small sample gee estimation of regression parameters for longitudinal data longitudinal clustered response data arise in many bio statistical applications which in general cannot be assumed to be independent generalized estimating equation gee is a widely used method to estimate marginal regression parameters for correlated responses the advantage of the gee is that the estimates of the regression parameters are asymptotically unbiased even if the correlation structure is misspecified although their small sample properties are not known in this paper two bias adjusted gee estimators of the regression parameters in longitudinal data are obtained when the number of subjects is small one is based on a bias correction and the other is based on a bias reduction simulations show that the performances of both the bias corrected methods are similar in terms of bias efficiency coverage probability average coverage length impact of misspecification of correlation structure and impact of cluster size on bias correction both these methods show superior properties over the gee estimates for small samples further analysis of data involving a small number of subjects also shows improvement in bias mse standard error and length of the confidence interval of the estimates by the two bias adjusted methods over the gee estimates for small to moderate sample sizes n 50 either of the bias corrected methods geebc and geebr can be used however the method geebc should be preferred over geebr as the former is computationally easier for large sample sizes the gee method can be used copyright 2014 john wiley sons ltd

continuous pole placement for delay equations in this paper we describe a stabilization method for linear time delay systems which extends the classical pole placement method for ordinary differential equations unlike methods based on finite spectrum assignment our method does not render the closed loop system finite dimensional but consists of controlling the rightmost eigenvalues because these are moved to the left half plane in a quasi continuous way we refer to our method as continuous pole placement we explain the method by means of the stabilization of a linear finite dimensional system in the presence of an input delay and illustrate its applicability to more general stabilization problems

on the analysis of very small samples of gaussian repeated measurements an alternative approach the analysis of very small samples of gaussian repeated measurements can be challenging first due to a very small number of independent subjects contributing outcomes over time statistical power can be quite small second nuisance covariance parameters must be appropriately accounted for in the analysis in order to maintain the nominal test size however available statistical strategies that ensure valid statistical inference may lack power whereas more powerful methods may have the potential for inflated test sizes therefore we explore an alternative approach to the analysis of very small samples of gaussian repeated measurements with the goal of maintaining valid inference while also improving statistical power relative to other valid methods this approach uses generalized estimating equations with a bias corrected empirical covariance matrix that accounts for all small sample aspects of nuisance correlation parameter estimation in order to maintain valid inference furthermore the approach utilizes correlation selection strategies with the goal of choosing the working structure that will result in the greatest power in our study we show that when accurate modeling of the nuisance correlation structure impacts the efficiency of regression parameter estimation this method can improve power relative to existing methods that yield valid inference copyright 2017 john wiley sons ltd

finite sample adjustments in estimating equations and covariance estimators for intracluster correlations bias corrected covariance estimators are introduced in the context of an estimating equations approach for intracluster correlations among binary outcomes simulation study results show that the bias corrected covariance estimators perform better than uncorrected sandwich estimators in terms of bias and coverage probabilities additionally introduction of a matrix based bias correction into the estimating equations considerably improves point and interval estimation for the intracluster correlations the methods are illustrated using data from a nested cross sectional cluster trial on reducing underage drinking

working correlation structure misspecification estimation and covariate design implications for generalised estimating equations performance the method of generalised estimating equations for regression modelling of clustered outcomes allows for specification of a working matrix that is intended to approximate the true correlation matrix of the observations we investigate the asymptotic relative efficiency of the generalised estimating equation for the mean parameters when the correlation parameters are estimated by various methods the asymptotic relative efficiency depends on three features of the analysis namely i the discrepancy between the working correlation structure and the unobservable true correlation structure ii the method by which the correlation parameters are estimated and iii the design by which we refer to both the structures of the predictor matrices within clusters and distribution of cluster sizes analytical and numerical studies of realistic data analysis scenarios show that choice of working covariance model has a substantial impact on regression estimator efficiency protection against avoidable loss of efficiency associated with covariance misspecification is obtained when a gaussian estimation pseudolikelihood procedure is used with an ar 1 structure

modelling of covariance structures in generalised estimating equations for longitudinal data summary when used for modelling longitudinal data generalised estimating equations specify a working structure for the within subject covariance matrices aiming to produce efficient parameter estimators however misspecification of the working covariance structure may lead to a large loss of efficiency of the estimators of the mean parameters in this paper we propose an approach for joint modelling of the mean and covariance structures of longitudinal data within the framework of generalised estimating equations the resulting estimators for the mean and covariance parameters are shown to be consistent and asymptotically normally distributed real data analysis and simulation studies show that the proposed approach yields efficient estimators for both the mean and covariance parameters

