models for longitudinal data with random effects and ar 1 errors abstract for longitudinal data on several individuals linear models that contain both random effects across individuals and autocorrelation in the within individual errors are studied a score test for autocorrelation in the within individual errors for the conditional independence random effects model is first developed an explicit maximum likelihood estimation procedure using the scoring method for the model with random effects and autoregressive ar 1 errors is then derived empirical bayes estimation of the random effects and prediction of future responses of an individual based on this random effects with ar 1 errors model are also considered a numerical example is presented to illustrate these methods

linear and nonlinear mixed effects models for censored hiv viral loads using normal independent distributions summary hiv rna viral load measures are often subjected to some upper and lower detection limits depending on the quantification assays hence the responses are either left or right censored linear and nonlinear mixed effects models with modifications to accommodate censoring are routinely used to analyze this type of data and are based on normality assumptions for the random terms however those analyses might not provide robust inference when the normality assumptions are questionable in this article we develop a bayesian framework for censored linear and nonlinear models replacing the gaussian assumptions for the random terms with normal independent ni distributions the ni is an attractive class of symmetric heavy tailed densities that includes the normal student s t slash and the contaminated normal distributions as special cases the marginal likelihood is tractable using approximations for nonlinear models and can be used to develop bayesian case deletion influence diagnostics based on the kullback leibler divergence the newly developed procedures are illustrated with two hiv aids studies on viral loads that were initially analyzed using normal censored mixed effects models as well as simulations

semiparametric nonlinear mixed effects models and their applications nonlinear mixed effects models nlmms and self modeling nonlinear regression semor models are often used to fit repeated measures data they use a common function shared by all subjects to model variation within each subject and some fixed and or random parameters to model variation between subjects the parametric nlmm may be too restrictive and the semiparametric semor model ignores correlations within each subject in this article we propose a class of semiparametric nonlinear mixed effects models snmms that extend nlmms semor models and many other existing models in a natural way a snmm assumes that the mean function depends on some parameters and nonparametric functions the parameters provide an interpretable data summary the nonparametric functions provide flexibility to allow the data to decide some unknown or uncertain components such as the shape of the mean response over time a second stage model with fixed and random effects is used to model the parameters smoothing splines are us

a hybrid estimator in nonlinear and generalised linear mixed effects models summary a hybrid method that combines laplace s approximation and monte carlo simulations to evaluate integrals in the likelihood function is proposed for estimation of the parameters in nonlinear mixed effects models that assume a normal parametric family for the random effects simulations show that these parametric estimates of fixed effects are close to the nonparametric estimates even though the mixing distribution is far from the assumed normal parametric family an asymptotic theory of this hybrid method for parametric estimation without requiring the true mixing distribution to belong to the assumed parametric family is developed to explain these results this hybrid method and its asymptotic theory are also extended to generalised linear mixed effects models

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

mixed effects nonlinear regression for unbalanced repeated measures repeated measures data such as clinical pharmacokinetic data growth data and dose response data are often inherently nonlinear with respect to a given response function and are frequently incomplete and or unbalanced nonlinear random effects models together with a variety of estimation procedures have been proposed for the analysis of such data this paper is concerned with a straightforward procedure for estimating and comparing the parameters of a generalized mixed effects nonlinear regression model the asymptotic properties of the proposed estimators are given and large sample tests of hypothesis provided the results are applied to in vitro data on the water transport kinetics of hemodialyzers used in the treatment of patients with chronic renal failure

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

a multivariate bayesian model for embryonic growth most longitudinal growth curve models evaluate the evolution of each of the anthropometric measurements separately when applied to a reference population this exercise leads to univariate reference curves against which new individuals can be evaluated however growth should be evaluated in totality that is by evaluating all body characteristics jointly recently cole et al suggested the superimposition by translation and rotation sitar model which expresses individual growth curves by three subject specific parameters indicating their deviation from a flexible overall growth curve this model allows the characterization of normal growth in a flexible though compact manner in this paper we generalize the sitar model in a bayesian way to multiple dimensions the multivariate sitar model allows us to create multivariate reference regions which is advantageous for prediction the usefulness of the model is illustrated on longitudinal measurements of embryonic growth obtained in the first semester of pregnancy collected in the ongoing rotterdam predict study further we demonstrate how the model can be used to find determinants of embryonic growth copyright 2015 john wiley sons ltd

proof of the gorenstein interval conjecture in low socle degree abstract roughly ten years ago the following gorenstein interval conjecture gic was proposed whenever 1 h 1 h i h e i h e 1 1 and 1 h 1 h i h e i h e 1 1 are both gorenstein hilbert functions for some 2 then 1 h 1 h i h e i h e 1 1 is also gorenstein for all 1 2 1 since an explicit characterization of which hilbert functions are gorenstein is widely believed to be hopeless the gic if true would at least provide the existence of a strong and very natural structural property for such basic functions in commutative algebra before now very little progress was made on the gic the main goal of this note is to prove the case e 5 in arbitrary codimension our arguments will be in part constructive and will combine several different tools of commutative algebra and classical algebraic geometry

