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

temporal process regression summary we consider regression for response and covariates which are temporal processes observed over intervals a functional generalised linear model is proposed which includes extensions of standard models in multi state survival analysis simple nonparametric esti mators of time indexed parameters are developed using working independence estimating equations and are shown to be uniformly consistent and to converge weakly to gaussian processes the procedure does not require smoothing or a markov assumption unlike approaches based on transition intensities the usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy the estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters providing novel goodness of fit tests the methodology s practical utility is illustrated in a data analysis

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

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

a moving blocks empirical likelihood method for longitudinal data summary in the analysis of longitudinal or panel data neglecting the serial correlations among the repeated measurements within subjects may lead to inefficient inference in particular when the number of repeated measurements is large it may be desirable to model the serial correlations more generally an appealing approach is to accommodate the serial correlations nonparametrically in this article we propose a moving blocks empirical likelihood method for general estimating equations asymptotic results are derived under sequential limits simulation studies are conducted to investigate the finite sample performances of the proposed methods and compare them with the elementwise and subject wise empirical likelihood methods of wang et al 2010 biometrika 97 79 93 and the block empirical likelihood method of you et al 2006 can j statist 34 79 96 an application to an aids longitudinal study is presented

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

generalized estimating equation model for binary outcomes with missing covariates summary this paper presents an approach to handling missing covariates in the generalized estimating equation gee model for binary outcomes when the probability of missingness depends on the observed outcomes and covariates the proposed method is to replace the missing quantities in the estimating function with consistent estimates in special cases the proposed model reduces to a weighted gee model for the completely observed units where the weight is the inverse of the probability of missingness our method can be viewed as an extension of the mean score method by reilly and pepe 1995 biometrika 82 299 314 to the gee context under certain regularity conditions the estimates of the regression coefficients obtained by the proposed method are consistent and asymptotically normally distributed the finite sample properties of the estimates are illustrated via computer simulations an application to the study of dementia among stroke patients is presented

bivariate latent variable models for clustered discrete and continuous outcomes abstract we use the concept of a latent variable to derive the joint distribution of a continuous and a discrete outcome and then extend the model to allow for clustered data the model can be parameterized in a way that allows one to write the joint distribution as a product of a standard random effects model for the continuous variable and a correlated probit model for the discrete variable this factorization suggests a convenient approach to parameter estimation using quasi likelihood techniques our approach is motivated by the analysis of developmental toxicity experiments for which a number of discrete and continuous outcomes are measured on offspring clustered within litters fetal weight and malformation data illustrate the results

parameterization of multivariate random effects models for categorical data alternative parameterizations and problems of identification and estimation of multivariate random effects models for categorical responses are investigated the issues are illustrated in the context of the multivariate binomial logit normal bln model introduced by coull and agresti 2000 biometrics 56 73 80 we demonstrate that the bln model is poorly identified unless proper restrictions are imposed on the parameters moreover estimation of bln models is unduly computationally complex in the first application considered by coull and agresti an identification problem results in highly unstable highly correlated parameter estimates and large standard errors a probit normal version of the specified bln model is demonstrated to be underidentified whereas the bln model is empirically underidentified identification can be achieved by constraining one of the parameters we show that a one factor probit model is equivalent to the probit version of the specified bln model and that a one factor logit model is empirically equivalent to the bln model estimation is greatly simplified by using a factor model

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

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

power analyses for longitudinal study designs with missing data existing methods for power analysis for longitudinal study designs are limited in that they do not adequately address random missing data patterns although the pattern of missing data can be assessed during data analysis it is unknown during the design phase of a study the random nature of the missing data pattern adds another layer of complexity in addressing missing data for power analysis in this paper we model the occurrence of missing data with a two state first order markov process and integrate the modelling information into the power function to account for random missing data patterns the markov model is easily specified to accommodate different anticipated missing data processes we develop this approach for the two most popular longitudinal models the generalized estimating equations gee and the linear mixed effects model under the missing completely at random mcar assumption for gee we also limit our consideration to the working independence correlation model the proposed methodology is illustrated with numerous examples that are motivated by real study designs

