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

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

a likelihood ratio test regarding two nested but oblique order restricted hypotheses abstract cadoret woolson and winokur 1977 considered two theories regarding the genetic makeup of patients suffering from unipolar affective disorder these two theories imply nested but oblique order restrictions on the parameters of a statistical model a likelihood ratio test for these two restrictions is studied and found to be dominated by a test that neglects some of the information in the model

a note on one sided tests with multiple endpoints summary testing problems with multivariate one sided alternative hypotheses are common in clinical trials with multiple endpoints in the case of comparing two treatments treatment 1 is often preferred if it is superior for at least one of the endpoints and not biologically inferior for the remaining endpoints bloch et al 2001 biometrics57 1039 1047 propose an intersection union test iut for this testing problem but their test does not utilize the appropriate multivariate one sided test in this note we modify their test by an alternative iut that does utilize the appropriate one sided test empirical and graphical evidence show that the proposed test is more appropriate for this testing problem

design of group sequential clinical trials with multiple endpoints abstract group sequential testing for randomized clinical trials designed with multiple endpoints is considered previously computed tables for single endpoints are still useful with a change in interpretation of certain parameters the advantage in setting sample size based on multiple endpoints is that the sample size required when a trial is designed using more than one endpoint is smaller than the sample size based on any single endpoint if the two calculations are undertaken using matching significance levels and powers an example is provided and possible extensions of the work are discussed

adjusting o brien s test to control type i error for the generalized nonparametric behrens fisher problem summaryo brien 1984 biometrics40 1079 1087 introduced a simple nonparametric test procedure for testing whether multiple outcomes in one treatment group have consistently larger values than outcomes in the other treatment group we first explore the theoretical properties of o brien s test we then extend it to the general nonparametric behrens fisher hypothesis problem when no assumption is made regarding the shape of the distributions we provide conditions when o brien s test controls its error probability asymptotically and when it fails we also provide adjusted tests when the conditions do not hold throughout this article we do not assume that all outcomes are continuous simulations are performed to compare the adjusted tests to o brien s test the difference is also illustrated using data from a parkinson s disease clinical trial

use of odds ratio or relative risk to measure a treatment effect in clinical trials with multiple correlated binary outcomes data from the ninds t pa stroke trial in clinical trials when a single outcome is not sufficient to describe the underlying concept of interest it may be necessary to compare treatment groups on multiple correlated outcomes a global test based on a logit link function provides an estimate of the odds ratio for assessing a common treatment effect among correlated binary outcomes in this paper we extend the use of generalized estimating equations gee to calculate a common relative risk from correlated binary outcomes based on a log link function in the context of global tests we discuss the equivalence and difference between logit and log links and their estimates we also derive a formula for calculating a common risk difference between two treatment groups based on multiple correlated binary outcomes with categorical covariates assuming the asymptotic equivalency between the logit and log linear links we discuss the statistical tools to be used in choosing between the logit and log links when models on different links yield contrasting results examples using data from the ninds t pa stroke trials are provided we conclude in a study of correlated binary outcomes that the choice of the logit or log link could be based on a comparison of goodness of link

bivariate latent variable models for clustered discrete and continuous outcomes abstract we use the concept of a latent variable to derive the joint distribution of a continuous and a discrete outcome and then extend the model to allow for clustered data the model can be parameterized in a way that allows one to write the joint distribution as a product of a standard random effects model for the continuous variable and a correlated probit model for the discrete variable this factorization suggests a convenient approach to parameter estimation using quasi likelihood techniques our approach is motivated by the analysis of developmental toxicity experiments for which a number of discrete and continuous outcomes are measured on offspring clustered within litters fetal weight and malformation data illustrate the results

testing marginal homogeneity against stochastic order in multivariate ordinal data summary many assessment instruments used in the evaluation of toxicity safety pain or disease progression consider multiple ordinal endpoints to fully capture the presence and severity of treatment effects contingency tables underlying these correlated responses are often sparse and imbalanced rendering asymptotic results unreliable or model fitting prohibitively complex without overly simplistic assumptions on the marginal and joint distribution instead of a modeling approach we look at stochastic order and marginal inhomogeneity as an expression or manifestation of a treatment effect under much weaker assumptions often endpoints are grouped together into physiological domains or by the body function they describe we derive tests based on these subgroups which might supplement or replace the individual endpoint analysis because they are more powerful the permutation or bootstrap distribution is used throughout to obtain global subgroup and individual significance levels as they naturally incorporate the correlation among endpoints we provide a theorem that establishes a connection between marginal homogeneity and the stronger exchangeability assumption under the permutation approach multiplicity adjustments for the individual endpoints are obtained via stepdown procedures while subgroup significance levels are adjusted via the full closed testing procedure the proposed methodology is illustrated using a collection of 25 correlated ordinal endpoints grouped into six domains to evaluate toxicity of a chemical compound

