moment based criteria for determining bioequivalence summary in comparative bioavailability trials the focus in the determination of bioequivalence has been to show that the average bioavailability on a test formulation is similar to the average bioavailability on a reference formulation it has been argued that this is not adequate to insure interchangeability of formulations we introduce a moment based criterion which ensures interchangeability for at least a certain proportion of the population an algorithm is given for determining the constant in the criterion an example is used to illustrate how the criterion may be applied

algorithm for generalized co complementarity problems in banach spaces abstract in this paper we introduce a new class of generalized co complementarity problems in banach spaces an iterative algorithm for finding approximate solutions of these problems is considered some convergence results for this iterative algorithm are derived and several existence results are obtained

use of the repeated cross over designs in assessing bioequivalence we consider applications of the repeated 2 x 2 cross over design to evaluating bioequivalence between the two formulations the repeated 2 x 2 cross over design allows us not only to assess bioequivalence on average bioavailability and to examine the subject by formulation interaction but also to obtain independent unbiased estimates of intrasubject variability one consequence of unequal intrasubject variabilities is that the sum of squares of intersubject residuals and the sum of squares of subject by formulation residuals are not independent we also discuss the relative merits of this design as compared to the standard 2 x 2 cross over design without repeated measurements in terms of precision and sample size with respect to the ratio of the number of subjects to the repeated measurements per subject we investigate other uses of the 2 x 2 cross over in examining the bioequivalence between the two different dosing regimens possible applications of other repeated cross over designs to bioequivalence for more than two formulations are explored a numerical example illustrates the proposed procedure

algorithm for generalized multivalued variational inequalities in hilbert spaces abstract we introduce a new class of generalized multivalued variational inequality problems 1 1 which contain several existing variational inequality problems as special cases an iterative algorithm for finding approximate solutions of problem 1 1 is considered several convergence results for this algorithm are derived and in particular several existence results of problems 1 1 are obtained we also introduce a new class of generalized multivalued complementarity problems 5 1 which also contain several known complementarity problems as special cases it is shown that problem 5 1 is equivalent to problem 1 1 of special type from which several existence results of problem 5 1 are obtained

tests for individual and population bioequivalence based on generalized p values the u s food and drug administration fda has proposed new regulations that address the prescribability and switchability of new formulations of already approved drugs these new criteria are known respectively as population and individual bioequivalence two methods have been proposed in the bioequivalence literature for assessing population and individual bioequivalence that calculate an upper 95 per cent confidence bound for the bioequivalence criterion in question and then test bio equivalence by comparing this bound to the limit established by the fda in this paper we propose applying the generalized test function gtf methodology of tsui and weerahandi journal of the american statistical association 1989 602 607 to this problem to produce tests based on a generalized p value gpv this methodology allows us to construct hypothesis tests in the presence of nuisance parameters using simulation we show that these tests perform well in comparison to the confidence interval methods and have superior power for assessing population bioequivalence copyright 2003 john wiley sons ltd

a practical approach for evaluating population and individual bioequivalence pharmacokinetic measurements provided by subjects to each of two formulations of a drug have a joint distribution that can be characterized by parameters reflecting scale and correlation as well as location the bioavailability of the formulations can be expressed in terms of the means of the marginal distributions their means and variances or the marginal means and variances and the joint correlation these expressions correspond respectively to average population and individual bioequivalence when the joint distribution of the measurements is bivariate normal current proposals for assessing the degree of bioequivalence of two formulations are based on statistics that are composites of variance components and squares of expected mean differences from a mixed linear model there are technical and practical issues associated with these proposals particularly that they require more complicated designs than the familiar 2 x 2 cross over this paper describes an alternative approach that can be applied with standard 2 x 2 cross over designs and that provides evaluations of population and individual bioequivalence that should be adequate for all practical clinical purposes the approach is based on easily computed correlation and regression coefficients whose statistical properties under normality are well known and for which non parametric and robust alternatives exist when normality cannot be assumed the approach yields conclusions consistent with those obtained by the current proposals when applied to data sets supplied by the fda in the cases where the conclusions do not match the new approach appears to be more consistent with the data