tests of homogeneity for generalized linear models abstract we propose two tests for testing homogeneity among clustered data adjusting for the effects of covariates the first is a score test for a generalized linear model with random effect in which the distribution of the response variable given the random effect is entirely defined in contrast to the likelihood ratio test however the score test does not require estimation of the parameters of a mixed effects model nor specification of the mixing distribution the second test is proposed in the framework of the generalized estimating equation gee approach in deriving this test we need only the specification of the marginal expectation and variance of the response variable and the fourth moment for the overdispersion term whereas for deriving the score test for mixed effects models the entire conditional distribution must be specified we demonstrate that the two tests are identical when the covariance matrix assumed in the gee approach is that of the random effects model in both approaches

a determinant based criterion for working correlation structure selection in generalized estimating equations in generalized estimating equations gee the correlation between the repeated observations on a subject is specified with a working correlation matrix correct specification of the working correlation structure ensures efficient estimators of the regression coefficients among the criteria used in practice for selecting working correlation structure rotnitzky jewell quasi information criterion qic and correlation information criterion cic are based on the fact that if the assumed working correlation structure is correct then the model based naive and the sandwich robust covariance estimators of the regression coefficient estimators should be close to each other the sandwich covariance estimator used in defining the rotnitzky jewell qic and cic criteria is biased downward and has a larger variability than the corresponding model based covariance estimator motivated by this fact a new criterion is proposed in this paper based on the bias corrected sandwich covariance estimator for selecting an appropriate working correlation structure in gee a comparison of the proposed and the competing criteria is shown using simulation studies with correlated binary responses the results revealed that the proposed criterion generally performs better than the competing criteria an example of selecting the appropriate working correlation structure has also been shown using the data from madras schizophrenia study

estimating a multivariate familial correlation using joint models for canonical correlations application to memory score analysis from familial hispanic alzheimer s disease study analysis of multiple traits can provide additional information beyond analysis of a single trait allowing better understanding of the underlying genetic mechanism of a common disease to accommodate multiple traits in familial correlation analysis adjusting for confounders we develop a regression model for canonical correlation parameters and propose joint modeling along with mean and scale parameters the proposed method is more powerful than the regression method modeling pairwise correlations because it captures familial aggregation manifested in multiple traits through maximum canonical correlation

flutter and thermal deflection suppression of composite plates using shape memory alloy an efficient finite element method for predicting critical temperature postbuckling deflection and flutter response is developed for composite plates embedded with prestrained shape memory alloy sma fibers the large panel deflection and the temperature dependent material properties of sma and composites are considered in the formulation the finite element formulation and solution procedure of the incremental updated lagrangian method for nonlinear temperature dependent material properties are presented briefly flow yaw angle at supersonic speed is also considered for the static solution the system equations are solved by using the newton raphson iteration method to determine aerothermal postbuckling deflection under the combined aerodynamic and thermal loads for the dynamic flutter response the modal method and time domain numerical integration are employed results show that the critical buckling temperature can be raised high enough so that buckling deflection can be eliminated and the desired flat and stable region in the temperature dynamic pressure operating domain can be enlarged greatly for composite surface panels embedded with sma for supersonic vehicle applications weight savings based on critical temperature when sma is used are also presented

on linear and quadratic estimating functions summary in some cases maximum quasi likelihood estimation which is at the core of glim can fail to give reasonable results a more general class of estimating functions is investigated which avoids such failure and also is not restricted to the particular forms of mean and variance function of glim

ring theoretical properties of affine cellular algebras abstract as a generalisation of graham and lehrer s cellular algebras affine cellular algebras have been introduced in 12 in order to treat affine hecke algebras of type a and affine versions of diagram algebras like affine temperley lieb algebras in a unifying fashion affine cellular algebras include kleshchev s graded quasihereditary algebras khovanov lauda rouquier algebras and various other classes of algebras in this paper we will study ring theoretical properties of affine cellular algebras we show that any affine cellular algebra a satisfies a polynomial identity furthermore we show that a can be embedded into its asymptotic algebra if the occurring commutative affine k algebras b j are reduced and the determinants of the swich matrices are non zero divisors as a consequence we show that the gelfand kirillov dimension of a is less than or equal to the largest krull dimension of the algebras b j and that equality holds in case all affine cell ideals are idempotent or if the krull dimension of the algebras b j is less than or equal to 1 special emphasis is given to the question when an affine cell ideal is idempotent generated by an idempotent or finitely generated