improving power in small sample longitudinal studies when using generalized estimating equations generalized estimating equations gee are often used for the marginal analysis of longitudinal data although much work has been performed to improve the validity of gee for the analysis of data arising from small sample studies little attention has been given to power in such settings therefore we propose a valid gee approach to improve power in small sample longitudinal study settings in which the temporal spacing of outcomes is the same for each subject specifically we use a modified empirical sandwich covariance matrix estimator within correlation structure selection criteria and test statistics use of this estimator can improve the accuracy of selection criteria and increase the degrees of freedom to be used for inference the resulting impacts on power are demonstrated via a simulation study and application example copyright 2016 john wiley sons ltd

stability test and stability conditions for delay differential systems in this paper a stability test method is demonstrated and explicit stability conditions for the first order retarded delay differential system with constant coefficients in the delay interval are obtained this paper shows that under suitable conditions a class of first order delay systems can be alternately stable as the delay h

delay independent stability of a special sequence of neutral difference differential equations with one delay abstract the equations of the title for the state x t r l x d a d a n x t b d b d b n x t t 0 d d dt 1 are considered in the parameter space a b b n p where p is defined by re a a is a second real point a a for a r a is the complex conjugate to a c the same notation applies for b by a tedious but elementary analysis the set s0 p for which 1 has for all t 0 only asymptotically stable solutions in the sense of liapunov is explicitly determined in the form b n a b 1 2 the stability boundary decays exponentially with respect to the multiplicity n of the zeros of the coefficient polynomials of 1 we generalize a theorem of f brauer j differential equations 69 1987 185 191 dealing with delay independent stability for characteristic equations of the form h z a z b z exp zt t 0 3 where a z b z are holomorphic in re z 0 in such a way that the neutral case can be dealt with

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

multivariate continuation ratio models connections and caveats summary we develop semiparametric estimation methods for a pair of regressions that characterize the first and second moments of clustered discrete survival times in the first regression we represent discrete survival times through univariate continuation indicators whose expectations are modeled using a generalized linear model in the second regression we model the marginal pairwise association of survival times using the clayton oakes cross product ratio clayton 1978 biometrika65 141 151 oakes 1989 journal of the american statistical association84 487 493 these models have recently been proposed by shih 1998 biometrics54 1115 1128 we relate the discrete survival models to multivariate multinomial models presented in heagerty and zeger 1996 journal of the american statistical society91 1024 1036 and derive a paired estimating equations procedure that is computationally feasible for moderate and large clusters we extend the work of guo and lin 1994 biometrics50 632 639 and shih 1998 to allow covariance weighted estimating equations and investigate the impact of weighting in terms of asymptotic relative efficiency we demonstrate that the multinomial structure must be acknowledged when adopting weighted estimating equations and show that a naive use of gee methods can lead to inconsistent parameter estimates finally we illustrate the proposed methodology by analyzing psychological testing data previously summarized by tenhave and uttal 1994 applied statistics43 371 384 and guo and lin 1994

regression analysis of overdispersed correlated count data with subject specific covariates a robust likelihood approach for the analysis of overdispersed correlated count data that takes into account cluster varying covariates is proposed we emphasise two characteristics of the proposed method that the correlation structure satisfies the constraints on the second moments and that the estimation of the correlation structure guarantees consistent estimates of the regression coefficients in addition we extend the mean specification to include within and between cluster effects the method is illustrated through the analysis of data from two studies in the first study cross sectional count data from a randomised controlled trial are analysed to evaluate the efficacy of a communication skills training programme the second study involves longitudinal count data which represent counts of damaged hand joints in patients with psoriatic arthritis motivated by this study we generalize our model to accommodate for a subpopulation of patients who are not susceptible to the development of damaged hand joints

weighted rank regression for clustered data analysis we consider ranked based regression models for clustered data analysis a weighted wilcoxon rank method is proposed to take account of within cluster correlations and varying cluster sizes the asymptotic normality of the resulting estimators is established a method to estimate covariance of the estimators is also given which can bypass estimation of the density function simulation studies are carried out to compare different estimators for a number of scenarios on the correlation structure presence absence of outliers and different correlation values the proposed methods appear to perform well in particular the one incorporating the correlation in the weighting achieves the highest efficiency and robustness against misspecification of correlation structure and outliers a real example is provided for illustration

working correlation structure identification in generalized estimating equations selecting an appropriate working correlation structure is pertinent to clustered data analysis using generalized estimating equations gee because an inappropriate choice will lead to inefficient parameter estimation we investigate the well known criterion of qic for selecting a working correlation structure and have found that performance of the qic is deteriorated by a term that is theoretically independent of the correlation structures but has to be estimated with an error this leads us to propose a correlation information criterion cic that substantially improves the qic performance extensive simulation studies indicate that the cic has remarkable improvement in selecting the correct correlation structures we also illustrate our findings using a data set from the madras longitudinal schizophrenia study