latent transition regression for mixed outcomes l etat sanitaire d un patient est une observation complexe souvent caracterisee par plusieurs mesures l etude des transitions entre etats sanitaires successifs necessite la prise en compte de multiples variables mesurees longitudinalement les difficultes d analyse de ces donnees longitudinales multivariees sont encore renforcees lorsque les resultats d interet combinent des variables continues categorielles et des effectifs pour resoudre ces difficultes nous proposons une approche entierement bayesienne de regression sur les transitions latentes qui permet l analyse conjointe d un melange de variables longitudinales quelle qu en soit la distribution l etat sanitaire est considere comme une variable categorielle latente et les autres variables de resultat sont traitees comme des mesures secondaires de l etat sanitaire latent avec des erreurs de mesure cette approche permet l estimation conjointe de la prevalence de chaque etat sanitaire et des probabilites de transition entre etats en fonction des covariables les resultats observes sont lies aux etats sanitaires sous jacents par des modeles de regression qui incluent des effets individuels pour tenir compte de la correlation residuelle entre mesures repetees au cours du temps et des covariables pour tenir compte des differentes mesures de l etat sanitaire latent nous illustrons la methode proposee avec des donnees provenant d une etude longitudinale sur le mal de dos

dominance of likelihood ratio tests under cone constraints it is well known that anomalies are sometimes observed when using the likelihood ratio test lrt for testing restricted hypotheses in a normal model this paper considers a general framework for these anomalies to occur we provide a condition that relates the null and the alternative hypotheses under which the dominance of the lrt is obtained conditions are also given which guarantee the equivalence between the lrt and a simpler test the situations of known and unknown variances are considered and examples are given to illustrate the results

constructing multiple test procedures for partially ordered hypothesis sets a popular method to control multiplicity in confirmatory clinical trials is to use a so called hierarchical or fixed sequence test procedure this requires that the null hypotheses are ordered a priori for example in order of clinical importance the procedure tests the hypotheses in this order using level tests until one is not rejected it then stops so that no subsequent hypotheses are eligible for rejection this procedure strongly controls the familywise error rate fwe that is to say the probability that any true hypotheses are rejected this paper describes a simple generalization of this approach in which the null hypotheses are partially ordered it is convenient to display the partial ordering in a directed acyclic graph dag we consider sequentially rejective procedures based on the partial ordering in which a hypothesis is tested only when all preceding hypotheses have been tested and rejected in general such procedures do not control the fwe but it is shown that when certain intersection hypotheses are added strong control of the fwe is obtained the purpose of the method is to construct inference strategies for confirmatory clinical trials that better reflect the trial objectives copyright 2007 john wiley sons ltd

new diffusion models in image processing we give some diffusion models which can be used for restoration in image processing the proposed model is a combination of fast growth with respect to low gradient and slow growth when the gradient is large the existence theorem for solutions and some numerical results are given

existence and geometric properties of solutions of a free boundary problem in potential theory let 0 g h l rn n 2 be two given density functions at least one of them bounded away from zero outside a compact set and g holder continuous we prove that for any compactly supported positive measure which is sufficiently concentrated e g has sufficiently high n 1 dimensional density there exists a bounded open set rn such that the newtonian potential of the measure hln ghn 1 agrees with that of outside some regularity of is obtained as well as several results on the geometry of example if h and g are constant then for any x the inward normal ray of at x if it exists intersects the closed convex hull of typeset by ams tex 1 2 bjorn gustafsson and henrik shahgholian

on the growth and stability of real analytic functions this paper addresses the issue of whether integrals of real analytic functions remain finite under small deformations an approach based on uniform estimates for certain classes of one dimensional integrals is introduced it is powerful enough to recover the stability properties of real integrals in two dimensions which follow from the work of karpushkin as well as produce new results in higher dimensions in dimension three the new stability results are sharp as shown by the well known example of varchenko

bootstrap inference on the boundary of the parameter space with application to conditional volatility models abstract it is a well established fact that with an unknown number of nuisance parameters at the boundary testing a null hypothesis on the boundary of the parameter space is infeasible in practice as the limiting distributions of standard test statistics are non pivotal in particular likelihood ratio statistics have limiting distributions which can be characterized in terms of quadratic forms minimized over cones where the shape of the cones depends on the unknown location of the possibly multiple model parameters not restricted by the null hypothesis we propose to solve this inference problem by a novel bootstrap which we show to be valid under general conditions irrespective of the presence of unknown nuisance parameters on the boundary that is the new bootstrap replicates the unknown limiting distribution of the likelihood ratio statistic under the null hypothesis and is bounded in probability under the alternative the new bootstrap approach which is very simple to implement is based on shrinkage of the parameter estimates used to generate the bootstrap sample toward the boundary of the parameter space at an appropriate rate as an application of our general theory we treat the problem of inference in finite order arch models with coefficients subject to inequality constraints extensive monte carlo simulations illustrate that the proposed bootstrap has attractive finite sample properties both under the null and under the alternative hypothesis