a bayesian population model with hierarchical mixture priors applied to blood count data abstract population pharmacokinetic and pharmacodynamic studies require analyzing nonlinear growth curves fit to multiple measurements from study subjects we propose a class of nonlinear population models with nonparametric second stage priors for analyzing such data the proposed models apply a flexible class of mixtures to implement the nonparametric second stage the discussion is based on a pharmacodynamic study involving longitudinal data consisting of hematologic profiles i e blood counts measured over time of cancer patients undergoing chemotherapy we describe a full posterior analysis in a bayesian framework this includes prediction of future observations profiles and end points for new patients estimation of the mean response function for observed individuals and inference on population characteristics the mixture model is specified and given a hyperprior distribution by means of a dirichlet processes prior on the mixing measure estimation is implemented by a combination of various m

infant growth modelling using a shape invariant model with random effects models for infant growth have usually been based on parametric forms commonly an exponential or similar model which have been shown to fit poorly especially during the first year of life an alternative approach is to use a non parametric model based on a shape invariant model sim where a single function is transformed by shifting and scaling to fit each subject in the model a regression spline is used as the function with log transformation of the data and a simplification of the sim obtained from the relationship with the exponential model all subjects are fitted as a nonlinear mixed effects model allowing the variation in the parameters between subjects to be determined methods for the inclusion of covariates in growth models based on sim are developed with parameters for time independent covariates included in the model by varying either the shape the size parameter or the growth parameter and time dependent co variates included by transforming the time axis to either increase or decrease the growth rate dependent on the co variate similar to methods used for accelerated failure time models the model is used to fit weight data for 602 infants measured from 0 to 2 years as part of the childhood asthma prevention study caps trial and to determine the effect of breastfeeding on infant weight

high dimensional maximum marginal likelihood item factor analysis by adaptive quadrature although the bock aitkin likelihood based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations a serious limitation of the method is its reliance on fixed point gauss hermite g h quadrature in the solution of the likelihood equations and likelihood ratio tests when the number of latent dimensions is large computational considerations require that the number of quadrature points per dimension be few but with large numbers of items the dispersion of the likelihood given the response pattern becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space in this paper we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data in particular we show that adaptive g h quadrature combined with mean and covariance adjustments at each iteration of an em algorithm produces an accurate fast converging solution with as few as two points per dimension evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions unlike an earlier application of adaptive gibbs sampling to this problem by meng and schilling the simulations also confirm the validity of the present method in calculating likelihood ratio chi square statistics for determining the number of factors required in the model finally we apply the method to a sample of real data from a test of teacher qualifications

compressed algebras artin algebras having given socle degrees and maximal length j emsalem and the author showed in 18 that a general polynomialf of degree j in the ring sp k yl yr has jr rl 1 linearly independent partial derivates of order i for i 0 1 t j 2 here we generalize the proof to show that the various partial derivates of s polynomials of specified degrees are as independent as possible given the room available using this result we construct and describe the varieties g e and z e parametrizing the graded and nongraded compressed algebra quotients a r i of the power series ring r k xl sxr having given socle type e these algebras are artin algebras having maximal length dimk a possible given the embedding degree r and given the socle type sequence e els s es where ei is the number of generators of the dual module a of a having degree i the variety z e is locally closed irreducible and is a bundle over g e fibred by affine spaces an whose dimension is known we consider the compressed algebras a new class of interesting algebras and a source of examples many of them are nonsmoothable have no deformation to k k for dimension reasons for some choices of the sequence e d buchsbaum d eisenbud and the author have shown that the graded compressed algebras of socle type e have almost linear minimal resolutions over r with ranks and degrees determined by e other examples have given type e dimk socle a and are defined by an ideal i with certain given numbers of generators in r

self modelling with random shift and scale parameters and a free knot spline shape function the shape invariant model is a semi parametric approach to estimating a function relationship from clustered data multiple observations on each of a number of individuals the common response curve shape over individuals is estimated by adjusting for individual scaling differences while pooling shape information in practice the common response curve is restricted to some flexible family of functions this paper introduces the use of a free knot spline shape function and reduces the number of parameters in the shape invariant model by assuming a random distribution on the parameters that control the individual scaling of the shape function new graphical diagnostics are presented parameter identifiability and estimation are discussed and an example is presented

flexible regression models with cubic splines we describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates this simple method can help prevent the problems that result from inappropriate linearity assumptions we compare restricted cubic spline regression to non parametric procedures for characterizing the relationship between age and survival in the stanford heart transplant data we also provide an illustrative example in cancer therapeutics