modeling instability in the control system for human respiration applications to infant non rem sleep mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system the phenomena collectively referred to as periodic breathing including cheyne stokes respiration and apneustic breathing have important medical implications the hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of f s grodins j gray a i norins r w jones j appl physiol 1 1954 283 308 in the early 1950s the purpose of this paper is to extend the model presented by m c k khoo r e kronauer k p strohl a s slutsky j appl physiol 53 3 1982 644 659 in 1991 to include variable delay in the feedback control loop and to study the phenomena of periodic breathing and apnea as they occur during quiet sleep in infant sleep respiration at around 4 months of age the nonlinear mathematical model consists of a feedback control system of five delay differential equations numerical simulations are performed to study instabilities in the control system and the occurrence of periodic breathing and apnea in the above case which is a time frame of high incidence of sudden infant death syndrome sids

meta analytical synthesis of regression coefficients under different categorization scheme of continuous covariates recently the number of clinical prediction models sharing the same regression task has increased in the medical literature however evidence synthesis methodologies that use the results of these regression models have not been sufficiently studied particularly in meta analysis settings where only regression coefficients are available one of the difficulties lies in the differences between the categorization schemes of continuous covariates across different studies in general categorization methods using cutoff values are study specific across available models even if they focus on the same covariates of interest differences in the categorization of covariates could lead to serious bias in the estimated regression coefficients and thus in subsequent syntheses to tackle this issue we developed synthesis methods for linear regression models with different categorization schemes of covariates a 2 step approach to aggregate the regression coefficient estimates is proposed the first step is to estimate the joint distribution of covariates by introducing a latent sampling distribution which uses one set of individual participant data to estimate the marginal distribution of covariates with categorization the second step is to use a nonlinear mixed effects model with correction terms for the bias due to categorization to estimate the overall regression coefficients especially in terms of precision numerical simulations show that our approach outperforms conventional methods which only use studies with common covariates or ignore the differences between categorization schemes the method developed in this study is also applied to a series of who epidemiologic studies on white blood cell counts

smooth bootstrap methods for analysis of longitudinal data in analysis of longitudinal data the variance matrix of the parameter estimates is usually estimated by the sandwich method in which the variance for each subject is estimated by its residual products we propose smooth bootstrap methods by perturbing the estimating functions to obtain bootstrapped realizations of the parameter estimates for statistical inference our extensive simulation studies indicate that the variance estimators by our proposed methods can not only correct the bias of the sandwich estimator but also improve the confidence interval coverage we applied the proposed method to a data set from a clinical trial of antibiotics for leprosy

a caveat concerning independence estimating equations with multivariate binary data clustered binary data occur commonly in both the biomedical and health sciences in this paper we consider logistic regression models for multivariate binary responses where the association between the responses is largely regarded as a nuisance characteristic of the data in particular we consider the estimator based on independence estimating equations iee which assumes that the responses are independent this estimator has been shown to be nearly efficient when compared with maximum likelihood ml and generalized estimating equations gee in a variety of settings the purpose of this paper is to highlight a circumstance where assuming independence can lead to quite substantial losses of efficiency in particular when the covariate design includes within cluster covariates assuming independence can lead to a considerable loss of efficiency in estimating the regression parameters associated with those covariates

generalized estimating equations to estimate the ordered stereotype logit model for panel data by modeling the effects of predictor variables as a multiplicative function of regression parameters being invariant over categories and category specific scalar effects the ordered stereotype logit model is a flexible regression model for ordinal response variables in this article we propose a generalized estimating equations gee approach to estimate the ordered stereotype logit model for panel data based on working covariance matrices which are not required to be correctly specified a simulation study compares the performance of gee estimators based on various working correlation matrices and working covariance matrices using local odds ratios estimation of the model is illustrated using a real world dataset the results from the simulation study suggest that gee estimation of this model is feasible in medium sized and large samples and that estimators based on local odds ratios as realized in this study tend to be less efficient compared with estimators based on a working correlation matrix for low true correlations the efficiency gains seem to be rather small and if the working covariance structure is too flexible the corresponding estimator may even be less efficient compared with the gee estimator assuming independence like for gee estimators more generally if the true correlations over time are high then a working covariance structure which is close to the true structure can lead to considerable efficiency gains compared with assuming independence

