aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

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

bayesian modeling of the covariance structure for irregular longitudinal data using the partial autocorrelation function in long term follow up studies irregular longitudinal data are observed when individuals are assessed repeatedly over time but at uncommon and irregularly spaced time points modeling the covariance structure for this type of data is challenging as it requires specification of a covariance function that is positive definite moreover in certain settings careful modeling of the covariance structure for irregular longitudinal data can be crucial in order to ensure no bias arises in the mean structure two common settings where this occurs are studies with outcome dependent follow up and studies with ignorable missing data outcome dependent follow up occurs when individuals with a history of poor health outcomes had more follow up measurements and the intervals between the repeated measurements were shorter when the follow up time process only depends on previous outcomes likelihood based methods can still provide consistent estimates of the regression parameters given that both the mean and covariance structures of the irregular longitudinal data are correctly specified and no model for the follow up time process is required for ignorable missing data the missing data mechanism does not need to be specified but valid likelihood based inference requires correct specification of the covariance structure in both cases flexible modeling approaches for the covariance structure are essential in this paper we develop a flexible approach to modeling the covariance structure for irregular continuous longitudinal data using the partial autocorrelation function and the variance function in particular we propose semiparametric non stationary partial autocorrelation function models which do not suffer from complex positive definiteness restrictions like the autocorrelation function we describe a bayesian approach discuss computational issues and apply the proposed methods to cd4 count data from a pediatric aids clinical trial 2015 the authors statistics in medicine published by john wiley sons ltd

longitudinal models for analysis of respiratory function we compare the results of fitting three longitudinal models two autoregressive models the serial correlation model and a damped autoregressive model and a compound symmetry model to data on a cohort of 1154 adult men in boston the serial correlation model assumes that the error terms are autocorrelated with correlation of the form t for visits t years apart while the damped autoregressive assumes that the correlation between error terms of observations t years apart is of the form t the compound symmetry model assumes that the errors are correlated with the same correlation regardless of how far apart observations are in time these three models are all related in that the serial correlation and compound symmetry models are particular cases of the damped autoregressive models that is 1 corresponds to the serial correlation model and 0 corresponds to the compound symmetry model for current smokers the damped autoregressive model provided a significantly better fit than either of the other two models p 0 001 for never smokers the damped autoregressive and compound symmetry models were almost identical with both providing a significantly better fit than the serial correlation model p 0 001

joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

an algorithm for the minimization of mixed l sub 1 and l sub 2 norms with application to bayesian estimation the regularizing functional approach is widely used in many estimation problems in practice the solution is defined as one minimum point of a suitable functional the main part of which accounts for the underlying physical model whereas the regularizing part represents some prior information about the unknowns in the bayesian interpretation one has a maximum a posteriori map estimator in which the main and regularizing parts are represented respectively by likelihood and prior distributions when either the prior or likelihood is a laplace distribution and the other is a gaussian distribution one is led to consider functionals that include both absolute and square norms the authors present a characterization of the minimum points of such functionals together with a descent type algorithm for numerical computations the results of monte carlo simulations are also reported

a semi discrete tailored finite point method for a class of anisotropic diffusion problems abstract this work proposes a tailored finite point method tfpm for the numerical solution of an anisotropic diffusion problem which has much smaller diffusion coefficient along one direction than the other on a rectangular domain the paper includes analysis on the differentiability of the solution to the given problem under some compatibility conditions it has detailed derivation for a semi discrete tfpm for the given problem this work also proves a uniform error estimate on the approximate solution numerical results show that the tfpm is accurate as well as efficient for the strongly anisotropic diffusion problem examples include those that do not satisfy compatibility and regularity conditions for the incompatible problems numerical experiments indicate that the method proposed can still offer good numerical approximations