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

anomalies of the likelihood ratio test for testing restricted hypotheses m m from lemma 3 4 in cohen kemperman and sackrowitz 1993 y y m m p m m m m m z g y c u since y y z g c also we have y y z g m h m m m p mq1 p m m u hence y y z g y c y c then y y z g m mq1 p y c u mq 1 m m m m by decomposing y s z q y y z and applying lemma 3 5 in 5 m m 5 2 5 m cohen kemperman and sackrowitz 1993 y y p y ru g z mq 1 m 5 2 m m 5 mq1 5 2 y p z ru but p y ru s p z ru therefore y g mq 1 mq1 mq1 5 mq1 5 2 z the main results in the corrected paper are given under more general

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

a scaled linear mixed model for multiple outcomes summary we propose a scaled linear mixed model to assess the effects of exposure and other covariates on multiple continuous outcomes the most general form of the model allows a different exposure effect for each outcome an important special case is a model that represents the exposure effects using a common global measure that can be characterized in terms of effect sizes correlations among different outcomes within the same subject are accommodated using random effects we develop two approaches to model fitting including the maximum likelihood method and the working parameter method a key feature of both methods is that they can be easily implemented by repeatedly calling software for fitting standard linear mixed models e g sas proc mixed compared to the maximum likelihood method the working parameter method is easier to implement and yields fully efficient estimators of the parameters of interest we illustrate the proposed methods by analyzing data from a study of the effects of occupational pesticide exposure on semen quality in a cohort of chinese men

good points and constructive resolution of singluarities in this paper we present a new algorithm of resolution of singularities over fields of characteristic zero making use of invariants that come from abhyankar s good point theory abl we also prove new properties on constructive or algorithmic desingularization let us explain what we mean by an algorithm of resolution consider a pair x w where w is a regular variety over a base field k of characteristic zero not necessarily irreducible i e w smooth over k and xc w is a closed non empty subscheme call c the class of all such pairs over different base fields k natural maps i x w1 x w arise within this class for instance if w1 w is a smooth map over a fixed field k or if w1 w arises from an arbitrary change of base field fix now a totally ordered set i and suppose assigned for each pair 7 x w of u a function p x i which is upper semi continuous and takes only finitely many values say 1 c r c i let max bp be the biggest ai and set

efficiency and power of tests for multiple binary outcomes abstract global tests provide a useful tool for comparing two or more groups with respect to multiple correlated outcomes we adapt and compare the performance of tests that have been suggested for use with multiple continuous outcomes to the case of multiple binary outcomes comparisons and guidelines are based on asymptotic relative efficiencies are s and simulations these results are illustrated using an application from teratology we extend the work of lefkopoulou and ryan to include general m group comparisons alternatives where group effects may differ for each outcome a concise form for this general class of score tests is derived to compute the are s for this class of tests we devise a useful characterization of the alternative space based on multivariate polar coordinates our findings indicate that the common outcome effect tests are efficient for a remarkably large range of circumstances a simple formula applies to compute the maximum number of unaffected outcomes that can be included i

a combined score test for binary and ordinal endpoints from clinical trials there is growing interest especially for trials in stroke in combining multiple endpoints in a single clinical evaluation of an experimental treatment the endpoints might be repeated evaluations of the same characteristic or alternative measures of progress on different scales often they will be binary or ordinal and those are the cases studied here in this paper we take a direct approach to combining the univariate score statistics for comparing treatments with respect to each endpoint the correlations between the score statistics are derived and used to allow a valid combined score test to be applied a sample size formula is deduced and application in sequential designs is discussed the method is compared with an alternative approach based on generalized estimating equations in an illustrative analysis and replicated simulations and the advantages and disadvantages of the two approaches are discussed

a permutational step up method of testing multiple outcomes this paper describes a permutational step up multiple comparison procedure to adjust the p values from k related hypotheses the method is applicable when two groups of subjects are being compared on each of k outcomes it is related to the step down method of westfall and young 1993 resampling based multiple testing examples and methods for p value adjustment volume 1 and troendle 1995 journal of the american statistical association 90 370 378 and is an alternative to the analytic step up method of dunnett and tamhane 1992 journal of the american statistical association 87 162 170 which requires a specific distribution and correlation structure by conditioning on the data observed this permutational method avoids any specific distribution or correlation assumption it is shown very generally that the method asymptotically controls the familywise probability of a type i error

extended maxt tests of one sided hypotheses in many statistical applications of one sided tests of multiple hypotheses researchers are often concerned not only with global tests of the intersection of individual hypotheses but also with multiple tests of individual hypotheses for example in clinical trial studies researchers often need to find out the efficacy of a treatment as well as the significance of each outcome measurement endpoint of the treatment this article proposes maxt type tests aiming at improving the global power of existing maxt tests our extended maxt tests are constructed by adding an extra component to the maximand set of existing maxt tests the added component is a weighted sum of other components some power properties relating to choices of weight are studied our simulation study shows that the proposed tests can considerably improve the global power of existing maxt tests and can also outperform many other global tests under some alternatives and or some nonnormal distributions furthermore it is shown that such g

