analytical methods for solving the time fractional swift hohenberg s h equation in this paper the most effective methods the homotopy perturbation method hpm and the differential transform method dtm are applied to obtain the approximate solutions of the nonlinear time fractional swift hohenberg s h equation the basic philosophy of these methods does not involve linearization weak nonlinearity assumptions or perturbation theory numerical solutions for various combinations of the parameters a eigenvalue parameter l length and a fractional index are obtained the solutions of the s h equation are useful for studies of shear thinning effects in non newtonian fluid flows at the end the solutions obtained are also presented graphically

the natural element method in solid mechanics the application of the natural element method nem 1 2 to boundary value problems in two dimensional small displacement elastostatics is presented the discrete model of the domain consists of a set of distinct nodes n and a polygonal description of the boundary in the natural element method the trial and test functions are constructed using natural neighbour interpolants these interpolants are based on the voronoi tessellation of the set of nodes n the interpolants are smooth c 1 everywhere except at the nodes where they are c 0 in one dimension nem is identical to linear nite elements the nem interpolant is strictly linear between adjacent nodes on the boundary of the convex hull which facilitates imposition of essential boundary conditions a methodology to model material discontinuities and non convex bodies cracks using nem is also described a standard displacement based galerkin procedure is used to obtain the discrete system of linear equations application of nem to various problems in solid mechanics which include the patch test gradient problems bimaterial interface and a static crack problem are presented excellent agreement with exact analytical solutions is obtained which exemplies the accuracy and robustness of nem and suggests its potential application in the context of other classes of problems crack growth plates and large deformations to name a few 1998 john wiley sons ltd

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

group sequential enrichment design incorporating subgroup selection an important component of clinical trials in drug development is the analysis of treatment efficacy in patient subgroups subpopulations because of concerns of multiplicity and of the small sample sizes often involved such analyses can present substantial statistical challenges and may lead to misleading conclusions as a confirmatory seamless phase ii iii design we propose an adaptive enrichment group sequential procedure whereby resources can be concentrated on subgroups most likely to respond to treatment stopping boundaries are defined through upper and lower spending functions the procedure is presented in terms of the efficient score enabling the analysis of normal binary or time to event data it addresses the dilution effect by eliminating populations at the first stage that appear likely to derive no therapeutic benefit it subsequently proceeds with the definitive assessment of treatment efficacy among the remaining pooled populations using a group sequential design the procedure provides strong protection of familywise type i error rate and we employ a bootstrap algorithm to obtain point and interval estimates that are adjusted for the selection bias we give examples to demonstrate how the design is used we make comparisons with adaptive two stage combination test procedures and with a group sequential test that does not account for the presence of subgroups numerical results show that the procedure has high power to detect subgroup specific effects and the use of multiple interim analysis points can lead to substantial sample size savings

the use of fitted higher order polynomial coefficients as covariates in the analysis of growth curves summary for orthogonal polynomials fitted to repeated measurements a computer simulation study is used to investigate the effect of selecting higher order coefficients as covariates in order to minimise the estimated variance of lower order coefficients under the assumption that the repeated measurements have certain autoregressive covariance structures it is seen that the gain in precision due to covariance adjustment can be largely illusory and that the resulting estimates of variances can grossly underestimate the true variances the consequences of using all higher order coefficients as covariates is also examined and seen to produce a gain in precision in certain circumstances in this paper we consider a particular approach to the analysis of data from repeated measurements experiments i e those in which a sequence of measurements of the same variable is taken from each experimental unit such experiments are commonly used for comparing the growth rates of organisms there are difficulties in the analysis of repeated measurements data due to the impossibility of randomly allocating time points which means that the assumptions of homogeneity of variance and uniform within unit correlation frequently do not hold we are concerned here with examples in which no theoretically based mathematical model is available to explain the observed response over time but in which the response can be adequately summarized by a polynomial of low degree wishart 1938 showed in the context of growth rate comparison how most of the difficulties can then be conveniently avoided by fitting the low degree polynomial to the repeated measurements from each unit and basing subsequent analysis on the fitted coefficients the analysis of the linear coefficients provides a comparison of average growth rate that of the quadratic coefficients a comparison of rate of change of growth rate and so on however in this process the information in the fitted higher order coefficients which do not show treatment differences is discarded in an attempt to make use of this information some or all of these higher order coefficients can be used as covariates a possibility first mentioned by leech and healy 1959 much of the theoretical groundwork for this approach was laid by rao 1965 1966 1967 in a multivariate context and an exposition of the technique is given by grizzle and allen 1969 one of the main problems in the approach lies in the choice of which covariates to use the use of all the covariates corresponds under normality to maximum likelihood estimation but it can easily be shown that this is sometimes very inefficient the selection of only those covariates that contribute significantly or to any extent to the precision is tempting but is likely to lead to an overestimate of this precision it can be argued that this is no different from the problem of covariate selection in standard analysis of covariance however in the present situation the potential covariates are transformed

an analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids abstract this paper presents an analysis of strong discontinuities in inelastic solids at finite strains solutions exhibiting this type of discontinuities characterized by a discontinuous displacement field are shown to make physical and mathematical sense in a classical multiplicative plasticity continuum model if the softening modulus is reinterpreted as a singular distribution physically the strainsoftening is localized along the discontinuity conditions for the appearance of strong discontinuities in the geometrically nonlinear range are characterized as it is the response of the material during localization in addition these analytical results are exploited in the design of a new class of finite element methods the proposed methods fall within the class of enhanced strain methods and lead to solutions independent of the mesh size and insensitive to mesh alignment without requiring any regularization of the solutions by numerical parameters like a characteristic length