generalized f complementarity problems in banach spaces abstract in this paper a class of generalized f complementarity problems and three classes of variational inequalities are introduced in real banach spaces and the equivalences among them are established under certain conditions several coercivity conditions are introduced for the existence of solutions of the generalized f complementarity problem under some suitable assumptions it is shown that each of these coercivity conditions is equivalent to the nonemptyness and boundedness of the solution set for the generalized f complementarity problem in infinite dimensional banach spaces and even the nonemptyness and compactness of the solution set for the generalized f complementarity problem in finite dimensional spaces the existence of least elements for the feasible set of the generalized f complementarity problem is also presented under suitable conditions

assessment of individual and population bioequivalence using the probability that bioavailabilities are similar in this paper a new method for the assessment of both individual and population bioequivalence is proposed that is derived from the approach of schall and luus 1993 statistics in medicine 12 1109 1124 the method considers the probability that the bioavailability of the test product is close to the bioavailability of the reference product against the probability that the bioavailability of the reference product is close to itself when administered on different occasions the method can accommodate unequal variances of the bioavailabilities of the test and reference formulations as well as formulation by subject interaction it is thus as general and comprehensive as the methods proposed by sheiner 1992 statistics in medicine 11 1777 1778 and schall and luus 1993 but the resulting bioequivalence criteria are simple and easy to interpret

sample sizes for bioequivalence studies in recent years a number of decision rules based on sound statistical principles have been proposed for deciding if a test formulation is bioequivalent to a reference formulation the decision rule based on confidence intervals has been accepted by regulatory agencies at least by the food and drug administration of the united states a useful property of this decision rule is that the regulatory agency need not require a certain sample size since the level of protection against wrongly deciding bioequivalence is set by the choice of the alpha level used to compute the confidence intervals the manufacturer claiming bioequivalence is concerned about sample size for sample size determines the probability of falsely deciding non bioequivalence when the test formulation does indeed have an acceptable relative bioavailability curves of probability of rejecting bioequivalence have been computed for error coefficient of variation of 10 20 and 30 per cent for relative bioavailability from 70 to 130 per cent and for protection levels of 90 and 95 per cent these curves can be used for choosing the sample size for a bioequivalence study

optimum allocation of treatments for welch s test in equivalence assessment as an extension of welch s test to equivalence trials in a matched pair design we determine the sample sizes n1 and n2 that maximize the power at given alternatives for a given total sample size n n1 n2 although the optimal allocation is obtained asymptotically when the ratio of the standard deviations in both treatment groups equals the ratio of the sample sizes numerical investigations show that this result does not hold for sample sizes where n or 25 for convenience we provide tables containing the optimal combinations for the bioequivalence problem as well as for the problem of testing a clinically relevant difference results indicate that for small sample sizes the optimum allocations of the treatments differ significantly in both testing problems although they are asymptotically identical in addition we provide simple approximate equations that can be used for the determination of the required sample sizes to control a preassigned probability of type ii error

comparison of different methods for decision making in bioequivalence assessment if the regulatory requirements are symmetrical the use of symmetrical confidence intervals as a decision rule for bioequivalence assessment leads as shown by simulations to better level properties and an inferior power compared to a rule based on shortest confidence intervals a choice between these two approaches will have to depend on a loss function for asymmetric regulatory requirements symmetrical confidence intervals should not be used however a decision can still be based on posterior probabilities pr theta epsilon r1 r2 x or shortest confidence intervals for purposes of inference presentation and interpretation of results we think that the use of symmetrical confidence intervals alone can be misleading and we therefore recommend that the posterior probabilities and densities or at least the shortest confidence intervals be given