a systematic review of the reporting of sample size calculations and corresponding data components in observational functional magnetic resonance imaging studies anecdotal evidence suggests that functional magnetic resonance imaging fmri studies rarely consider statistical power when setting a sample size this raises concerns since undersized studies may fail to detect effects of interest and encourage data dredging although sample size methodology in this field exists implementation requires specifications of estimated effect size and variance components we therefore systematically evaluated how often estimates of effect size and variance components were reported in observational fmri studies involving clinical human participants published in six leading journals between january 2010 and december 2011 a random sample of 100 eligible articles was included in data extraction and analyses two independent reviewers assessed the reporting of sample size calculations and the data components required to perform the calculations in the fmri literature one article 1 reported sample size calculations the reporting of parameter estimates for effect size 8 between subject variance 4 within subject variance 1 and temporal autocorrelation matrix 0 was uncommon three articles 3 reported cohen s d or f effect sizes the majority 83 reported peak or average t z or f statistics the inter rater agreement was very good with a prevalence adjusted bias adjusted kappa pabak value greater than 0 88 we concluded that sample size calculations were seldom reported in fmri studies moreover omission of parameter estimates for effect size between and within subject variances and temporal autocorrelation matrix could limit investigators ability to perform power analyses for new studies we suggest routine reporting of these quantities and recommend strategies for reducing bias in their reported values

a parametric family of correlation structures for the analysis of longitudinal data in epidemiological settings we are often faced with numerous short time series and a parsimonious parametrization of the correlation structure is desired in order to optimize the efficiency of the estimation procedure we propose a damped exponential correlation structure for modeling multivariate gaussian outcomes the correlation between two observations separated by s units of time is modeled as gamma s theta where gamma is the correlation between elements separated by one s unit and theta is a damping parameter for theta 0 theta 1 and theta infinity the correlation structures of compound symmetry first order autoregressive and first order moving average processes are obtained although the ar 2 dependency structure and the combination of random effects and ar 1 errors are not special cases of the proposed parametric family these structures can be well approximated within the family for short time series maximum likelihood methods for parameter estimation and interpretations of intermediate models 0 less than theta less than 1 are discussed in the context of modeling pulmonary function in an adult population in the netherlands and t cell subsets in homosexual men infected with human immunodeficiency virus type i

an approach to the analysis of repeated measurements summary a linear model for repeated measurements is proposed in which the correlation structure within each time sequence of measurements includes parameters for measurement error variation between experimental units and serial correlation within units an approach to data analysis is presented which involves preliminary analysis by ordinary least squares use of the empirical semi variogram of residuals to suggest a suitable correlation structure and formal inference using likelihood based methods applications to two biological data sets are described

a modified pseudolikelihood approach for analysis of longitudinal data we consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters in general inconsistent estimators of the covariance variance correlation parameters will be produced when the working correlation matrix is misspecified which may result in great loss of efficiency of the mean parameter estimators albeit the consistency is preserved we consider using different working correlation models for the variance and the mean parameters in particular we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified the designated working correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters simulation studies indicate that the proposed algorithm performs very well we also applied different estimation procedures to a data set from a clinical trial for illustration

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

total variation time flow with quantile regression for image restoration abstract we analyze the time flow of a modification of the standard rudin osher fatemi rof model for image restoration using the bounded variation semi norm the solution of the minimization problem is a restored image in the space of bounded variation bv recovered from a corrupted one and the functional may include a more general term s x u rather than the l 2 penalty term as used in the original rof model these new models were introduced in osher and scherzer 2004 31 in particular we will examine the non differentiable quantile regression penalty we will define an appropriate weak solution of the time flow and in particular prove the existence in l 0 b v and l stability