fully nonlinear singularly perturbed equations and asymptotic free boundaries in this paper we study one phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials e u with e 0 we establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non degenerated for problems governed by concave equations we establish uniform weak geometric properties of approximating level surfaces we also provide a thorough analysis of the free boundary problem obtained as a limit as the e parameter term goes to zero we find the precise jumping condition of limiting solutions through the phase transition which involves a subtle homogenization process of the governing fully nonlinear operator in particular for rotational invariant operators f d2u we show the normal derivative of limiting function is constant along the interface smoothness properties of the free boundary are also addressed

linearly divergent treatment effects in clinical trials with repeated measures efficient analysis using summary statistics in many randomized clinical trials with repeated measures of a response variable one anticipates a linear divergence over time in the difference between treatments this paper explores how to make an efficient choice of analysis based on individual patient summary statistics with the objective of estimating the mean rate of treatment divergence the simplest choice of summary statistic is the regression coefficient of response on time for each subject slope the gains in statistical efficiency imposed by adjusting for the observed pre treatment levels or even better the estimated intercepts are clarified in the process we develop the optimal linear summary statistic for any repeated measures design with assumed known covariance structure and shape of true mean treatment difference over time statistical power considerations are explored and an example from an asthma trial is used to illustrate the main points

maximal averages over hypersurfaces and the newton polyhedron abstract using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques l p boundedness theorems for p 2 are obtained for maximal averages over hypersurfaces in r n for n 2 these estimates are sharp in various situations including the convex hypersurfaces of finite line type considered by several authors as a corollary we also give a generalization of the result of sogge and stein that for some finite p the maximal operator corresponding to a hypersurface whose gaussian curvature does not vanish to infinite order is bounded on l p for some finite p analogous estimates are proven for fourier transforms of surface measures and these are sharp for the same hypersurfaces as the maximal operators

newton polygons and local integrability of negative powers of smooth functions in the plane let f x y be any smooth real valued function with f 0 0 0 for a sufficiently small neighborhood u of the origin we study the number sup u f x f it is known that sometimes this number can be expressed in a natural way using the newton polygon of f we provide necessary and sufficient conditions for this newton polygon characterization to hold the behavior of the integral at the supremal e is also analyzed

a simple multivariate test for one sided alternatives abstract a simple test for multivariate normal data is proposed that has good power for alternatives where the mean vector tends to be positive the test rejects if the quadratic form of the sample mean vector exceeds its 2 critical value and the sum of the elements of the mean vector exceeds zero the proposed test is shown to have type i error rate equal to whether or not the covariance matrix is known tight bounds on the power of the proposed test are provided and the test is compared to three likelihood ratio tests

simultaneous inference on treatment effects in survival studies with factorial designs a clinical trial with a 2 2 factorial design involves randomization of subjects to treatment a or a and within each group further randomization to treatment b or b under this design one can assess the effects of treatments a and b on a clinical endpoint using all patients one may additionally compare treatment a treatment b or combination therapy ab to a b with multiple comparisons however it may be desirable to control the overall type i error especially for regulatory purposes because the subjects overlap in the comparisons the test statistics are generally correlated by accounting for the correlations one can achieve higher statistical power compared to the conventional bonferroni correction herein we derive the correlation between any two stratified or unstratified log rank statistics for a 2 2 factorial design with a survival time endpoint such that the overall type i error for multiple treatment comparisons can be properly controlled in addition we allow for adjustment of prognostic factors in the treatment comparisons and conduct simultaneous inference on the effect sizes we use simulation studies to show that the proposed methods perform well in realistic situations we then provide an application to a recently completed randomized controlled clinical trial on alcohol dependence finally we discuss extensions of our approach to other factorial designs and multiple endpoints

oscillatory integral decay sublevel set growth and the newton polyhedron extending some resolution of singularities methods greenblatt in j funct anal 255 8 1957 1994 2008 of the author a generalization of a well known theorem of varchenko funct anal appl 18 3 175 196 1976 relating decay of oscillatory integrals to the newton polyhedron is proven they are derived from analogous results for sublevel integrals proven here varchenko s theorem requires a certain nondegeneracy condition on the faces of the newton polyhedron on the phase in this paper it is shown that the estimates of varchenko s theorem also hold for a significant class of phase functions for which this nondegeneracy condition does not hold thus in problems where one wants to switch coordinates to a coordinate system where varchenko s estimates are valid one has greater flexibility some additional estimates are also proven for more degenerate situations including some too degenerate for the newton polyhedron to give the optimal decay in the sense of varchenko

