a is a closed and complemented subspace of x while a is right fredholm if a 1 and n a is a complemented subspace of x set l x for the set of all left fredholm operators on x and r x for the set of all right fredholm operators on x for a 2 b x denote by asc a dsc a the ascent the descent of a 2 b x i e the smallest non negative integer n such that n a n n a n 1 r a n r a n 1 if such n does not exist then asc a 1 dsc a 1 an operator a2 b x is upper semi browder if it is upper semi fredholm with finite ascent while a is lower semi browder if it is lower semi fredholm with finite descent letb x b x denote the set of all upper lower semi browder operators we shall say that an operator a2 b x is left browder if it is left fredholm with finite ascent while a is right browder if it is right fredholm with finite descent letbl x br x denote the set of all left right browder operators an operator a2 b x is browder if it is fredholm with finite ascent and finite descent the set of all browder operators on x is denoted byb x clearly b x b x b x bl x br x

tests for qualitative treatment by centre interaction using a pushback procedure in multicentre clinical trials using a common protocol the centres are usually regarded as being a fixed factor thus allowing any treatment by centre interaction to be omitted from the error term for the effect of treatment however we feel it necessary to use the treatment by centre interaction as the error term if there is substantial evidence that the interaction with centres is qualitative instead of quantitative to make allowance for the estimated uncertainties of the centre means we propose choosing a reference value for example the median of the ordered array of centre means and converting the individual centre results into standardized deviations from the reference value the deviations are then reordered and the results pushed back by amounts appropriate for the corresponding order statistics in a sample from the relevant distribution the pushed back standardized deviations are then restored to the original scale the appearance of opposite signs among the destandardized values for the various centres is then taken as substantial evidence of qualitative interaction procedures are presented using in any combination i gaussian or student s t distribution ii order statistic medians or outward 90 per cent points of the corresponding order statistic distributions iii pooling or grouping and pooling the internally estimated standard deviations of the centre means the use of the least conservative combination student s t outward 90 per cent points grouping and pooling is recommended

quotients of absolute galois groups which determine the entire galois cohomology for a prime power q pd and a field f containing a root of unity of order q we show that the galois cohomology ring h g f mathbb z q is determined by a quotient g f 3 of the absolute galois group gf related to its descending q central sequence conversely we show that g f 3 is determined by the lower cohomology of gf this is used to give new examples of pro p groups which do not occur as absolute galois groups of fields

asymptotic behavior of robust estimators of regression and scale parameters with fixed carriers for the linear regression model y i mathbf x i mathbf beta varepsilon i with fixed mathbf x i s the asymptotic normality of hat mathbf beta hat sigma which minimizes the huber dutter loss function sum sigma rho y i mathbf x i mathbf beta sigma a n sigma is established under rather general conditions

two new tests associated with analysis of variance summary two new tests are described in connexion with analysis of variance the first is for the presence of qualitative interaction between a treatment and an intrinsic factor both with qualitatively defined levels such an interaction is said to be present if there are two levels of the treatment factor and two levels of the intrinsic factor such that the treatment difference has opposite signs at the two levels of the intrinsic factor the second test is for the presence of a treatment effect which can be tested via any of a hierarchical sequence u1 u1 u2 u1 ur of sums of squares an allowance for selection is calculated

reconstructing function fields from rational quotients of mod ell galois groups in this paper we develop the main step in the global theory for the mod ell analogue of bogomolov s program in birational anabelian geometry for higher dimensional function fields over algebraically closed fields more precisely we show how to reconstruct a function field k of transcendence degree ge 5 over an algebraically closed field up to inseparable extensions from the mod ell abelian by central galois group of k endowed with the collection of mod ell rational quotients

detecting qualitative interaction a bayesian approach differences in treatment effects between centers in a multi center trial may be important these differences represent treatment by subgroup interaction peto defines qualitative interaction qi to occur when the simple treatment effect in one subgroup has a different sign than in another subgroup this interaction is important interaction where the treatment effects are of the same sign in all subgroups is called quantitative and is often not important because the treatment recommendation is identical in all cases a hierarchical model is used here with exchangeable mean responses to each treatment between subgroups the posterior probability of qi and the corresponding bayes factor are proposed as a diagnostic and as a test statistic the model is motivated by two multi center trials with binary responses the frequentist power and size of the test using the bayes factor are examined and compared with two other commonly used tests the impact of imbalance between the sample sizes in each subgroup on power is examined and the test based on the bayes factor typically has better power for unbalanced designs especially for small sample sizes an exact test based on the bayes factor is also suggested assuming the hierarchical model the bayes factor provides a concise summary of the evidence for or against qi it is shown by example that it is easily adapted to summarize the evidence for clinically meaningful qi defined as the simple effects being of opposite signs and larger in absolute value than a minimal clinically meaningful effect

