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

misspecified proportional hazard models summary the properties of an estimator based on a proportional hazards model are investigated when the model is incorrect the estimator from the partial likelihood is shown to be consistent for a parameter that is defined implicitly the results are used to investigate the effects on estimation if the true model is accelerated failure time or if covariates are omitted from the proportional hazards model

nonlinear sor and mesh redistribution this article concerns the numerical approximations for the problems of minimizing strict convex functionals the functionals are discretized by the finite element methods and then solved by the nonlinear sor with an optimal mesh the redistribution approach is considered to generate the optimal mesh with an optimal mesh the number of unknowns can be dramatically reduced and this generally gives a computationally efficient model numerical results of the model catenoid problem and a geodesics problem illustrating the efficiency of the optimal mesh approach in the nonlinear sor will be presented

steady shallow ice sheets as obstacle problems well posedness and finite element approximation we formulate steady shallow ice sheet flow as an obstacle problem the unknown being the ice upper surface and the obstacle being the underlying bedrock topography this generates a free boundary defining the ice sheet extent the obstacle problem is written as a variational inequality subject to the positive ice thickness constraint the corresponding pde is a highly nonlinear elliptic equation which generalizes the p laplacian equation our formulation also permits variable ice softness basal sliding and elevation dependent surface mass balance existence and uniqueness are shown in restricted cases which we may reformulate as a convex minimization problem in the general case we show existence by applying a fixed point argument using continuity results from that argument we construct a numerical solution by solving a sequence of obstacle p laplacian like problems by finite element approximation as a real application we compute the steady state shape of the greenland ice sheet in a steady pre

a posteriori finite element error control for the p laplace problem three a posteriori error estimates are discussed for the p laplace problem as a model equation for nonquadratic growth of order p 2 the standard residual based error estimate is addressed with emphasis on involved constants determined as local eigenvalues up to errors in their numerical computation they provide a guaranteed upper bound for the w1 p seminorm and a weighted w1 2 seminorm of u uh under the major assumption that nabla u h is positive a second sharper a posteriori estimate is based on the numerical solution of local interface problems averaging techniques for a posteriori error control on the flux error motivate a third estimator the results are presented for any shape regular but unstructured triangulation into triangles they partly rely on a positive weight function rho nabla u p 2 nabla u h p 2 2 similar to the notion of quasi norms due to barrett and liu numerical experiments indicate the surprising accuracy of our averaging error estimators confirm our estimates and allow for a comparison

a posteriori error control for distributed elliptic optimal control problems with control constraints discretized by hp finite elements abstract a distributed elliptic control problem with control constraints is considered which is formulated as a three field problem and consists of two variational equations for the state and the co state variables as well as of a variational inequality for the control variable two discretization approaches with h p finite elements are discussed in the first discretization approach all variables state co state and control are discretized a semi smooth newton method is introduced for solving the resulting algebraic system in the second discretization approach only the state and the co state variables are discretized whereas the control is determined by projection a simple fixed point scheme is presented for the iterative solution of this approach the main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates which enables h p adaptive mesh refinements in particular the estimates can be applied to the iteration solutions so that they can be used as a stopping criterion of the iterative solution schemes in several numerical experiments the order of convergence of the adaptive discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied

estimation and comparison of changes in the presence of information right censoring by modeling the censoring process abstract in estimating and comparing the rates of change of a continuous variable between two groups the unweighted averages of individual simple least squares estimates from each group are often used under a linear random effects model when all individuals have completed observations at identical time points these statistics are maximum likelihood estimates for the expected rates of change however with censored of missing data these estimates are no longer efficient when compared to generalized least squares estimates when in addition the right censoring process is dependent upon the individual rates of change i e informative right censoring the generalized least squares estimates will be biased likelihood ratio test for informativeness of the censoring process and maximum likelihood estimates for the expected rates of change and the parameters of the right censoring process are developed under a linear random effect models with a probit model for the right censoring process in realistic situations we illustrate that the bias in estimating group rate of change and the reduction of power in comparing group difference could be substantial when strong dependency of the right censoring process on individual rates of change is ignored author

higher order fem for the obstacle problem of the p laplacian a variational inequality approach abstract we consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p laplacian differential operator for p 1 we prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity moreover we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions all our results contain the known results for the linear case of p 2 we present numerical results on the improved convergence rates of adaptive schemes mesh size adaptivity with and without polynomial degree adaptation for the singular case of p 1 5 and for the degenerated case of p 3