power and sample size calculations for longitudinal studies comparing rates of change with a time varying exposure existing study design formulas for longitudinal studies have assumed that the exposure is time invariant we derived sample size formulas for studies comparing rates of change by exposure when the exposure varies with time within a subject focusing on observational studies where this variation is not controlled by the investigator two scenarios are considered one assuming that the effect of exposure on the response is acute and the other assuming that it is cumulative we show that accurate calculations can often be obtained by providing the intraclass correlation of exposure and the exposure prevalence at each time point when comparing rates of change studies with a time varying exposure are in general less efficient than studies with a time invariant one we provide a public access program to perform the calculations described in the paper http www hsph harvard edu faculty spiegelman optitxs html copyright 2009 john wiley sons ltd

marginal analyses of longitudinal data with an informative pattern of observations we consider solutions to generalized estimating equations with singular working correlation matrices of which the estimator of diggle et al 2007 is a special case we give explicit conditions for consistent estimation when the pattern of observations may be informative in such cases simulations reveal reduced bias and reduced mean squared error compared with existing alternatives a study of peritoneal dialysis is used to illustrate the methodology

second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems abstract we consider a coupled system of singularly perturbed first order ordinary differential equations the equations have small positive parameters of different magnitudes associated with them a decomposition of the exact solution into smooth and singular components is constructed that is useful for the analysis of the numerical method we apply the hodie high order difference approximation with identity expansions technique to construct a second order finite difference scheme and combine this with standard backward euler difference scheme in a special way on a piecewise uniform shishkin mesh to solve the system numerically it is proved that the method is second order up to a logarithmic factor convergent in the maximum norm uniformly in both perturbation parameters numerical experiments are conducted to support the theoretical results

model selection techniques for the covariance matrix for incomplete longitudinal data in longitudinal studies with incomplete data where the number of time points can become numerous it is often advantageous to model the covariance matrix we describe several covariance models for example mixed models compound symmetry ar 1 type models and combination models that offer parsimonious alternatives to unstructured sigma we evaluate each covariance model with longitudinal data concerning cholesterol as the repeated outcome measure we discuss strategies for deciding the best model and show a graphical technique for judging goodness of fit of covariance models

a parameter uniform numerical method for a system of singularly perturbed ordinary differential equations in this paper an initial value problem for a system of singularly perturbed ordinary differential equations is considered a parameter robust computational method is constructed and it is proved that it gives essentially first order parameter uniform convergence in the maximum norm numerical results are presented in support of the theory

parametric maximum flow algorithms for fast total variation minimization this report studies the global minimization of anisotropically discretized total variation tv energies with an l p in particular l 1 and l 2 fidelity term using parametric maximum flow algorithms to minimize s t cut representations of these energies the tv l 2 model also known as the rudin osher fatemi rof model is suitable for restoring images contaminated by gaussian noise while the tv l 1 model is able to remove impulsive noise from grayscale images and perform multiscale decompositions of them preliminary numerical results on large scale two dimensional ct and three dimensional brain mr images are presented to illustrate the effectiveness of these approaches

maximum likelihood estimation for incomplete repeated measures experiments under an arma covariance structure a stochastic model is presented for the analysis of incomplete repeated measures experiments the general linear model is used to relate the response measures to other variables which are thought to account for inherent variation an autoregressive moving average arma time series representation is used to model disturbance terms maximum likelihood estimation procedures are considered and the properties of these estimators are derived it is concluded that while the assumptions underpinning the arma covariance models may be somewhat restrictive they provide a useful inferential vehicle particularly in the presence of missing values

some optimal designs for repeated measurements with autoregressive errors summary a number of individuals are observed at equally spaced time points different individuals behave independently but the observations on the same individual have the autocorrelation structure of a first order autoregressive process we are concerned with the optimal allocation of treatments assuming that at each time point a new selection must be made conditions for optimality are given for models without nuisance parameters and also with covariates and with block effects