the subject by formulation interaction as a criterion of interchangeability of drugs summary the primary purpose of a bioequivalence study is often to meet the regulatory requirement of similar average bioavailabilities this is usually done by estimating the mean difference between formulations in a crossover design however a small difference as well as a rather narrow confidence interval might be obtained also if the subjects respond differently to the formulations i e if there is a nonnegligible subject by formulation interaction hence it can be argued that this interaction should be part of the criterion of interchangeability to be able to study this interaction the ordinary simple crossover design is not appropriate instead a design where each formulation is administered at least twice to each subject is desirable such a design is obtained by repeating a simple crossover layout to evaluate the impact of the subject by formulation interaction the ratio of the variance components of the interaction and the error is estimated this approach is illustrated by an example two different formulations of a drug are considered bioequivalent if their average bioavailabilities are similar the assessment of this similarity is usually based on an interval estimate of the mean difference between the formulations however this focus on means is an insufficient criterion since a small difference as well as a rather narrow confidence interval might be obtained also if the formulation differences show a considerable variation among subjects the impact on drug interchangeability of this so called subject byformulation interaction is discussed for example by hwang et al 1978 in order to estimate this interaction they suggest a design where each formulation is administered at least twice to each subject in this paper we briefly discuss how to design and analyze a bioequivalence study in order to evaluate the impact of the subject by formulation interaction as a measure of the magnitude of the interaction we propose the ratio of the variance components of the interaction and the error the ideas of the paper are illustrated with data from a comparative study of two generic formulations of furosemide

uniform k theory and poincare duality for uniform k homology abstract we revisit spakula s uniform k homology construct the external product for it and use this to deduce homotopy invariance of uniform k homology we define uniform k theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry we further construct a cap product with uniform k homology and prove poincare duality between uniform k theory and uniform k homology on spinc manifolds of bounded geometry

an improvement of commonly used tests in bioequivalence assessment if equivalence of two treatments has to be proved by an experiment with normally distributed observations and fixed sample size there are two usual methods on the one hand there is the double t test and on the other hand a test procedure suggested by anderson hauck which is uniformly more powerful than the first one but only capable of keeping asymptotically to the nominal level based on these two methods a new test is presented which is uniformly more powerful than the double t test and also keeps the preassigned level

maximal and reduced roe algebras of coarsely embeddable spaces in 7 gong wang and yu introduced a maximal or universal version of the roe c algebra associated to a metric space we study the relationship between this maximal roe algebra and the usual version in both the uniform and non uniform cases the main result is that if a uniformly discrete bounded geometry metric space x coarsely embeds into a hilbert space then the canonical map between the maximal and usual uniform roe algebras induces an isomorphism on k theory we also give a simple proof that if x has property a then the maximal and usual uniform roe algebras are the same these two results are natural coarse geometric analogues of certain well known implications of a t menability and amenability for group c algebras the techniques used are e theoretic building on work of higson kasparov and trout 11 12 and yu 28

exact inference about the within subject variability in 2 2 crossover trials abstract the comparison of primary interest in a 2 2 crossover trial typically concerns the effect of the treatments say a and b on the mean response level this article deals with another important aspect namely the within subject response variability under a and b differences in drug formulation and or administration may lead to considerable differences in within subject variability whatever the difference in terms of mean level consideration of both these aspects may therefore be of considerable importance for the judgement of the treatments it is shown that although there are no within subject treatment replications it is possible to make various exact inferences about the a b ratio of within subject variances and about the a b difference in mean level simultaneously and marginally these inferences are semiparametric in that no distributional assumption is made about the between subject variability whereas a normality assumption is used for the within subject variability the infere

numerical validation of solutions of linear complementarity problems this paper proposes a validation method for solutions of linear complementarity problems the validation procedure consists of two sufficient conditions that can be tested on a digital computer if the first condition is satisfied then a given multidimensional interval centered at an approximate solution of the problem is guaranteed to contain an exact solution if the second condition is satisfied then the multidimensional interval is guaranteed to contain no exact solution this study is based on the mean value theorem for absolutely continuous functions and the reformulation of linear complementarity problems as nonsmooth nonlinear systems of equations

a statistical assessment of clinical equivalence an observed confidence distribution is proposed as a measure of strength of evidence for practically equivalent efficacies of two treatments the concept is independent of prior opinions about relevant sizes of a difference in efficacy it also avoids retrospective power calculations for trials with missed recruitment goals and allows combination of the confidence from independent experiments