power spectra and dynamical invariants for delay differential and difference equations abstract the mackey glass and ikeda delay differential equations ddes are models of feedback systems in which attractor dimension is proportional to the delay power spectra from these models are investigated analytically and numerically as the delay increases much beyond the system response time an exponentially decaying spectrum with a near periodic superimposed modulation appears it is shown that each peak in this modulation is associated with a mode of oscillation predicted by linear stability analysis around the fixed point the number of such modes within a characteristic autocorrelation time of the solution agrees with the lyapunov dimension of the attractor the disappearance of this modulation at higher delay values is also explained the decay rate of the spectrum in the mid to high frequency range is found to agree with the sum of the positive lyapunov exponents i e with the metric entropy realistic models with a distribution of delays display similar spectra as their memory kernel narrows into a delta function for very large delay the dde approaches an infinite dimensional continuous time difference equation ctde with each point on a delay interval evolving independently and according to a chaotic discrete time map ctde spectra which have many features of finite delay dde spectra can be computed analytically if the spectrum of this map is known for constant initial functions they have a 1 f 2 form at low frequencies our analysis suggests that the spectrum of the chaotic map is at the origin of the peak shapes in dde spectra at low frequency our findings highlight the relevance of the linear prrperties of ddes to the characterization of their nonlinear properties

information ratio test for model misspecification in quasi likelihood inference in this article we focus on the circumstances in quasi likelihood inference that the estimation accuracy of mean structure parameters is guaranteed by correct specification of the first moment but the estimation efficiency could be diminished due to misspecification of the second moment we propose an information ratio ir statistic to test for model misspecification of the variance covariance structure through a comparison between two forms of information matrix the negative sensitivity matrix and the variability matrix we establish asymptotic distributions of the proposed ir test statistics we also suggest an approximation to the asymptotic distribution of the ir statistic via a perturbation resampling method moreover we propose a selection criterion based on the ir test to select the best fitting variance covariance structure from a class of candidates through simulation studies it is shown that the ir statistic provides a powerful statistical tool to detect different scenarios of misspecific

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

multivariate mean parameter estimation by using a partly exponential model summary a class of partly exponential models is proposed for the regression analysis of multivariate response data the class is parameterized in terms of the response mean and a general shape parameter it includes the generalized linear error model and exponential dispersion models as special cases maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations and maximum likelihood estimates of mean and shape parameters are asymptotically independent some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix

model diagnostic tests for selecting informative correlation structure in correlated data in the generalized method of moments approach to longitudinal data analysis unbiased estimating functions can be constructed to incorporate both the marginal mean and the correlation structure of the data increasing the number of parameters in the correlation structure corresponds to increasing the number of estimating functions thus building a correlation model is equivalent to selecting estimating functions this paper proposes a chi squared test to choose informative unbiased estimating functions we show that this methodology is useful for identifying which source of correlation it is important to incorporate when there are multiple possible sources of correlation this method can also be applied to determine the optimal working correlation for the generalized estimating equation approach

stability of a competition model with two stage structure a population model with stage structure which describes the competition of two species for space is considered the model is established under the assumptions that reproductive rates of two species are affected by competition and that the space occupied by one type of juveniles cannot be neglected estimates on the length of delays for which positive equilibrium of the model remains stable are given sufficient conditions are also derived that ensure global stability of positive equilibrium

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

a note on implementation of decaying product correlation structures for quasi least squares this note implements an unstructured decaying product matrix via the quasi least squares approach for estimation of the correlation parameters in the framework of generalized estimating equations the structure we consider is fairly general without requiring the large number of parameters that are involved in a fully unstructured matrix it is straightforward to show that the quasi least squares estimators of the correlation parameters yield feasible values for the unstructured decaying product structure furthermore subject to conditions that are easily checked the quasi least squares estimators are valid for longitudinal bernoulli data we demonstrate implementation of the structure in a longitudinal clinical trial with both a continuous and binary outcome variable

stability conditions for a class of delay differential systems in this paper necessary and sufficient conditions of asymptotic stability independent of delay of a certain class of both the retarded and neutral types of delay differential systems are obtained the stability conditions reduce to positivity of a quartic equation conditions for the latter in explicit form are given by the authors in an earlier publication in addition sufficient conditions for stability for a general order delay differential system independent of delay are also obtained such conditions are ascertained from the fact that a sufficient condition for any algebraic equation to have no positive real roots is that all its coefficients be positive

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

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

miscellanea on the efficiency of regression estimators in generalised linear models for longitudinal data summary liang zeger 1986 introduced a generalised estimating equations approach based on a working correlation matrix to obtain consistent and efficient estimators of regression parameters in the class of generalised linear models for repeated measures data as demonstrated by crowder 1995 because of the uncertainty of definition of the working correlation matrix the liang zeger approach may in some cases lead to a complete breakdown of the estimation of the regression parameters in this paper we show that even though the liang zeger approach in many situations yields consistent estimators for the regression parameters these estimators are usually inefficient as compared to the regression estimators obtained by using the independence estimating equations approach