the use of an autoregressive model for the analysis of longitudinal data in epidemiologic studies korn and whittemore1 2 have presented methods for analyzing longitudinal data where the number of observations per individual is large relative to the number of variables considered for each subject however this is often not the case in epidemiologic studies since one usually collects data at relatively few time points and the quantity of data collected for each individual at each time point is typically extensive we present here an autoregressive model for analyzing longitudinal data of this type for the case of a continuous outcome variable some of the important features of this model are that one can 1 in the same analysis consider both independent variables that are time dependent and those that are fixed over time 2 partially use data for an individual where some examinations are missing 3 assess relationships between changes in outcome and exposure over short periods of time 4 use ordinary multiple regression methods anderson3 has considered this type of model but to our knowledge the model has never been applied to biostatistical problems we illustrate these methods with data from a longitudinal study that seeks to identify the role of personal cigarette smoking on changes in pulmonary function in children

trees and tracking epidemiologists have used the term tracking to connote an individual s maintenance of relative rank of some longitudinally measured characteristic over a given time span to assess the extent to which an attribute tracks we have first to summarize individual growth curves and second to quantify the notion of maintenance of relative rank both in the face of random error a sequence of papers appearing in 1981 provided differing methodologies for appraising tracking here we take a different approach to tracking by using regression trees for longitudinal data the above two concerns are simultaneously addressed in that the procedure identifies subgroups defined in terms of covariates within which the collection of growth curves is homogeneous after reviewing the existing approaches to tracking we describe the tree structured methodology and present an illustrative example pertaining to lung function growth in children

digital filters as absolute norm regularizers a variational formulation of the digital filtering problem based on a certain regularizing functional is presented in particular the special case where the regularizing functional is defined by means of l sub 1 norm is considered and a characterization of its minima is presented it is shown that for a certain class of signals one solution to the regularization problem can be obtained by means of a suitable recursive median filter some examples and a possible extension of the variational approach are also discussed

robust statistical modeling using the t distribution abstract the t distribution provides a useful extension of the normal for statistical modeling of data sets involving errors with longer than normal tails an analytical strategy based on maximum likelihood for a general model with multivariate t errors is suggested and applied to a variety of problems including linear and nonlinear regression robust estimation of the mean and covariance matrix with missing data unbalanced multivariate repeated measures data multivariate modeling of pedigree data and multivariate nonlinear regression the degrees of freedom parameter of the t distribution provides a convenient dimension for achieving robust statistical inference with moderate increases in computational complexity for many models estimation of precision from asymptotic theory and the bootstrap is discussed and graphical methods for checking the appropriateness of the t distribution are presented

a property of the minimum vectors of a regularizing functional defined by means of the absolute norm we consider a regularizing functional defined by means of the l sub 1 norm where the regularization is obtained using first differences as is well known such a functional can be put in relation with recursive median filters of appropriate window length we show that at least one of the minima is reached at a vector whose components have values over the same discrete set of the given signal this suggests a simple method to refine the approximate solution to the regularization problem which can be obtained with recursive median filters of increasing order we also report an example of application where the refinement method is employed for a signal detection problem

stochastic analysis for the recursive median filter process vector probability measure functions density function for recursively median filtered signals are found when the underlying input binary sequences are either independent identically distributed i i d or markov chains the results are parametric in the window size of the filter and in the probability distribution of the input sequence using statistical threshold decomposition the same results are found for discrete alphabet random sequences that are either i i d or markov chains some examples illustrating the efficacy of the recursive median filter relative to the nonrecursive implementation are presented in particular the breakdown probabilities are tabulated for both recursive and nonrecursive median filters

regression analysis in the presence of heterogeneous intraclass correlations summary in the statistical analysis of twinship and familial data one often encounters the need for regression methods that control for the possibly different aggregation of the twins and the members of the families according to their type e g monozygotic dizygotic for twinship data we present the maximum likelihood solution for the regression analysis of data in the presence of heterogeneous intraclass correlations this work extends previous results for the case of a homogeneous intraclass correlation an application of the methods for the analysis of twinship data is included