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

linear mixed effects modeling approach to fmri group analysis conventional group analysis is usually performed with student type t test regression or standard an c ova in which the variance covariance matrix is presumed to have a simple structure some correction approaches are adopted when assumptions about the covariance structure is violated however as experiments are designed with different degrees of sophistication these traditional methods can become cumbersome or even be unable to handle the situation at hand for example most current fmri software packages have difficulty analyzing the following scenarios at group level 1 taking within subject variability into account when there are effect estimates from multiple runs or sessions 2 continuous explanatory variables covariates modeling in the presence of a within subject repeated measures factor multiple subject grouping between subjects factors or the mixture of both 3 subject specific adjustments in covariate modeling 4 group analysis with estimation of hemodynamic response hdr function by multiple basis functions 5 various cases of missing data in longitudinal studies and 6 group studies involving family members or twins here we present a linear mixed effects modeling lme methodology that extends the conventional group analysis approach to analyze many complicated cases including the six prototypes delineated above whose analyses would be otherwise either difficult or unfeasible under traditional frameworks such as an c ova and general linear model glm in addition the strength of the lme framework lies in its flexibility to model and estimate the variance covariance structures for both random effects and residuals the intraclass correlation icc values can be easily obtained with an lme model with crossed random effects even at the presence of confounding fixed effects the simulations of one prototypical scenario indicate that the lme modeling keeps a balance between the control for false positives and the sensitivity for activation detection the importance of hypothesis formulation is also illustrated in the simulations comparisons with alternative group analysis approaches and the limitations of lme are discussed in details

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

simple generalized estimating equations gees and weighted generalized estimating equations wgees in longitudinal studies with dropouts guidelines and implementation in r missing data are a common problem in clinical and epidemiological research especially in longitudinal studies despite many methodological advances in recent decades many papers on clinical trials and epidemiological studies do not report using principled statistical methods to accommodate missing data or use ineffective or inappropriate techniques two refined techniques are presented here generalized estimating equations gees and weighted generalized estimating equations wgees these techniques are an extension of generalized linear models to longitudinal or clustered data where observations are no longer independent they can appropriately handle missing data when the missingness is completely at random gee and wgee or at random wgee and do not require the outcome to be normally distributed our aim is to describe and illustrate with a real example in a simple and accessible way to researchers these techniques for handling missing data in the context of longitudinal studies subject to dropout and show how to implement them in r we apply them to assess the evolution of health related quality of life in coronary patients in a data set subject to dropout copyright 2016 john wiley sons ltd

correlated bivariate continuous and binary outcomes issues and applications increasingly multiple outcomes are collected in order to characterize treatment effectiveness or to evaluate the impact of large policy initiatives often the multiple outcomes are non commensurate e g measured on different scales the common approach to inference is to model each outcome separately ignoring the potential correlation among the responses we describe and contrast several full likelihood and quasi likelihood multivariate methods for non commensurate outcomes we present a new multivariate model to analyze binary and continuous correlated outcomes using a latent variable we study the efficiency gains of the multivariate methods relative to the univariate approach for complete data all approaches yield consistent parameter estimates when the mean structure of all outcomes depends on the same set of covariates efficiency gains by adopting a multivariate approach are negligible in contrast when the mean outcomes depend on different covariate sets large efficiency gains are realized three real examples illustrate the different approaches

analysis of longitudinal data unbalanced over time summary the paper considers modelling estimating and diagnostically verifying the response process generating longitudinal data with emphasis on association between repeated measures from unbalanced longitudinal designs our model is based on separate specifications of the moments for the mean standard deviation and correlation with different components possibly sharing common parameters we propose a general class of correlation structures that comprise random effects measurement errors and a serially correlated process these three elements are combined via flexible time varying weights whereas the serial correlation can depend flexibly on the mean time and lag when the measurement schedule is independent of the response process our estimation procedure yields consistent and asymptotically normal estimates for the mean parameters even when the standard deviation and correlation are misspecified and for the standard deviation parameters even when the correlation is misspecified a generic diagnostic method is developed for verifying the models for the mean standard deviation and in particular the correlation which is applicable even when the data are severely unbalanced the methodology is illustrated by an analysis of data from a longitudinal study that was designed to characterize pulmonary growth in girls