repeated measures in clinical trials analysis using mean summary statistics and its implications for design this paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments quite often the data for each patient may be effectively summarized by a pre treatment mean and a post treatment mean analysis of covariance is the method of choice and its superiority over analysis of post treatment means or analysis of mean changes is quantified as regards both reduced variance and avoidance of bias using a simple model for the covariance structure between time points quantitative consideration is also given to practical issues in the design of repeated measures studies the merits of having more than one pre treatment measurement are demonstrated and methods for determining sample sizes in repeated measures designs are provided several examples from clinical trials are presented and broad practical recommendations are made the examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials the analysis using summary statistics makes no such assumption however allowance in design for alternative non equal correlation structures can and should be made when necessary

uniformly more powerful tests in a one sided multivariate problem abstract we consider a testing problem in which a normal mean vector is at the origin under the null hypothesis and in a cone under the alternative and the covariance matrix is unknown perlman studied the likelihood ratio lr test and found that it is biased and hence does not compare favorably with hotelling s t2 we point out that certain tests of perlman are uniformly more powerful as well as simpler than the lr test these are the corresponding lr tests when the alternative is replaced with a half space that contains the cone we prove that these tests are unbiased by developing some basic results on stochastic ordering critical values for these tests are given simulation results suggest that the new tests are also uniformly more powerful than hotelling s t2 for the special case when the cone is the positive orthant we discuss an issue about invariant test finally we discuss the assumption of unknown covariance matrix and conclude that the new tests are more useful for small samples than for

a multi dimensional resolution of singularities with applications to analysis we formulate a resolution of singularities algorithm for analyzing the zero sets of real analytic functions in dimensions geq 3 rather than using the celebrated result of hironaka the algorithm is modeled on a more explicit and elementary approach used in the contemporary algebraic geometry literature as an application we define a new notion of the height of real analytic functions compute the critical integrability index and obtain sharp growth rate of sublevel sets this also leads to a characterization of the oscillation index of scalar oscillatory integrals with real analytic phases in all dimensions

accurate critical constants for the one sided approximate likelihood ratio test of a normal mean vector when the covariance matrix is estimated summary tang gnecco and geller 1989 biometrika76 577 583 proposed an approximate likelihood ratio alr test of the null hypothesis that a normal mean vector equals a null vector against the alternative that all of its components are nonnegative with at least one strictly positive this test is useful for comparing a treatment group with a control group on multiple endpoints and the data from the two groups are assumed to follow multivariate normal distributions with different mean vectors and a common covariance matrix the homoscedastic case tang et al derived the test statistic and its null distribution assuming a known covariance matrix in practice when the covariance matrix is estimated the critical constants tabulated by tang et al result in a highly liberal test to deal with this problem we derive an accurate small sample approximation to the null distribution of the alr test statistic by using the moment matching method the proposed approximation is then extended to the heteroscedastic case the accuracy of both the approximations is verified by simulations a real data example is given to illustrate the use of the approximations

the analysis of multiple endpoints in clinical trials treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall type i error rate one option is to select a single primary endpoint for formal statistical inference but this is not always feasible another approach is to apply bonferroni correction i e multiply each p value by the total number of endpoints its conservatism for correlated endpoints is examined for multivariate normal data a third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics quantitative binary and survival endpoints are all considered within this general framework two examples are presented and the relative merits of the proposed strategies are discussed

multiple testing in clinical trials the basic ideas of multiple testing are outlined and the problem of how to control the probability of erroneous decisions is discussed the main emphasis is on the concept of the multiple level of significance controlling the experiment or family error in the strong sense which can be achieved by applying the principle of closed tests various practical situations encountered in multiple testing in clinical trials are considered more than one end point more than two treatments such as comparisons with a single control comparisons using ordered alternatives all pairwise comparisons and contrast methods and more than one trial tests based on global statistics the union intersection principle and other criteria are discussed the application of the multiple test concept in sequential sampling is investigated finally some comments are made on multiple power multiple confidence intervals and directed decisions

repeated significance testing for a general class of statistics used in censored survival analysis abstract the asymptotic joint distribution of a class of sequentially computed statistics used in survival analysis is derived when the individuals under study enter serially and are subject to random loss to follow up these results are used for constructing repeated significance tests over real time

an elementary coordinate dependent local resolution of singularities and applications abstract an elementary classical analysis resolution of singularities method is developed extensively using explicit coordinate systems the algorithm is designed to be applicable to subjects such as oscillatory integrals and critical integrability exponents as one might expect the trade off for such an elementary method is a weaker theorem than hironaka s work h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero i ann of math 2 79 1964 109 203 h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero ii ann of math 2 79 1964 205 326 or its subsequent simplications and extensions such as e bierstone p milman canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant invent math 128 2 1997 207 302 s encinas o villamayor good points and constructive resolution of singularities acta math 181 1 1998 109 158 j kollar resolution of singularities seattle lectures preprint a n varchenko newton polyhedra and estimates of oscillatory integrals funct anal appl 18 3 1976 175 196 nonetheless the methods of this paper can be used to prove a variety of theorems of interest in analysis as illustration two consequences are given first and most notably a general theorem regarding the existence of critical integrability exponents are established secondly a quick proof of a well known inequality of lojasiewicz s lojasiewicz ensembles semi analytiques inst hautes etudes sci bures sur yvette 1964 is given the arguments here are substantially different from the general algorithms such as h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero i ann of math 2 79 1964 109 203 h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero ii ann of math 2 79 1964 205 326 or the elementary arguments of e bierstone p milman semianalytic and subanalytic sets inst hautes etudes sci publ math 67 1988 5 42 and h sussman real analytic desingularization and subanalytic sets an elementary approach trans amer math soc 317 2 1990 417 461 the methods here have as antecedents the earlier work of the author m greenblatt a direct resolution of singularities for functions of two variables with applications to analysis j anal math 92 2004 233 257 phong and stein d h phong e m stein the newton polyhedron and oscillatory integral operators acta math 179 1997 107 152 and varchenko a n varchenko newton polyhedra and estimates of oscillatory integrals funct anal appl 18 3 1976 175 196