flexible longitudinal linear mixed models for multiple censored responses data in biomedical studies and clinical trials repeated measures are often subject to some upper and or lower limits of detection hence the responses are either left or right censored a complication arises when more than one series of responses is repeatedly collected on each subject at irregular intervals over a period of time and the data exhibit tails heavier than the normal distribution the multivariate censored linear mixed effect mlmec model is a frequently used tool for a joint analysis of more than one series of longitudinal data in this context we develop a robust generalization of the mlmec based on the scale mixtures of normal distributions to take into account the autocorrelation existing among irregularly observed measures a damped exponential correlation structure is considered for this complex longitudinal structure we propose an exact estimation procedure to obtain the maximum likelihood estimates of the fixed effects and variance components using a stochastic approximation of the em algorithm this approach allows us to estimate the parameters of interest easily and quickly as well as to obtain the standard errors of the fixed effects the predictions of unobservable values of the responses and the log likelihood function as a byproduct the proposed method is applied to analyze a set of aids data and is examined via a simulation study

sample size determination for constrained longitudinal data analysis the longitudinal data analysis model proposed by liang and zeger sankhy indian j stat ser b 2000 62 134 148 uses the baseline as well as postbaseline values as the dependent variables and the baseline mean responses are constrained to be the same across treatment groups due to randomization compared with the conventional longitudinal analysis of covariance this approach can correctly estimate the variance of within group mean changes and achieve the specified coverage probabilities general results on the sample size and power calculations for this model in the presence of missing data are obtained the sample size relationship between the constrained and unconstrained longitudinal data analysis is established simple expressions for sample size calculation are obtained for the compound symmetry and first order autoregressive correlation structures the sensitivity of the sample size requirement to the configuration of correlation structure and retention pattern is assessed the performance of several ad hoc approximations for longitudinal data analysis sample size calculation is evaluated simulation studies are conducted to assess the validity of the proposed sample size formulas with deviation from normality the sample size formulas are also illustrated in detail using real clinical trial data copyright 2008 john wiley sons ltd

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

statistical threshold decomposition for recursive and nonrecursive median filters the statistical analysis of recursive nonlinear filters is generally difficult the analysis of recursive median filters has been limited to the trivial cases of signals with a small number of quantization levels and to small window sizes a block state description of recursively filtered signals is developed and by applying this description to threshold decomposition closed form expressions for the statistics of recursive median filters are obtained in this case the number of quantization levels and the window size do not increase the analysis complexity since the output statistics depend on the distribution of a single threshold filtered binary signal the statistical decomposition is also developed for nonrecursive median filter operations yielding a connection from classical order statistics to the threshold decomposition approach finally some statistical properties are derived for recursively median filtered signals

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

layer adapted methods for quasilinear singularly perturbed delay differential problems abstract in this work we consider a class of initial value problems for quasilinear singularly perturbed first order delay differential equations to solve this class of problems numerically we consider two finite difference schemes the backward euler scheme and a high order hybrid scheme which is a blend of the trapezoidal scheme and the backward euler scheme we derive general convergence results for both the schemes based on which a number of layer adapted meshes can be constructed and analyzed as consequences of these results we establish uniform convergence of the schemes on certain layer adapted meshes numerical experiments confirm our theoretical findings

a numerical method for a system of singularly perturbed reaction diffusion equations a dirichlet problem for a system of two coupled singularly perturbed reaction diffusion ordinary differential equations is examined a numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed numerical results are presented which illustrate the theoretical results

median filtering by threshold decomposition median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to further passes of the median filter in this paper it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals filtering each binary signal to a root with a binary median filter and then reversing the decomposition this equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals since the effects of median filters on binary signals are well understood this technique is a powerful new tool