an em algorithm for nonlinear random effects models the pharmaceutical industry is currently interested in the population approach and population models also known as mixed effects models and random effects models depending on the precise form population models are useful in that they can account for both withinand between individual sources of variability and serial correlation within individual observations when analyzing unbalanced repeated measures data the modelling of population pharmacodynamic or pharmacokinetic profiles typically involves nonlinear random effects models each individual s observations are modelled by identical up to unknown parameter values nonlinear regression models with the distribution of the observations or a transformation of the observations about expected responses taken to be normal with the degree of variability described by a variance model between individual variability is modelled by a population distribution for the individual regression parameter values random effects in a parametric analysis the population distribution is taken to be normal the parameters of which along with the parameters of the variance model are known as the population parameters maximum likelihood estimation of the population parameters for nonlinear random effects models was pioneered by beal and sheiner 1979 and since then a number of algorithms have appeared for approximate maximum likelihood including steimer et al 1984 lindstrom and bates 1990 beal and sheiner 1992 and mentre and gomeni 1995 all of these algorithms are approximate in some way for a summary see beal and sheiner 1992 wolfinger 1993 pinheiro and bates 1994 and davidian and giltinan 1995 in this paper an em algorithm for exact maximum likelihood estimation is introduced an em algorithm obtaining maximum likelihood estimates for linear random effects models was introduced by dempster laird and rubin 1977 laird and ware 1982 lindstrom and bates 1988 jennrich and schluchter 1986 and liu and rubin 1994 all describe hybrid em algorithms for the linear random effects model a true em algorithm for the linear model is described by jamshidian and jennrich 1993 mentre and gomeni 1995 describe an approximate em algorithm for nonlinear random effects models and from the algorithm given in this paper it can be seen clearly how their approximations arise the present algorithm uses monte carlo methods to perform the e step a strategy previously adopted in an altogether different model by guo and thompson 1994 guo and thompson require a gibbs sampler that is a markov chain monte carlo method for their e step but the present algorithm uses independent samples in section 2 of this paper the nonlinear random effects model is described section 3 gives the em algorithm without random effect covariates while section 4 gives the modified algorithm in the

shape invariant modeling of circadian rhythms with random effects and smoothing spline anova decompositions medical studies often collect physiological and or psychological measurements over time from multiple subjects to study dynamics such as circadian rhythms under the assumption that the expected response functions of all subjects are the same after shift and scale transformations shape invariant models have been applied to analyze this kind of data the shift and scale parameters provide efficient and interpretable data summaries while the common shape function is usually modeled nonparametrically to provide flexibility however due to the deterministic nature of the shift and scale parameters potential correlations within a subject are ignored furthermore the shape of the common function may depend on other factors such as disease in this article we propose shape invariant mixed effects models a second stage model with fixed and random effects is used to model individual shift and scale parameters a second stage smoothing spline anova model is used to study potential covariate effects on the common shape function we apply our methods to a real data set to investigate disease effects on circadian rhythms of cortisol a hormone that is affected by stress we find that patients with cushing s syndrome lost circadian rhythms and their 24 hour means were elevated to very high levels patients with major depression had the same circadian shape and phases as normal subjects however their 24 hour mean levels were elevated and amplitudes were dampened for some patients

direct regression models for longitudinal rates of change comparing rates of growth or rates of change across covariate defined subgroups is a primary objective for many longitudinal studies in the special case of a linear trend over time the interaction between a covariate and time will characterize differences in longitudinal rates of change however in the presence of a non linear longitudinal trajectory the standard mean regression approach does not permit parsimonious description or inference regarding differences in rates of change therefore we propose regression methodology for longitudinal data that allows a direct structured comparison of rates across subgroups even in the presence of a non linear trend over time our basic longitudinal rate regression method assumes a proportional difference across covariate groups in the rate of change across time but this assumption can be relaxed rates are compared relative to a generally specified time trend for which we discuss both parametric and non parametric estimating approaches we develop mixed model longitudinal methodology that explicitly characterizes subject to subject variation in rates as well as a marginal estimating equation based method in addition we detail a score test to detect violations of the proportionality assumption and we allow time varying rate effects as a natural generalization simulation results demonstrate potential gains in power for the longitudinal rate regression model relative to a linear mixed effects model in the presence of a non linear trend in time we apply our method to a study of growth among infants born to hiv infected mothers and conclude with a discussion of possible extensions for our methods

reducible family of height three level algebras let r k x 1 x r be the polynomial ring in r variables over an infinite field k and let m be the maximal ideal of r here a level algebra will be a graded artinian quotient a of r having socle soc a 0 m in a single degree j the hilbert function h a h 0 h 1 h j gives the dimension h i dim k a i of each degree i graded piece of a for 0 i j the embedding dimension of a is h 1 and the type of a is dim k soc a here h j the family levalg h of level algebra quotients of r having hilbert function h forms an open subscheme of the family of graded algebras or via macaulay duality of a grassmannian we show that for each of the hilbert functions h 1 3 4 4 and h 2 1 3 6 8 9 3 the family levalg h has several irreducible components theorems 2 3 a 2 4 we show also that these examples each lift to points however in the first example an irreducible betti stratum for artinian algebras becomes reducible when lifted to points theorem 2 3 b we show that the second example is the first in an infinite sequence of examples of type three hilbert functions h c in which also the number of components gets arbitrarily large theorem 2 10 the first case where the phenomenon of multiple components can occur i e the lowest embedding dimension and then the lowest type is that of dimension three and type two examples of this first case have been obtained by the authors unpublished and also by j o kleppe

