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

on topological dynamics of turing machines abstract we associate to a turing machine two dynamical systems which we call turing machine with moving tape tmt and turing machine with moving head tmh tmt are equivalent to generalized shifts of moore 1990 and they include two sided full shifts tmh are shift commuting maps of two sided sofic systems in both classes we characterize systems with the shadowing property show that a bijective expansive tmt is conjugate to a subshift of finite type and that topological entropy of every tmh is zero we conjecture that every tmt has a periodic point

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 multivariate binary data with alternating logistic regressions marginal models for multivariate binary data permit separate modelling of the relationship of the response with explanatory variables and the association between pairs of responses when the former is the scientific focus a first order generalized estimating equation method liang zeger 1986 is easy to implement and gives efficient estimates of regression coefficients although estimates of the association among the binary outcomes can be inefficient when the association model is a focus simultaneous modelling of the responses and all pairwise products prentice 1988 using second order estimating equations gives more efficient estimates of association parameters as well however this procedure can become computationally infeasible as the cluster size gets large this paper proposes an alternative approach alternating logistic regressions for simultaneously regressing the response on explanatory variables as well as modelling the association among responses in terms of pairwise odds ratios this algorithm iterates between a logistic regression using first order generalized estimating equations to estimate regression coefficients and a logistic regression of each response on others from the same cluster using an appropriate offset to update the odds ratio parameters for clusters of size n alternating logistic regression involves evaluation and inversion of matrices of order n2 rather than n4 as required for second order generalized estimating equations the alternating logistic regression estimates are shown to be reasonably efficient relative to solutions of second order equations in a few problems the new method is illustrated with an analysis of neuropsychological tests on patients with epileptic seizures

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

bivariate categorical data analysis using normal linear conditional multinomial probability model bivariate multinomial data such as the left and right eyes retinopathy status data are analyzed either by using a joint bivariate probability model or by exploiting certain odds ratio based association models however the joint bivariate probability model yields marginal probabilities which are complicated functions of marginal and association parameters for both variables and the odds ratio based association model treats the odds ratios involved in the joint probabilities as working parameters which are consequently estimated through certain arbitrary working regression models also this later odds ratio based model does not provide any easy interpretations of the correlations between two categorical variables on the basis of pre specified marginal probabilities in this paper we develop a bivariate normal type linear conditional multinomial probability model to understand the correlations between two categorical variables the parameters involved in the model are consistently estimated using the optimal likelihood and generalized quasi likelihood approaches the proposed model and the inferences are illustrated through an intensive simulation study as well as an analysis of the well known wisconsin diabetic retinopathy status data copyright 2014 john wiley sons ltd

working covariance model selection for generalized estimating equations we investigate methods for data based selection of working covariance models in the analysis of correlated data with generalized estimating equations we study two selection criteria gaussian pseudolikelihood and a geodesic distance based on discrepancy between model sensitive and model robust regression parameter covariance estimators the gaussian pseudolikelihood is found in simulation to be reasonably sensitive for several response distributions and noncanonical mean variance relations for longitudinal data application is also made to a clinical dataset assessment of adequacy of both correlation and variance models for longitudinal data should be routine in applications and we describe open source software supporting this practice

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

practical issues in using generalized estimating equations for inference on transitions in longitudinal data what is being estimated generalized estimating equations gees are commonly used to estimate transition models when the markov assumption does not hold but first order transition probabilities are still of interest the transition inference is sensitive to the choice of working correlation in this paper we consider a random process transition model as the true underlying data generating mechanism which characterizes subject heterogeneity and complex dependence structure of the outcome process in a very flexible way we formally define two types of transition probabilities at the population level naive transition probabilities that average across all the transitions and population average transition probabilities that average the subject specific transition probabilities through asymptotic bias calculations and finite sample simulations we demonstrate that the unstructured working correlation provides unbiased estimators of the population average transition probabilities while the independence working correlation provides unbiased estimators of the naive transition probabilities for population average transition estimation we demonstrate that the sandwich estimator fails for unstructured gee and recommend the use of either jackknife or bootstrap variance estimates the proposed method is motivated by and applied to the next generation health study where the interest is in estimating the population average transition probabilities of alcohol use in adolescents

estimated estimating equations semiparametric inference for clustered and longitudinal data summary we introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions this estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi likelihood regression where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link and variance covariance is an unknown smooth function of the marginal means we propose to estimate the nonparametric link and variance covariance functions via smoothing methods whereas the regression parameters are obtained via the estimated estimating equations these are score equations that contain nonparametric function estimates the proposed estimated estimating equations approach is motivated by its flexibility and easy implementation moreover if data follow a generalized linear mixed model with either a specified or an unspecified distribution of random effects and link function the model proposed emerges as the corresponding marginal population average version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model without the need to specify any other components of this generalized linear mixed model among marginal models the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations applications of estimated estimating equations include diagnostics and link selection the asymptotic distribution of the proposed estimators for the model parameters is derived enabling statistical inference practical illustrations include poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses

the foundations of finite sample estimation in stochastic processes generally in the literature on stochastic processes estimation is investigated in terms of asymptotic properties in this paper we establish some finite sample optimal estimation let y1 y2 be a discrete stochastic process since we will be discussing results for samples of finite size n we restrict the process to r in the following the term parameter is used in the same broad sense as by godambe thompson 1984 let y be a class of probability distributions f on r and 0 0 f f e jz be a real parameter let hi be a real function of y 1 yi and 0 be such that

a comparison of bias corrected empirical covariance estimators with generalized estimating equations in small sample longitudinal study settings data arising from longitudinal studies are commonly analyzed with generalized estimating equations previous literature has shown that liberal inference may result from the use of the empirical sandwich covariance matrix estimator when the number of subjects is small therefore two different approaches have been used to improve the validity of inference first many different small sample corrections to the empirical estimator have been offered in order to reduce bias in resulting standard error estimates second critical values can be obtained from a t distribution or an f distribution with approximated degrees of freedom although limited studies on the comparison of these small sample corrections and degrees of freedom have been published there is a need for a comprehensive study of currently existing methods in a wider range of scenarios therefore in this manuscript we conduct such a simulation study finding two methods to attain nominal type i error rates more consistently than other methods in a variety of settings first a recently proposed method by westgate and burchett 2016 statistics in medicine 35 3733 3744 that specifies both a covariance estimator and degrees of freedom and second an average of two popular corrections developed by mancl and derouen 2001 biometrics 57 126 134 and kauermann and carroll 2001 journal of the american statistical association 96 1387 1396 with degrees of freedom equaling the number of subjects minus the number of parameters in the marginal model

assessing robustness of generalised estimating equations and quadratic inference functions in the presence of data contamination or outliers some empirical studies have indicated that the two methods of generalised estimating equations and quadratic inference functions appear to have rather different robustness behaviour this paper presents a theoretical investigation from the perspective of the influence function to identify the causes for the difference we show that quadratic inference functions lead to bounded influence functions and the corresponding m estimator has a redescending property but the generalised estimating equation approach does not we also illustrate that unlike generalised estimating equations quadratic inference functions can still provide consistent estimators even if part of the data is contaminated we conclude that the quadratic inference function is a preferable method to the generalised estimating equation as far as robustness is concerned this conclusion is supported by simulations and real data examples

improving generalised estimating equations using quadratic inference functions summary generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified however under such misspecification the estimator of the regression parameter can be inefficient in this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter and that remains optimal even if the working correlation structure is misspecified the idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices a representation that is valid for the working correlations most commonly used both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations this approach also provides a chi squared inference function for testing nested models and a chi squared regression misspecification test furthermore the test statistic follows a chi squared distribution asymptotically whether or not the working correlation structure is correctly specified

tests of hypotheses in overdispersed poisson regression and other quasi likelihood models abstract test statistics for evaluating the significance of added variables in a regression equation are developed for mixed poisson models where the structural parameter that determines the mean variance relationship var 2 is estimated by the method of moments and the regression coefficients are estimated by quasi likelihood the formulas presented for test statistics and related estimating equations are applicable generally to quasi likelihood models specified by an arbitrary mean value function x together with a variance function v that contains one or more unknown parameters two versions of the wald and score tests are investigated one calculated from the usual model based covariance matrix whose validity depends on correct specification of the variance function and another using an empirical covariance matrix that has a more general asymptotic justification monte carlo simulations demonstrate that the quasi likelihood method of moments ql m procedures yield ap

akaike s information criterion in generalized estimating equations correlated response data are common in biomedical studies regression analysis based on the generalized estimating equations gee is an increasingly important method for such data however there seem to be few model selection criteria available in gee the well known akaike information criterion aic cannot be directly applied since aic is based on maximum likelihood estimation while gee is nonlikelihood based we propose a modification to aic where the likelihood is replaced by the quasi likelihood and a proper adjustment is made for the penalty term its performance is investigated through simulation studies for illustration the method is applied to a real data set