a free boundary problem for the heat equation arising in flame propagation we introduce a new free boundary problem for the heat equation of interest in combustion theory it is obtained in the description of laminar flames as an asymptotic limit for high activation energy the problem asks for the determination of a domain in space time q c rn x 0 t and a function u x t 0 defined in q such that ut au in q u takes certain initial conditions u x 0 uo x for x e qo oqfn t 0 and two conditions are satisfied on the free boundary r a2 n t 0 u o and u1 1 where u1 denotes the derivative of u along the spatial exterior normal to r we approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data

a minimum problem with free boundary in orlicz spaces abstract we consider the optimization problem of minimizing g u u 0 d x in the class of functions w 1 g with u 0 w 0 1 g for a given 0 0 and bounded w 1 g is the class of weakly differentiable functions with g u d x the conditions on the function g allow for a different behavior at 0 and at we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface also we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the lipschitz regularity of the weak solutions and the c 1 regularity of their free boundaries near flat free boundary points

a composite design for transition from a preliminary to a full scale study in drug development a sequence of studies are conducted to evaluate effectiveness or efficacy and safety such as a phase ii study to assess pharmacological activity or safety that is then followed by a definitive phase iii study to assess clinical effectiveness rather than conducting separate successive studies we describe a design in which the patients enrolled in a preliminary e g phase ii study are continued into a subsequent full scale e g phase iii study this design also applies to a study that uses an internal pilot with a preliminary assessment of efficacy or safety the combined preliminary to full scale design potentially reduces the total numbers of patients required and the total duration of the program the design allows a futility or safety stopping boundary for the preliminary study result that is specified in terms of a lower critical z value z l and the pursuant type ii error probability incurred under a specified alternative hypothesis of a beneficial effect or no toxicity at that stage this boundary also leads to a deflation of the type i error probability for the final test at the completion of the full scale study such that a critical value for the final test z f can be determined that provides the desired level of the type i error probability exactly thus it is possible to determine sample sizes at the two stages and critical values z l and z f that provide specified type i and ii error probabilities for the combined study we describe the design using large sample normally distributed z tests at the two phases including a test for means or proportions or survival times or combinations thereof such as a test for means at phase ii followed by a test for proportions at phase iii we compare the properties of this design versus the conduct of two successive studies and explore the factors that influence the operating characteristics of the design we also discuss the practical considerations in the implementation of the design

conditional likelihood ratio test for a nonnegative normal mean vector abstract the conditional likelihood ratio test is derived for significance of a multivariate mean having nonnegative components this test is shown to be uniformly more powerful than the unconditional likelihood ratio test derived by perlman the computation involved in the new test is a straightforward programming task simulation results suggest that this test is also uniformly more powerful than a half space test proposed by tang and hotelling s t 2 test the consistency invariance and unbiasedness of the new test are established and the test is illustrated with data from a randomized clinical trial

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

simple hironaka resolution in characteristic zero building upon work of villamayor and bierstone milman we give a proof of the canonical hironaka principalization and desingularization the idea of homogenized ideals introduced in the paper gives a priori the canonicity of algorithm and radically simplifies the proof

sharp l estimates for one dimensional oscillatory integral operators with c phase in this paper sharp l 2 estimates are proved for general local one dimensional oscillatory integral operators with smooth phase the proof involves a coarse resolution of singularities in conjunction with a stopping time argument and extensions of the analytic methods of d h phong and e m stein

resampling based methods for the analysis of multiple endpoints in clinical trials we consider multivariate tests for comparing two treatments with multiple endpoints the test decision is drawn from the simultaneous consideration of the univariate tests for the single endpoints a general class of these tests called cut off tests can be given which however can lead to highly conservative procedures because the dependencies among the endpoints are not taken into account in applying resampling based methods considerable improvements for these tests can be achieved resampling based cut off tests are proposed which are sensitive against a treatment difference in a single endpoint in a subgroup of endpoints or in all endpoints the results of monte carlo simulations demonstrate that a remarkable gain in statistical power as compared to the crude simultaneous consideration can be reached in particular for the multivariate one sided test situation the proposed tests can be recommended as an example the application of the tests is demonstrated by data from a clinical trial