the estimation of prediction error having constructed a data based estimation rule perhaps a logistic regression or a classification tree the statistician would like to know its performance as a predictor of future cases there are two main theories concerning prediction error 1 penalty methods such as cp akaike s information criterion and stein s unbiased risk estimate that depend on the covariance between data points and their corresponding predictions and 2 cross validation and related nonparametric bootstrap techniques this article concerns the connection between the two theories a rao blackwell type of relation is derived in which nonparametric methods such as cross validation are seen to be randomized versions of their covariance penalty counterparts the model based penalty methods offer substantially better accuracy assuming that the model is believable

affine cellularity of affine yokonuma hecke algebras abstract we establish an explicit algebra isomorphism between the affine yokonuma hecke algebra y r n q and a direct sum of matrix algebras with coefficients in tensor products of affine hecke algebras of type a as an application of this result we show that y r n q is affine cellular in the sense of koenig and xi and further prove that it has finite global dimension when the parameter q is not a root of the poincare polynomial as another application we also recover the modular representation theory of y r n q previously obtained in 7

a combined score test for binary and ordinal endpoints from clinical trials there is growing interest especially for trials in stroke in combining multiple endpoints in a single clinical evaluation of an experimental treatment the endpoints might be repeated evaluations of the same characteristic or alternative measures of progress on different scales often they will be binary or ordinal and those are the cases studied here in this paper we take a direct approach to combining the univariate score statistics for comparing treatments with respect to each endpoint the correlations between the score statistics are derived and used to allow a valid combined score test to be applied a sample size formula is deduced and application in sequential designs is discussed the method is compared with an alternative approach based on generalized estimating equations in an illustrative analysis and replicated simulations and the advantages and disadvantages of the two approaches are discussed

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

accounting for covariates observed post randomization for discrete and continuous repeated measures data adjusting for covariates observed post randomization is a difficult issue in randomized clinical trials one approach is to enter these covariates as time dependent covariates in a general linear model however this approach fails to account for interactions between the primary end point and confounding variable as they evolve over time in this paper we adopt the view that since the confounder is observed following randomization it should be treated as an outcome and analysed accordingly we consider the repeated measures design where both the primary end point and the confounding measure are observed repeatedly over patient follow up a generalized estimating equation model is applied to allow each set of repeated measures to be modelled in terms of important explanatory variables seemingly unrelated regression combines the two models into an overall framework for analysis inference is then performed by imposing restrictions on the confounder regression parameters to reflect the behaviour profile of interest estimation and identification procedures are described and the methodology is illustrated with an example

functional association models for multivariate survival processes we consider multivariate temporal processes that are continuously observed within overlapping time windows the intended application is in censored multistate and multivariate survival settings where point processes are continuously observed these data differ from other functional data like longitudinal data which are discretely observed at irregular times existing functional approaches to survival processes use intensity models which require smoothing and depend critically on the choice of smoothing parameters similarly to discretely observed data in this article we study functional mean and association regression models for the point processes with unspecified time varying coefficients the continuous observation scheme is exploited the coefficients may be estimated nonparametrically by extending generalized estimating equations to continuously observed data the estimators automatically converge at the parametric rate without smoothing unlike with discretely observed data uniform consistenc

performance of analytical methods for overdispersed counts in cluster randomized trials sample size degree of clustering and imbalance many different methods have been proposed for the analysis of cluster randomized trials crts over the last 30 years however the evaluation of methods on overdispersed count data has been based mostly on the comparison of results using empiric data i e when the true model parameters are not known in this study we assess via simulation the performance of five methods for the analysis of counts in situations similar to real community intervention trials we used the negative binomial distribution to simulate overdispersed counts of crts with two study arms allowing the period of time under observation to vary among individuals we assessed different sample sizes degrees of clustering and degrees of cluster size imbalance the compared methods are i the two sample t test of cluster level rates ii generalized estimating equations gee with empirical covariance estimators iii gee with model based covariance estimators iv generalized linear mixed models glmm and v bayesian hierarchical models bayes hm variation in sample size and clustering led to differences between the methods in terms of coverage significance power and random effects estimation glmm and bayes hm performed better in general with bayes hm producing less dispersed results for random effects estimates although upward biased when clustering was low gee showed higher power but anticonservative coverage and elevated type i error rates imbalance affected the overall performance of the cluster level t test and the gee s coverage in small samples important effects arising from accounting for overdispersion are illustrated through the analysis of a community intervention trial on solar water disinfection in rural bolivia