estimation of growth parameters from multiple recapture data this article develops a method for analysis of growth data with multiple recaptures when the initial ages for all individuals are unknown the existing approaches either impute the initial ages or model them as random effects assumptions about the initial age are not verifiable because all the initial ages are unknown we present an alternative approach that treats all the lengths including the length at first capture as correlated repeated measures for each individual optimal estimating equations are developed using the generalized estimating equations approach that only requires the first two moment assumptions explicit expressions for estimation of both mean growth parameters and variance components are given to minimize the computational complexity simulation studies indicate that the proposed method works well two real data sets are analyzed for illustration one from whelks dicathais aegaota and the other from southern rock lobster jasus edwardsii in south australia

intra cluster correlation selection for cluster randomized trials in this paper we give focus to cluster randomized trials also known as group randomized trials which randomize clusters or groups of subjects to different trial arms such as intervention or control outcomes from subjects within the same cluster tend to exhibit an exchangeable correlation measured by the intra cluster correlation coefficient icc our primary interest is to test if the intervention has an impact on the marginal mean of an outcome using recently developed methods we propose how to select a working icc structure with the goal of choosing the structure that results in the smallest standard errors for regression parameter estimates and thus the greatest power for this test specifically we utilize small sample corrections for the estimation of the covariance matrix of regression parameter estimates this matrix is incorporated within correlation selection criteria proposed in the generalized estimating equations literature to choose one of multiple working icc structures under consideration we demonstrate the potential power and utility of this approach when used in cluster randomized trial settings via a simulation study and application example and we discuss practical considerations for its use in practice copyright 2016 john wiley sons ltd

analysis of repeated bouts of measurements in the framework of generalized estimating equations we consider a longitudinal study of interstitial cystitis ic in women in which the time between bouts of repeated measurements is large relative to the within bout separation in time our outcome of interest is the number of nocturnal voids that we model via quasi least squares qls in the framework of generalized estimating equations gee to account for potential intra subject correlation we directly apply a banded toeplitz correlation structure that previously was only implemented in an ad hoc approach using gee we describe this structure its appropriateness for data from the ic study and the results of our analysis we then demonstrate that correct specification of the underlying correlation structure versus incorrectly applying a simpler structure can prevent substantial losses in efficiency in estimation of the regression parameter these comparisons involve the limiting values of the estimates of the correlation parameters which are not consistent for the misspecification scenarios considered here we therefore obtain the limiting values of the qls estimates when the structure is incorrectly specified

stability conditions for the general linear difference differential equation with constant coefficients and one constant delay abstract for the application low degree cases of the equations of the title are of special interest lag independent stability conditions are derived for all systems up to degree two inclusive some necessary stability conditions for large lags are expounded in the non neutral case left half planes depending on the lag and containing all zeros of the characteristic function are obtained a sufficient lag independent condition for the whole class considered is found special attention is devoted to the neutral case for that case a collection of disks covering all zeros of the characteristic function is given the result is illustrated by the example of the general neutral characteristic function with coalescing zeros finally zero free regions around the zeros of the coefficient polynomials are derived to derive the results conformal mappings are studied

stability analysis and stability conditions in the delay intervals for second order delay systems abstract in tills paper the stability and the stability conditions of a number of second order retarded delay linear differential systems are discussed the paper shows especially that the system which has stationary oscillation when without delay can become asymptotically stable in the large in certain suitable delay intervals

improved methods for the marginal analysis of longitudinal data in the presence of time dependent covariates generalized estimating equations gees are commonly used for the marginal analysis of longitudinal data in order to obtain consistent regression parameter estimates these estimating equations must be unbiased however in the presence of certain types of time dependent covariates these equations can be biased unless they incorporate the independence working correlation structure moreover in this case regression parameter estimation can be very inefficient because not all valid moment conditions are incorporated within the corresponding estimating equations therefore approaches based on the generalized method of moments or quadratic inference functions have been proposed in order to utilize all valid moment conditions however we have found in previous studies as well as the current study that such methods will not always provide valid inference and can also be improved upon in terms of finite sample regression parameter estimation therefore we propose both a modified gee approach and a method selection strategy in order to ensure valid inference with the goal of improving regression parameter estimation in a simulation study and application example we compare existing and proposed methods and demonstrate that our modified gee approach performs well and the correlation information criterion has good accuracy with respect to selecting the best approach in terms of regression parameter estimation copyright 2017 john wiley sons ltd

using second order generalized estimating equations to model heterogeneous intraclass correlation in cluster randomized trials in cluster randomized trials it is commonly assumed that the magnitude of the correlation among subjects within a cluster is constant across clusters however the correlation may in fact be heterogeneous and depend on cluster characteristics accurate modeling of the correlation has the potential to improve inference we use second order generalized estimating equations to model heterogeneous correlation in cluster randomized trials using simulation studies we show that accurate modeling of heterogeneous correlation can improve inference when the correlation is high or varies by cluster size we apply the methods to a cluster randomized trial of an intervention to promote breast cancer screening