adaptive stack filtering under the mean absolute error criterion an adaptive filter algorithm is developed for the class of stack filters which is a class of nonlinear filters obeying a weak superposition property the adaptation algorithm can be interpreted as a learning algorithm for a group of decision making units the decisions of which are subject to a set of constraints called the stacking constraints under a rather weak statistical assumption on the training inputs the decision strategy adopted by the group which evolves according to the proposed learning algorithm is shown to converge to an optimal strategy in the sense that it corresponds to an optimal stack filter under the mean absolute error criterion this adaptive algorithm requires only increment decrement and comparison operations and only local interconnections between the learning units implementation of the algorithm in hardware is therefore very feasible an example is provided to show how the adaptive stack filtering algorithm can be used in an application in image processing

solving a partially singularly perturbed initial value problem on shishkin meshes a coupled first order system of one singularly perturbed and one non perturbed ordinary differential equation with prescribed initial conditions is considered a shishkin piecewise uniform mesh is constructed and used in conjunction with a classical finite difference operator to form a new numerical method for solving this problem it is proved that the numerical approximations generated by this method are essentially first order convergent in the maximum norm at all points of the domain uniformly with respect to the singular perturbation parameter numerical results are presented in support of the theory

analysis of serially correlated data using quasi least squares summary quasi least squares qls a marginal statistical approach via generalized estimating equations that is described in the balanced data setting by chaganty 1997 journal of statistical planning and inference 63 39 54 allows for application of a wide range of working correlation structures when analyzing serially correlated data we extend the application of qls to serially correlated unequally spaced and unbalanced data using three useful working correlation models the firstorder autoregressive ar 1 the markov and the generalized markov structure described by nuniez anton and woodworth 1994 biometrics 50 445 456 we compare qls and the original formulation of the generalized estimating equation approach gee for these structures demonstrating that i infeasibility of the gee correlation parameter estimates can be a problem ii it is difficult to obtain consistent moment estimates of the correlation parameters for the generalized markov structure and iii the use of qls can lead to reduced mean square error of the estimate of the regression parameter for small samples of moderately correlated data to choose between alternative correlation models we propose a criterion that is based on the principle of generalized least squares finally data for which the generalized markov structure is appropriate are analyzed to demonstrate the use of qls in selecting a suitable working correlation structure and identifying in this paper we apply a statistical method based on the generalized estimating equation approach of liang and zeger 1986 to the analysis of longitudinal data that may be difficult to analyze using other established methods we consider repeated measures data collected by taking measurements of an outcome variable and associated covariates on each of a group of independent subjects our primary data analysis goal is to identify important covariates and to explain their effect on the marginal mean of the outcome variable while also accounting for the correlation among observations on each subject accomplishing this research objective may be difficult due to certain conditions that are typical in longitudinal studies the timing and total number of measurements taken may vary from subject to subject so that the data may be unbalanced and unequally spaced the outcome variable may not be normally distributed the intrasubject correlation may be described using a time dependent pattern for example the correlation between two measurements may decrease as they become more

marginal and dynamic regression models for longitudinal data regression models for longitudinal data often employ random effects and serial correlation to account for residual variation between and within subjects most of these models are marginal models separating the mean and covariance parameters this paper discusses the use of dynamic models in which a lagged response serves as a predictor and compares these to marginal models regression parameters have a different interpretation in dynamic models as they describe changes in response levels rather than the levels themselves lagged predictors are also useful with longitudinal data explicitly quantifying the effect of previous levels of risk factors these models are explored through analysis of data from the childhood respiratory study modelling lung function fev1 levels as a function of age height sex and smoking status in children measured over a five year period copyright 2001 john wiley sons ltd