misspecifying the covariance structure in a linear mixed model under mar drop out misspecification of the covariance structure in a linear mixed model lmm can lead to biased population parameters estimates under mar drop out in our motivating example of modeling cd4 cell counts during untreated hiv infection random intercept and slope lmms are frequently used in this article we evaluate the performance of lmms with specific covariance structures in terms of bias in the fixed effects estimates under specific mar drop out mechanisms and adopt a bayesian model comparison criterion to discriminate between the examined approaches in real data applications we analytically show that using a random intercept and slope structure when the true one is more complex can lead to seriously biased estimates with the degree of bias depending on the magnitude of the mar drop out under misspecified covariance structure we compare in terms of induced bias the approach of adding a fractional brownian motion bm process on top of random intercepts and slopes with the approach of using splines for the random effects in general the performance of both approaches was satisfactory with the bm model leading to smaller bias in most cases a simulation study is carried out to evaluate the performance of the proposed bayesian criterion in identifying the model with the correct covariance structure overall the proposed method performs better than the aic and bic criteria under our specific simulation setting the models under consideration are applied to real data from the cascade study the most plausible model is identified by all examined criteria

slope estimation in the presence of informative right censoring modeling the number of observations as a geometric random variable summary a method is proposed for the estimation of rate of change from incomplete longitudinal data where the number of observations made for each subject is assumed to vary depending on the level of the response variable the proposed method involves a random slope model in which the number of observations is modeled as a geometric distribution with its mean dependent on the individual subject s rate of change the method adjusts for informative right censoring and provides estimates of the slopes of individual subjects as well as of the population under noninformative right censoring these estimators of the slopes are equivalent to bayes estimators fearn 1975 biometrika 62 89 100 the simulation study demonstrates that in cases where the censoring process is informative the proposed estimator is more efficient than either the unweighted or weighted estimator of slope the method is illustrated by the analysis of renal transplant data

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

a note on discrete solutions of the plateau problem on demontre des theoremes pour la convergence de solutions discretes du probleme de plateau sous l hypothese que le contour de jordan soit rectifiable

the obstacle problem for the p laplacian via optimal stopping of tug of war games we present a probabilistic approach to the obstacle problem for the p laplace operator the solutions are approximated by running processes determined by tug of war games plus noise and letting the step size go to zero not unlike the case when brownian motion is approximated by random walks rather than stopping the process when the boundary is reached the value function is obtained by maximizing over all possible stopping times that are smaller than the exit time of the domain

convergence analysis of trial free boundary methods for the two dimensional filtration problem in this paper we present a scheme of convergence analysis of trial free boundary methods for the two dimensional filtration or dam problem for the purpose we present a new variational principle of the filtration problem this variational principle is defined on the set of admissible domains candidates of the solution in the dam under mild assumptions on the configuration of the dam we may assume that all admissible domains are mapped from the unit disk by conformal mappings thus proving convergence of trial free boundaries is reduced to proving convergence of the conformal mappings on the unit disk and it is done using a method in the theory of minimal surfaces numerical examples are given

discrete solution of the plateau problem and its convergence definition d une solution discrete du probleme de plateau et demonstration des theoremes de convergence de la solution discrete lorsque la frontiere satisfait une certaine condition exemple numerique

averaging techniques yield reliable a posteriori finite element error control for obstacle problems the reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second order partial differential equations this paper establishes related reliable and efficient a posteriori error estimates for the energy norm error of an obstacle problem on unstructured grids as a model example for variational inequalities the surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms consistency terms and a multiplicative constant

hp interpolation of nonsmooth functions and an application to hp a posteriori error estimation the quasi interpolation operators of clement and scott zhang type are generalized to the setting of the hp fem finite element method new polynomial lifting and inverse estimates are presented the classical residual based a posteriori error estimator is generalized to the hp fem

linear convergence of an adaptive finite element method for the p laplacian equation we study an adaptive finite element method for the p laplacian like pdes using piecewise linear continuous functions the error is measured by means of the quasi norm of barrett and liu we provide residual based error estimators without a gap between the upper and lower bound we show linear convergence of the algorithm which is similar to the one of morin nochetto and siebert all results are obtained without extra marking for the oscillation