stability in a scalar differential equation with multiple distributed time delays abstract we consider a linear scalar delay differential equation dde consisting of two arbitrary distributed time delays we formulate necessary conditions for stability of the trivial solution which are independent of the distributions for the case of one discrete and one gamma distributed delay we give an explicit description of the region of stability of the trivial solution and discuss how this depends on the model parameters

hopf bifurcation in differential equations with delay for tumor immune system competition model this paper deals with the qualitative analysis of the solutions to a model that refers to the competition between the immune system and an aggressive host such as a tumor the model which describes this competition is governed by a system of differential equations with one delay it is shown that the dynamics depends crucially on the time delay parameter by using the time delay as a parameter of bifurcation the analysis is focused on the hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state the obtained results depict the oscillations given by simulations see m galach int j appl math comput sci 13 2003 pp 395 406 which are observed in reality see d kirschner and j c panetta j math biol 37 1998 pp 235 252 it is suggested to examine by laboratory experiments how to employ these results for control of tumor growth

stability with respect to the delay on a paper of k l cooke and p van den driessche a proposal for the correction of the main result of the title s paper is made some accompanying comments on other points are given so a simpler central result is derived an explicit representation of the set of stable delays is achieved

effects of variance function misspecification in analysis of longitudinal data the approach of generalized estimating equations gee is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within subject correlations the variance is often assumed to be a known function of the mean this article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses our numerical studies indicate that 1 correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified 2 misspecification of the variance function impacts much more on estimators for within cluster covariates than for cluster level covariates and 3 if the variance function is misspecified correct choice of the correlation structure may not necessarily improve estimation efficiency we illustrate impacts of different variance functions using a real data set from cow growth

robust estimating functions and bias correction for longitudinal data analysis robust methods are useful in making reliable statistical inferences when there are small deviations from the model assumptions the widely used method of the generalized estimating equations can be robustified by replacing the standardized residuals with the m residuals if the pearson residuals are assumed to be unbiased from zero parameter estimators from the robust approach are asymptotically biased when error distributions are not symmetric we propose a distribution free method for correcting this bias our extensive numerical studies show that the proposed method can reduce the bias substantially examples are given for illustration

efficiency of regression estimates for clustered data statistical methods for clustered data such as generalized estimating equations gee and generalized least squares gls require selecting a correlation or convariance structure to specify the dependence between observations within a cluster valid regression estimates can be obtained that do not depend on correct specification of the true correlation but inappropriate specifications can result in a loss of efficiency we derive general expressions for the asymptotic relative efficiency of gee and gls estimators under nested correlation structures efficiency is shown to depend on the covariate distribution the cluster sizes the response variable correlation and the regression parameters the results demonstrate that efficiency is quite sensitive to the between and within cluster variation of the covariates and provide useful characterizations of models for which upper and lower efficiency bounds are attained efficiency losses for simple working correlation matrices such as independence can be large even for small to moderate correlations and cluster sizes

correlation structure and variable selection in generalized estimating equations via composite likelihood information criteria the method of generalized estimating equations gee is popular in the biostatistics literature for analyzing longitudinal binary and count data it assumes a generalized linear model for the outcome variable and a working correlation among repeated measurements in this paper we introduce a viable competitor the weighted scores method for generalized linear model margins we weight the univariate score equations using a working discretized multivariate normal model that is a proper multivariate model because the weighted scores method is a parametric method based on likelihood we propose composite likelihood information criteria as an intermediate step for model selection the same criteria can be used for both correlation structure and variable selection simulations studies and the application example show that our method outperforms other existing model selection methods in gee from the example it can be seen that our methods not only improve on gee in terms of interpretability and efficiency but also can change the inferential conclusions with respect to gee copyright 2016 john wiley sons ltd

absolute stability in delay equations abstract an equilibrium of a delay equation is said to be absolutely stable if it is asymptotically stable for all delays this is equivalent to a certain characteristic equation s having all its roots in the left half plane for all delays we obtain a simple necessary and sufficient condition for absolute stability of a class of delay equations

