statistical models for autocorrelated count data a generalized linear mixed model is an increasingly popular choice for the modelling of correlated non normal responses in a regression setting a number of methods are currently available for fitting a generalized linear mixed model including monte carlo markov chain maximum likelihood algorithms approximate maximum likelihood pql iterative bias correction and others of interest in this paper is to compare the parameter estimation of the various methods in the modelling of a count data set the incidence of polio in the usa over the period 1970 1983 using a longlinear generalized linear mixed model with an autoregressive correlation structure despite the fact that all of these methods are considered valid modelling techniques we find that parameter estimates and standard errors differ substantially between analyses particularly in the estimation of the parameters describing the random effects distribution a small simulation study is helpful in understanding some of these differences the methods lead to reasonably similar predictions for future observations with small differences observed in some monthly counts

model comparison of generalized linear mixed models generalized linear mixed models glmms have been widely appreciated in biological and medical research maximum likelihood estimation has received a great deal of attention comparatively not much has been done on model comparison or hypotheses testing in this article we propose a path sampling procedure to compute the observed data log likelihood function so that the bayesian information criterion bic can be applied to model comparison or hypothesis testing advantages of the proposed path sampling procedure are discussed two medical data sets are analysed for providing illustrative examples of the proposed methodology

autoregressive spatial smoothing and temporal spline smoothing for mapping rates this article proposes generalized additive mixed models for the analysis of geographic and temporal variability of mortality rates this class of models accommodates random spatial effects and fixed and random temporal components spatiotemporal models that use autoregressive local smoothing across the spatial dimension and b spline smoothing over the temporal dimension are developed the objective is the identification of temporal trends and the production of a series of smoothed maps from which spatial patterns of mortality risks can be monitored over time regions with consistently high rate estimates may be followed for further investigation the methodology is illustrated by analysis of british columbia infant mortality data

a mixed mover stayer model for spatiotemporal two state processes studies of recurring infection or chronic disease often collect longitudinal data on the disease status of subjects two state transitional models are useful for analysis in such studies where at any point in time an individual may be said to occupy either a diseased or disease free state and interest centers on the transition process between states here two additional features are present the data are spatially arranged and it is important to account for spatial correlation in the transitional processes corresponding to different subjects in addition there are subgroups of individuals with different mechanisms of transitions these subgroups are not known a priori and hence group membership must be estimated covariates modulating transitions are included in a logistic additive framework inference for the resulting mixture spatial markov regression model is not straightforward we develop here a monte carlo expectation maximization algorithm for maximum likelihood estimation and a markov chain monte carlo sampling scheme for summarizing the posterior distribution in a bayesian analysis the methodology is applied to a study of recurrent weevil infestation in british columbia forests

nonlinear multilevel models with an application to discrete response data summary a procedure is proposed for the analysis of multilevel nonlinear models using a linearization the case of log linear models for discrete response data is studied in detail nonlinear models arise in a number of circumstances notably when modelling discrete data in this paper we consider the multilevel nonlinear model as in linear multilevel models we shall consider the general case where any of the model coefficients can be random at any level and where the random parameters may also be specified functions of the fixed parameter estimates discussed by h goldstein r prosser and j rasbash in an as yet unpublished report in the next two sections we set out the model and define notation this is followed by a section on estimation and then some examples

a semiparametric method for estimating population size for capture recapture experiments with random covariates in continuous time a semiparametric estimation procedure is proposed to model capture recapture data with the aim of estimating the population size for a closed population individuals covariates are possibly time dependent and missing at noncaptured times and may be measured with error a set of estimating equations ees based on covariate process and capture recapture data is constructed to estimate the relevant parameters and the population size these ees can be solved by an algorithm similar to an em algorithm simulation results show that the proposed procedures work better than the naive estimate in some cases they are even better than ideal estimates for which the true values of covariates are available for all captured subjects over the entire experimental period we apply the method to a capture recapture experiment on the bird species prinia flaviventris in hong kong