comparing linear and nonlinear mixed model approaches to cosinor analysis the cosinor model used for variables governed by circadian and other biological rhythms is a nonlinear model in the amplitude and acrophase parameters that has a linear representation upon transformation with linear cosinor analysis amplitude and acrophase for each harmonic can be computed as nonlinear functions of the estimated linear regression coefficients here a flexible mixed model approach to cosinor analysis is considered where the fixed effect parameters may enter nonlinearly as acrophase and amplitude for each harmonic or linearly after transformation to regression coefficients in addition the random effects may enter nonlinearly as subject specific deviations from the acrophases and amplitudes or linearly as subject specific deviations from the regression coefficients it is also possible for the fixed effects to enter nonlinearly while the random effects enter linearly additionally we evaluate whether including higher order linear harmonic terms as random effects that is rao khatri covariance adjustment improves precision applying the delta method to nonlinear functions of the parameters from linear mixed cosinor models to obtain approximate variances produces results that are often identical to results from nonlinear mixed models consequently traditional linear cosinor analysis can often be used to estimate and compare the nonlinear parameters of interest that is amplitudes and acrophases via the delta method this is advantageous since the nonlinear mixed model may have convergence difficulties for more complex models however for some multiple group analyses the linear cosinor transformation should not be used and we clarify when the two methods are equivalent and when they differ

hilbert functions and level algebras abstract the authors consider certain quotients a r i of the polynomial ring r k x1 xr over an arbitrary field k they first determine upper and lower bounds on the hilbert functions of any algebra having the form a r v where v is the largest ideal of r agreeing in degrees at least j with the ideal v generated by a vector subspace v rj of degree j forms these bounds extend the macaulay bounds theorem 1 4 a level artinian quotient of a r i has socle soc a 0 m m x1 xr in a single degree j the authors next determine the extremal hilbert functions hmax t j r and hmin t j r that occur for level graded artin algebra quotients of r having socle degree j and type t dimk soc a and they describe the extremal strata theorem 1 8 they next give a natural upper bound for the hilbert function of level algebra quotients of the coordinate ring o z of a punctual subscheme z p n n r 1 in terms of the hilbert function h z and j t this bound was known and known to be sharp in the case z is locally gorenstein and t 1 finally they show that there are no level algebras of hilbert function t 1 3 4 5 6 2 proposition 2 7 given that there are smooth punctual subschemes z of p 2 with hilbert function h z 1 3 4 5 6 6 this result shows that the sharpness of the natural upper bound in the case where z is gorenstein and t 1 does not extend to level algebra quotients of type 2 even when z is smooth the extremality results for the hilbert functions of level algebras and some counterexamples use the extremality theorem of f h s macaulay 1927 proc london math soc 26 531 555 and g gotzmann s 1978 math z 158 61 70 results on the hilbert scheme

a segregated finite element solution for non isothermal flow abstract a segregated finite element algorithm for the solution of the supg formulation of the incompressible steady state navier stokes equations for non isothermal flow is presented in this paper the method features equal order interpolation for all the flow variables the simpler and simplest algorithms are employed and the sets of non symmetric linear equations are solved by means of the preconditioned conjugate gradient squared solver whilst the preconditioned conjugate gradient solver is used to solve the sets of symmetric linear equations three cases are used as a basis for the study i e the flow in a square cavity the flow between parallel plates and natural convection in a square cavity the effect that the choice of the values of the relaxation parameters has on the number of iterations which is needed to obtain a converged solution is investigated in each case there is an optimum combination which is amongst others a function of the grid and the flow parameters a non optimum choice may lead either to an unnecessary number of iterations or the solution becoming unstable from the results it is concluded that the method performs well when the optimum combination of the values for the relaxation parameters is used however much can still be done to improve the robustness of the method

interval conjectures for level hilbert functions abstract we conjecture that the set of all hilbert functions of artinian level algebras enjoys a very natural form of regularity which we call the interval conjecture ic if for some positive integer 1 h 1 h i h e and 1 h 1 h i h e are both level h vectors then 1 h 1 h i h e is also level for each integer 0 1 in the gorenstein case i e when h e 1 we also supply the gorenstein interval conjecture gic which naturally generalizes the ic and basically states that the same property simultaneously holds for any two symmetric entries say h i and h e i of a gorenstein h vector these conjectures are inspired by the research performed in this area over the last few years a series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all gorenstein or of all level hilbert functions therefore our conjectures would at least provide the existence of a very strong and natural form of order in the structure of such important and complicated sets we are still far from proving the conjectures at this point however we will already solve a few interesting cases especially when it comes to the ic in this paper among them that of gorenstein h vectors of socle degree 4 that of level h vectors of socle degree 2 and that of non unimodal level h vectors of socle degree 3 and any given codimension