galois groups of maximal p extensions let p be an odd prime and f a field of characteristic different from p containing a primitive pth root of unity assume that the galois group g of the maximal p extension of f has a finite normal series with abelian factor groups then the commutator subgroup of g is abelian moreover g has a normal abelian subgroup with pro cyclic factor group if in addition f contains a primitive p 2 th root of unity then g has generators x y i i i with relations y i y j y j y i and xy i x 1 y i q 1 where q 0 or q p n for some n 1

tests against qualitative interaction exact critical values and robust tests summary consider a study to evaluate treatment a with a placebo in two or more groups of patients if treatment a is beneficial to one group of patients and harmful to another then we say that there is qualitative interaction or crossover interaction between patient groups and the treatments gail and simon 1985 biometrics41 361 372 developed a large sample procedure for this testing problem their test has received favorable coverage in the literature in this article we obtain corresponding exact finite sample results for normal error distribution and provide a table of critical values the test statistic is similar to the familiar f ratio and its p value is equal to a weighted sum of tail areas of f distributions the computations to implement this are simple a simulation study shows that the exact critical values provided here for normal error distribution are preferable to the asymptotic critical values for a wide range of error distributions we also develop tests that are power robust against long tailed error distributions our robust test uses m estimators instead of the least squares estimators we show that the efficiency robustness of the m estimator translates to power robustness of the corresponding test therefore our robust tests are better if outliers are expected a simulation study illustrates the substantial power advantages of our robust tests

uniformly more powerful tests for hypotheses concerning linear inequalities and normal means abstract this article considers some hypothesis testing problems regarding normal means in these problems the hypotheses are defined by linear inequalities on the means we show that in certain problems the likelihood ratio test lrt is not very powerful we describe a test that has the same size as the lrt and is uniformly more powerful the test is easily implemented since its critical values are standard normal percentiles the increase in power with the new test can be substantial for example the new test s power is 1 2 times bigger 10 times bigger for 05 than the lrt s power for some parameter points in a simple example specifically let x x 1 xp p 2 be a multivariate normal random vector with unknown mean 1 p and known nonsingular covariance matrix we consider testing the null hypothesis h 0 b i 0 for some i 1 k versus the alternative hypothesis h 1 b i 0 for all i 1 k here b 1 b k k 2 are specified p dimensional ve

bayesian subset analysis as a means of assessing the importance of variation in treatment effect among patient subsets we derived posterior distributions for subset specific treatment effects the effects are represented by combinations of terms for treatment and treatment by covariate interaction effects in familiar regression models exchange ability among the interactions is a key assumption thus the results are of interest primarily in the context of examining a collection of subsets with no definite a priori distinction relative to treatment effect exchangeability leads to a shrinking of the posterior distributions of the interaction terms toward the natural origin of 0 offsetting the tendency of the estimated effects to disperse the method is applied to parameter estimates from a proportional hazards regression analysis of survival data from a clinical trial invoking the approximate multivariate normal distribution of the estimates no subjective prior distributions are required vague priors are used for all of the regression coefficients except the treatment by covariate interactions which are assumed to follow a normal distribution

ruston riesz and perturbation classes abstract we determine the perturbation classes of fredholm and weyl elements as well the commuting perturbation classes of fredholm weyl and browder elements with respect to unbounded banach algebra homomorphism t among other things we use the ruston elements of mouton mouton and raubenheimer also we investigate the class of polynomially almost t null and the class of polynomially t riesz elements

solvable absolute galois groups are metabelian we show that solvable absolute galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion free procyclic group in particular solvable absolute galois groups are metabelian moreover any field with solvable absolute galois group g admits a non trivial henselian valuation unless each sylow subgroup of g is either procyclic or isomorphic to z2 z 2z a complete classification of solvable absolute galois groups up to isomorphism is given