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

variance component models with binary response interviewer variability summary interviewer variability in a binary response is an example of a problem requiring variance component estimation in a non normal family the maximum likelihood estimation procedure is derived and used to examine some binary items on a large questionnaire this raises some interesting questions about the use of unbalanced ano va methods with these data two interviewers asking the same question of the same respondent can obtain different responses such interviewer effects create problems in the design and analysis of survey questionnaires collins 1980 gives a review of these problems together with a variety of partial remedies these include pilot studies and careful design to eliminate difficulties with wording of questions and instructions on the questionnaires in spite of these precautions interviewer effects remain questionnaire items are usually categorical and occasionally binary but usually for analysis the questionnaire item is reduced to a binary response it is the variability of the population of interviewers that is of interest and not just the individual interviewers thus this variability is best modelled as a random effect rather than a fixed effect the number of interviews completed by each interviewer varies so that the design is unbalanced thus we have an unbalanced variance component model with a categorical or binary response the intraclass correlation the ratio of the interviewer variance to the sum of all variances in the model is of great interest as a measure of interviewer variability in estimating the variance components the categorical or binary nature of the response is usually ignored and the analysis carried out using analysis of variance this is not maximum likelihood but analysis of variance does have some desirable properties when the error variables are non normal the rule of thumb generally applied is that the analysis of variance is reasonably accurate as long as the proportions in each of the categories of the response are between 0 1 and 0 9 scheffe 1959 p 343 discusses the effects of non normality in a one way balanced model very little theoretical work has been

the statistical performance of an mcf 7 cell culture assay evaluated using generalized linear mixed models and a score test biological assays often utilize experimental designs where observations are replicated at multiple levels and where each level represents a separate component of the assay s overall variance statistical analysis of such data usually ignores these design effects whereas more sophisticated methods would improve the statistical power of assays this report evaluates the statistical performance of an in vitro mcf 7 cell proliferation assay e screen by identifying the optimal generalized linear mixed model glmm that accurately represents the assay s experimental design and variance components our statistical assessment found that 17 oestradiol cell culture assay data were best modelled with a glmm configured with a reciprocal link function a gamma error distribution and three sources of design variation plate to plate well to well and the interaction between plate to plate variation and dose the gamma distributed random error of the assay was estimated to have a coefficient of variation cov 3 2 per cent and a variance component score test described by x lin found that each of the three variance components were statistically significant the optimal glmm also confirmed the estrogenicity of five weakly oestrogenic polychlorinated biphenyls pcbs 17 49 66 74 and 128 based on information criteria the optimal gamma glmm consistently out performed equivalent naive normal and log normal linear models both with and without random effects terms because the gamma glmm was by far the best model on conceptual and empirical grounds and requires only trivially more effort to use we encourage its use and suggest that naive models be avoided when possible copyright 2006 john wiley sons ltd

comparison of statistical methods for analysis of clustered binary observations when correlated observations are obtained in a randomized controlled trial the assumption of independence among observations within cluster likely will not hold because the observations share the same cluster e g clinic physician or subject further the outcome measurements of interest are often binary the objective of this paper is to compare the performance of four statistical methods for analysis of clustered binary observations namely 1 full likelihood method 2 penalized quasi likelihood method 3 generalized estimating equation method 4 fixed effects logistic regression method the first three methods take correlations into account in inferential processes whereas the last method does not type i error rate power bias and standard error are compared across the four statistical methods through computer simulations under varying effect sizes intraclass correlation coefficients number of clusters and number of observations per cluster including large numbers 20 and 100 of observations per cluster the results show that the performance of the full likelihood and the penalized quasi likelihood methods is superior for analysis of clustered binary observations and is not necessarily inferior to that of the fixed effects logistic regression fit even when within cluster correlations are zero