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

inference for clustered data using the independence loglikelihood we use the properties of independence estimating equations to adjust the independence loglikelihood function in the presence of clustering the proposed adjustment relies on the robust sandwich estimator of the parameter covariance matrix which is easily calculated the methodology competes favourably with established techniques based on independence estimating equations we provide some insight as to why this is so the adjustment is applied to examples relating to the modelling of wind speed in europe and annual maximum temperatures in the u k

a comparison of two bias corrected covariance estimators for generalized estimating equations summarymancl and derouen 2001 biometrics57 126 134 and kauermann and carroll 2001 jasa96 1387 1398 proposed alternative bias corrected covariance estimators for generalized estimating equations parameter estimates of regression models for marginal means the finite sample properties of these estimators are compared to those of the uncorrected sandwich estimator that underestimates variances in small samples although the formula of mancl and derouen generally overestimates variances it often leads to coverage of 95 confidence intervals near the nominal level even in some situations with as few as 10 clusters an explanation for these seemingly contradictory results is that the tendency to undercoverage resulting from the substantial variability of sandwich estimators counteracts the impact of overcorrecting the bias however these positive results do not generally hold for small cluster sizes e g 10 their estimator often results in overcoverage and the bias corrected covariance estimator of kauermann and carroll may be preferred the methods are illustrated using data from a nested cross sectional cluster intervention trial on reducing underage drinking

generalized estimating equations for correlated binary data using the odds ratio as a measure of association summary moment methods for analyzing repeated binary responses have been proposed by liang zeger 1986 and extended by prentice 1988 in their generalized estimating equations both liang zeger 1986 and prentice 1988 estimate the parameters associated with the expected value of an individual s vector of binary responses as well as the correlations between pairs of binary responses because the odds ratio has many desirable properties and some investigators may find the odds ratio is easier to interpret we discuss modelling the association between binary responses at pairs of times with the odds ratio we then modify the estimating equations of prentice to estimate the odds ratios in simulations the parameter estimates for the logistic regression model for the marginal probabilities appear slightly more efficient when using the odds ratio parameterization

what can go wrong when ignoring correlation bounds in the use of generalized estimating equations the analysis of repeated measure or clustered data is often complicated by the presence of correlation further complications arise for discrete responses where the marginal probability dependent frechet bounds impose feasibility limits on the correlation that are often more restrictive than the positive definite range some popular statistical methods such as generalized estimating equations gee ignore these bounds and as such can generate erroneous estimates and lead to incorrect inferential results in this paper we discuss two alternative strategies i using qic to select a data driven correlation value within the frechet bounds and ii the use of likelihood based latent variable modeling such as multivariate probit to get around the problem all together we provide two examples of the repercussions of incorrectly using existing gee software in the presence of correlated binary responses

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

a serially correlated gamma frailty model for longitudinal count data a poisson gamma model is introduced to account for between subjects heterogeneity and within subjects serial correlation occurring in longitudinal count data the model extends the usual time constant shared frailty approach to allow time varying serially correlated gamma frailty whilst retaining standard marginal assumptions a composite likelihood approach to estimation and testing for serial correlation is proposed the work is motivated by a clinical trial on patient controlled analgesia where the number of analgesic doses taken by hospital patients in successive time intervals following abdominal surgery is recorded

marginal analysis of incomplete longitudinal binary data a cautionary note on locf imputation in recent years there has been considerable research devoted to the development of methods for the analysis of incomplete data in longitudinal studies despite these advances the methods used in practice have changed relatively little particularly in the reporting of pharmaceutical trials in this setting perhaps the most widely adopted strategy for dealing with incomplete longitudinal data is imputation by the last observation carried forward locf approach in which values for missing responses are imputed using observations from the most recently completed assessment we examine the asymptotic and empirical bias the empirical type i error rate and the empirical coverage probability associated with estimators and tests of treatment effect based on the locf imputation strategy we consider a setting involving longitudinal binary data with longitudinal analyses based on generalized estimating equations and an analysis based simply on the response at the end of the scheduled follow up we find that for both of these approaches imputation by locf can lead to substantial biases in estimators of treatment effects the type i error rates of associated tests can be greatly inflated and the coverage probability can be far from the nominal level alternative analyses based on all available data lead to estimators with comparatively small bias and inverse probability weighted analyses yield consistent estimators subject to correct specification of the missing data process we illustrate the differences between various methods of dealing with drop outs using data from a study of smoking behavior

on the accuracy of efficiency of estimating equation approach summary liang and zeger 1986 biometrika73 13 22 introduced a generalized estimating equation gee approach based on a working correlation matrix to obtain efficient estimators of regression parameters in the class of generalized linear models for repeated measures data as demonstrated by crowder 1995 biometrika82 407 410 because of uncertainty of the 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 after taking this comment of crowder into account recently sutradhar and das 1999 biometrika86 459 465 examined the loss of efficiency of the regression estimators due to misspecification of the correlation structures but their study was confined to the regression estimation with cluster level covariates as in the original paper of liang and zeger in this paper we study this efficiency loss problem for the generalized regression models with within cluster covariates by utilizing the approach of sutsadhar and das 1999