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

maximum likelihood computations with repeated measures application of the em algorithm abstract the purpose of this article is to consider the use of the em algorithm dempster laird and rubin 1977 for both maximum likelihood ml and restricted maximum likelihood reml estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure anderson 1973 several models and methods of analysis have been proposed in recent years for repeated measures data ware 1985 presented an overview because the em algorithm is a general purpose iterative method for computing ml estimates with incomplete data it has often been used in this particular setting dempster et al 1977 andrade and helms 1984 jennrich and schluchter 1985 there are two apparently different approaches to using the em algorithm in this setting in one application each experimental unit is observed under a standard protocol specifying measurements at each of n occasions or under n conditions and incompleteness implies that the number of measurements actua

a general approach for two stage analysis of multilevel clustered non gaussian data in this article we propose a two stage approach to modeling multilevel clustered non gaussian data with sufficiently large numbers of continuous measures per cluster such data are common in biological and medical studies utilizing monitoring or image processing equipment we consider a general class of hierarchical models that generalizes the model in the global two stage gts method for nonlinear mixed effects models by using any square root n consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model and by extending the stage 2 model to accommodate random effects from multiple levels of clustering the second stage model is a standard linear mixed effects model with normal random effects but the cluster specific distributions conditional on random effects can be non gaussian this methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest for estimation of the population parameters we propose a conditional restricted maximum likelihood creml approach and establish the asymptotic properties of the creml estimators the proposed general approach is illustrated using quartiles as cluster specific parameters estimated in the first stage and applied to the data example from a collagen fibril development study we demonstrate using simulations that in samples with small numbers of independent clusters the creml estimators may perform better than conditional maximum likelihood estimators which are a direct extension of the estimators from the gts method

british 1990 growth reference centiles for weight height body mass index and head circumference fitted by maximum penalized likelihood to update the british growth reference anthropometric data for weight height body mass index weight height2 and head circumference from 17 distinct surveys representative of england scotland and wales 37 700 children age range 23 weeks gestation to 23 years were analysed by maximum penalized likelihood using the lms method this estimates the measurement centiles in terms of three age sex specific cubic spline curves the l curve box cox power to remove skewness m curve median and s curve coefficient of variation a two stage fitting procedure was developed to model the age trends in median weight and height and simulation was used to estimate confidence intervals for the fitted centiles the reference converts measurements to standard deviation scores sds that are very close to normally distributed the means medians and skewness for the four measurements are effectively zero overall with standard deviations very close to one and only slight evidence of positive kurtosis beyond 2 sds the ability to express anthropometry as sds greatly simplifies growth assessment

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

a case study in exploratory functional data analysis geometrical features of the internal carotid artery this pilot study is a product of the aneurisk project a scientific program that aims at evaluating the role of vascular geometry and hemodynamics in the pathogenesis of cerebral aneurysms by means of functional data analyses we explore the aneurisk dataset to highlight the relations between the geometric features of the internal carotid artery expressed by its radius profile and centerline curvature and the aneurysm location after introducing a new similarity index for functional data we eliminate ancillary variability of vessel radius and curvature profiles through an iterative registration procedure we then reduce data dimension by means of functional principal components analysis last a quadratic discriminant analysis of functional principal components scores allows us to discriminate patients with aneurysms in different districts

unbalanced repeated measures models with structured covariance matrices the question of how to analyze unbalanced or incomplete repeated measures data is a common problem facing analysts we address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within subject covariances models that can be fit include standard univariate and multivariate models with incomplete data random effects models and models with time series and factor analytic error structures we describe newton raphson and fisher scoring algorithms for computing maximum likelihood estimates and generalized em algorithms for computing restricted and unrestricted maximum likelihood estimates an example fitting several models to a set of growth data is included

a non unimodal codimension 3 level h vector 1 3 6 10 15 21 28 27 27 28 is a level h vector this example answers negatively the open question as to whether all codimension 3 level h vectors are unimodal moreover using the same simple technique we are able to construct level algebras of codimension 3 whose h vectors have exactly n maxima for any positive integer n these non unimodal h vectors in particular provide examples of codimension 3 level algebras not enjoying the weak lefschetz property wlp their existence was also an open problem before in the second part of the paper we further investigate this fundamental property and show that there even exist codimension 3 level algebras of type 3 without the wlp