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

pseudo almost periodic solutions of some differential equations ii we first present some results about pseudo almost periodic functions then we use these results to show some theorems about existence and uniqueness of the pseudo almost periodic solutions of the linear and the quasi linear nonautonomous equations finally we set up some theorems of the existence of pseudo almost periodic solutions implying the existence of almost periodic solutions of general ordinary differential equation systems

a generalized estimating equations approach for spatially correlated binary data applications to the analysis of neuroimaging data this paper proposes a generalized estimating equations approach for the analysis of spatially correlated binary data when there are large numbers of spatially correlated observations on a moderate number of subjects this approach is useful when the scientific focus is on modeling the marginal mean structure proper modeling of the spatial correlation structure is shown to provide large efficiency gains along with precise standard error estimates for inference on mean structure parameters generalized estimating equations for estimating the parameters of both the mean and spatial correlation structure are proposed the use of semivariogram models for parameterizing the correlation structure is discussed and estimation of the sample semivariogram is proposed as a technique for choosing parametric models and starting values for generalized estimating equations estimation the methodology is illustrated with neuroimaging data collected as part of the national institute of neurological disorders and stroke ninds stroke data bank a simulation study demonstrates the importance of accurate modeling of the spatial correlation structure in data with large numbers of spatially correlated observations such as those found in neuroimaging studies

semiparametric mean covariance regression analysis for longitudinal data efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure existing approaches usually focus on modeling the mean with specification of certain covariance structures which may lead to inefficient or biased estimators of parameters in the mean if misspecification occurs in this article we propose a data driven approach based on semiparametric regression models for the mean and the covariance simultaneously motivated by the modified cholesky decomposition a regression spline based approach using generalized estimating equations is developed to estimate the parameters in the mean and the covariance the resulting estimators for the regression coefficients in both the mean and the covariance are shown to be consistent and asymptotically normally distributed in addition the nonparametric functions in these two structures are estimated at their optimal rate of convergence simulation studies and a real data analysis show

stability and bifurcations for the chronic state in marchuk s model of an immune system in the paper we present known and new results concerning stability and the hopf bifurcation for the positive steady state describing a chronic disease in marchuk s model of an immune system we describe conditions guaranteeing local stability or instability of this state in a general case and for very strong immune system we compare these results with the results known in the literature we show that the positive steady state can be stable only for very specific parameter values stability analysis is illustrated by mikhailov s hodographs and numerical simulations conditions for the hopf bifurcation and stability of arising periodic orbit are also studied these conditions are checked for arbitrary chosen realistic parameter values numerical examples of arising due to the hopf bifurcation periodic solutions are presented

improving the correlation structure selection approach for generalized estimating equations and balanced longitudinal data generalized estimating equations are commonly used to analyze correlated data choosing an appropriate working correlation structure for the data is important as the efficiency of generalized estimating equations depends on how closely this structure approximates the true structure therefore most studies have proposed multiple criteria to select the working correlation structure although some of these criteria have neither been compared nor extensively studied to ease the correlation selection process we propose a criterion that utilizes the trace of the empirical covariance matrix furthermore use of the unstructured working correlation can potentially improve estimation precision and therefore should be considered when data arise from a balanced longitudinal study however most previous studies have not allowed the unstructured working correlation to be selected as it estimates more nuisance correlation parameters than other structures such as ar 1 or exchangeable therefore we propose appropriate penalties for the selection criteria that can be imposed upon the unstructured working correlation via simulation in multiple scenarios and in application to a longitudinal study we show that the trace of the empirical covariance matrix works very well relative to existing criteria we further show that allowing criteria to select the unstructured working correlation when utilizing the penalties can substantially improve parameter estimation copyright 2014 john wiley sons ltd

on the use of a working correlation matrix in using generalised linear models for repeated measures summary in a seminal paper liang zeger 1986 extended the use of generalised linear models to repeated measures data they based the analysis on specifications for the means and variances of the observations as usual for generalised linear models but showed how specifications for the correlations between measurements made on the same unit could be avoided by using a working correlation matrix in some cases the parameters involved in this matrix are subject to an uncertainty of definition which can lead to a breakdown of the asymptotic properties of the estimators

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

delayed stochastic resonance with a 1 d ring abstract we study here a simple one dimensional ring composed of stochastic binary elements with delayed interaction with this model we can analytically compute interspike interval histograms and show how resonance or rhythmic behaviors arise with only noise and delay

consistent model selection and data driven smooth tests for longitudinal data in the estimating equations approach summary model selection for marginal regression analysis of longitudinal data is challenging owing to the presence of correlation and the difficulty of specifying the full likelihood particularly for correlated categorical data the paper introduces a novel bayesian information criterion type model selection procedure based on the quadratic inference function which does not require the full likelihood or quasi likelihood with probability approaching 1 the criterion selects the most parsimonious correct model although a working correlation matrix is assumed there is no need to estimate the nuisance parameters in the working correlation matrix moreover the model selection procedure is robust against the misspecification of the working correlation matrix the criterion proposed can also be used to construct a data driven neyman smooth test for checking the goodness of fit of a postulated model this test is especially useful and often yields much higher power in situations where the classical directional test behaves poorly the finite sample performance of the model selection and model checking procedures is demonstrated through monte carlo studies and analysis of a clinical trial data set