design and analysis of group sequential tests based on the type i error spending rate function summary lan demets 1983 devised a method of constructing discrete group sequential boundaries by using the type i error spending rate function it is extended so as to generate asymmetric as well as symmetric two sided boundaries for clinical trials the design aspect of this procedure is explored in terms of the maximum sample size needed to achieve a desired level of power and the expected stopping times under both null and alternative hypotheses finally these properties are employed in search of appropriate type i error spending rate functions for differing situations

a unifying family of group sequential test designs summary currently the design of group sequential clinical trials requires choosing among several distinct design categories design scales and strategies for determining stopping rules this approach can limit the design selection process so that clinical issues are not fully addressed this paper describes a family of designs that unifies previous approaches and allows continuous movement among the previous categories this unified approach facilitates the process of tailoring the design to address important clinical issues the unified family of designs is constructed from a generalization of a four boundary group sequential design in which the shape and location of each boundary can be independently specified methods for implementing the design using error spending functions are described examples illustrating the use of the design family are also presented

repeated significance testing in longitudinal clinical trials summary when longitudinal clinical trials are monitored multiplicity from repeated significance testing as well as from repeated measures has to be accounted for properly to control the overall type i error this often involves a multidimensional integration procedure to compute group sequential boundaries we establish an independent increments structure of sequentially computed test statistics based on the generalised estimating equations of liang zeger 1986 for longitudinal data this simplifies the computational procedure for group sequential boundaries to one involving recursive one dimensional integrations and allows the use of standard methodology for group sequential tests we also apply the error spending function approach of lan demets 1983 by defining information fraction in this setting

use of logrank tests and group sequential methods at fixed calendar times this paper presents simulations to determine the operating characteristics of several group sequential boundaries when applied to the repeated analysis of survival data at equal intervals of calendar time because the group sequential boundaries of pocock 1977 biometrika 64 191 199 and o brien and fleming 1979 biometrics 35 549 556 were constructed on the assumption that each interim analysis provides an equal increment of statistical information these boundaries are not theoretically appropriate for interim analysis at prescheduled calendar times nonetheless our simulations show that these boundaries yield size and power near nominal levels for repeated logrank analyses at equal intervals of calendar time

sequential testing in clinical trials with repeated measurements summary in some clinical trials the number of subjects on each treatment is fixed and at successive equally spaced time points each subject yields a measurement which reflects response to treatment a model is presented for such repeated measurements data a sequential procedure is proposed for testing the null hypothesis of no difference in mean effect between two treatments in this procedure the maximum number of measurements per subject and absorption probabilities for intermediate significance tests are preassigned to yield a required approximate overall significance level the jth significance test follows the jth measurement on each subject and the test depends on values of model parameters estimated from all the data available so far simulations show that the proposed procedure yields overall significance levels close to 5 when required for data from a variety of

reproducing kernel method for the numerical solution of the 1d swift hohenberg equation abstract the swift hohenberg equation is a nonlinear partial differential equation of fourth order that models the formation and evolution of patterns in a wide range of physical systems we study the 1d swift hohenberg equation in order to demonstrate the utility of the reproducing kernel method the solution is represented in the form of a series in the reproducing kernel space and truncating this series representation we obtain the n term approximate solution in the first approach we aim to explain how to construct a reproducing kernel method without using gram schmidt orthogonalization as orthogonalization is computationally expensive this approach will therefore be most practical for obtaining numerical solutions gram schmidt orthogonalization is later applied in the second approach despite the increased computational time as this approach will prove theoretically useful when we perform a formal convergence analysis of the reproducing kernel method for the swift hohenberg equation we demonstrate the applicability of the method through various test problems for a variety of initial data and parameter values

sample size calculation for planning group sequential longitudinal trials procedures are developed in this paper for sample size calculations for planning a group sequential longitudinal trial with various correlation structures using a test statistic based on generalized estimating equations lee kim and tsiatis and group sequential boundaries based on type i error spending functions lan and demets

an efficient augmented finite element method for arbitrary cracking and crack interaction in solids summary this paper presents an augmentation method that enables bilinear finite elements to efficiently and accurately account for arbitrary multiple intra elemental discontinuities in 2d solids the augmented finite element method a fem employs four internal nodes to account for the crack displacements due to an intra elemental weak or strong discontinuity and it permits repeated elemental augmentation to include multiple interactive cracks it thus enables a unified treatment of damage evolution from a weak discontinuity to a strong discontinuity and to multiple interactive cohesive cracks all within a single bilinear element that employs standard external nodal dofs only a novel elemental condensation procedure has been developed to solve the internal nodal dofs as functions of the external nodal dofs for any irreversible piece wise linear cohesive laws it leads to a fully condensed elemental equilibrium equation with mathematical exactness eliminating the need for nonlinear equilibrium iterations at elemental level the new a fem s high fidelity simulation capabilities to interactive cohesive crack formation and propagation in homogeneous and heterogeneous solids have been demonstrated through several challenging numerical examples it is shown that the proposed a fem empowered by the new elemental condensation procedure is numerically very efficient accurate and robust copyright 2014 john wiley sons ltd