finite element approximation of h surfaces in this paper a piecewise linear finite element approximation of h surfaces or surfaces with constant mean curvature spanned by a given jordan curve in r3 is considered it is proved that the finite element h surfaces converge to the exact h surfaces under the condition that the jordan curve is rectifiable several numerical examples are given

the approximation theory for the p version finite element method and application to non linear elliptic pdes approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements estimates for the rate of convergence in sobolev spaces w m q omega q in 1 infty are given the results are applied to estimate the rate of convergence when the p version finite element method is used to approximate the alpha laplacian it is shown that the rate of convergence of the p version is always at least that of the h version measured in terms of number of degrees of freedom used if the solution is very smooth then the p version attains an exponential rate of convergence if the solution has certain types of singularity the rate of convergence of the p version is twice that of the h version the analysis generalises the work of babuska and others to the case q not 2 in addition the approximation theoretic results find immediate application for some types of spectral and spectral element methods

a posteriori fe error control for p laplacian by gradient recovery in quasi norm a posteriori error estimators based on quasi norm gradient recovery are established for the finite element approximation of the p laplacian on unstructured meshes the new a posteriori error estimators provide both upper and lower bounds in the quasi norm for the discretization error the main tools for the proofs of reliability are approximation error estimates for a local approximation operator in the quasi norm

adaptive fe be coupling for strongly nonlinear transmission problems with coulomb friction we analyze an adaptive finite element boundary element procedure for scalar elastoplastic interface problems involving friction where a nonlinear uniformly monotone operator such as the p laplacian is coupled to the linear laplace equation on the exterior domain the problem is reduced to a boundary domain variational inequality a discretized saddle point formulation of which is then solved using the uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme the galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of l p and l 2 sobolev spaces

quasi norm interpolation error estimates for the piecewise linear finite element approximation of p laplacian problems in this work new interpolation error estimates have been derived for some well known interpolators in the quasi norms the estimates are found to be essential to obtain the optimal a priori error bounds under the weakened regularity conditions for the piecewise linear finite element approximation of a class of degenerate equations in particular by using these estimates we can close the existing gap between the regularity required for deriving the optimal error bounds and the regularity achievable for the smooth data for the 2 d and 3 d p laplacian

longitudinal and time to drop out joint models can lead to seriously biased estimates when the drop out mechanism is at random missing data are common in longitudinal studies likelihood based methods ignoring the missingness mechanism are unbiased provided missingness is at random mar under not at random missingness mnar joint modeling is commonly used often as part of sensitivity analyses in our motivating example of modeling cd4 count trajectories during untreated hiv infection cd4 counts are mainly censored due to treatment initiation with the nature of this mechanism remaining debatable here we evaluate the bias in the disease progression marker s change over time slope of a specific class of joint models termed shared random effects models srems under mar drop out and propose an alternative srem model our proposed model relates drop out to both the observed marker s data and the corresponding random effects in contrast to most srems which assume that the marker and the drop out processes are independent given the random effects we analytically calculate the asymptotic bias in two srems under specific mar drop out mechanisms showing that the bias in marker s slope increases as the drop out probability increases the performance of the proposed model and other commonly used srems is evaluated under specific mar and mnar scenarios through simulation studies under mar the proposed model yields nearly unbiased slope estimates whereas the other srems yield seriously biased estimates under mnar the proposed model estimates are approximately unbiased whereas those from the other srems are moderately to heavily biased depending on the parameterization used the examined models are also fitted to real data and results are compared discussed in the light of our analytical and simulation based findings

on summary measures analysis of the linear mixed effects model for repeated measures when data are not missing completely at random subjects often drop out of longitudinal studies prematurely yielding unbalanced data with unequal numbers of measures for each subject a simple and convenient approach to analysis is to develop summary measures for each individual and then regress the summary measures on between subject covariates we examine properties of this approach in the context of the linear mixed effects model when the data are not missing completely at random in the sense that drop out depends on the values of the repeated measures after conditioning on fixed covariates the approach is compared with likelihood based approaches that model the vector of repeated measures for each individual methods are compared by simulation for the case where repeated measures over time are linear and can be summarized by a slope and intercept for each individual our simulations suggest that summary measures analysis based on the slopes alone is comparable to full maximum likelihood when the data are missing completely at random but is markedly inferior when the data are not missing completely at random analysis discarding the incomplete cases is even worse with large biases and very poor confidence coverage