definition of individual bioequivalence occasion to occasion versus mean switchability two moment based scaled definitions of individual bioequivalence are discussed based on a mixed effects linear model their evaluations respectively lead to an unweighted theta 11 and a parametric theta 15 metric the two metrics are estimated with respect to study design and two estimation methods results show that the two ibe metrics perform equivalently in the fully replicated design in the semi replicated design the definition of theta 11 may not be valid while the evaluation of theta 15 results in a reduction of the weights in the mean difference and switchability components of the metric percentage rejection rates in the latter design indicate that theta 11 is more conservative than theta 15 this is because there is an increase of about 15 per cent in the producer risk in theta 11 relative to theta 15 compared to a 7 per cent increase in the consumer risk in theta 15 relative to theta 11 a further disadvantage of the design is that there is a 33 per cent loss in the subject by treatment variance efficiency which is reflected in a similar amount of decreased sensitivity to departures from perfect bioequivalence even when more subjects are used to equalize the number of exposure occasions in the two designs it is concluded that a mean switchability criterion may be more appropriate from an interpretability perspective the bootstrap resampling method used to evaluate individual bioequivalence based on theta 11 may need to be bias corrected and that the semi replicated design should be used cautiously

uniform k homology theory we define a uniform version of analytic k homology theory for separable proper metric spaces furthermore we define an index map from this theory into the k theory of uniform roe c algebras analogous to the coarse assembly map from analytic k homology into the k theory of roe c algebras we show that our theory has a mayer vietoris sequence we prove that for a torsion free countable discrete group the direct limit of the uniform k homology of the rips complexes of limd k u pd is isomorphic to k top l the left hand side of the baum connes conjecture with coefficients in l in particular this provides a computation of the uniform k homology groups for some torsion free groups as an application of uniform k homology we prove a criterion for amenability in terms of vanishing of a fundamental class in spirit of similar criteria in uniformly finite homology and k theory of uniform roe algebras

a serre swan theorem for bundles of bounded geometry abstract the serre swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions this theorem is here generalized to manifolds of bounded geometry in this context it states that the category of hilbert bundles of bounded geometry is equivalent to the category of operator modules over the operator algebra of continuously differentiable functions which vanish at infinity operator modules are generalizations of hilbert c modules where the category of c algebras has been replaced by a more flexible category of involutive algebras of bounded operators the operator algebras operator modules play an important role in the study of the unbounded kasparov product

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

an index theorem on open manifolds ii on decrit et on applique un nouveau theoreme d indice pour des operateurs elliptiques sur certaines varietes non compactes on calcule un indice a valeur reelle pour des operateurs de type dirac sur des varietes non compactes de geometrie bornee

bayesian inference for ratios of coefficients in a linear model a variety of statistical problems e g slope ratio and parallel line bioassay calibration bioequivalence can be viewed as questions of inference on the ratio of two coefflcients in a suitably constructed linear model this paper develops a bayesian approach for the general formulation of the problem using a class of priors which involve the unknown ratio explicitly the posterior distribution of the ratio is obtained analytically and its properties are investigated especially the sensitivity to the choice of the prior examples are given of applications to slope ratio bioassay comparison of the mean effects of two drugs and a bioequivalence problem

unit treatment interaction and its practical consequences summary most statistical characterizations of a treatment effect focus on the average effect of the treatment over an entire population however average effects may provide inadequate information sometimes misleading information when a substantial unit treatment interaction is present in the population it is even possible that a nonnegligible proportion of the individuals in the population experience an unfavorable treatment effect even though the treatment might appear to be beneficial when considering population averages this paper examines the extent to which information about unit treatment interaction can be extracted using observed data from a two treatment completely randomized experiment a method for utilizing the information from an available covariate is proposed although unit treatment interaction is a nonidentifiable quantity we show that mathematical bounds for it can be estimated from observed data these bounds lead to estimated bounds for the probability of an unfavorable treatment effect maximum likelihood estimators of the bounds and their corresponding large sample distributions are given the use of the estimated bounds is illustrated in a clinical trials data example