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

goodness of fit tests for gee modeling with binary responses analysis of data with repeated measures is often accomplished through the use of generalized estimating equations gee methodology although methods exist for assessing the adequacy of the fitted models for uncorrelated data with likelihood methods it is not appropriate to use these methods for models fitted with gee methodology we propose model based and robust empirically corrected goodness of fit tests for gee modeling with binary responses based on partitioning the space of covariates into distinct regions and forming score statistics that are asymptotically distributed as chi square random variables with the appropriate degrees of freedom the null distribution and the statistical power of the proposed goodness of fit tests were assessed using simulated data the proposed goodness of fit tests are illustrated by two examples using data from clinical studies

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

efficiency of generalized estimating equations for binary responses summary using standard correlation bounds we show that in generalized estimation equations gees the so called working correlation matrix r for analysing binary data cannot in general be the true correlation matrix of the data methods for estimating the correlation param eter in current gee software for binary responses disregard these bounds to show that the gee applied on binary data has high efficiency we use a multivariate binary model so that the covariance matrix from estimating equation theory can be compared with the inverse fisher information matrix but r should be viewed as the weight matrix and it should not be confused with the correlation matrix of the binary responses we also do a comparison with more general weighted estimating equations by using a matrix cauchy schwarz inequality our analysis leads to simple rules for the choice of in an exchangeable or autoregressive ar 1 weight matrix r based on the strength of dependence between the binary variables an example is given to illustrate the assessment of dependence and choice of

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

multiplicity positive solutions to periodic problems for first order impulsive differential equations this paper is devoted to studying the existence of single and multiple positive solutions to periodic problems for first order impulsive differential equations in addition we apply our existence theorems to nonlinear periodic problems with singularity at the origin our proofs are based on krasnoselskii fixed point theorem in cones

quasi periodic configurations and undecidable dynamics for tilings infinite words and turing machines we describe turing machines tilings and infinite words as dynamical systems and analyze some of their dynamical properties it is known that some of these systems do not always have periodic configurations we prove that they always have quasi periodic configurations and we quantify quasi periodicity we then study the decidability of dynamical properties for these systems in analogy to rice s theorem for computable functions we derive a theorem that characterizes dynamical system properties that are undecidable as an illustration of this result we prove that topological entropy is undecidable for turing machines and for tilings

maintaining the validity of inference in small sample stepped wedge cluster randomized trials with binary outcomes when using generalized estimating equations stepped wedge cluster trials are an increasingly popular alternative to traditional parallel cluster randomized trials such trials often utilize a small number of clusters and numerous time intervals and these components must be considered when choosing an analysis method a generalized linear mixed model containing a random intercept and fixed time and intervention covariates is the most common analysis approach however the sole use of a random intercept applies a constant intraclass correlation coefficient structure which is an assumption that is likely to be violated given stepped wedge trials swts have multiple time intervals alternatively generalized estimating equations gee are robust to the misspecification of the working correlation structure although it has been shown that small sample adjustments to standard error estimates and the use of appropriate degrees of freedom are required to maintain the validity of inference when the number of clusters is small in this article we show using an extensive simulation study based on a motivating example and a more general design the use of gee can maintain the validity of inference in small sample swts with binary outcomes furthermore we show which combinations of bias corrections to standard error estimates and degrees of freedom work best in terms of attaining nominal type i error rates

logistic regression for dependent binary observations summary the likelihood of a set of binary dependent outcomes with or without explanatory variables is expressed as a product of conditional probabilities each of which is assumed to be logistic the models are called regressive logistic models they provide a simple but relatively unknown parametrization of the multivariate distribution they have the theoretical and practical advantage that they can be analyzed and fitted as in logistic regression for independent outcomes and with the same computer programs the paper is largely expository and is intended to motivate the development and usage of the regressive logistic models the discussion includes serially dependent outcomes equally predictive outcomes more specialized patterns of dependence multidimensional tables and three examples this paper seeks to promote the development and usage of a simple but relatively unknown approach to the specification of models for dependent binary outcomes that are useful in situations where it is apt to describe the dependence in terms of conditional probabilities the joint probability is decomposed into a product of successive conditional probabilities each of which is assumed to be univariate logistic since the probability of an observation is conditioned on all preceding observations the models will be referred to as regressive logistic models two broad situations are distinguished in this paper the first concerning sequences of binary outcomes possibly with measures on associated explanatory variables can usually be supported by simple patterns of dependence these will be discussed in sections 2 5 the second broad situation concerns dependent binary outcomes crossclassified into multidimensional tables that admit a fuller parametrization this is discussed in section 6 for all the regressive logistic models discussed here the parametrization is such that jointly dependent binary observations with or without explanatory variables can be fitted with existing computer programs for the standard logistic regression for independent observations some of the possibilities of the approach are illustrated with real data in section 8