adjusting for measurement error to assess health effects of variability in biomarkers longitudinal studies of health effects often relate individuals biomarker levels to disease progression repeated measurements also provide an opportunity to assess within individual biomarker variability and it is reasonable to postulate that this measure might provide additional information about a particular outcome variable given the existing precedent for application of adjustment methods to account for measurement error in subject specific average levels of a covariate this concept motivates the application of such methods to incorporate variability as well in this paper we investigate the nature of the relationship between the decline of cd4 cell count induced by infection with human immunodeficiency virus and cd4 level and variability prior to infection we first describe the distribution of repeated cd4 measurements prior to infection using a model that accounts both for random average levels and random subject specific variance components based on this model we define true unobservable random variables that correspond to prior level and stability we perform a linear regression analysis using these latent variables as covariates by means of a full maximum likelihood approach we compare the resulting parameter estimates with those based on regressions employing sample based estimates of pre infection levels and variances and empirical bayes estimates of these quantities although the final inferences are similar to those based on the unadjusted analysis we find that the magnitude of association with prior level decreases while that with prior stability increases stratified analyses indicate that smoking status affects the relationship between prior cd4 level and initial cd4 decline we point out advantages associated with the maximum likelihood approach in this particular application

recursive median filters of increasing order a variational approach for a certain class of signals it is known that the recursive median filters provide a minimum of a related regularizing functional defined by means of the l sub 1 norm we generalize such results and we show that a good approximation to the minima of such a functional can be obtained for any signal by means of successive applications of recursive median filters of increasing order

an em algorithm fitting first order conditional autoregressive models to longitudinal data abstract an em algorithm fits a state space formulation of the longitudinal regression model in which a continuous response depends on the lagged response and both time dependent and time independent covariates the baseline response depends only on covariates the model handles both missing data and gaussian measurement error on both response and continuous covariates the e step uses the kalman filter and associated filtering algorithms to update the unknown true response and predictor series for the observed data the m step uses standard closed form gaussian results standard errors come from the supplemented em sem algorithm the model accurately fits 6 years of pulmonary function measurements on 158 children with many missing observations

an adaptive moving grid method for a system of singularly perturbed initial value problems a system of first order singularly perturbed initial value problems is considered the system is discretized by a backward euler difference scheme for which a priori error analysis in the maximum norm is constructed it is shown from the a priori error bound that there exists a mesh with n subintervals that gives optimal error bound of o n 1 which is robust with respect to the perturbation parameters a partly heuristic argument based on a priori error analysis leads to a suitable monitor function based on an a posteriori error bound a first order rate of convergence independent of all perturbation parameters is established a linear and a nonlinear examples are tested and the numerical results are provided to demonstrate the effectiveness of our adaptive moving grid method

on the analysis of mixed longitudinal growth data mixed longitudinal growth data consists of several observations on a characteristic over a limited age range for each individual in a study this data is then combined to model growth over the total age range of all individuals in the study the limited data collected on each individual precludes a subject specific approach to modelling so that a population based approach must be used here we propose a method for the analysis of mixed longitudinal data using linear models constructed from cubic splines to model both the mean and variance curves of an observed characteristic the method is illustrated on growth in height and head circumference for children in a recently collected mixed longitudinal data set concerning the growth of victorian schoolchildren where particular concern was the timing of growth spurts and increases in variability

stack filters and the mean absolute error criterion a method to determine the stack filter which minimizes the mean absolute error between its output and a desired signal given noisy observations of this desired signal is presented specifically an optimal window width b stack filter can be determined with a linear program with o b2 sup b variables this algorithm is efficient since the number of different inputs to a window width b filter is m sup b if the filter has m valued input and the number of stack filters grows faster than 2 raised to the 2 sup b 2 power it is shown that optimal stack filtering under the mean absolute error criterion is analogous to optimal linear filtering under the mean squared error criterion both linear filters and stack filters are defined by superposition properties both classes are implementable and both have tractable procedures for finding the optimal filter under an appropriate error criterion

statistical analysis for the convergence rate of signals to median filter roots the convergence behavior of the median filter is analyzed a concept called convergence point is used to characterize the convergence rates the results show that any independently and identically distributed i i d signal will become nearly stable after three or four passes of median filter with a large probability simulations are provided to illustrate the theoretical developments