on the inefficiency of the adaptive design for monitoring clinical trials summary adaptive designs which allow the sample size to be modified based on sequentially computed observed treatment differences have been advocated recently for monitoring clinical trials although such methods have a great deal of appeal on the surface we show that such methods are inefficient and that one can improve uniformly on such adaptive designs using standard group sequential tests based on the sequentially computed likelihood ratio test statistic

analytical and numerical results for the swift hohenberg equation the problem of the swift hohenberg equation is considered in this paper using homotopy analysis method ham the series solution is developed and its convergence is discussed and documented here for the first time in particular we focus on the roles of the eigenvalue parameter a and the length parameter l on the large time behaviour of the solution for a given time t we obtain analytical expressions for eigenvalue parameter a and length l which show how different values of these parameters may lead to qualitatively different large time profiles also the results are presented graphically the results obtained reveal many interesting behaviors that warrant further study of the equations related to non newtonian fluid phenomena especially the shear thinning phenomena shear thinning reduces the wall shear stress

arbitrarily smooth generalized finite element approximations this paper presents a procedure to build ck k arbitrarily large generalized finite element fe shape functions defined on non structured finite element meshes the functions have the same support as corresponding global fe lagrangian shape functions meshes with both convex and non convex clouds set of elements sharing a vertex node can be used the so called r functions are used to build ck fe based partition of unity functions with non convex support a technique to combine c0 lagrangian fe shape functions with the proposed ck partition of unity is presented the technique allows the use of ck generalized fe shape functions in parts of the computational domain where their high smoothness is required as in the case of problems with distributional boundary conditions and the less computationally demanding c0 generalized fe shape functions elsewhere in the domain a linear elasticity problem with a concentrated moment is solved using the proposed ck generalized fe method higher order distributional boundary conditions can also be handled by the method a detailed convergence analysis is presented for this class of problems as well as for problems in energy space the integrability of the functions using standard gauss legendre rules is also investigated

detecting treatment differences in group sequential longitudinal studies with covariate adjustment in longitudinal studies comparing two treatments over a series of common follow up measurements there may be interest in determining if there is a treatment difference at any follow up period when there may be a non monotone treatment effect over time to evaluate this question jeffries and geller 2015 examined a number of clinical trial designs that allowed adaptive choice of the follow up time exhibiting the greatest evidence of treatment difference in a group sequential testing setting with gaussian data the methods are applicable when a few measurements were taken at prespecified follow up periods here we test the intersection null hypothesis of no difference at any follow up time versus the alternative that there is a difference for at least one follow up time results of jeffries and geller 2015 are extended by considering a broader range of modeled data and the inclusion of covariates using generalized estimating equations testing procedures are developed to determine a set of follow up times that exhibit a treatment difference that accounts for multiplicity in follow up times and interim analyses

exact statistical inference for group sequential trials summary this paper considers clinical trials comparing two treatments with dichotomous responses where the data are examined periodically for early evidence of treatment difference the existing group sequential methods for such trials are based on the large sample normal approximation to the joint distribution of the estimators of treatment difference over interim analyses we demonstrate through extensive numerical studies that for small and even moderate sized trials these approximate procedures may lead to tests with supranominal size mainly when unpooled estimators of variance are used and confidence intervals with under nominal coverage probability we then study exact methods for group sequential testing repeated interval estimation and interval estimation following sequential testing the new procedures can accommodate any treatment allocation rules an example using real data is provided in clinical trials with sequential patient entry fixed sample size designs are sometimes unjustified on ethical grounds and fully sequential procedures are often impractical one compromise is the group sequential design under which multiple looks are performed during the trial it is widely recognized that the use of the standard single stage f1xedsample size test or confidence interval at each look would lead to an overall type i error or noncoverage rate significantly higher than the nominal level a number of methods have been proposed to determine proper critical values for maintaining a prespecified overall significance level in sequential testing to construct repeated confidence intervals with a given overall coverage probability and to obtain a valid confidence interval following a sequential trial an excellent review on the existing methodology was given by jennison and turnbull 1989 in this paper we are mainly concerned with the commonly used group sequential trials comparing two treatments a and b with relatively quick dichotomous responses success and failure the data are reviewed periodically with cumulative mk subjects receiving treatment a and nk subjects receiving treatment b by the kth look let random variables xk and yk denote the numbers of successes out of the msk and nk subjects respectively we assume that the maximum number of looks k is fixed in advance though some of the procedures developed here can be extended to the case where k is not fixed in advance

sequential rank tests with repeated measurements in clinical trials abstract for comparing responses in two groups of subjects observed repeatedly we propose a group sequential procedure based on linear rank statistics the asymptotic normality of the sequentially computed linear rank statistics is obtained and construction of the group sequential boundaries is based on this distribution theory by virtue of this asymptotic approximation the proposed procedure can be applied to interim analyses with either continuous or discrete repeated measurements even for staggered patient entry simulation results suggest the theory is approximately correct it can also be useful for testing the equality of two changes and rates of change as well as the equality of two means of the responses this procedure is illustrated with a real example