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

analysing longitudinal count data with overdispersion summary in many biomedical studies longitudinal count data comprise repeated responses and a set of multidimensional covariates for a large number of individuals when the response variable in such models is subject to overdispersion the overdispersion parameter influences the marginal variance in such cases the overdispersion parameter plays a significant role in efficient estimation of the regression parameters this raises the need for joint estimation of the regression parameters and the overdispersion parameter the longitudinal correlations being nuisance parameters in this paper we develop a generalised estimating equations approach based on a general autocorrelation structure for the repeated overdispersed data the asymptotic properties of the estimators of the main parameters are discussed and the estimation methodology is illustrated by analysing data on epileptic seizure counts

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

on the robust variance estimator in generalised estimating equations summary the variance matrix of the estimated regression coefficient in the liang zeger generalised estimating equation approach can be consistently estimated by the so called sandwich or robust estimator in this note we propose a modification to the liang zeger prescription for implementing the robust variance estimator analytical and numerical evidence shows the superior performance of our proposal

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients abstract we consider the following nonlinear impulsive delay differential equation with variable coefficients and forcing term x t a t x t f t x e t t 0 t t k x t k x t k i k x t k k n using new methods different from those in tang et al appl math comput 131 2002 373 nieto j comput appl math lett 15 2002 489 franco and nieto j comput appl math 88 1998 144 nieto j math anal appl 205 1997 423 he and yu j math anal appl 272 2002 67 we establish stability theorems for when e t 0 under the assumption that there is bk such that bkx2 x x ik x x2 for k n and x r theorems 2 1 2 4 the existence results for three positive periodic solutions and nonexistence results for periodic solution of when e t 0 under the assumption exp 0 t a u d u 0 t k 1 b k and bk 1 for all k theorems 3 1 3 2 and the existence results for periodic solution of at resonance which is caused by impulses i e exp 0 t a u d u 0 t k 1 b k under the assumption bk 1 for all k theorems 4 2 4 3 some results obtained improve and generalize the known theorems and some other results are new examples are presented to illustrate the main results

conditional score functions some optimality results summary the conditional score function has previously been shown to generate the optimal estimating equation for a parameter of interest when the conditioning statistic is complete and sufficient for the nuisance parameters godambe 1976 the present paper generalizes these results to partial likelihood factorizations and then examines the nature of the problem when the appropriate conditioning statistics depend on the parameter of interest in this case globally optimal estimating functions are impossible a weaker criterion of optimal weighting leads to a class of estimated conditional score functions which satisfy an information equality the purpose of this paper is to develop and justify an approach based on conditional likelihood to estimation in the presence of nuisance parameters the arguments focus on the case when there is a complete and sufficient statistic s 0 for the nuisance parameters when the parameter of interest 0 is fixed an extended notion of completeness will be presented in the context of partial likelihoods the problem takes on two separate natures depending on whether the statistic s 0 truly depends on 0 or not if s s 0 is free of 0 then the conditional likelihood of the data given s has been shown to generate optimal inferential procedures giving a best estimating equation godambe 1976 and a best asymptotically normal estimator in a mixture setting lindsay 1980 if s 0 depends on 0 then it will be shown that the best estimating equation also a conditional likelihood score will generally be only locally optimal leaving no single best strategy consideration of an optimal weighting strategy yields estimators which have second order properties similar to the conditional likelihoods of the first type

a bias correction for covariance estimators to improve inference with generalized estimating equations that use an unstructured correlation matrix generalized estimating equations gees are routinely used for the marginal analysis of correlated data the efficiency of gee depends on how closely the working covariance structure resembles the true structure and therefore accurate modeling of the working correlation of the data is important a popular approach is the use of an unstructured working correlation matrix as it is not as restrictive as simpler structures such as exchangeable and ar 1 and thus can theoretically improve efficiency however because of the potential for having to estimate a large number of correlation parameters variances of regression parameter estimates can be larger than theoretically expected when utilizing the unstructured working correlation matrix therefore standard error estimates can be negatively biased to account for this additional finite sample variability we derive a bias correction that can be applied to typical estimators of the covariance matrix of parameter estimates via simulation and in application to a longitudinal study we show that our proposed correction improves standard error estimation and statistical inference

binary regression using an extended beta binomial distribution with discussion of correlation induced by covariate measurement errors abstract the beta binomial distribution is extended to allow negative correlations among binary variates within an experimental unit regression models are proposed for both the binary variate response rate and for the pairwise correlation between binary variates and corresponding likelihood estimation procedures are described binary regression problems are also considered in which measurement errors in the regression variables represent the sole source of overdispersion some corresponding response rate estimation procedures are described and illustrated

