analysing repeated measurements with possibly missing observations by modelling marginal distributions suppose that subjects are observed repeatedly over a common set of time points with possibly time dependent covariates and possibly missing observations at each time point we model the marginal distribution of the response variable and the effect of the covariates on that distribution using a class of quasi likelihood models studied in mccullagh and nelder no parametric model of dependence of the repeated observations of the subject is assumed for large samples the quasi likelihood estimates of the time specific regression coefficients over the set of predetermined time points are shown to be approximately jointly normal this coupled with various inference procedures provides a global picture about the effects of the covariates on the response variable over the entire study period a lack of fit test for testing the adequacy of the assumed quasi likelihood model is also provided all the methods considered here are illustrated with real life examples

collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers a numerical scheme is proposed to solve singularly perturbed two point boundary value problems with a turning point exhibiting twin boundary layers the scheme comprises a b spline collocation method on a uniform mesh which leads to a tridiagonal linear system asymptotic bounds are established for the derivative of the analytical solution of a turning point problem the analysis is done on a uniform mesh which permits its extension to the case of adaptive meshes which may be used to improve the solution the design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem some relevant numerical examples are also illustrated to verify computationally the theoretical aspects

nonparametric validation of similar distributions and assessment of goodness of fit summary in this paper the problem of assessing the similarity of two cumulative distribution functions f and g is considered an asymptotic test based on an x trimmed version of mallows distance r f g between f and g is suggested thus demonstrating the similarity of f and g within a preassigned r f g neighbourhood at a controlled type i error rate the test proposed is applied to the validation of goodness of fit and for the nonparametric assessment of bioequivalence it is shown that r f g can be interpreted as average and population equivalence our approach is illustrated by various examples

parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers this article presents a numerical method to solve singularly perturbed turning point problems exhibiting two exponential boundary layers classical finite difference schemes do not yield parameter uniform convergent results on a uniform mesh in general robust computational techniques for boundary layers chapman hall london crc press boca raton fl 2000 in order to overcome this difficulty we propose an appropriate piecewise uniform shishkin mesh and apply the classical finite difference schemes on this mesh error estimates are derived by decomposing the solution into smooth and singular components the present method is layer resolving as well as parameter uniform convergent numerical examples are presented to show the applicability and efficiency of the method

a finite volume element formulation and error analysis for the non stationary conduction convection problem abstract the non stationary conduction convection problem including the velocity vector field and the pressure field as well as the temperature field is studied with a finite volume element fve method a fully discrete fve formulation and the error estimates between the fully discrete fve solutions and the accuracy solution are provided it is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that the fve method is feasible and efficient for finding the numerical solutions of the non stationary conduction convection problem and is one of the most effective numerical methods by comparing the results of the numerical simulations of the fve formulation with those of the numerical simulations of the finite element method and the finite difference scheme for the non stationary conduction convection problem

combining dependent tests with incomplete repeated measurements summary in comparing the effectiveness of two treatments repeated measurements of the same characteristic are often taken under two or more distinct conditions for each experimental subject a locally best linear combination of individual test statistics obtained under different conditions is proposed to test the equality of the two treatments the test procedure allows different patterns of missing observations for the two groups to be compared special cases such as combining wilcoxon tests t tests and 2 x 2 tables are discussed in detail with real examples

nonconforming mixed finite element approximation to the stationary navier stokes equations on anisotropic meshes abstract the main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary navier stokes equations under anisotropic meshes that is the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure the convergence analysis is presented and the optimal error estimates both in a broken h 1 norm for the velocity and in an l 2 norm for the pressure are derived on anisotropic meshes

superconvergent analysis of a nonconforming mixed finite element method for non stationary conduction convection problem abstract in this paper a backward euler mixed finite element mfe approximation scheme is presented for the non stationary conduction convection problem in which the constrained nonconforming rotated cnr q 1 element is used to approximate the velocity u the temperature t and the q 0 element to the pressure p based on the characters of these elements and some special skills i e mean value skill and a new transforming skill with respect to the superclose results of order o h 2 for u t in the broken h 1 norm and p in l 2 norm are deduced respectively here h is the subdivision parameter and the time step furthermore the global superconvergent results are obtained through the interpolation postprocessing technique finally numerical results are provided to confirm the theoretical analysis

the p version of the finite element method for a singularly perturbed boundary value problem the p version of the finite element method is applied to solve the singularly perturbed two point boundary value problem with or without turning point with the special choice of mesh points global error estimates are derived in some cases the exponential rate of convergence is obtained some numerical results are given to show the performance of the proposed method