uniformly more powerful one sided tests for hypotheses about linear inequalities let x have a multivariate p dimensional normal distribution p 2 with unknown mean and known nonsingular covariance consider testing h 0 b i 0 for some i 1 k versus h 1 b i 0 for all i 1 k where b 1 b k k 2 are known vectors that define the hypotheses for any 0 1 2 we construct a size test that is uniformly more powerful than the size a likelihood ratio test lrt the proposed test is an intersection union test other authors have presented uniformly more powerful tests under restrictions on the covariance matrix and on the hypothesis being tested our new test is uniformly more powerful than the lrt for all known nonsingular covariance matrices and all hypotheses so our results show that in a very general class of problems the lrt can be uniformly dominated

rigid elements valuations and realization of witt rings to the theory over the residue class field of the valuation this theorem generalizes to any 2 henselian valuation cf kl sect 12 21 let v be the valuation let a e k f o 1 be such that v a is not divisible by two the key idea behind the proof is that the quadratic form 1 a only represents elements in u a such an element a is called

a comparison of the power of two tests for qualitative interactions qualitative or crossover interactions arise when a new treatment compared with a control treatment is beneficial in some subsets of patients and harmful in other subsets we present a new range test for crossover interactions and compare it with the likelihood ratio test developed by gail and simon the range test has greater power when the new treatment is harmful in only a few subsets whereas the likelihood ratio test has greater power when the new treatment is harmful in several subsets we provide power tables for both tests to facilitate sample size calculations for designing experiments to detect qualitative interactions and for interpreting the results of clinical trials

a bayesian subgroup analysis with a zero enriched polya urn scheme we introduce a new approach to inference for subgroups in clinical trials we use bayesian model selection and a threshold on posterior model probabilities to identify subgroup effects for reporting for each covariate of interest we define a separate class of models and use the posterior probability associated with each model and the threshold to determine the existence of a subgroup effect as usual in bayesian clinical trial design we compute frequentist operating characteristics and achieve the desired error probabilities by choosing an appropriate threshold s for the posterior probabilities copyright 2010 john wiley sons ltd

bayesian multiple testing for two sample multivariate endpoints dans les essais cliniques ou l on s interesse a plusieurs variables il arrive souvent que des tests simultanes portent sur des resultats correles entre eux et concernent des hypotheses elles meme correlees entre elles cet article dans un cadre bayesien propose d utiliser comme loi de probabilite a priori des effets traitement une distribution multivariee ou la combinaison des parametres permet simultanement d affecter une probabilite strictement positive a l absence d effet traitement pour chacune des hypotheses et de modeliser differents types de correlations la correlation entre les effets traitements standardises c est a dire rapportes aux ecarts types des variables concernees la correlation entre les variables dichotomiques de presence ou d absence de l effet traitement la correlation enfin entre les statistiques de test conditionnellement aux effets reels nous developpons ensuite dans le cas de l etude multivariee de deux echantillons une methode de comparaisons multiples qui nous permet de calculer pour des variables prealablement specifiees les probabilites a posteriori d une absence d effet traitement sont aussi discutees dans le contexte d un exemple clinique reel les demarches relatives au choix de la probabilite a priori ainsi que les problemes de robustesse

on essentially semi regular linear relations abstract the characterization of bounded essentially semi regular operators in terms of kato decomposition of finite type was studied by several authors in this paper we extend this characterization to the case of essentially semi regular linear relations we also give other characterization of such linear relations further we apply the obtained results to analyse the stability of the class of essentially semi regular linear relations under additional operator perturbations finally as an application we get some useful connections between the fredholm spectrum and the essentially semi regular spectrum of linear relations

a recursive description of pro p galois groups abstract in this note we study a modified version of the elementary type conjecture for pro p galois groups to be precise let g p f be the galois group of the maximal galois p extension of a field f containing a primitive p th root of unity under some natural assumptions concerning valuation rings of f and also orderings of f when p 2 we prove that g p f can be obtained from suitable closed subgroups using a finite number of free pro p products and semidirect products