some covariance models for longitudinal count data with overdispersion a family of covariance models for longitudinal counts with predictive covariates is presented these models account for overdispersion heteroscedasticity and dependence among repeated observations the approach is a quasi likelihood regression similar to the formulation given by liang and zeger 1986 biometrika 73 13 22 generalized estimating equations for both the covariate parameters and the variance covariance parameters are presented large sample properties of the parameter estimates are derived the proposed methods are illustrated by an analysis of epileptic seizure count data arising from a study of progabide as an adjuvant therapy for partial seizures

on repeated measures analysis with misspecified covariance structure in recent years various sophisticated methods have been developed for the analysis of repeated measures or longitudinal data the more traditional approach based on a normal likelihood function has been shown to be unsatisfactory in the sense of yielding asymptotically biased estimates when the covariance structure is misspecified more recent methodology based on generalized linear models and quasi likelihood estimation has gained widespread acceptance as generalized estimating equations however this also has theoretical problems in this paper a suggestion is made for improving the asymptotic behaviour of estimators by using the older approach implemented via gaussian estimation the resulting estimating equations include the quasi score function as one component so the methodology proposed can be viewed as a combination of gaussian estimation and generalized estimating equations which has a firmer asymptotic basis than either alone has

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

a covariance estimator for gee with improved small sample properties summary in this paper we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations gee hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small resampling methods such as the jackknife and bootstrap have been suggested for covariance estimation when the number of clusters is small a drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near zero cell counts caused by resampling we propose a bias corrected covariance estimator that avoids this problem in a small simulation study we compare the bias corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10 40 subjects with equal and unequal cluster sizes of 16 64 observations the bias corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2 3 times the nominal level the methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease

estimating equations for association structures this paper investigates generalized estimating equations for association parameters which are frequently of interest in family studies with emphasis on covariance estimation separate link functions are used to connect the mean the scale and the correlation to linear predictors involving possibly different sets of covariates and separate estimating equations are proposed for the three sets of parameters simulations show that the robust sandwich variance estimator and the jackknife variance estimator for the correlation parameters are generally close to the empirical variance for the sample size of 50 clusters the results contradict ziegler et al and kastner and ziegler where the sandwich estimator obtained from the software mareg was shown to be unsuitable for practical usage the problem appears to arise because the mareg variance estimator does not account for variability in estimation of the scale parameters but may be valid with fixed scale we also find that the formula for the approximate jackknife variance estimator in ziegler et al is deficient resulting in systematic deviations from the fully iterated jackknife variance estimator a general jackknife formula is provided and performs well in numerical studies data from a study on the genetics of alcoholism is used to illustrate the importance of reliable variance estimation in biomedical applications

deciding stability and mortality of piecewise affine dynamical systems in this paper we study problems such as given a discrete time dynamical system of the form x t 1 f x t where f r n r n is a piecewise affine function decide whether all trajectories converge to 0 we show in our main theorem that this attractivity problem is undecidable as soon as n greater than or equal to2 the same is true of two related problems stability is the dynamical system globally asymptotically stable and mortality do all trajectories go through 0 we then show that am activity and stability become decidable in dimension 1 for continuous functions c 2001 elsevier science b v all rights reserved

variances are not always nuisance parameters summary in classical problems e g comparing two populations fitting a regression surface etc variability is a nuisance parameter the term nuisance parameter is meant here in both the technical and the practical sense however there are many instances where understanding the structure of variability is just as central as understanding the mean structure the purpose of this article is to review a few of these problems i focus in particular on two issues a the determination of the validity of an assay and b the issue of the power for detecting health effects from nutrient intakes when the latter are measured by food frequency questionnaires i will also briefly mention the problems of variance structure in generalized linear mixed models robust parameter design in quality technology and the signal in microarrays in these and other problems treating variance structure as a nuisance instead of a central part of the modeling effort not only leads to inefficient estimation of means but also to misleading conclusions

analysing ordinal longitudinal survey data generalised estimating equations approach summary longitudinal survey data may comprise ordinal polytomous repeated observations and a set of multidimensional covariates for a large number of individuals one of the main goals of the longitudinal survey is then to describe the marginal expectation of the ordinal polytomous outcome variable as a function of the covariates while accounting for the structural as well as longitudinal correlations the structural correlations come from the polytomous nature of the response variable and the longitudinal correlations from the repetition of the polytomous responses over time in this paper we develop a generalised estimating equations approach based on autocorrelation structure to deal with multivariate polytomous longitudinal survey data the univariate and bivariate analyses being special cases the regression estimators are shown to be consistent for the corresponding regression parameters the methods are illustrated by using the survey of labour and income dynamics data from statistics canada

a note on the efficiency of sandwich covariance matrix estimation the sandwich estimator also known as robust covariance matrix estimator heteroscedasticity consistent covariance matrix estimate or empirical covariance matrix estimator has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified surprisingly though there has been little discussion of properties of the sandwich method other than consistency we investigate the sandwich estimator in quasi likelihood models asymptotically and in the linear case analytically we show that under certain circumstances when the quasi likelihood model is correct the sandwich estimate is often far more variable than the usual parametric variance estimate the increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the p