generalized linear models with random effects salamander mating revisited in recent years much effort has been devoted to extending regression methodology to non gaussian data where responses are not independent these methods for dependent responses are suitable for data from longitudinal studies or nested designs however use of these methods for crossed designs seems to have serious limitations due to the intensive computations involved because of the intractable nature of the joint distribution in this paper we cast the problem in a bayesian framework and use a monte carlo method the gibbs sampler to avoid current computational limitations the flexibility of this approach is illustrated by analyzing the interesting salamander mating data reported by mccullagh and nelder 1989 generalized linear models 2nd edition london chapman and hall

bias analysis and simex approach in generalized linear mixed measurement error models abstract we consider generalized linear mixed models glmms for clustered data when one of the predictors is measured with error when the measurement error is additive and normally distributed and the error prone predictor is itself normally distributed we show that the observed data also follow a glmm but with a different fixed effects structure from the original model a different and more complex random effects structure and restrictions on the parameters this characterization enables us to compute the biases that result in common glmms when one ignores measurement error for instance in one common situation the biases in parameter estimates become larger as the number of observations within a cluster increases both for regression coefficients and for variance components parameter estimation is described using the simex method a relatively new functional method that makes no assumptions about the structure of the unobservable predictors simulations and an example illustrate the results

robust estimation in generalized linear mixed models generalized linear mixed models glmms are widely used to analyse non normal response data with extra variation but non robust estimators are still routinely used we propose robust methods for maximum quasi likelihood and residual maximum quasi likelihood estimation to limit the influence of outlying observations in glmms the estimation procedure parallels the development of robust estimation methods in linear mixed models but with adjustments in the dependent variable and the variance component the methods proposed are applied to three data sets and a comparison is made with the nonparametric maximum likelihood approach when applied to a set of epileptic seizure data the methods proposed have the desired effect of limiting the influence of outlying observations on the parameter estimates simulation shows that one of the residual maximum quasi likelihood proposals has a smaller bias than those of the other estimation methods we further discuss the equivalence of two glmm formulations when the response variable follows an exponential family their extensions to robust glmms and their comparative advantages in modelling are described some possible modifications of the robust glmm estimation methods are given to provide further flexibility for applying the method

maximum likelihood variance components estimation for binary data abstract we consider a class of probit normal models for binary data and describe ml and reml estimation of variance components for that class as well as best prediction for the realized values of the random effects ml estimates are calculated using an em algorithm for complicated models em includes a gibbs step the computations are illustrated through two examples

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

hierarchical generalized linear models we consider hierarchical generalized linear models which allow extra error components in the linear predictors of generalized linear models the distribution of these components is not restricted to be normal this allows a broader class of models which includes generalized linear mixed models we use a generalization of henderson s joint likelihood called a hierarchical or h likelihood for inferences from hierarchical generalized linear models this avoids the integration that is necessary when marginal likelihood is used under appropriate conditions maximizing the h likelihood gives fixed effect estimators that are asymptotically equivalent to those obtained from the use of marginal likelihood at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors an adjusted profile h likelihood is shown to give the required generalization of restricted maximum likelihood for the estimation of dispersion components a scaled deviance test for the goodness of fit a model selection criterion for choosing between various dispersion models and a graphical method for checking the distributional assumption of random effects are proposed the ideas of quasi likelihood and extended quasi likelihood are generalized to the new class we give examples of the poisson gamma binomial beta and gamma inverse gamma hierarchical generalized linear models a resolution is proposed for the apparent difference between population averaged and subject specific models a unified framework is provided for viewing and extending many existing methods

estimation in generalized linear models with random effects summary a conceptually very simple but general algorithm for the estimation of the fixed effects random effects and components of dispersion in generalized linear models with random effects is proposed conditions are described under which the algorithm yields approximate maximum likelihood or quasi maximum likelihood estimates of the fixed effects and dispersion components and approximate empirical bayes estimates of the random effects the algorithm is applied to two data sets to illustrate the estimation of components of dispersion and the modelling of overdispersion