point and interval estimation following a sequential clinical trial summary the usual methodology employed in analysis after a sequential clinical trial is based on orderings of the possible samples resulting from the design however this approach lacks flexibility for use in wider applications in this paper two estimation techniques not based on orderings are considered and modified to obtain improved accuracy a bias adjusted maximum likelihood estimate together with a new and general method for setting confidence limits are discussed the realistic scenario of group sequential monitoring is assumed and methods for exact estimation are given accuracy of the methodology after a triangular test and an o brien fleming test are demonstrated through simulation

the dual test safeguarding p value combination tests for adaptive designs many modern adaptive designs apply an analysis where p values from different stages are weighted together to an overall hypothesis test one merit of this combination approach is that the design can be made very flexible however combination tests violate the sufficiency and conditionality principles as a consequence combination tests may lead to absurd conclusions such as proving a positive effect while the average effect is negative we explore the possibility of modifying the test so that such illogical conclusions are no longer possible the dual test requires both the weighted combination test and a nave test ignoring the adaptations to be statistically significant the result is that the flexibility and type i error level control of the combination test are preserved while the nave test adds a safeguard against unconvincing results the dual test is by construction at least as conservative as the combination test however many design changes will not lead to any power loss a typical situation where the combination approach can be used is two stage sample size reestimation ssr for this case we give a complete specification of all sample size modifications for which the two tests are equally powerful we also study the overall power loss for some suggested ssr rules rules based on conditional power generally lead to ignorable power loss while a decision analytic approach exhibits clear discrepancies between the two tests

nonparametric rank based methods for group sequential monitoring of paired censored survival data summary this research gives methods for nonparametric sequential monitoring of paired censored survival data in the two sample problem using paired weighted log rank statistics with adjustments for dependence in survival and censoring outcomes the joint asymptotic closed form distribution of these sequentially monitored statistics has a dependent increments structure simulations validating operating characteristics of the proposed methods highlight power and size consequences of ignoring even mildly correlated data a motivating example is presented via the early treatment diabetic retinopathy study

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

modeling of failure in composites by x fem and level sets within a multiscale framework abstract composites or multi phase materials are characterized by a distinct heterogeneous microstructure the failure modes of these materials are governed by several micromechanical effects like debonding phenomena and matrix cracks the overall mechanical behavior of composites in the linear as well as the nonlinear regime is not only governed by the material properties of the components and their bonds but also by the material layout in the present contribution the material structure is resolved and modeled on a small scale allowing to deal with these effects for the numerical simulation we apply a combination of the extended finite element method x fem n moes j dolbow t belytschko a finite element method for crack growth without remeshing int j numer methods engrg 46 1999 131 150 and the level set method lsm s osher j sethian fronts propagating with curvature dependent speed algorithms based on hamilton jacobi formulations j comput phys 79 1988 12 49 in the x fem the finite element approximation is enriched by appropriate functions through the concept of partition of unity the geometry of material interfaces and cracks is described by the lsm the combination of both x fem and lsm turns out to be very natural since the enrichment can be described and even constructed in terms of level set functions in order to project the material behavior modeled on a small scale onto the large or structural scale we employ the variational multiscale method vmm t hughes g feijoo l mazzei j b quiney the variational multiscale method a paradigm for computational mechanics comput methods appl mech engrg 166 1998 3 24 this concept is based on an additive split of the displacement field into large and small scale parts for an efficient solution of the discrete problem we postulate that the small scale displacements are locally supported in order to achieve this objective one has to assume appropriate constraint conditions it can be shown that the applied numerical model allows a considerable flexibility concerning the variation of the material design and consequently of the mechanical behavior of a composite

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

group sequential tests for delayed responses with discussion summary group sequential methods are used routinely to monitor clinical trials and to provide early stopping when there is evidence of a treatment effect a lack of an effect or concerns about patient safety in many studies the response of clinical interest is measured some time after the start of treatment and there are subjects at each interim analysis who have been treated but are yet to respond we formulate a new form of group sequential test which gives a proper treatment of these pipeline subjects these tests can be applied even when the continued accrual of data after the decision to stop the trial is unexpected we illustrate our methods through a series of examples we define error spending versions of these new designs which handle unpredictable group sizes and provide an information monitoring framework that can accommodate nuisance parameters such as an unknown response variance by studying optimal versions of our new designs we show how the benefits of lower expected sample size that are normally achieved by a group sequential test are reduced when there is a delay in response the loss of efficiency for larger delays can be ameliorated by incorporating data on a correlated short term end point fitting a joint model for the two end points but still making inferences on the original longer term end point we derive p values and confidence intervals on termination of our new tests

estimation of conditional and marginal odds ratios using the prognostic score introduced by hansen in 2008 the prognostic score pgs has been presented as the prognostic analogue of the propensity score pps pps based methods are intended to estimate marginal effects most previous studies evaluated the performance of existing pgs based methods adjustment stratification and matching using the pgs in situations in which the theoretical conditional and marginal effects are equal i e collapsible situations to support the use of pgs framework as an alternative to the pps framework applied researchers must have reliable information about the type of treatment effect estimated by each method we propose four new pgs based methods each developed to estimate a specific type of treatment effect we evaluated the ability of existing and new pgs based methods to estimate the conditional treatment effect cte the marginal average treatment effect on the whole population ate and the marginal average treatment effect on the treated population att when the odds ratio a non collapsible estimator is the measure of interest the performance of pgs based methods was assessed by monte carlo simulations and compared with pps based methods and multivariate regression analysis existing pgs based methods did not allow for estimating the ate and showed unacceptable performance when the proportion of exposed subjects was large when estimating marginal effects pps based methods were too conservative whereas the new pgs based methods performed better with low prevalence of exposure and had coverages closer to the nominal value when estimating cte the new pgs based methods performed as well as traditional multivariate regression copyright 2016 john wiley sons ltd