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

robust analysis of generalized linear mixed models the method of maximum likelihood ml is widely used for analyzing generalized linear mixed models glmm s a full maximum likelihood analysis requires numerical integration techniques for calculation of the log likelihood and to avoid the computational problems involving irreducibly high dimensional integrals several maximum likelihood algorithms have been proposed in the literature to estimate the model parameters by approximating the log likelihood function although these likelihood algorithms are useful for fitting the glmm s efficiently under strict model assumptions they can be highly influenced by the presence of unusual data points in this article the author develops a technique for finding robust maximum likelihood rml estimates of the model parameters in glmm s which appears to be useful in downweighting the influential data points when estimating the parameters the asymptotic properties of the robust estimators are investigated under some regularity conditions small simulations are

estimating and interpolating a markov chain from aggregate data given aggregated longitudinal data generated by a markov chain which may be nonhomogeneous the problem considered is that of modelling estimating and interpolating the logarithms of partial odds and hence the transition probabilities by partial odds is meant the probability of a transition to another state divided by the probability of no transition a result establishing asymptotic normality leads to vector weighted least squares estimation of parameterised partial odds using standard regression methods it is shown how to obtain estimates of one step transition probabilities from widely or irregularly spaced data the methods are illustrated on an example concerning competing causes of death

a bias corrected covariance estimate for improved inference with quadratic inference functions the method of quadratic inference functions qif is an increasingly popular method for the analysis of correlated data because of its multiple advantages over generalized estimating equations gee one advantage is that it is more efficient for parameter estimation when the working covariance structure for the data is misspecified in the qif literature the asymptotic covariance formula is used to obtain standard errors we show that in small to moderately sized samples these standard error estimates can be severely biased downward therefore inflating test size and decreasing coverage probability we propose adjustments to the asymptotic covariance formula that eliminate finite sample biases and as shown via simulation lead to substantial improvements in standard error estimates inference and coverage the proposed method is illustrated in application to a cluster randomized trial and a longitudinal study furthermore qif and gee are contrasted via simulation and these applications

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

the existence of periodic solutions to certain impulsive differential equations abstract for autonomous ordinary differential equations it is shown that there always exist impulsive forcing patterns which permit the existence of periodic solutions however when the differential equation is nonautonomous proving the existence of impulsive periodic solutions appears more complex the authors study here an impulsive differential system which appears to have no periodic solutions continuous manifolds of initial vectors are found for which it is proven that the trajectory emanating from each initial vector is unstable hence nonperiodic it is also established that a particular type of impulsive periodic solution the so called simple periodic motion cannot exist by changing the type of impulse applied some periodic solutions are obtained one of which though not physically meaningful is perhaps mathematically significant

an improved quadratic inference function for parameter estimation in the analysis of correlated data generalized estimating equations gee are commonly employed for the analysis of correlated data however the quadratic inference function qif method is increasing in popularity because of its multiple theoretical advantages over gee we base our focus on the fact that the qif method is more efficient than gee when the working covariance structure for the data is misspecified it has been shown that because of the use of an empirical weighting covariance matrix inside its estimating equations the qif method s realized estimation performance can potentially be inferior to gee s when the number of independent clusters is not large we therefore propose an alternative weighting matrix for the qif which asymptotically is an optimally weighted combination of the empirical covariance matrix and its model based version which is derived by minimizing its expected quadratic loss use of the proposed weighting matrix maintains the large sample advantages the qif approach has over gee and as shown via simulation improves small sample parameter estimation we also illustrated the proposed method in the analysis of a longitudinal study copyright 2012 john wiley sons ltd

a finite sample correction for the variance of linear efficient two step gmm estimators monte carlo studies have shown that estimated asymptotic standard errors of the efficient two step generalized method of moments gmm estimator can be severely downward biased in small samples the weight matrix used in the calculation of the efficient two step gmm estimator is based on initial consistent parameter estimates in this paper it is shown that the extra variation due to the presence of these estimated parameters in the weight matrix accounts for much of the difference between the finite sample and the usual asymptotic variance of the two step gmm estimator when the moment conditions used are linear in the parameters this difference can be estimated resulting in a finite sample corrected estimate of the variance in a monte carlo study of a panel data model it is shown that the corrected variance estimate approximates the finite sample variance well leading to more accurate inference

modelling and generating correlated binary variables summary many applications use simple parametric models for the correlation structure of binary responses which are observed in clusters the usual approach to use correlation models appropriate for normally distributed responses suffers from two drawbacks when the marginal probabilities within the clusters differ first as it does not explicitly take into account constraints on the second moments which must be satisfied for binary responses it may fail to model realistically the range of correlations present in the data secondly computer simulation of observations from these models is very difficult we present an alternative class of correlation models which reflect the binary nature of the responses and allow for simple simulation some key wor ds binary variable computer simulation correlation structure generalised estimating equation