a large deviation principle for join the shortest queue we consider a join the shortest queue model which is as follows there are k single fifo servers and m arrival processes the customers from a given arrival process can be served only by the servers from a certain subset of all servers the actual destination is the server with the smallest weighted queue length the arrival processes are assumed to obey a large deviation principle while service is exponential a large deviation principle is established for the queue length process the action functional is expressed in terms of solutions to mathematical programming problems the large deviation limit point is identified as a weak solution to a system of idempotent equations uniqueness of the weak solution follows by trajectorial uniqueness

a nonstationary negative binomial time series with time dependent covariates enterococcus counts in boston harbor boston harbor has a history of poor water quality including contamination by enteric pathogens we conduct a statistical analysis of data collected by the massachusetts water resources authority mwra between 1996 and 2002 to evaluate the effects of court mandated improvements in sewage treatment motivated by the ineffectiveness of standard poisson mixture models and their zero inflated counterparts we propose a new negative binomial model for time series of enterococcus counts in boston harbor where nonstationarity and autocorrelation are modeled using a nonparametric smooth function of time in the predictor without further restrictions this function is not identifiable in the presence of time dependent covariates consequently we use a basis orthogonal to the space spanned by the covariates and use penalized quasi likelihood pql for estimation we conclude that enterococcus counts were greatly reduced near the nut island treatment plant nitp outfalls following the transfer of wastewaters fro

an assessment of estimation methods for generalized linear mixed models with binary outcomes two main classes of methodology have been developed for addressing the analytical intractability of generalized linear mixed models likelihood based methods and bayesian methods likelihood based methods such as the penalized quasi likelihood approach have been shown to produce biased estimates especially for binary clustered data with small clusters sizes more recent methods using adaptive gaussian quadrature perform well but can be overwhelmed by problems with large numbers of random effects and efficient algorithms to better handle these situations have not yet been integrated in standard statistical packages bayesian methods although they have good frequentist properties when the model is correct are known to be computationally intensive and also require specialized code limiting their use in practice in this article we introduce a modification of the hybrid approach of capanu and begg 2011 biometrics 67 371 380 as a bridge between the likelihood based and bayesian approaches by employing bayesian estimation for the variance components followed by laplacian estimation for the regression coefficients we investigate its performance as well as that of several likelihood based methods in the setting of generalized linear mixed models with binary outcomes we apply the methods to three datasets and conduct simulations to illustrate their properties simulation results indicate that for moderate to large numbers of observations per random effect adaptive gaussian quadrature and the laplacian approximation are very accurate with adaptive gaussian quadrature preferable as the number of observations per random effect increases the hybrid approach is overall similar to the laplace method and it can be superior for data with very sparse random effects copyright 2013 john wiley sons ltd

structured additive regression for categorical space time data a mixed model approach motivated by a space time study on forest health with damage state of trees as the response we propose a general class of structured additive regression models for categorical responses allowing for a flexible semiparametric predictor nonlinear effects of continuous covariates time trends and interactions between continuous covariates are modeled by penalized splines spatial effects can be estimated based on markov random fields gaussian random fields or two dimensional penalized splines we present our approach from a bayesian perspective with inference based on a categorical linear mixed model representation the resulting empirical bayes method is closely related to penalized likelihood estimation in a frequentist setting variance components corresponding to inverse smoothing parameters are estimated using approximate restricted maximum likelihood in simulation studies we investigate the performance of different choices for the spatial effect compare the empirical bayes approach to competing methodology and study the bias of mixed model estimates as an application we analyze data from the forest health survey