a semi parametric bayesian approach to average bioequivalence bioequivalence assessment is an issue of great interest development of statistical methods for assessing bioequivalence is an important area of research for statisticians bioequivalence is usually determined based on the normal distribution we relax this assumption and develop a semi parametric mixed model for bioequivalence data the proposed method is quite flexible and practically meaningful our proposed method is based on a mixture normal distribution and a non parametric bayesian approach using a dirichlet process mixture prior a numerical example illustrates the use of our procedure

some bayesian and non bayesian procedures for the analysis of comparative experiments and for small area estimation computational aspects frequentist properties and relationships abstract the estimation of a treatment contrast from experimental data and the estimation of a small area mean are special cases of the prediction of the realization of a linear combination of fixed and random effects in a possibly unbalanced two part mixed linear model in this article a bayesian approach to point and interval prediction is presented and its computational requirements are examined differences between the bayesian approach and the traditional classical approach are discussed in general terms and in addition in terms of two examples taken from the literature 1 the comparison of drug formulations in a bioavailability trial westlake and 2 the estimation of corn crop areas using satellite data battese harter and fuller some deficiencies in the classical approach are pointed out and the bayesian approach is considered from a frequentist perspective it is shown via a monte carlo study that for certain noninformative choices of the prior distribution the frequentist prop

consistency of the bootstrap procedure in individual bioequivalence recently new concepts have been proposed for assessing bioequivalence of two drug formulations namely the so called population and individual bioequivalence using moment based and probability based measures for evaluating the proposed bioequivalence concepts criteria have been formulated to decide whether two formulations should be regarded as bioequivalent or not this decision has of course to be based on an adequate statistical method where the food and drug administration fda guidance 1997 recommends the use of a bootstrap percentile interval in this paper we discuss theoretical properties such as consistency and accuracy of the recommended bootstrap intervals we focus our investigations on the concept of individual bioequivalence and here especially on the scaled versions of the moment based as well as the probability based measures as recommended by the fda as estimates for the former we consider those obtained from an according analysis of variance and restricted maximum likelihood estimators under mixed effect models where an unbiased estimator of the latter can be derived from the corresponding relative frequencies

equivalence and interval testing for lehmann s alternative abstract equivalence and interval tests for lehmann s alternative that extend the well known savage test for one sided hypotheses are proposed the proposed tests are shown to be unbiased with a strictly unimodal power function provided the sample sizes in both treatment groups are equal by means of a numerical investigation of the bias in the case of unequal sample sizes that are not too far apart the suggested tests still turn out to provide practicable solutions because the computational effort to perform the suggested tests is considerable tables containing the critical values are displayed to perform these tests easily a numerical analysis of the power function of the interval test establishes this procedure as a powerful tool for detection of a significantly relevant difference in the small sample case in contrast to the case of interval testing the fact arises that the performance of a powerful equivalence study under lehmann s alternative requires an extensive amount of data because the p

a proposal for interpreting and reporting negative studies an issue of continuing interest is the interpretation and reporting of negative studies namely studies that do not find statistically significant differences the most common approach is the design power method which determines irrespective of the observed difference what differences the study could have been expected to detect we propose an alternative approach the application of equivalence testing methods where we define equivalence to mean that the actual difference lies within some specified limits this approach in contrast to the design power approach provides a way of quantifying with p values what was actually determined from the study instead of saying what the study may or may not have accomplished with some degree of certainty power for example a possible outcome of the equivalence testing approach is the conclusion at the 5 per cent level that two means or proportions do not differ by more than some specified amount the equivalence testing approach applies to any study design we illustrate the method with a cancer clinical trial and an epidemiologic case control study in addition for those studies in which one cannot specify limits a priori we propose the use of equivalence curves to summarize and present the study results

an iterative method for a system of linear complementarity problems with perturbations and interval data in this paper we introduce a total step method for solving a system of linear complementarity problems with perturbations and interval data it is applied to two interval matrices a and b and two interval vectors b and c we prove that the sequence generated by the total step method converges to x y which includes the solution set for the system of linear complementarity problems defined by any fixed email protected a email protected b email protected b and email protected c we also consider a modification of the method and show that if we start with two interval vectors containing the limits then the iterates contain the limits we close our paper with two examples which illustrate our theoretical results