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

a least squares galerkin petrov nonconforming mixed finite element method for the stationary conduction convection problem abstract in this paper a least squares galerkin petrov nonconforming mixed finite element method lsgpnmfm is proposed and analyzed for the stationary conduction convection problem we use p 2 nonconforming as approximation space for the velocity the linear element for the pressure space and the quadratic element for the temperature space the mixed finite element spaces x h and m h need not satisfy inf sup condition the existence uniqueness and convergence of the discrete solution are presented and error estimates of optimal order are derived in the case of sufficient viscosity

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

a computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers singularly perturbed turning point problems tpps for second order ordinary differential equations des exhibiting twin boundary layers are considered in order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods in this method the given internal domain of definition of differential equation is divided into four subintervals and we solve the differential equation in each interval and combine the solution the implementation of the method on parallel architectures is mentioned error estimates are derived and numerical examples are given

continuous and numerical analysis of a multiple boundary turning point problem a singularly perturbed boundary value problem with a multiple turning point at a boundary is considered a representation of the solution is given and it is used in the construction of a uniform finite difference scheme the scheme is a first order exponentially fitted one an improved modification on a special discretization mesh is given

analysis of repeated ordered categorical outcomes with possibly missing observations and time dependent covariates abstract this article describes a method for comparing responses in two groups of subjects observed repeatedly at a common set of observation times when the response is an ordered categorical outcome the method which allows both time dependent covariates and missing observations consists of two analytic steps in the first step the data are analyzed separately at each occasion using a regression model chosen from the class of models for ordinal data proposed by mccullagh 1980 the joint asymptotic distribution of the estimates of these occasion specific regression coefficients and a consistent estimator of their asymptotic covariance matrix are obtained without imposing any parametric model of dependence on the repeated observations in the second step this asymptotic distribution together with appropriate simultaneous inference procedures is used to characterize the overall difference between groups and the variation in group differences over time the missing data process may differ between gro

some general methods for the analysis of categorical data in longitudinal studies this paper is concerned with the analysis of multivariate categorical data from epidemiologic and clinical studies with longitudinal designs an expository discussion of pertinent hypotheses for such situations is provided within the context of two relevant data sets appropriate large sample tests of these hypotheses are developed through the application of weighted least squares to generate wald statistics these procedures are illustrated with extensive analyses of one of these data sets in some situations the resulting cross classification of the response variables leads to extremely sparse frequency data especially when the number of subjects is not large for such repeated measurement designs in which a single variable is measured repeatedly over time this paper considers the use of a generalized mantel haenszel strategy for tests of marginal homogeneity symmetry these randomization model methods are illustrated for data in which the repeated measurement variable is reported on an ordinal scale this paper also focuses on the available computing software to implement these methods within the version 5 release of the sas system the randomization model approach can be implemented within the freq procedure and a broad range of models and hypotheses can be investigated within the catmod procedure

numerical analysis of singularly perturbed delay differential turning point problem abstract in this paper we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts similar boundary value problems are encountered while simulating several real life processes for instance first exit time problem in the modelling of neuronal variability a rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem in the development of numerical methods for constructing an approximation to the solution of the problem a special type of mesh is generated to tackle the delay term along with the turning point then to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term an exponential fitting parameter is used several numerical examples are presented to support the theory developed in the paper

testing for omitted variables and non linearity in regression models for longitudinal data when fitting regression models to investigate the relationship between an outcome variable and independent variables of primary interest there is often concern whether omitted variables or assuming a different functional relationship could have changed the conclusion or interpretation of the results in longitudinal studies of aging the concern with omitted variables is well known in the context of cohort and period effects which refer to unmeasured variables systematically related to the individual s year of birth and secular trends in outcome respectively we present and compare three approaches to detecting omitted confounders and non linearity in the random effects model for longitudinal data laird and ware 1982 with random slope and intercept across individuals the first approach compares simple unweighted within and between regression coefficients the second is the hausman specification test for regression models and the third approach involves testing directly the significance of functions of individual specific covariate means means i in the random effects regression model this last approach is motivated by the models that arise when cohort or period effects are ignored we compare the three approaches and illustrate their application