affine cellularity of bln algebras abstract we show that the bln algebra which was introduced by mcgerty is affine cellular in the sense of koenig and xi in fact we establish a stronger property namely that the affine cell ideals are generated by idempotents this particularly implies that the global dimension of the bln algebra is finite for affine type a we obtain that the affine q schur algebra u d n n when d n is affine cellular and has finite global dimension

a simple estimator for a shared frailty regression model summary we propose a simple estimation procedure for a proportional hazards frailty regression model for clustered survival data in which the dependence is generated by a positive stable distribution inferences for the frailty parameter can be obtained by using output from cox regression analyses the computational burden is substantially less than that of the other approaches to estimation the large sample behaviour of the estimator is studied and simulations show that the approximations are appropriate for use with realistic sample sizes the methods are motivated by studies of familial associations in the natural history of diseases their practical utility is illustrated with sib pair data from beaver dam wisconsin

a comparison of generalized linear mixed model procedures with estimating equations for variance and covariance parameter estimation in longitudinal studies and group randomized trials response data in longitudinal studies and group randomized trials are gathered on units that belong to clusters within which data are usually positively correlated therefore estimates and confidence intervals for intraclass correlation or variance components are helpful when designing a longitudinal study or group randomized trial data simulated from both study designs are used to investigate the estimation of variance and covariance parameters from the following procedures for continuous outcomes restricted maximum likelihood reml and estimating equations ee for binary outcomes restricted pseudo likelihood repl and estimating equations ee we evaluate these procedures to see which provide valid and precise estimates as well as correct standard errors for the intraclass correlation coefficient or variance components reml seems the better choice for estimating terms related to correlation for models with normal outcomes especially in group randomized trial situations results for reml and ee are mixed when outcomes are continuous and non normal with binary outcomes neither repl nor ee provides satisfactory estimation or inference in longitudinal study situations while repl is preferable for group randomized trials

an appraisal of methods for the analysis of longitudinal categorical data with mar drop outs a number of methods for analysing longitudinal ordinal categorical data with missing at random drop outs are considered two are maximum likelihood methods maxlik which employ marginal global odds ratios to model associations the remainder use weighted or unweighted generalized estimating equations gee two of the gee use cholesky decomposed standardized residuals to model the association structure while another three extend methods developed for longitudinal binary data in which the association structures are modelled using either gaussian estimation multivariate normal estimating equations or conditional residuals simulated data sets were used to discover differences among the methods in terms of biases variances and convergence rates when the association structure is misspecified the methods were also applied to a real medical data set two of the gee methods referred to as cond and ml norm in this paper and by their originators were found to have relatively good convergence rates and mean squared errors for all sample sizes 80 120 300 considered and one more referred to as mgee in this paper and by its originators worked fairly well for all but the smallest sample size 80

model selection criterion for causal parameters in structural mean models based on a quasi likelihood summary structural mean models smms have been proposed for estimating causal parameters in the presence of nonignorable non compliance in clinical trials to obtain a valid causal estimate we must impose several assumptions one of these is the correct specification of the structural model building on pan s work 2001 biometrics 57 120 125 on developing a model selection criterion for generalized estimating equations we propose a new approach for model selection of smms based on a quasi likelihood we provide a formal model selection criterion that is an extension of akaike s information criterion using subset selection of baseline covariates our method allows us to understand whether the treatment effect varies across the available baseline covariate levels and or to quantify the treatment effect on a specific covariates level to target specific individuals to maximize treatment benefit we present simulation results in which our method performs reasonably well compared to other testing methods in terms of both the probability of selecting the correct model and the predictive performances of the individual treatment effects we use a large randomized clinical trial of pravastatin as a motivation