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 generalized linear model with nested strata of extra poisson variation on etend la definition de quasivraisemblance pour tenir compte de plusieurs strates de variation emboitees dans un modele lineaire generalise

conditional logistic regression models for correlated binary data summary a class of conditional logistic regression models for clustered binary data is considered this includes the polychotomous logistic model of rosner 1984 as a special case properties such as the joint distribution and pairwise odds ratio are investigated a class of easily computed estimating functions is introduced which is shown to have high efficiency compared to the computationally intensive maximum likelihood approach an example on chronic obstructive pulmonary disease among sibs is presented for illustration

on the presence of periodic configurations in turing machines and in counter machines a configuration of a turing machine is given by a tape content together with a particular state of the machine petr k rka has conjectured that every turing machine when seen as a dynamical system on the space of its configurations has at least one periodic orbit in this paper we provide an explicit counterexample to this conjecture we also consider counter machines and prove that in this case the problem of determining if a given machine has a periodic orbit in configuration space is undecidable

generalized least squares taylor series linearization and fisher s scoring in multivariate nonlinear regression in this article we consider a general multivariate nonlinear regression setting in which the marginal mean and variance covariance structure share a common set of regression parameters estimation is carried out via iteratively reweighted generalized least squares irgls that entails repeated application of taylor series linearization and estimated generalized least squares egls under normality this irgls procedure is equivalent to fisher s method of scoring and hence maximum likelihood estimation mle however estimates from this procedure are also shown to minimize a bias corrected generalized least squares objective function that does not require the assumption of normality under fairly mild regularity conditions the resulting estimates are consistent asymptotically normal and under normality assumptions asymptotically efficient the estimates are compared against those obtained as solutions to the usual generalized estimating equations gee using both simulation and numerical examples

unbiased estimating equations from working correlation models for irregularly timed repeated measures the method of generalized estimating equations gees has been criticized recently for a failure to protect against misspecification of working correlation models which in some cases leads to loss of efficiency or infeasibility of solutions however the feasibility and efficiency of gee methods can be enhanced considerably by using flexible families of working correlation models we propose two ways of constructing unbiased estimating equations from general correlation models for irregularly timed repeated measures to supplement and enhance gee the supplementary estimating equations are obtained by differentiation of the cholesky decomposition of the working correlation or as score equations for decoupled gaussian pseudolikelihood the estimating equations are solved with computational effort equivalent to that required for a first order gee full details and analytic expressions are developed for a generalized markovian model that was evaluated through simulation large sample sandwich standard errors for working correlation parameter estimates are derived and shown to have good performance the proposed estimating functions are further illustrated in an analysis of repeated measures of pulmonary function in children

a family of multivariate binary distributions for simulating correlated binary variables with specified marginal means and correlations summary we introduce a family of multivariate binary distributions with certain conditional linear property this family is particularly useful for efficient and easy simulation of correlated binary variables with a given marginal mean vector and correlation matrix

impulsive differential equations this paper deals with the periodic solutions problem for impulsive differential equations by using lyapunov s second method and the contraction mapping principle some conditions ensuring the existence and global attractiveness of unique periodic solutions are derived which are given from impulsive control and impulsive perturbation points of view as an application the existence and global attractiveness of unique periodic solutions for hopfield neural networks are discussed finally two numerical examples are provided to demonstrate the effectiveness of the proposed results

the effect of cluster size imbalance and covariates on the estimation performance of quadratic inference functions generalized estimating equations gee are commonly used for the analysis of correlated data however use of quadratic inference functions qifs is becoming popular because it increases efficiency relative to gee when the working covariance structure is misspecified although shown to be advantageous in the literature the impacts of covariates and imbalanced cluster sizes on the estimation performance of the qif method in finite samples have not been studied this cluster size variation causes qif s estimating equations and gee to be in separate classes when an exchangeable correlation structure is implemented causing qif and gee to be incomparable in terms of efficiency when utilizing this structure and the number of clusters is not large we discuss how covariates and cluster size imbalance can cause qif rather than gee to produce estimates with the larger variability this occurrence is mainly due to the empirical nature of weighting qif employs rather than differences in estimating equations classes we demonstrate qif s lost estimation precision through simulation studies covering a variety of general cluster randomized trial scenarios and compare qif and gee in the analysis of data from a cluster randomized trial copyright 2012 john wiley sons ltd

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

a 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