towards a generalization of a discrete strong discontinuity approach a generalized discrete strong discontinuity approach gsda is presented in which non homogeneous jumps are embedded in parent finite elements the new formulation presents similar kinematics as for interface elements accurately simulating both rigid body motion and stretching induced by the opening of a discontinuity this generalized discrete strong discontinuity approach is built within the framework of strong embedded discontinuity formulations as a consequence it is mesh independent and avoids the progressive remeshing needed with interface elements the gsda can be viewed as a generalization of other embedded formulations for which particular assumptions are adopted regarding both the jump transmission and the variational principle structural examples including mode i mode ii and mixed mode fracture are computed and compared with other embedded formulations and experimental results

interim analyses in 2 x 2 crossover trials a method is presented for performing interim analyses in long term 2 x 2 crossover trials with serial patient entry the analyses are based on a linear statistic that combines data from individuals observed for one treatment period with data from individuals observed for both periods the coefficients in this linear combination can be chosen quite arbitrarily but we focus on variance based weights to maximize power for tests regarding direct treatment effects the type i error rate of this procedure is controlled by utilizing the joint distribution of the linear statistics over analysis stages methods for performing power and sample size calculations are indicated a two stage sequential design involving simultaneous patient entry and a single between period interim analysis is considered in detail the power and average number of measurements required for this design are compared to those of the usual crossover trial the results indicate that while there is minimal loss in power relative to the usual crossover design in the absence of differential carry over effects the proposed design can have substantially greater power when differential carry over effects are present the two stage crossover design can also lead to more economical studies in terms of the expected number of measurements required due to the potential for early stopping attention is directed toward normally distributed responses

generalized gaussian quadrature rules on arbitrary polygons in this paper we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons these quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required we have used this algorithm for the construction of symmetric and non symmetric quadrature rules over convex and concave polygons while in the case of symmetric quadratures our results are comparable to available rules the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions to demonstrate the efficiency of the new quadrature rules we have tested them for the integration of rational polygonal shape functions over a regular hexagon for a relative error of 10 8 in the computation of stiffness matrix entries one needs at least 198 evaluation points when the region is partitioned whereas 85 points suffice with our quadrature rule copyright 2009 john wiley sons ltd

sequential comparison of changes with repeated measurements data abstract there are many clinical trials in which the patients enter sequentially and a response variable is measured repeatedly over time for each patient a group sequential procedure is proposed for comparing the rates of change between two treatment groups some existing procedures for testing the equality of means between two treatment groups with repeated measurements data such as those proposed by armitage stratton and worthington and by geary can be interpreted as special cases of the proposed procedure under a linear mixed effects model the asymptotic joint distribution of the sequentially computed statistics is derived construction of the group sequential boundaries is based on this distribution theory the extension to nonnormal distributions of errors is also discussed the proposed procedure allows for using either the sequential method of slud and wei or that of lan and demets this procedure is then applied to repeated bone density measurements on a sample of 74 middle aged women and

a review of methods for futility stopping based on conditional power conditional power cp is the probability that the final study result will be statistically significant given the data observed thus far and a specific assumption about the pattern of the data to be observed in the remainder of the study such as assuming the original design effect or the effect estimated from the current data or under the null hypothesis in many clinical trials a cp computation at a pre specified point in the study such as mid way is used as the basis for early termination for futility when there is little evidence of a beneficial effect brownian motion can be used to describe the distribution of the interim z test value the corresponding b value and the cp values under a specific assumption about the future data a stopping boundary on the cp value specifies an equivalent boundary on the b value from which the probability of stopping for futility can then be computed based on the planned study design sample size and duration and the assumed true effect size this yields expressions for the total type i and ii error probabilities as the probability of stopping increases the probability of a type i error alpha decreases from the nominal desired level e g 0 05 while the probability of a type ii error beta increases from the level specified in the study design thus a stopping boundary on the b value can be determined such that the inflation in type ii error probability is controlled at a desired level an iterative procedure is also described that determines a stopping boundary on the b value and a final test critical z value with specified type i and ii error probabilities the implementation in conjunction with a group sequential analysis for effectiveness is also described

xfem based fictitious domain method for linear elasticity model with crack reduction of computational cost of solutions is a key issue for crack identification or crack propagation problems one of the solutions is to avoid remeshing the domain when the crack position changes or when the crack extends to avoid remeshing we propose a new finite element approach for the numerical simulation of discontinuities of displacements generated by cracks inside elastic media the approach is based on a fictitious domain method originally developed for dirichlet conditions for the poisson problem and for the stokes problem which is adapted to the neumann boundary conditions of crack problems the crack is represented by level set functions numerical tests are made with a mixed formulation to emphasize the accuracy of the method as well as its robustness with respect to the geometry enforced by a stabilization technique in particular an inf sup condition is theoretically proven for the latter a realistic simulation with a uniformly pressurized fracture inside a volcano is given for i

a phase field model for rate independent crack propagation robust algorithmic implementation based on operator splits the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field following our recent work c miehe f welschinger m hofacker thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations international journal for numerical methods in engineering doi 10 1002 nme 2861 on phase field type fracture we propose in this paper a new variational framework for rate independent diffusive fracture that bases on the introduction of a local history field it contains a maximum reference energy obtained in the deformation history which may be considered as a measure for the maximum tensile strain obtained in history it is shown that this local variable drives the evolution of the crack phase field the introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology in particular it allows for the construction of a new algorithmic treatment of diffusive fracture here we propose an extremely robust operator split scheme that successively updates in a typical time step the history field the crack phase field and finally the displacement field a regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added the proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids we demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples