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

affine cellular algebras abstract graham and lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras many classes of finite dimensional algebras including various hecke algebras and diagram algebras have been shown to be cellular and the theory due to graham and lehrer successfully has been applied to these algebras we will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field affine hecke algebras of type a and infinite dimensional diagram algebras like the affine temperley lieb algebras are shown to be examples of our definition the isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties in this way representation theory of non commutative algebras is linked with commutative algebra moreover conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification these conditions are shown to be satisfied for the affine hecke algebra of type a if the quantum parameter is not a root of the poincare polynomial

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

latent variables measurement error and methods for analysing longitudinal binary and ordinal data we explore structural equations with latent variables for modelling between individual variability and measurement error in the analysis of longitudinal binary and ordinal data the structural equation formulation provides insight into the assumptions and differences in interpretation of methods that are popular for longitudinal data analysis introducing the concept of continuous latent variables makes it clear that marginal and cluster specific models differ because their predicted variables are scaled to different standard deviations and that adjustment for measurement error in the outcome involves a change in scale as well we apply both structural equation modelling and common longitudinal modelling approaches to data from a study of sleep disorders in the process we compare results from marginal modelling using an sas gee routine karim and zeger 1988 qu s gauss program qu 1992 for generalized mixed models using gee the mixor package for cluster specific mixed effects models hedeker and gibbons 1994 and liscomp for structural models muthen 1988

statistical inference in mixed models and analysis of twin and family data analysis of data from twin and family studies provides the foundation for studies of disease inheritance the development of advanced theory and computational software for general linear models has generated considerable interest for using mixed effect models to analyze twin and family data as a computationally more convenient and theoretically more sound alternative to the classical structure equation modeling despite the long history of twin and family data analysis some fundamental questions remain unanswered we addressed two important issues one is to determine the necessary and sufficient conditions for the identifiability in the mixed effects models for twin and family data the other is to derive the asymptotic distribution of the likelihood ratio test which is novel due to the fact that the standard regularity conditions are not satisfied we considered a series of specific yet important examples in which we demonstrated how to formulate mixed effect models to appropriately reflect the data and our key idea is the use of the cholesky decomposition finally we applied our method and theory to provide a more precise estimate of the heritability of two data sets than the previously reported estimate

a note on extended quasi likelihood abstract we study the method of extended quasi likelihood estimation and inference of a variance function recently proposed by nelder pregibon 1987 the estimates are inconsistent in general and the test levels can be biased but in many cases such as the exponential family the inconsistency and bias will not be a major concern extended quasi likelihood is compared with carroll ruppert s 1982 pseudo likelihood method which gives consistent estimates and when slightly modified asymptotically unbiased tests we quantify the showing in this instance that the two estimates are closely related and may be asymptotically equivalent in many important cases however in some cases outside the exponential family an asymptotic bias can persist heteroscedastic regression model

a robust and unified framework for estimating heritability in twin studies using generalized estimating equations the heritability of a phenotype measures the proportion of trait variance due to genetic factors in a population in the past 50 years studies with monozygotic and dizygotic twins have estimated heritability for 17 804 traits 1 thus twin studies are popular for estimating heritability researchers are often interested in estimating heritability for non normally distributed outcomes such as binary counts skewed or heavy tailed continuous traits in these settings the traditional normal ace model nace and falconer s method can produce poor coverage of the true heritability therefore we propose a robust generalized estimating equations gee2 framework for estimating the heritability of non normally distributed outcomes the traditional nace and falconer s method are derived within this unified gee2 framework which additionally provides robust standard errors although the traditional falconer s method cannot adjust for covariates the corresponding gee2 falconer can incorporate mean and variance level covariate effects e g let heritability vary by sex or age given a non normally distributed outcome the gee2 models are shown to attain better coverage of the true heritability compared to traditional methods finally a scenario is demonstrated where nace produces biased estimates of heritability while falconer remains unbiased therefore we recommend gee2 falconer for estimating the heritability of non normally distributed outcomes in twin studies