gee for multinomial responses using a local odds ratios parameterization in this article we propose a generalized estimating equations gee approach for correlated ordinal or nominal multinomial responses using a local odds ratios parameterization our motivation lies upon observing that i modeling the dependence between correlated multinomial responses via the local odds ratios is meaningful both for ordinal and nominal response scales and ii ordinary gee methods might not ensure the joint existence of the estimates of the marginal regression parameters and of the dependence structure to avoid ii we treat the so called working association vector as a nuisance parameter vector that defines the local odds ratios structure at the marginalized contingency tables after tabulating the responses without a covariate adjustment at each time pair to estimate and simultaneously approximate adequately possible underlying dependence structures we employ the family of association models proposed by goodman in simulations the parameter estimators with the proposed gee method for a marginal cumulative probit model appear to be less biased and more efficient than those with the independence working model especially for studies having time varying covariates and strong correlation

the stability chart for the linearized cushing equation with a discrete delay and with gamma distributed delays in a proper setting the problem of the title amounts to the determination of all pairs d t for which the function z z 1 exp zt d and the functions fn z z 1 1 ztn n d n n d t 0 have all their zeros in re z 0 of the complex z plane correct versions of results and proofs found in papers of k l cooke and j m grossman j math anal appl 86 1982 592 627 and s p blythe r m nisbeth w s c gurney and n macdonald j math anal appl 109 1985 388 396 are given the argumentation in the proofs of the present paper differs from that of the studies mentioned in contrast to blythe et al the stability behaviours of both models are seen as similar stability charts and stability conditions for some other classes of systems are also given in addition to the gamma distribution for the lags other distributions which are similar in shape but differ greatly in their stability behaviour are also considered

analyzing spatially distributed binary data using independent block estimating equations summary we estimate the relation between binary responses and corresponding covariate vectors both observed over a large spatial lattice we assume a hierarchical generalized linear model with probit link function partition the lattice into blocks and adopt the working assumption of independence between the blocks to obtain an easily solved estimating equation standard errors are obtained using the sandwich estimator together with window subsampling sherman 1996 journal of the royal statistical society series b 58 509 523 we apply this to a large data set describing long term vegetation growth together with two other approximate likelihood approaches pairwise composite likelihood cl and estimation under a working assumption of independence the independence and cl methods give similar point estimates and standard errors while the independent block approach gives considerably smaller standard errors as well as more easily interpretable point estimates we present numerical evidence suggesting this increased efficiency may hold more generally

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

efficient estimation for rank based regression with clustered data rank based inference is widely used because of its robustness this article provides optimal rank based estimating functions in analysis of clustered data with random cluster effects the extensive simulation studies carried out to evaluate the performance of the proposed method demonstrate that it is robust to outliers and is highly efficient given the existence of strong cluster correlations the performance of the proposed method is satisfactory even when the correlation structure is misspecified or when heteroscedasticity in variance is present finally a real dataset is analyzed for illustration

a profile likelihood approach for longitudinal data analysis summary inappropriate choice of working correlation structure in generalized estimating equations gee could lead to inefficient parameter estimation while impractical normality assumption in likelihood approach would limit its applicability in longitudinal data analysis in this article we propose a profile likelihood method for estimating parameters in longitudinal data analysis via maximizing the estimated likelihood the proposed method yields consistent and efficient estimates without specifications of the working correlation structure nor the underlying error distribution both theoretical and simulation results confirm the satisfactory performance of the proposed method we illustrate our methodology with a diastolic blood pressure data set

gaussian estimation for correlated binomial data summary the mean variance structure of correlated binomial data is discussed and whittle s 1961 gaussian estimation is suggested as a useful general method in this context comparisons are made with other monrent methods in current use

a comparison of several approaches for choosing between working correlation structures in generalized estimating equation analysis of longitudinal binary data the method of generalized estimating equations gee models the association between the repeated observations on a subject with a patterned correlation matrix correct specification of the underlying structure is a potentially beneficial goal in terms of improving efficiency and enhancing scientific understanding we consider two sets of criteria that have previously been suggested respectively for selecting an appropriate working correlation structure and for ruling out a particular structure s in the gee analysis of longitudinal studies with binary outcomes the first selection criterion chooses the structure for which the model based and the sandwich based estimator of the covariance matrix of the regression parameter estimator are closest while the second selection criterion chooses the structure that minimizes the weighted error sum of squares the rule out criterion deselects structures for which the estimated correlation parameter violates standard constraints for binary data that depend on the marginal means in addition we remove structures from consideration if their estimated parameter values yield an estimated correlation structure that is not positive definite we investigate the performance of the two sets of criteria using both simulated and real data in the context of a longitudinal trial that compares two treatments for major depressive episode practical recommendations are also given on using these criteria to aid in the efficient selection of a working correlation structure in gee analysis of longitudinal binary data