optimal testing of multiple hypotheses with common effect direction we present a theoretical basis for testing related endpoints typically it is known how to construct tests of the individual hypotheses but not how to combine them into a multiple test procedure that controls the familywise error rate using the closure method we emphasize the role of consonant procedures from an interpretive as well as a theoretical viewpoint surprisingly even if each intersection test has an optimality property the overall procedure obtained by applying closure to these tests may be inadmissible we introduce a new procedure which is consonant and has a maximin property under the normal model the results are then applied to proactive a clinical trial designed to investigate the effectiveness of a glucose lowering drug on macrovascular outcomes among patients with type 2 diabetes

a free boundary problem for the p x laplacian abstract we consider the optimization problem of minimizing 1 p x u p x x u 0 d x in the class of functions w 1 p with u 0 w 0 1 p for a given 0 0 and bounded w 1 p is the class of weakly differentiable functions with u p x d x we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface

an approximate likelihood ratio test for a normal mean vector with nonnegative components with application to clinical trials summary a new multivariate statistic designed to be sensitive to the alternative hypothesis that a mean vector lies in the positive orthant rather than to all alternatives or to alternatives on a line o brien 1984 is proposed the statistic is shown to approximate the likelihood ratio test statistic for such alternatives kud5 1963 perlman 1969 its null distribution is a special case of the chi bar squared distribution which allows critical values to be obtained such a statistic may be used in two armed clinical trials where we are interested in showing that one treatment is better than another with respect to several responses as an example a clinical trial is reanalyzed with stronger conclusions than had been drawn originally

testing hypotheses about an identified treatment when there are multiple endpoints abstract the problem of comparing an identified treatment with k other treatments is considered in a multivariate setting many formulations of composite alternative hypotheses are possible for example one might wish to examine whether the identified treatment is superior to the other treatments on all components of the response vector i e is uniformly best or whether the identified treatment is better than each treatment on at least one component i e is admissible for testing whether the identified treatment is uniformly best the known optimality of the min test in the univariate case is extended to the multivariate case if the distribution is multivariate normal then the min test is shown to be a likelihood ratio test for testing whether the identified treatment is admissible a min test based on the bonferroni inequality is suggested for the multivariate normal with unknown covariance matrix the likelihood ratio test is also a min test but it has less stable power characteristics than

order restricted inference for multivariate binary data with application to toxicology in many applications researchers collect multivariate binary response data under two or more naturally ordered experimental conditions in such situations one is often interested in using all binary outcomes simultaneously to detect an ordering among the experimental conditions to make such comparisons we develop a general methodology for testing for the multivariate stochastic order between k 2 multivariate binary distributions our proposed test uses order restricted estimators which according to our simulation study are more efficient than the unrestricted estimators in terms of their mean squared error we compared the power of the proposed test with that of several alternative tests including procedures that combine individual univariate tests for order such as union intersection tests and a bonferroni based test we also compared the proposed test with an unrestricted hotelling t2 type test our simulations suggest that the proposed method competes well with these alternatives the gain i

asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints summary the analysis of moment structural models has become an important tool of investigation in behavioural educational and economic studies the chi squared largesample test is routinely employed to assess the goodness of fit of the model considered however in order to invoke the standard asymptotic distribution theory certain regularity conditions have to be met here we consider the case where the population value of the parameter vector is a boundary point of the feasible region we show that in this case the asymptotic distribution of test statistic is a mixture of chi squared distributions the problem of finding the corresponding weights is discussed

hausdorff measure estimates and lipschitz regularity in inhomogeneous nonlinear free boundary problems in this paper we prove a hausdorff measure estimate for the free boundaries of subsolutions of fully nonlinear and quasilinear equations of the type f d 2u x geqq f x and rm div a x nabla u geqq mu where f in l q q n and is a signed radon measure with some appropriate growth condition gradient estimates for nonnegative harmonic functions with bounded normal derivatives along the boundary obtained by caffarelli and salsa geometric approach to free boundary problems 2005 are extended to the context of inhomogeneous problems involving fully nonlinear and p laplace equations as an application lipschitz regularity is obtained for one phase solutions of inhomogeneous nonlinear free boundary problems