random effects and variances as a synthesis of nonlinear regression analyses of mitochondrial electron transport summary this cross disciplinary paper develops random effects and variances methods especially for small samples within effects and applies them to mitochondrial enzyme kinetics a new force flux kinetic model is introduced for the alternative oxidase of plant mitochondria the results of 42 experiments are published and the force flux function is fitted by nonlinear regression for each experiment the results of the regression analyses are synthesized by a random effects model specifying random nonlinear regression coefficients and random variances within experiments distributed in an inverse gamma distribution the paper shows how random nonlinear regression coefficients analysis can summarize the results of a long series of similar biological experiments and sharpen the analysis of the results of each a large data set often consists of a family of components each of which can be fitted with the same model there is a within component variation for each compo nent and random effects variation between components the interpretation of the data requires not only regression analysis of each component but also a subsequent synthesis of the results of such analysis into an overall summary achieved by specifying random regression coefficients and random error variances in the example which follows the individual samples are so small that random variances are specified and contracted predictors of them are computed which are more accurate than the individual estimates they are required for weights in the analysis the residual maximum likelihood estimates of the random effects variance matrices are sometimes singular a lagrange multiplier method is introduced to compute them the strength of the integrated analysis comes however at the price of a complex set of decisions as to how the different components of the computation are to be fitted together the paper shows how this is done by a flow diagram for the computations as well as describing each individual component

goodness of fit in generalized nonlinear mixed effects models in recent years generalized linear and nonlinear mixed effects models have proved to be powerful tools for the analysis of unbalanced longitudinal data to date much of the work has focused on various methods for estimating and comparing the parameters of mixed effects models very little work has been done in the area of model selection and goodness of fit particularly with respect to the assumed variance covariance structure in this paper we present a goodness of fit statistic which can be used in a manner similar to the r2 criterion in linear regression for assessing the adequacy of an assumed mean and variance covariance structure in addition we introduce an approximate pseudo likelihood ratio test for testing the adequacy of the hypothesized convariance structure these methods are illustrated and compared to the usual normal theory likelihood methods akaike s information criterion and the likelihood ratio test using three examples simulation results indicate the pseudo likelihood ratio test compares favorably with the standard normal theory likelihood ratio test but both procedures are sensitive to departures from normality

comparison of two longitudinal growth models for preschool children summary two models for growth in length and weight during early childhood are compared for goodness of fit and other qualities one model is linear in its parameters while the second is nonlinear both models are fitted to the data for length and weight of each child in a large sample which has fairly complete and reliable records of measurements the results are used to study issues such as model reliability efficiency and precision overall and age specific goodness of fit and adequacy of assumptions 1 lntroduction one reason for fitting models to growth data is that an appropriate curve will conveniently summarize the information provided by the observations of an individual child thus a large number of observations collected over time on an individual can be represented by a few parameters hoel 1964 has shown that it is more efficient to compare values of these parameters than to compare mean heights at a series of ages by the appropriate multivariate techniques from smooth curves velocity and acceleration at any time point within the observed age range can be obtained directly by differentiation although the number and spacing of measurements may vary considerably between individuals a curve can still be fitted for each individual analyses of the original data might require age adjustment of the observations to equally spaced target ages or interpolation for missing observations one of the earliest statistical analyses of growth data was performed with a polynomial model wishart 1938 much research has concentrated on polynomial models because they are simple to fit and unlike nonlinear models have the property that the meanparameter curve is equivalent to the population mean curve merrell 1931 however polynomials of high degrees are often needed to fit data which contain observations prior to one year of age and no biological meaning can be attached to their parameters desirable features of a growth model include simplicity of the fitting procedure biological interpretability of the model parameters and model parsimony however it is essential that a model should provide a satisfactory fit to the data of individuals and desirable that the fitting procedure should be robust to missing observations to variability in age about the target ages and to measurement error for these are common occurrences in longitudinal growth studies precision efficiency and reliability are related to the ability of a model to use all the information in the data to provide parameter estimates and fitted curves with narrow confidence intervals and bands and thus to differentiate well between individuals finally since one purpose of estimating model parameters is to use them in

accelerated degradation tests modeling and analysis high reliability systems generally require individual system components having extremely high reliability over long periods of time short product development times require reliability tests to be conducted with severe time constraints frequently few or no failures occur during such tests even with acceleration thus it is difficult to assess reliability with traditional life tests that record only failure times for some components degradation measures can be taken over time a relationship between component failure and amount of degradation makes it possible to use degradation models and data to make inferences and predictions about a failure time distribution this article describes degradation reliability models that correspond to physical failure mechanisms we explain the connection between degradation reliability models and failure time reliability models acceleration is modeled by having an acceleration model that describes the effect that temperature or another accelerating variable has on

the weak and strong lefschetz properties for artinian k algebras abstract let a i 0ai be a standard graded artinian k algebra where chark 0 then a has the weak lefschetz property if there is an element l of degree 1 such that the multiplication l a i a i 1 has maximal rank for every i and a has the strong lefschetz property if l d a i a i d has maximal rank for every i and d the main results obtained in this paper are the following 1 every height three complete intersection has the weak lefschetz property our method surprisingly uses rank two vector bundles on p 2 and the grauert mulich theorem 2 we give a complete characterization including a concrete construction of the hilbert functions that can occur for k algebras with the weak or strong lefschetz property and the characterization is the same one 3 we give a sharp bound on the graded betti numbers achieved by our construction of artinian k algebras with the weak or strong lefschetz property and fixed hilbert function this bound is again the same for both properties some hilbert functions in fact force the algebra to have the maximal betti numbers 4 every artinian ideal in k x y possesses the strong lefschetz property this is false in higher codimension