pattern mixture models for multivariate incomplete data consider a random sample on variables x1 xv with some values of xv missing selection models specify the distribution of x1 xv over respondents and nonrespondents to xv and the conditional distribution that xv is missing given x1 xv in contrast pattern mixture models specify the conditional distribution of x 1 xv given that xv is observed or missing respectively and the marginal distribution of the binary indicator for whether or not xv is missing for multivariate data with a general pattern of missing values the literature has tended to adopt the selection modeling approach see for example little and rubin here pattern mixture models are proposed for this more general problem pattern mixture models are chronically underidentified in particular for the case of univariate nonresponse mentioned above there are no data on the distribution of xv given x1 xv 1 in the stratum with xv missing thus the models require restrictions or prior information to identify the paramet

estimation and comparison of changes in the presence of informative right censoring conditional linear model a general linear regression model for the usual least squares estimated rate of change slope on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups two noniterative estimators for the group slope means the linear minimum variance unbiased lmvub estimator and the linear minimum mean squared error lmmse estimator are proposed under this conditional model in realistic situations we illustrate that the lmvub and lmmse estimators derived under a simple linear regression model are quite competitive compared to the pseudo maximum likelihood estimator pmle derived by modeling the censoring probabilities generalizations to polynomial response curves and general linear models are also described

on two methods for approximating minimal surfaces in parametric form etude de deux methodes d approximation des surfaces minimales sous forme parametrique minimisation de l aire et de l energie convergence transformation conforme approchee exemples

biorthogonal basis functions in hp adaptive fem for elliptic obstacle problems abstract in this paper the discretization of a non symmetric elliptic obstacle problem with h p adaptive h 1 conforming finite elements is discussed for this purpose a higher order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions the h p adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper the use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi smooth newton scheme which can be modified to ensure global convergence h p adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the h 1 conforming finite element discretization this is typically done by using so called connectivity matrices in this paper a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions finally the applicability of the theoretical findings is demonstrated with several numerical experiments

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

hp fem convergence for unilateral contact problems with tresca friction in plane linear elastostatics abstract this paper is concerned with the convergence of the h p version of the finite element method h p fem for some nonsmooth unilateral problems in linear elastostatics we consider in particular the deformation of an elastic body unilaterally supported by a rigid foundation admitting tresca friction given friction along the rigid foundation solely subjected to body forces and surface tractions without being fixed along some part of its boundary for the discretization of the unilateral constraint and the nonsmooth friction functional we employ gauss lobatto quadrature we show convergence of the h p fem approximations for mechanically definite problems without imposing any regularity assumption moreover we treat the coercive case when the body is fixed along some part of the boundary based on an abstract cea falk estimate and operator interpolation arguments we establish an a priori error estimate in the energy norm under a reasonable regularity assumption

quasi norm local error estimators for p laplacian in this paper we extend the quasi norm techniques used in a priori error estimation of finite element approximation of degenerate nonlinear systems in order to carry out an improved a posteriori error analysis for the p laplacian we derive quasi norm a posteriori error estimators of residual type which are shown to provide improved upper and lower bounds on the discretization error for sufficiently regular solutions these estimators are further shown to be equivalent on the discretization error in a quasi norm numerical results demonstrating these a posteriori estimators are also presented

a posteriori error estimators for obstacle problems another look we show that a posteriori estimators for the obstacle problem are easily obtained from the theory for linear equations the theory would be even simpler if the lagrange multiplier does not have a nonconforming contribution as it has in actual finite element computations

newton raphson and em algorithms for linear mixed effects models for repeated measures data abstract we develop an efficient and effective implementation of the newton raphson nr algorithm for estimating the parameters in mixed effects models for repeated measures data we formulate the derivatives for both maximum likelihood and restricted maximum likelihood estimation and propose improvements to the algorithm discussed by jennrich and schluchter 1986 to speed convergence and ensure a positive definite covariance matrix for the random effects at each iteration we use matrix decompositions to develop efficient and computationally stable implementations of both the nr algorithm and an em algorithm laird and ware 1982 for this model we compare the two methods em vs nr in terms of computational order and performance on two sample data sets and conclude that in most situations a well implemented nr algorithm is preferable to the em algorithm or em algorithm with aitken s acceleration the term repeated measures refers to experimental designs where there are several individuals and several