parametric bootstrap and penalized quasi likelihood inference in conditional autoregressive models this paper discusses a variety of conditional autoregressive car models for mapping disease rates beyond the usual first order intrinsic car model we illustrate the utility and scope of such models for handling different types of data structures to encourage their routine use for map production at statistical and health agencies a simple algorithm for fitting such models is presented this is derived from penalized quasi likelihood pql inference which uses an analogue of best linear unbiased estimation for the regional risk ratios and restricted maximum likelihood for the variance components we offer the practitioner here the use of the parametric bootstrap for inference it is more reliable than standard maximum likelihood asymptotics for inference purposes since relevant hypotheses for the mapping of rates lie on the boundary of the parameter space we illustrate the parametric bootstrap test of the practically relevant and important simplifying hypothesis that there is no spatial autocorrelation although the parametric bootstrap requires computational effort it is straightforward to implement and offers a wealth of information relating to the estimators and their properties the proposed methodology is illustrated by analysing infant mortality in the province of british columbia in canada

joint selection in mixed models using regularized pql abstractthe application of generalized linear mixed models presents some major challenges for both estimation due to the intractable marginal likelihood and model selection as we usually want to jointly select over both fixed and random effects we propose to overcome these challenges by combining penalized quasi likelihood pql estimation with sparsity inducing penalties on the fixed and random coefficients the resulting approach referred to as regularized pql is a computationally efficient method for performing joint selection in mixed models a key aspect of regularized pql involves the use of a group based penalty for the random effects sparsity is induced such that all the coefficients for a random effect are shrunk to zero simultaneously which in turn leads to the random effect being removed from the model despite being a quasi likelihood approach we show that regularized pql is selection consistent that is it asymptotically selects the true set of fixed and random effects in the setti

bias correction in generalised linear mixed models with a single component of dispersion summary general expressions are derived for the asymptotic biases in three approximate estimators of regression coefficients and variance component for small values of the variance component in generalised linear mixed models with canonical link function and a single source of extraneous variation the estimators involve first and second order laplace expansions of the integrated likelihood and a related procedure known as penalised quasilikelihood numerical studies of a series of matched pairs of binary outcomes show that the first order estimators of the variance component are seriously biased easily computed correction factors produce satisfactory estimators of small variance components comparable to those obtained with a second order laplace expansion and markedly improve the asymptotic performance for larger values for a series of matched pairs of binomial observations the variance correction factors rapidly approach one as the binomial denominators increase these results greatly extend the range of parameter values for which the approximate estimation procedures have satisfactory asymptotic properties

conditional second order generalized estimating equations for generalized linear and nonlinear mixed effects models generalized linear and nonlinear mixed effects models are used extensively in health care research including applications in pharmacokinetics clinical trials and epidemiology because the underlying model may be nonlinear in the random effects there will generally be no closed form expression for the marginal likelihood or indeed for the marginal moments consequently estimation is often carried out either using numerical integration techniques or by approximating the marginal likelihood and or marginal moments using first order expansion methods an advantage of the first order methods is that they do not necessarily require specification of a conditional like lihood to estimate the regression parameters of interest however in many cases they may not take full advantage of the fact that the conditional variance depends on both fixed and random effects in this article we propose using conditional second order generalized estimating equations cgee2 to estimate both fixed and random effects param

aberrant crypt foci and semiparametric modeling of correlated binary data motivated by the spatial modeling of aberrant crypt foci acf in colon carcinogenesis we consider binary data with probabilities modeled as the sum of a nonparametric mean plus a latent gaussian spatial process that accounts for short range dependencies the mean is modeled in a general way using regression splines the mean function can be viewed as a fixed effect and is estimated with a penalty for regularization with the latent process viewed as another random effect the model becomes a generalized linear mixed model in our motivating data set and other applications the sample size is too large to easily accommodate maximum likelihood or restricted maximum likelihood estimation reml so pairwise likelihood a special case of composite likelihood is used instead we develop an asymptotic theory for models that are sufficiently general to be used in a wide variety of applications including but not limited to the problem that motivated this work the splines have penalty parameters that must converge to zero asymptotically we derive theory for this along with a data driven method for selecting the penalty parameter a method that is shown in simulations to improve greatly upon standard devices such as likelihood crossvalidation finally we apply the methods to the data from our experiment acf we discover an unexpected location for peak formation of acf