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

statistical tests with accurate size and power for balanced linear mixed models the convenience of linear mixed models for gaussian data has led to their widespread use unfortunately standard mixed model tests often have greatly inflated test size in small samples many applications with correlated outcomes in medical imaging and other fields have simple properties which do not require the generality of a mixed model alternately stating the special cases as a general linear multivariate model allows analysing them with either the univariate or multivariate approach to repeated measures unirep multirep even in small samples an appropriate unirep or multirep test always controls test size and has a good power approximation in sharp contrast to mixed model tests hence mixed model tests should never be used when one of the unirep tests uncorrected huynh feldt geisser greenhouse box conservative or multirep tests wilks hotelling lawley roy s pillai bartlett apply convenient methods give exact power for the uncorrected and box conservative tests simulations demonstrate that new power approximations for all four unirep tests eliminate most inaccuracy in existing methods in turn free software implements the approximations to give a better choice of sample size two repeated measures power analyses illustrate the methods the examples highlight the advantages of examining the entire response surface of power as a function of sample size mean differences and variability

modeling nonstationary longitudinal data an important theme of longitudinal data analysis in the past two decades has been the development and use of explicit parametric models for the data s variance covariance structure a variety of these models have been proposed of which most are second order stationary a few are flexible enough to accommodate nonstationarity i e nonconstant variances and or correlations that are not a function solely of elapsed time between measurements we review five nonstationary models that we regard as most useful 1 the unstructured covariance model 2 unstructured antedependence models 3 structured antedependence models 4 autoregressive integrated moving average and similar models and 5 random coefficients models we evaluate the relative strengths and limitations of each model emphasizing when it is inappropriate or unlikely to be useful we present three examples to illustrate the fitting and comparison of the models and to demonstrate that nonstationary longitudinal data can be modeled effectively and in some cases quite parsimoniously in these examples the antedependence models generally prove to be superior and the random coefficients models prove to be inferior we conclude that antedependence models should be given much greater consideration than they have historically received

sample size calculations for studies with correlated ordinal outcomes correlated ordinal response data often arise in public health studies sample size power calculations are a crucial step in designing such studies to ensure an adequate sample to detect a significant effect here we extend rochon s method of sample size estimation with a repeated binary response to the ordinal case the proposed sample size calculations are based on an analysis with generalized estimating equations gee and inference with the wald test simulation results demonstrate the merit of the proposed power calculations analysis of an arthritis clinical trial is used for illustration

influence diagnostics for linear longitudinal models abstract influence diagnostics are important for analyzing cross sectional regression studies because they allow the analyst to understand the impact of individual observations on the estimated regression model in this article we consider the role of influence diagnostics in subject specific longitudinal models diagnostics are proposed under both fixed and random subject effects our approach is based on subject deletion which in this setting involves deleting a group of correlated observations we develop partial influence statistics to understand the combined impact of observations from a subject on population parameters simple computational formulas make the procedures feasible finally we illustrate the use of our new influence statistics by examining a dataset to model a taxpayer s charitable givings

local estimation of age dependent variance components from longitudinal twin data summary in the study of longitudinal twin and family data interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function various parametric models for covariance structures have been proposed but e g in studies of children where growth spurts occur at various ages it is difficult to a prioridetermine an appropriate parametric model for the covariance structure in particular there is a general lack of the visualization procedures such as lowess that are invaluable in the initial stages of constructing a parametric model for a mean function here we use kernel smoothing to modify a cross sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data the methods are proposed t o be exploratory as an aid to parametric modeling rather than inferential although approximate asymptotic standard errors are derived in the appendix

minimum mean absolute error stack filtering with structural constraint and goals a theory for the structural behavior of stack filters is developed this theory provides a test which can determine if a given stack filter has any root signals a method for classifying the root signal behavior of any stack filter found to have roots and perhaps most important a method for designing stack filters with specific root signals or other structural behavior this theory of root signals for stack filters is then combined with the theory of minimum mean absolute error stack filtering this unified theory allows the designer to pick a filter which minimizes noise subject to constraints on its structural behavior