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

mixed stabilized finite element methods in nonlinear solid mechanics part ii strain localization this paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining a standard local isotropic rankine damage model with strain softening is used as exemplary constitutive model both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated from this analysis the issue of local discretization error in the pre localization regime is deemed as the main difficulty to be overcome in the discrete problem focus is placed on low order finite elements with continuous strain and displacement fields triangular p1p1 and quadrilateral q1q1 although the presented approach is very general numerical examples show that the resulting procedure is remarkably robust it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh bias dependence

finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions a new finite element formulation aimed at the solution of problems involving strain localization is presented the proposed formulation incorporates displacement interpolated embedded localization lines results are shown to converge to an exact solution when the mesh is refined and also to be quite insensitive to mesh distortions

complementarity problem arising from static growth of multiple cracks and mls based numerical manifold method abstract the full response of a brittle structure containing multiple cracks to loading under the servo control is of vital importance in the evaluation of properties of the structure during crack growth the fracture toughness condition at all the crack tips as well as the equilibrium condition should be obeyed leading to a nonlinear complementarity problem ncp the vector valued function in the ncp depends implicitly on the cracking increments which in turn determine the stress field the stress field can be obtained through solving a mixed variational problem since the degrees of freedom in the discrete variational problem vary with cracking not only in magnitude but also in number the jacobian matrix of the ncp is hard to compute and it cannot be expected to be solved by the newton methods that involve the calculation of jacobian matrices therefore a well scaled projection contraction algorithm is designed the proposed procedure is able to simulate growth of multiple cracks in a natural way allowing the crack tips to stop anywhere with no sensitivity to node configuration or cracking increments through the analysis of some examples that have been widely tested many interesting and profound phenomena are found which have never been revealed

confidence intervals for secondary parameters following a sequential test in sequential studies formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study however the completed data set may warrant a more detailed analysis involving the estimation of parameters corresponding to effects that were not used to determine when to stop and yet correlated with those that were this paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used the method assumes that information accumulates on the primary and secondary parameters at proportional rates this requirement will be valid in many potential applications but only in limited situations in survival analysis

exact confidence intervals following a group sequential trial a comparison of methods summary with interest in designing group sequential clinical trials increasing methodology for analysing the data arising in such trials is needed we discuss several approaches to constructing confidence intervals for the parameter of interest in a group sequential trial after the trial has ended we compare these met1fods to one already in the literature and point out interesting differences

modelling the covariance structure of repeated measurements models for repeated measurements used by wilson hebel and sherwin 1981 biometrics 37 553565 and mansour nordheim and rutledge 1985 biometrics 41 287 294 are considered these two models are generalized by the a p e k model obtained by introducing an extra parameter k a more general class of models described by kenward unpublished ph d thesis university of reading 1981 is briefly discussed simulation results are presented in which model parameters are estimated by maximum likelihood and likelihood ratios are used as criteria for choosing between models a set of blood pressure data is used to illustrate model fitting

an improved approximation for calculation of confidence intervals after a sequential clinical trial a method for the calculation of confidence intervals following a sequential clinical trial is proposed which is more accurate than methods based solely on continuous monitoring assumptions but is not as computationally laborious as previous methods suggested for group sequential trials the calculation takes account of the excess of the sample path over the boundary at the final inspection of the trial which may be substantial after group sequential monitoring and is ignored when continuous monitoring is assumed accuracy of such confidence intervals after a triangular test is investigated by simulation of coverage probabilities individual confidence limit probabilities and p value curves

interim analyses with repeated measurements in a sequential clinical trial summary in comparing two treatments suppose that patients are available for the study sequentially repeated measurements of the same characteristic are collected serially over time for each study patient sequential procedures are proposed to monitor the trial for early evidence of treatment difference or harmful side effects based on such repeated measurements all the methods are illustrated with a real life example

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

p value adjustment in sequential clinical trials many randomized clinical trials utilize a group sequential monitoring procedure for assessing treatment efficacy during the course of the trial for a group sequential trial in which the null hypothesis is ultimately rejected the nominal p value generally overstates the statistical significance of the result and some adjustment is required several orderings of the sample space have been proposed for performing this adjustment each with unique operating characteristics in this article we compare four proposed methods with respect to the degree to which each captures the strength of evidence against the null hypothesis implied by the data we conclude that the ordering of the sample space induced by the observed z score has the most desirable operating characteristics