procedures for comparing samples with multiple endpoints summary five procedures are considered for the comparison of two or more multivariate samples these procedures include a newly proposed nonparametric rank sum test and a generalized least squares test also considered are the following tests ordinary least squares hotelling s p and a bonferroni per experiment error rate approach applications are envisaged in which each variable represents a qualitatively different measure of response to treatment the null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others in all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test and is recommended for general use alternatively the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples a convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy when the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable appropriate parametric and nonparametric one way analysis of variance anova procedures and their properties are well known often however efficacy is measured by more than one variable although univariate methods for assessing each characteristic individually are useful in this setting there is often the additional need for a single overall objective probability statement that addresses the question of whether or not the experimental therapy is efficacious this need is particularly acute when the medical question is controversial and the sample size is small although a large literature exists on the comparison of multivariate samples the procedures tend to be infrequently used in practice as pointed out by meier 1975 p 523 the standard analysis for the comparison of two multivariate samples which is based on hotelling s t2 statistic addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest specifically the t2 procedure addresses the question are one or more of the treatments different making no distinction between variables that change favorably and variables that change unfavorably a second approach would be to assign per experiment error rates to each of the univariate tests by using bonferroni s inequality i e by multiplying each univariate p value by the number of variables studied the desired overall probability statement is then given by the minimum of the per experiment univariate p values although this approach may be useful in some settings it may lack power for alternatives in which most or all measures of efficacy are improved this will be of particular concern when the number k of endpoints studied is large relative to sample size in fact statistical significance will not be possible in some instances this problem will be exacerbated when the measures of efficacy are highly

bivariate modelling of clustered continuous and ordered categorical outcomes simultaneous observation of continuous and ordered categorical outcomes for each subject is common in biomedical research but multivariate analysis of the data is complicated by the multiple data types here we construct a model for the joint distribution of bivariate continuous and ordinal outcomes by applying the concept of latent variables to a multivariate normal distribution the approach is then extended to allow for clustering of the bivariate outcomes the model can be parameterized in a way that allows writing the joint distribution as a product of a standard random effects model for the continuous variable and a correlated cumulative probit model for the ordinal outcome this factorization suggests convenient parameter estimation using estimating equations foetal weight and malformation data from a developmental toxicity experiment illustrate the results

global rank tests for multiple possibly censored outcomes clinical trials often collect multiple outcomes on each patient as the treatment may be expected to affect the patient on many dimensions for example a treatment for a neurological disease such as als is intended to impact several dimensions of neurological function as well as survival the assessment of treatment on the basis of multiple outcomes is challenging both in terms of selecting a test and interpreting the results several global tests have been proposed and we provide a general approach to selecting and executing a global test the tests require minimal parametric assumptions are flexible about weighting of the various outcomes and are appropriate even when some or all of the outcomes are censored the test we propose is based on a simple scoring mechanism applied to each pair of subjects for each endpoint the pairwise scores are then reduced to a summary score and a rank sum test is applied to the summary scores this can be seen as a generalization of previously proposed nonparametric global tests e g o brien 1984 we discuss the choice of optimal weighting schemes based on power and relative importance of the outcomes as the optimal weights are generally unknown in practice we also propose an adaptive weighting scheme and evaluate its performance in simulations we apply the methods to analyze the impact of a treatment on neurological function and death in an als trial

algebraic estimates stability of local zeta functions and uniform estimates for distribution functions a method of algebraic estimates is developed and used to study the stability properties of integrals of the form r b jf z j i dv under small deformations of the function f the estimates are described in terms of a stratiflcation of the space of functionsfr z jp z j jq z j g by algebraic varieties on each of which the size of the integral of r z is given by an explicit algebraic expression the method gives an independent proof of a result on stability of tian in 2 dimensions as well as a partial extension of this result to 3 dimensions in arbitrary dimensions combined with a key lemma of siu it establishes the continuity of the mapping c r b jf z c j i dv1 dvn when f z c is a holomorphic function of z c in particular the leading pole is semicontinuous in f strengthening also an earlier result of lichtin

a rank test for bivariate time to event outcomes when one event is a surrogate in many clinical settings improving patient survival is of interest but a practical surrogate such as time to disease progression is instead used as a clinical trial s primary endpoint a time to first endpoint e g death or disease progression is commonly analyzed but may not be adequate to summarize patient outcomes if a subsequent event contains important additional information we consider a surrogate outcome very generally as one correlated with the true endpoint of interest settings of interest include those where the surrogate indicates a beneficial outcome so that the usual time to first endpoint of death or surrogate event is nonsensical we present a new two sample test for bivariate interval censored time to event data where one endpoint is a surrogate for the second less frequently observed endpoint of true interest this test examines whether patient groups have equal clinical severity if the true endpoint rarely occurs the proposed test acts like a weighted logrank test on the surrogate if it occurs for most individuals then our test acts like a weighted logrank test on the true endpoint if the surrogate is a useful statistical surrogate our test can have better power than tests based on the surrogate that naively handles the true endpoint in settings where the surrogate is not valid treatment affects the surrogate but not the true endpoint our test incorporates the information regarding the lack of treatment effect from the observed true endpoints and hence is expected to have a dampened treatment effect compared with tests based on the surrogate alone published 2016 this article is a u s government work and is in the public domain in the usa

real analytic desingularization and subanalytic sets an elementary approach we give a proof of a theorem on desingularization of real analytic functions which is a weaker version of h hironaka s result but has the advantage of being completely self contained and elementary and not involving any machinery from algebraic geometry we show that the basic facts about subanalytic sets can be proved from this result