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models generalized linear models are used for regression analysis in a number of cases including categorical responses where the classical assumptions are violated the statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator we present mild general conditions which respectively assure weak or strong consistency or asymptotic normality most of the previous work has been concerned with natural link functions in this case our normality condition though obtained by a different approach is closely related to a condition of haberman 1977a examples show how the general conditions reduce to weak requirements for special exponential families further for regressors with a compact range sufficient conditions are given which do not involve the unknown parameter and are therefore easy to check in practice responses with a bounded range e g categorical responses and stochastic regressors also are treated

comparison of methods for the analysis of longitudinal interval count data longitudinal studies are often concerned with estimating the recurrence rate of a non fatal event in many cases only the total number of events occurring during successive time intervals is known we compared a mixed poisson gamma regression method proposed by thall and a quasi likelihood method proposed by zeger and liang for the analysis of such data in the case where the mean was correctly specified using simulation techniques with large samples both methods produced similar standard errors in most situations except in the case of time dependent covariates with non poisson gamma data where they were seriously underestimated by the thall method a simple method for discriminating between the variance forms of the two methods is described the findings are applied to the analyses of clinical trials of non melanoma skin cancer and familial polyposis this study extends the findings of breslow concerning variance misspecification in overdispersed poisson and quasi likelihood models to the longitudinal setting

detection of outlying data in bioavailability bioequivalence studies this paper considers the problem of detecting outlying data in bioavailability bioequivalence studies we define outlying subjects as those whose responses in bioavailability to all formulations differ from the rest of the subjects we also define an outlying observation as the response in bioavailability of a subject to a particular formulation which is grossly different from the average bioavailability of that formulation calculated from all subjects we propose two test procedures the first based on two sample hotelling t2 is to detect possible outlying subjects the second based on residuals from formulation means is to identify possible outlying observations within subjects both procedures take into account the covariance structure of the responses to formulations dependence of test statistics and multiplicity of test procedures we apply the monte carlo or bootstrap simulation to evaluate the sampling distributions of test statistics an example from a 3 way crossover bioequivalence study illustrates the two procedures

variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer s abstract a numerical method based on cubic spline with nonuniform grid is given for the singularly perturbed two point boundary value problems having turning point the scheme derived in this paper is second order accurate numerical examples are given to support the predicted theory

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

on the accuracy of nonconforming quadrilateral q 1 element approximation for the navier stokes problem this paper deals with the nonconforming quadrilateral q1 element approximation of the navier stokes equation by using the so called velocity pressure mixed formulation a convergence condition is presented for the nonconforming stokes element it is shown that the condition is not only sufficient for the convergence but also in some sense is necessary the quasi optimal maximum norm error estimates of the velocity and its first derivatives and of the pressure are derived for the nonconforming quadrilateral q1 element for the stationary navier stokes problem the analysis is based on the weighted inf sup condition and the technique of weighted sobolev norm

p1 nonconforming quadrilateral finite element methods for second order elliptic problems a p1 nonconforming quadrilateral finite element is introduced for second order elliptic problems in two dimensions unlike the usual quadrilateral nonconforming finite elements which contain quadratic polynomials or polynomials of degree greater than 2 our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges one of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements an optimal rate of convergence is obtained also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral an extension of the p1 nonconforming element to three dimensions is also given finally several numerical results are reported to confirm the effective nature of the proposed new element

numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability the objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations spdde which arise in computational neuroscience in particular in the modeling of neuronal variability the mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with shifts the problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution exponentially fitted finite difference scheme based on il in allen southwell fitting is used on a specially designed mesh some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme effect of the shifts on the layer structure is illuminated for the considered examples

sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem a number of results exist in the literature for singularly perturbed differential equations without turning points in particular a number of difference schemes have been proposed that satisfy a stronger than normal convergence criteria known as uniform convergence this guarantees that the schemes model the boundary layers well we wish to examine whether these schemes will also be uniformly convergent if the equation has turning points to this end we derive sufficient conditions for uniform convergence which are satisfied not only by these schemes but by a more general class of schemes we show that the rate of convergence is determined by a characteristic parameter of the problem which may be less than one we confirm these theoretical results by numerical calculations

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1

analyzing repeated measures on generalized linear models via the bootstrap the analysis of longitudinal data for which the response variables have nonnormal error distributions previously has been complex and or dependent on restrictive assumptions in this paper simple methods are introduced for the class of generalized linear models glms regressions are fit to the data at each observation time functions of the resulting coefficients may be bootstrapped or the coefficients combined through closed form estimation of their covariances application is made to a data set on xerophthalmia in indonesian children