longitudinal clinical trials with adaptive choice of follow up time in longitudinal studies comparing two treatments with a maximum follow up time there may be interest in examining treatment effects for intermediate follow up times one motivation may be to identify the time period with greatest treatment difference when there is a non monotone treatment effect over time another motivation may be to make the trial more efficient in terms of time to reach a decision on whether a new treatment is efficacious or not here we test the composite null hypothesis of no difference at any follow up time versus the alternative that there is a difference at at least one follow up time the methods are applicable when a few measurements are taken over time such as in early longitudinal trials or in ancillary studies suppose the test statistic z t k will be used to test the hypothesis of no treatment effect at a fixed follow up time t k in this context a common approach is to perform a pilot study on n1 subjects and evaluate the treatment effect at the fixed time points t1 t k and choose t as the value of t k for which z t k is maximized having chosen t a second trial can be designed in a setting with group sequential testing we consider several adaptive alternatives to this approach that treat the pilot and second trial as a seamless combined entity and evaluate type i error and power characteristics the adaptive designs we consider typically have improved power over the common separate trial approach

sequential monitoring for comparison of changes in a response variable in clinical studies the spending function approach proposed by lan and demets 1983 biometrika 70 659 663 for sequential monitoring of clinical trials is applied to situations where comparison of changes in a continuous response variable between two groups is the primary concern death loss to follow up and missed visits could cause follow up measurements to be right censored or missing for some participants furthermore the probability of being censored may be dependent on the parameter value of the response variable informative censoring we propose to compare treatment effects by comparing areas under the expected response change curves between the two groups when the response curves are linear as a function of time in both groups this comparison is equivalent to comparing the rates of change in the response variable covariances of the sequential test statistics are derived conditions for having independent increments are presented for studies designed to evaluate long term treatment effects spending functions obtained by shifting the usual spending functions kim and demets 1987 biometrika 74 149 154 to the right and then rescaling to the remaining interval are also proposed such a shifted spending function is applied to the monitoring plan for the lung health study anthonisen 1989 american review of respiratory diseases 140 871 872

efficient group sequential tests with unpredictable group sizes summary we derive group sequential tests to minimize expected sample size a parametric family of group sequential tests is proposed containing tests which are nearly optimal in terms of minimizing expected sample size amongst all group sequential tests these parametric tests can be implemented when group sizes are unequal and unpredictable they are shown to achieve error rates close to their nominal level in a wide range of situations and to be efficient when compared with other tests based on the same group sizes and achieving the same error rates we consider group sequential procedures for choosing between two hypotheses concerning the mean of a normal distribution with known variance the group sequential approach pocock 1977 offers a practicable alternative to fully sequential tests which require accumulating data to be examined as each new observation is recorded it is also desirable to ensure the maximum sample size needed in a long term experiment is not much greater than that of the corresponding fixed sample size test and we shall take the specification of the maximum allowable sample size and the number of groups of observations as our starting point in deriving sequential procedures in 2 we formulate the testing problem and discuss its relevance to clinical trials in 3 we describe the derivation of boundaries which are optimal in the sense of minimizing various expected sample sizes subject to a fixed number of equally sized groups and a fixed maximum sample size in 4 we introduce a parametric family of group sequential tests containing tests which are nearly optimal by the criteria of 3 these tests are easily adapted to allow unequal and unpredictable group sizes we show in 5 that they remain efficient when actual group sizes are quite different from those anticipated programs used for the calculations are available from the author

type i error in sample size re estimations based on observed treatment difference sample size re estimation based on an observed difference can ensure an adequate power and potentially save a large amount of time and resources in clinical trials one of the concerns for such an approach is that it may inflate the type i error however such a possible inflation has not been mathematically quantified in this paper the mathematical mechanism of this inflation is explored for two sample normal tests a conditional type i error function based on normal data is derived this function not only provides the quantification but also gives mathematical mechanisms of possible inflation in the type i error due to the sample size re estimation theoretically based on their decision rules certain upper and lower bounds people can calculate this function and exactly visualize the changes in type i error computer simulations are performed to ensure the results if there are no bounds for the adjustment the inflation is evident if proper adjusting rules are used the inflation can be well controlled in some cases the type i error can even be reduced the trade off is to give up some unrealistic power we investigated several scenarios in which the mechanisms to change the type i error are different our simulations show that similar results may apply to other distributions copyright 2001 john wiley sons ltd

on the solutions of fractional swift hohenberg equation with dispersion in this article the approximate solutions of the non linear swift hohenberg equation with fractional time derivative in the presence of dispersive term have been obtained the fractional derivative is described in caputo sense time fractional nonlinear partial differential equations in the presence of dispersion and bifurcation parameters have been computed numerically to predict hydrodynamic fluctuations at convective instability for different particular cases and results are depicted through graphs

design and monitoring of multi arm multi stage clinical trials two arm group sequential designs have been widely used for over 40 years especially for studies with mortality endpoints the natural generalization of such designs to trials with multiple treatment arms and a common control mams designs has however been implemented rarely while the statistical methodology for this extension is clear the main limitation has been an efficient way to perform the computations past efforts were hampered by algorithms that were computationally explosive with the increasing interest in adaptive designs platform designs and other innovative designs that involve multiple comparisons over multiple stages the importance of mams designs is growing rapidly this article provides break through algorithms that can compute mams boundaries rapidly thereby making such designs practical for designs with efficacy only boundaries the computational effort increases linearly with number of arms and number of stages for designs with both efficacy and futility boundaries the computational effort doubles with successive increases in number of stages