unequally spaced longitudinal data with ar 1 serial correlation this paper discusses longitudinal data analysis when each subject is observed at different unequally spaced time points observations within subjects are assumed to be either uncorrelated or to have a continuous time first order autoregressive structure possibly with observation error the random coefficients are assumed to have an arbitrary between subject covariance matrix covariates can be included in the fixed effects part of the model exact maximum likelihood estimates of the unknown parameters are computed using the kalman filter to evaluate the likelihood which is then maximized with a nonlinear optimization program an example is presented where a large number of subjects are each observed at a small number of observation times hypothesis tests for selecting the best model are carried out using wald s test on contrasts or likelihood ratio tests based on fitting full and restricted models

asymptotic bias in the linear mixed effects model under non ignorable missing data mechanisms summary in longitudinal studies missingness of data is often an unavoidable problem estimators from the linear mixed effects model assume that missing data are missing at random however estimators are biased when this assumption is not met in the paper theoretical results for the asymptotic bias are established under non ignorable drop out drop in and other missing data patterns the asymptotic bias is large when the drop out subjects have only one or no observation especially for slope related parameters of the linear mixed effects model in the drop in case intercept related parameter estimators show substantial asymptotic bias when subjects enter late in the study eight other missing data patterns are considered and these produce asymptotic biases of a variety of magnitudes

a p version finite element method for nonlinear elliptic variational inequalities in 2d this article introduces and analyzes a p version fem for variational inequalities resulting from obstacle problems for some quasi linear elliptic partial differential operators we approximate the solution by controlling the obstacle condition in images of the gauss lobatto points we show existence and uniqueness for the discrete solution up from the p version for the obstacle problem we prove the convergence of up towards the solution with respect to the energy norm and assuming some additional regularity for the solution we derive an a priori error estimate in numerical experiments the p version turns out to be superior to the h version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation

on quasi norm interpolation error estimation and a posteriori error estimates for p laplacian in this paper we establish a series of interpolation error estimates for several widely used averaging interpolators in some quasi norms these estimates are among the key ingredients in our improved a posteriori error analysis for the p laplacian the quasi norm interpolation error estimates are used to derive a posteriori error estimators for some conforming and nonconforming finite element approximation of the p laplacian which are shown to provide upper and lower bounds for the discretization error

quasi norm error bounds for the finite element approximation of a non newtonian flow we consider the finite element approximation of a non newtonian flow where the viscosity obeys a general law including the carreau or power law for sufficiently regular solutions we prove energy type error bounds for the velocity and pressure these bounds improve on existing results in the literature a key step in the analysis is to prove abstract error bounds initially in a quasi norm which naturally arises in degenerate problems of this type

on the numerical approximation of unstableminimal surfaces with polygonal boundaries this work is concerned with the approximation and the numerical computation of polygonal minimal surfaces in bbb r q q ge 2 polygonal minimal surfaces correspond to the critical points of shiffman s function theta since this function is analytic polygonal minimal surfaces can be characterized by means of the second derivative of theta we present a finite element approximation of quasiminimal surfaces together with an error estimate in this way we obtain discrete approximations theta h of theta and f h of nabla theta in particular we prove that the discrete functions converge uniformly on certain compact subsets this will be the main tool for proving existence and convergence of discrete minimal surfaces in neighbourhoods of non degenerate minimal surfaces in the numerical part of this paper we compute numerical approximations of polygonal minimal surfaces by use of newton s method applied to f h

coupling regularization and adaptive hp bem for the solution of a delamination problem in this paper we couple regularization techniques with the adaptive hp version of the boundary element method hp bem for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions as a model example we treat the delamination of composite structures with a contaminated interface layer this problem has a weak formulation in terms of a nonsmooth variational inequality the resulting hemivariational inequality is first regularized and then discretized by an adaptive hp bem we give conditions for the uniqueness of the solution and provide an a priori error estimate furthermore we prove the very first a posteriori error estimate for the nonsmooth variational problem utilizing a novel mixed regularized formulation thus enabling hp adaptivity various numerical experiments illustrate the behavior strengths and limitations of the proposed high order approximation scheme