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

analyzing bivariate ordinal data using a global odds ratio abstract a moment method is proposed for regression analysis of bivariate ordered categorical data using the global odds ratio as the measure of association for modeling the margins this method utilizes the stochastic ordering implicit in the data this method allows for covariate effects in modeling the association between responses an application of the proposed model is illustrated using the ophthalmological data from the wisconsin epidemiologic study of diabetic retinopathy for identifying risk factors among younger onset diabetics this model is also extended to a longitudinal data setting with more than two repeated measures

genotype adjusted familial correlation analysis using three generalized estimating equations we analyze familial correlation of a memory score from caribbean hispanic families that have multiple family members affected with alzheimer s disease adjusting for having at least one apoe e4 allele as well as other confounders to enhance the efficiency of correlation model this paper proposes an alternative approach for three generalized estimating equations proposed by yan and fine statist med 2004 23 859 874 the efficiency of correlation model is evaluated through the asymptotic relative efficiency computation and simulations copyright 2008 john wiley sons ltd

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

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

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

affine cellularity of affine hecke algebras of rank two we show that affine hecke algebras of rank two with generic parameters are affine cellular in the sense of koenig xi

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

nonlinear flutter of composite panels under yawed supersonic flow using finite elements a finite element formulation is presented for the effects of arbitrary flow direction on the large amplitude supersonic flutter of composite panels the von karman large deflection plate theory is used to account for large amplitude limit cyde oscillations quasisteady first order piston theory aerodynamics is employed for aerodynamic loading and first order shear deformation theory is used for laminated composite panels an efficient solution procedure is presented by using the modal transformation to reduce the number of nonlinear panel flutter equations and then applying the linearized updated mode with nonlinear time function approximation to the reduced nonlinear panel flutter modal equations a modal participation is defined and the minimum number of linear modes for accurate and converged limit cycle response can be ensured examples are given for isotropic and composite panels at yawed supersonic flow

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

informative estimation and selection of correlation structure for longitudinal data identifying an informative correlation structure is important in improving estimation efficiency for longitudinal data we approximate the empirical estimator of the correlation matrix by groups of known basis matrices that represent different correlation structures and transform the correlation structure selection problem to a covariate selection problem to address both the complexity and the informativeness of the correlation matrix we minimize an objective function that consists of two parts the difference between the empirical information and a model approximation of the correlation matrix and a penalty that penalizes models with too many basis matrices the unique feature of the proposed estimation and selection of correlation structure is that it does not require the specification of the likelihood function and therefore it is applicable for discrete longitudinal data we carry out the proposed method through a groupwise penalty strategy which is able to identify more complex structures the

supersonic flutter analysis of stiffened laminated plates subject to thermal load abstract flutter analysis of stiffened laminated plates has been performed the purpose of this study is to analyze the flutter characteristics of stiffened plates subject to thermal load the first order shear deformable plate and timoshenko beam theories are used for the finite element modelling of a skin panel and stiffeners respectively the von karman non linear strain displacement relation is used to account for large deflections first order piston theory is used for modelling aerodynamic loads the newton raphson iteration method and complex eigenvalue solver with the lum ntf approximation method are used to obtain the postbuckled deflection and flutter information respectively the guyan reduction method and mode tracing procedure are employed for an efficient analysis the effects of various parameters such as the stiffening scheme lamination angle boundary conditions and temperature gradient on flutter characteristics are investigated through some numerical examples from these examples it can be shown that the selection of a proper stiffening scheme results in great improvements of flutter characteristics of laminated panels without introducing considerable weight penalty

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

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

fast forward selection for generalized estimating equations with a large number of predictor variables summary we propose a new variable selection criterion designed for use with forward selection algorithms the score information criterion sic the proposed criterion is based on score statistics which incorporate correlated response data the main advantage of the sic is that it is much faster to compute than existing model selection criteria when the number of predictor variables added to a model is large this is because sic can be computed for all candidate models without actually fitting them a second advantage is that it incorporates the correlation between variables into its quasi likelihood leading to more desirable properties than competing selection criteria consistency and prediction properties are shown for the sic we conduct simulation studies to evaluate the selection and prediction performances and compare these as well as computational times with some well known variable selection criteria we apply the sic on a real data set collected on arthropods by considering variable selection on a large number of interactions terms consisting of species traits and environmental covariates