t fredholm analysis and application to operator theory abstract in this paper we study the general setting of fredholm theory relative to a banach algebra homomorphism t mainly we prove some perturbation results on the t browder spectrum as applications we investigate the class of polynomially fredholm perturbation operators on a banach space we obtain stability results under commuting operator perturbations that belong to this class

on the birational anabelian program initiated by bogomolov i recall that a program initiated by bogomolov in 1990 aims at reconstructing function fields k k with td k k 1 and k algebraically closed from the maximal pro l abelian by central galois group pi hbox to1pt rm c phantom g k of k where l is any prime number char k in this paper we complete that program in the case k is an algebraic closure of a finite field

the galois action on geometric lattices and the mod ell i om this paper studies the galois action on a special lattice of geometric origin which is related to mod ell abelian by central quotients of geometric fundamental groups of varieties as a consequence we formulate and prove the mod ell abelian by central variant strengthening of a conjecture due to ihara oda matsumoto

approximate bayesian evaluation of multiple treatment effects summary we propose an approximate bayesian method for comparing an experimental treatment to a control based on a randomized clinical trial with multivariate patient outcomes overall treatment effect is characterized by a vector of parameters corresponding to effects on the individual patient outcomes we partition the parameter space into four sets where respectively the experimental treatment is superior to the control the control is superior to the experimental the two treatments are equivalent and the treatment effects are discordant we compute posterior probabilities of the parameter sets by treating an estimator of the parameter vector like a random variable in the bayesian paradigm the approximation may be used in any setting where a consistent asymptotically normal estimator of the parameter vector is available the method is illustrated by application to a breast cancer data set consisting of multiple time to event outcomes with covariates and to count data arising from a cross classification of response infection and treatment in an acute leukemia trial

smoothing balanced single error term analysis of variance we present an approach to smoothing balanced single error term analysis of variance anova descended from smith that also allows spatial temporal or spatiotemporal smoothing the approach addresses unreplicated designs masked contrasts in effects with many degrees of freedom and subgroup analysis demonstrated using a study of denture lining materials our approach is bayesian but can be viewed as a way to generate frequentist procedures a simulation experiment compares four priors unsmoothed anova and dropping nonsignificant interactions three priors have advantages when some interactions are absent dropping nonsignificant interactions has serious flaws we contrast our approach with the approaches of nobile green and gelman

robust tests of inequality constraints and one sided hypotheses in the linear model summary for the linear regression model with independent and identically distributed errors robust tests of various hypotheses on the regression parameter are developed by robust we mean robustness of size and power against long tailed error distributions which may not be symmetric the proposed test statistic denoted by fm resembles the usual f statistic the two main results are i under the null hypothesis the asymptotic distribution of fm and that of the least squares based f statistic are the same and ii the pitman asymptotic efficiency of fm relative to f is equal to the asymptotic efficiency of the corresponding m estimator relative to the least squares estimator an example and simulation results illustrate that fm has desirable robustness properties compared to f

subgroup finding via bayesian additive regression trees we provide a bayesian decision theoretic approach to finding subgroups that have elevated treatment effects our approach separates the modeling of the response variable from the task of subgroup finding and allows a flexible modeling of the response variable irrespective of potential subgroups of interest we use bayesian additive regression trees to model the response variable and use a utility function defined in terms of a candidate subgroup and the predicted response for that subgroup subgroups are identified by maximizing the expected utility where the expectation is taken with respect to the posterior predictive distribution of the response and the maximization is carried out over an a priori specified set of candidate subgroups our approach allows subgroups based on both quantitative and categorical covariates we illustrate the approach using simulated data set study and a real data set copyright 2017 john wiley sons ltd

bayesian subset analysis in a colorectal cancer clinical trial subset analysis is the examination of treatment comparisons within groups of patients with restricted levels of patient characteristics such analyses are vulnerable to multiplicity effects we examine the problem in the context of a proportional hazards model with terms for treatment each of several dichotomous covariates representing the patient characteristics of interest and treatment by covariate interaction effects parametrically a subset specific treatment effect is equal to the treatment effect term plus a linear combination of the interaction terms we present bayesian point and interval estimates under the assumption that the interaction terms are exchangeable and the prior distributions for the other regression parameters are locally uniform this produces a shrinking of the estimated interaction effects towards zero thereby discounting them and dealing in a natural way with multiplicity we illustrate the method using results of a recent north central cancer treatment group mayo clinic study in advanced colorectal cancer