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

a segregated spectral finite element method for the 2d transient incompressible navier stokes equations abstract in this paper the spectral element approximation and the velocity pressuredecoupling method implementing the simple semi implicit method for pressure linked equations algorithm are first combined to form a high order segregated scheme for the solution of the two dimensional transient incompressible navier stokes equations in contrast to previous segregated finite element methods based on the simple algorithm the pressure equation is derived from the continuity equation using the element matrices to ensure convergence high order element basis functions are adopted which greatly reduces the number of nodal points used in the calculation the validation test that has an analytical solution demonstrates the high accuracy and convergence rate of the method the flow in a lid driven cavity with different inclination angles and the flow over a backward facing step are investigated to further illustrate the performance of the scheme the computed results are in excellent agreement with the benchmark solutions the almost periodic solution for the flow of re 10000 in a lid driven square cavity is also captured by the present scheme

design in nonlinear mixed effects models optimization using the fedorov wynn algorithm and power of the wald test for binary covariates we extend the methodology for designs evaluation and optimization in nonlinear mixed effects models with an illustration of the decrease of human immunodeficiency virus viral load after antiretroviral treatment initiation described by a bi exponential model we first show the relevance of the predicted standard errors ses given by the computation of the population fisher information matrix using the r function pfim in comparison to those computed with the stochastic approximation expectation maximization algorithm implemented in the monolix software we then highlight the usefulness of the fedorov wynn fw algorithm for designs optimization compared to the simplex algorithm from the predicted se of pfim we compute the predicted power of the wald test to detect a treatment effect as well as the number of subjects needed to achieve a given power using the fw algorithm we investigate the influence of the design on the power and show that for optimized designs with the same total number of samples the power increases when the number of subjects increases and the number of samples per subject decreases a simulation study is also performed with the nlme function of r to confirm this result and show the relevance of the predicted powers compared to those observed by simulation

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

some alternatives to asymptotic tests for the analysis of pharmacogenetic data using nonlinear mixed effects models nonlinear mixed effects models allow investigating individual differences in drug concentration profiles pharmacokinetics and responses pharmacogenetics focuses on the genetic component of this variability two tests often used to detect a gene effect on a pharmacokinetic parameter are 1 the wald test assessing whether estimates for the gene effect are significantly different from 0 and 2 the likelihood ratio test comparing models with and without the genetic effect because those asymptotic tests show inflated type i error on small sample size and or with unevenly distributed genotypes we develop two alternatives and evaluate them by means of a simulation study first we assess the performance of the permutation test using the wald and the likelihood ratio statistics second for the wald test we propose the use of the f distribution with four different values for the denominator degrees of freedom we also explore the influence of the estimation algorithm using both the first order conditional estimation with interaction linearization based algorithm and the stochastic approximation expectation maximization algorithm we apply these methods to the analysis of the pharmacogenetics of indinavir in hiv patients recruited in the cophar2 anrs 111 trial results of the simulation study show that the permutation test seems appropriate but at the cost of an additional computational burden one of the four f distribution based approaches provides a correct type i error estimate for the wald test and should be further investigated

discriminant analysis for longitudinal data with multiple continuous responses and possibly missing data multiple outcomes are often used to properly characterize an effect of interest this article discusses model based statistical methods for the classification of units into one of two or more groups where for each unit repeated measurements over time are obtained on each outcome we relate the observed outcomes using multivariate nonlinear mixed effects models to describe evolutions in different groups due to its flexibility the random effects approach for the joint modeling of multiple outcomes can be used to estimate population parameters for a discriminant model that classifies units into distinct predefined groups or populations parameter estimation is done via the expectation maximization algorithm with a linear approximation step we conduct a simulation study that sheds light on the effect that the linear approximation has on classification results we present an example using data from a study in 161 pregnant women in santiago chile where the main interest is to predict normal versus abnormal pregnancy outcomes