regression analysis of correlated ordinal data using orthogonalized residuals semi parametric regression models for the joint estimation of marginal mean and within cluster pairwise association parameters are used in a variety of settings for population averaged modeling of multivariate categorical outcomes recently a formulation of alternating logistic regressions based on orthogonalized marginal residuals has been introduced for correlated binary data unlike the original procedure based on conditional residuals its covariance estimator is invariant to the ordering of observations within clusters in this article the orthogonalized residuals method is extended to model correlated ordinal data with a global odds ratio and shown in a simulation study to be more efficient and less biased with regards to estimating within cluster association parameters than an existing extension to ordinal data of alternating logistic regressions based on conditional residuals orthogonalized residuals are used to estimate a model for three correlated ordinal outcomes measured repeatedly in a longitudinal clinical trial of an intervention to improve recovery of patients perception of altered sensation following jaw surgery

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

efficient semiparametric mean association estimation for longitudinal binary responses semiparametric methods for longitudinal data with association within subjects have recently received considerable attention however existing methods for semiparametric longitudinal binary regression modeling i mainly concern mean structures with association parameters treated as nuisance ii generally require a correct specification of the covariance structure for misspecified covariance structure may lead to inefficient mean parameter estimates and iii usually run into computation and estimation problems when the time points are irregularly and possibly subject specific in this article we propose a semiparametric logistic regression model which simultaneously takes into account both the mean and response association structures via conditional log odds ratio for multivariate longitudinal binary outcomes our main interest lies in efficient estimation of both the marginal and association parameters the estimators of the parameters are obtained via the profile kernel approach we evaluate the proposed methodology through simulation studies and apply it to a real dataset both theoretical and empirical results demonstrate that the proposed method yields highly efficient estimators and performs satisfactorily copyright 2012 john wiley sons ltd

variance components testing in the longitudinal mixed effects model this article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by laird and ware 1982 biometrics 38 963 974 our discussion of the large sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by self and liang 1987 journal of the american statistical association 82 605 610

tests of independence treatment heterogeneity dose related trend with exchangeable binary data abstract existing methods of testing for treatment effects for clustered binary data include the beta binomial quasilikelihood gee procedures all of these methods revolve around the mean response and the second order correlation however these two parameters alone do not fully determine the effect of treatment this article develops nonparametric likelihood ratio procedures to test for independence heterogeneity dose related trend in dose response studies involving exchangeable binary data the hypotheses of independence heterogeneity trend are expressed in terms of joint probabilities of similar responses among cluster mates constrained maximum likelihood estimates of these probabilities are computed and used to construct test statistics unlike the test statistics for independence and heterogeneity the asymptotic distribution of the likelihood ratio test for trend is not exactly a chi square however an upper bound of its p value is obtained by using a chi squared distribution a set of clustere

fsem functional structural equation models for twin functional data the aim of this article is to develop a novel class of functional structural equation models fsems for dissecting functional genetic and environmental effects on twin functional data while chara

bivariate modelling of clustered continuous and ordered categorical outcomes simultaneous observation of continuous and ordered categorical outcomes for each subject is common in biomedical research but multivariate analysis of the data is complicated by the multiple data types here we construct a model for the joint distribution of bivariate continuous and ordinal outcomes by applying the concept of latent variables to a multivariate normal distribution the approach is then extended to allow for clustering of the bivariate outcomes the model can be parameterized in a way that allows writing the joint distribution as a product of a standard random effects model for the continuous variable and a correlated cumulative probit model for the ordinal outcome this factorization suggests convenient parameter estimation using estimating equations foetal weight and malformation data from a developmental toxicity experiment illustrate the results

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

model selection for generalized estimating equations accommodating dropout missingness the generalized estimating equation gee has been a popular tool for marginal regression analysis with longitudinal data and its extension the weighted gee approach can further accommodate data that are missing at random mar model selection methodologies for gee however have not been systematically developed to allow for missing data we propose the missing longitudinal information criterion mlic for selection of the mean model and the mlic for correlation mlicc for selection of the correlation structure in gee when the outcome data are subject to dropout monotone missingness and are mar our simulation results reveal that the mlic and mlicc are effective for variable selection in the mean model and selecting the correlation structure respectively we also demonstrate the remarkable drawbacks of naively treating incomplete data as if they were complete and applying the existing gee model selection method the utility of proposed method is further illustrated by two real applications involving missing longitudinal outcome data

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

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

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