qualitative interactions in multifactor studies in clinical trials qualitative interaction or crossover interaction is said to occur when one treatment is superior for some sets of patients and the alternative treatment is superior for other subsets here we propose a definition of no qualitative interaction with respect to a single continuous covariate which implies that one treatment is superior to the other treatment over a prespecified range of the covariate further in studies involving patients cross classified by two or more prognostic factors we define a marginal qualitative interaction with respect to a treatment factor and a prognostic factor by averaging in some sense over the remaining prognostic factors in some situations the methodology of gail and simon 1985 biometrics 41 361 372 is shown to be appropriate their procedure assumes independent estimates of treatment effect for each subset of patients and this may not be appropriate for the generalizations we propose therefore we generalize the procedure of gail and simon to the case of two correlated estimates of treatment effect providing a table of critical values results for one sided tests for the case of j correlated estimates are also obtained we also present an example illustrating our procedure

how vague is vague a simulation study of the impact of the use of vague prior distributions in mcmc using winbugs there has been a recent growth in the use of bayesian methods in medical research the main reasons for this are the development of computer intensive simulation based methods such as markov chain monte carlo mcmc increases in computing power and the introduction of powerful software such as winbugs this has enabled increasingly complex models to be fitted the ability to fit these complex models has led to mcmc methods being used as a convenient tool by frequentists who may have no desire to be fully bayesian often researchers want the data to dominate when there is no prior information and thus attempt to use vague prior distributions however with small amounts of data the use of vague priors can be problematic the results are potentially sensitive to the choice of prior distribution in general there are fewer problems with location parameters the main problem is with scale parameters with scale parameters not only does one have to decide the distributional form of the prior distribution but also whether to put the prior distribution on the variance standard deviation or precision we have conducted a simulation study comparing the effects of 13 different prior distributions for the scale parameter on simulated random effects meta analysis data we varied the number of studies 5 10 and 30 and compared three different between study variances to give nine different simulation scenarios one thousand data sets were generated for each scenario and each data set was analysed using the 13 different prior distributions the frequentist properties of bias and coverage were investigated for the between study variance and the effect size the choice of prior distribution was crucial when there were just five studies there was a large variation in the estimates of the between study variance for the 13 different prior distributions with a large number of studies the choice of prior distribution was less important the effect size estimated was not biased but the precision with which it was estimated varied with the choice of prior distribution leading to varying coverage intervals and potentially to different statistical inferences again there was less of a problem with a larger number of studies there is a particular problem if the between study variance is close to the boundary at zero as mcmc results tend to produce upwardly biased estimates of the between study variance particularly if inferences are based on the posterior mean the choice of vague prior distribution can lead to a marked variation in results particularly in small studies sensitivity to the choice of prior distribution should always be assessed

a graphical method to assess treatment covariate interactions using the cox model on subsets of the data we introduce the subpopulation treatment effect pattern plot stepp method designed to facilitate the interpretation of estimates of treatment effect derived from different but potentially overlapping subsets of clinical trial data in particular we consider sequences of subpopulations defined with respect to a covariate and obtain confidence bands for the collection of treatment effects here obtained from the cox proportional hazards model associated with the sequences the method is aimed at determining whether the magnitude of the treatment effect changes as a function of the values of the covariate we apply stepp to a breast cancer clinical trial data set to evaluate the treatment effect as a function of the oestrogen receptor content of the primary tumour

the kernel unipotent conjecture and the vanishing of massey products for odd rigid fields abstract a major difficult problem in galois theory is the characterization of profinite groups which are realizable as absolute galois groups of fields recently the kernel n unipotent conjecture and the vanishing n massey conjecture for n 3 were formulated these conjectures evolved in the last forty years as a byproduct of the application of topological methods to galois cohomology we show that both of these conjectures are true for odd rigid fields this is the first case of a significant family of fields where both of the conjectures are verified besides fields whose galois groups of p maximal extensions are free pro p groups we also prove the kernel unipotent conjecture for demushkin groups of rank 2 and establish various filtration results for free pro p groups provide examples of pro p groups which do not have the kernel n unipotent property compare various zassenhaus filtrations with the descending p central series and establish new type of automatic galois realization