evaluation by simulation of tests based on non linear mixed effects models in pharmacokinetic interaction and bioequivalence cross over trials we propose tests based on non linear mixed effects models nlmem in pharmacokinetic interaction and bioequivalence cross over trials comparing two treatments or two formulations to compare the logarithm of the area under the curve auc using these models two approaches are studied in the first one concentration data are analysed globally with and without the estimation of a treatment effect and in the second one they are analysed separately in each treatment group with the estimation of the individual parameters four tests for comparison of the logarithm auc between two treatment arms are studied a likelihood ratio test lrt a wald test and two tests parametric and non parametric comparing the individual empirical bayes eb estimates these tests are adapted to the case of equivalence except the lrt which does not have any simple extension we evaluate by simulation of the type i error and the power for both comparison and equivalence tests they are compared to the standard tests recommended by the fda and the emea based on non compartmental nc auc trials for a usual pk model are simulated under h0 and several h1 using s plus software and analysed with the nlme function different configurations of the number of subjects n 12 24 and 40 and of the number of samples per subject j 10 5 and 3 are studied the type i error of lrt and wald comparison test in the 5000 replications of interaction cross over trials is found to be 20 9 per cent and 21 7 per cent respectively in the original design n 12 j 10 which is far superior to 5 per cent and decreases when n increases when n is fixed is found to increase with j power is satisfactory for both tests after correction of the significance threshold results of eb and nc tests are similar with satisfactory powers and a type i error close to 5 per cent except when j 3 for eb tests similar results are obtained for equivalence tests except for eb and nc student tests which are not of a great interest nc tests keep their place when the number of samples per subject j is large but nlmem seem useful for cross over studies performed in special populations where j limited the evaluation by monte carlo simulations of empirical threshold seems however necessary because of the inflation of the type i error copyright 2005 john wiley sons ltd

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

a bayesian approach to nonlinear random effects models nonlinear random effects models are considered from the bayesian point of view the method of analysis follows closely that of lindley and smith 1972 journal of the royal statistical society series b 34 1 42 the numerical method is related to the em algorithm

optimal design in random effects regression models summary an approach is proposed to optimal design of experiments for estimating randomeffects regression models the population designs are defined by the number of subjects and the individual designs to be performed cost functions associated with individual designs are incorporated for a given maximal cost an algorithm is proposed for finding the statistical population design that maximises the determinant of the fisher information matrix of the population parameters the fisher information matrix is formulated for linear models and normal distributions the approach is applied to the design of an optimal experiment in toxicokinetics using a first order linearisation of the model several cost functions and designs of various orders are studied an example illustrates the optimal population designs and the increased efficiency of some optimal designs over more standard designs

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

bayesian approach for a nonlinear growth model nonlinear least squares methods are currently used for fitting a well known growth model namely the jenss model to the length measurements of a child followed throughout the first six years of life an empirical bayes approach is developed for fitting the model and the prior distribution of the growth model parameters is estimated from a large sample of least squares parameters an expression which is proportional to the posterior distribution is derived so that the posterior mode can be estimated given the observations on a child this posterior mode provides bayes estimates of the jenss curve parameters for the child

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

estimating data transformations in nonlinear mixed effects models a routine practice in the analysis of repeated measurement data is to represent individual responses by a mixed effects model on some transformed scale for example for pharmacokinetic growth and other data both the response and the regression model are typically transformed to achieve approximate within individual normality and constant variance on the new scale however the choice of transformation is often made subjectively or by default with adoption of a standard choice such as the log we propose a mixed effects framework based on the transform both sides model where the transformation is represented by a monotone parametric function and is estimated from the data for this model we describe a practical fitting strategy based on approximation of the marginal likelihood inference is complicated by the fact that estimation of the transformation requires modification of the usual standard errors for estimators of fixed effects however we show that under conditions relevant to common applications this complication is asymptotically negligible allowing straightforward implementation via standard software

estimation in forest yield models using composite link functions with random effects a sigmoidal forest yield model is estimated from longitudinal measurements of sample plots where measurement times and intervals vary the projection form of the yield model is fitted as a generalized linear model glm with composite link functions and an offset this allows age and additional covariates to be easily incorporated in a more flexible nonlinear structure than that provided by single link glms also existing algorithms for fitting generalized linear mixed models glmms are easily adapted to fit the projection model thereby allowing incorporation of random plot effects estimation for the composite link glmm is described as well as an alternative to the glmm in which random effects are incorporated as linear components on the same scale as the response this model is referred to as an additive generalized linear mixed model aglmm unlike the glmm the fixed effect parameter estimates in the aglmm are population average pa the aglmm is fitted using a modification of a subject specific ss algorithm using marginal expectation the random effects in the aglmm may have a natural interpretation or may provide a means of generating a marginal covariance structure as is the case with linear mixed models a simple example of an aglmm for binomial data and logit link is given in both this and the forest yield example a single artificial random effect is constructed to give exchangeable correlation for observations within subjects and a comparison of fit of the aglmm and glmm is made using a conditional quasi deviance if the fit is similar and pa parameter estimates are required the aglmm is preferred

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

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 relative off set orthogonality convergence criterion for nonlinear least squares an orthogonality convergence criterion using relative offset is proposed this criterion is compared to currently used criteria and its advantages are discussed