group sequential distribution free methods for the analysis of multivariate observations many studies involve the collection of multivariate observations such as repeated measures on two groups of subjects who are recruited over time i e with staggered entry of subjects various marginal distribution free multivariate methods have been proposed for the analyses of such multivariate observations where some measures may be missing at random using the multivariate u statistic of wei and johnson 1985 biometrika 72 359 364 we describe the group sequential analysis of such a study where the multivariate observations are observed sequentially both within and among subjects we describe a multivariate generalization of the hodges and lehmann 1963 annals of mathematical statistics 34 598 611 estimator of a location shift that can be obtained via the multivariate u statistic with the mann whitney wilcoxon kernel we then describe large sample group sequential interval estimators and tests based on an aggregate estimate of the location shift combined over all of the repeated measures we also describe how the same steps could be employed to perform a group sequential analysis based on any one of the variety of marginal multivariate methods that have been proposed these methods are applied to a real life example

two sample repeated significance tests based on the modified wilcoxon statistic abstract the asymptotic distribution theory of sequentially computed modified wilcoxon scores is developed for two sample survival data with random staggered entry and random loss to follow up the asymptotic covariance indicates generally dependent modified wilcoxon increments contradicting the authors reading of jones and whitehead 1979 a repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory the early stopping properties of this procedure are illustrated by a prostate cancer example

simultaneous group sequential analysis of rank based and weighted kaplan meier tests for paired censored survival data this research sequentially monitors paired survival differences using a new class of nonparametric tests based on functionals of standardized paired weighted log rank pwlr and standardized paired weighted kaplan meier pwkm tests during a trial these tests may alternately assume the role of the more extreme statistic by monitoring pemax the maximum between the absolute values of the standardized pwlr and pwkm one combines advantages of rank based rb and non rb paired testing paradigms simulations show that monitoring treatment differences using pemax maintains type i error and is nearly as powerful as using the more advantageous of the two tests in proportional hazards ph as well as non ph situations hence pemax preserves power more robustly than individually monitored pwlr and pwkm while maintaining a reasonably simple approach to design and analysis of results an example from the early treatment diabetic retinopathy study etdrs is given

a group sequential holm procedure with multiple primary endpoints we propose a group sequential holm procedure when there are multiple primary endpoints this method addresses multiplicities arising from multiple primary endpoints and from multiple analyses in a group sequential design it has been shown to be a closed testing procedure and preserves the familywise error rate in the strong sense when multiple endpoints are the only concern without an interim analysis the method simplifies to the weighted holm procedure the proposed method is more powerful than the parallel group sequential method and avoids the need to anticipate the testing order as in the fixed sequence testing scheme

sequential methods for parametric survival models summary the asymptotic sequential distribution is derived for score tests and maximum likelihood estimates in parametric survival models in the presence of nuisance parameters where individuals enter over time and the data are subject to right censoring it is shown that these sequential tests and estimates have an independent increments structure this allows the use of standard group sequential methods for early stopping

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

pattern selection of solutions of the swift hohenberg equation abstract we study the large time behaviour of solutions u x t of the swift hohenberg equation on a one dimensional domain 0 l focusing in particular on the role of the eigenvalue parameter and the length l of the domain on the selection of the limiting profiles v we show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles and prove the existence of a collection n of disjoint intervals which depend on the value of such that if l n then solutions all converge to the trivial solution v 0 we show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches we establish global bounds for the solutions u x t and identify the global attractor finally we derive an estimate about the shape of the final state

sequential monitoring of clinical trials the role of information and brownian motion sequential monitoring has been a topic of major interest in clinical trials methodology over the past two decades this paper presents a unified conceptual framework for sequential monitoring that covers a wide variety of monitoring procedures in a wide variety of clinical trial settings the central elements of this framework consist of a suitable concept of statistical information and a scheme for using this concept as a basis for summarizing the accumulating results of a trial in a standardized form through a stochastic process that can be shown to approximate classical brownian motion the ideas are developed in a simple step by step fashion and illustrated by several practical examples

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

exact confidence bounds following adaptive group sequential tests summary we provide a method for obtaining confidence intervals point estimates and p values for the primary effect size parameter at the end of a two arm group sequential clinical trial in which adaptive changes have been implemented along the way the method is based on applying the adaptive hypothesis testing procedure of muller and schafer 2001 to a sequence of dual tests derived from the stage wise adjusted confidence interval of tsiatis rosner and mehta 1984 in the non adaptive setting this confidence interval is known to provide exact coverage in the adaptive setting exact coverage is guaranteed provided the adaptation takes place at the penultimate stage in general however all that can be claimed theoretically is that the coverage is guaranteed to be conservative nevertheless extensive simulation experiments supported by an empirical characterization of the conditional error function demonstrate convincingly that for all practical purposes the coverage is exact and the point estimate is median unbiased no procedure has previously been available for producing confidence intervals and point estimates with these desirable properties in an adaptive group sequential setting the methodology is illustrated by an application to a clinical trial of deep brain stimulation for parkinson s disease

rational spline nonstandard finite difference scheme for the solution of time fractional swift hohenberg equation abstract in this paper we introduce a new scheme based on rational spline function and nonstandard finite difference technique to solve the time fractional swift hohenberg equation in the sense of riemann liouville derivative via fourier method the method is convergent and unconditionally stable also we investigated the existence and uniqueness of the proposed method numerical results are demonstrated to validate the applicability and the theoretical results