estimation of generalized linear latent variable models generalized linear latent variable models gllvms as defined by bartholomew and knott enable modelling of relationships between manifest and latent variables they extend structural equation modelling techniques which are powerful tools in the social sciences however because of the complexity of the log likelihood function of a gllvm an approximation such as numerical integration must be used for inference this can limit drastically the number of variables in the model and can lead to biased estimators we propose a new estimator for the parameters of a gllvm based on a laplace approximation to the likelihood function and which can be computed even for models with a large number of variables the new estimator can be viewed as an m estimator leading to readily available asymptotic properties and correct inference a simulation study shows its excellent finite sample properties in particular when compared with a well established approach such as lisrel a real data example on the measurement of wealth for the computation of multidimensional inequality is analysed to highlight the importance of the methodology

hierarchical bayesian spatiotemporal analysis of revascularization odds using smoothing splines hierarchical bayesian models are proposed for over dispersed longitudinal spatially correlated binomial data this class of models accounts for correlation among regions by using random effects and allows a flexible modelling of spatiotemporal odds by using smoothing splines the aim is i to develop models which will identify temporal trends of odds and produce smoothed maps including regional effects ii to specify markov chain monte carlo mcmc inference for fitting such models iii to study the sensitivity of such bayesian binomial spline spatiotemporal analyses to prior assumptions and iv to compare mechanisms for assessing goodness of fit an analysis of regional variation for revascularization odds of patients hospitalized for acute coronary syndrome in quebec motivates and illustrates the methods developed

standard errors of prediction in generalized linear mixed models abstract the unconditional mean squared error of prediction umsep is widely used as a measure of prediction variance for inferences concerning linear combinations of fixed and random effects in the classical normal theory mixed model but the umsep is inappropriate for generalized linear mixed models where the conditional variance of the random effects depends on the data when the random effects describe variation between independent small domains and domain specific prediction is of interest we propose a conditional mean squared error of prediction cmsep as a general measure of prediction variance the cmsep is shown to be the sum of the conditional variance and a positive correction that accounts for the sampling variability of parameter estimates we derive a second order correct estimate of the cmsep that consists of three components a a plug in estimate of the conditional variance b a plug in estimate of a taylor series approximation to the correction term and c a bootstrap estimate of

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

a type of restricted maximum likelihood estimator of variance components in generalised linear mixed models summary the maximum likelihood estimator of the variance components in a linear model can be biased downwards restricted maximum likelihood reml corrects this problem by using the likelihood of a set of residual contrasts and is generally considered superior however this original restricted maximum likelihood definition does not directly extend beyond linear models we propose a reml type estimator for generalised linear mixed models by correcting the bias in the profile score function of the variance components the proposed estimator has the same consistency properties as the maximum likelihood estimator if the number of parameters in the mean and variance components models remains fixed however the estimator of the variance components has a smaller finite sample bias a simulation study with a logistic mixed model shows that the proposed estimator is effective in correcting the downward bias in the maximum likelihood estimator

binary time series modeling with application to adhesion frequency experiments repeated adhesion frequency assay is the only published method for measuring the kinetic rates of cell adhesion cell adhesion plays an important role in many physiological and pathological processes traditional analysis of adhesion frequency experiments assumes that the adhesion test cycles are independent bernoulli trials this assumption often can be violated in practice motivated by the analysis of repeated adhesion tests a binary time series model incorporating random effects is developed a goodness of fit statistic is introduced to assess the adequacy of distribution assumptions on the dependent binary data with random effects the asymptotic distribution of the goodness of fit statistic is derived and its finite sample performance is examined through a simulation study application of the proposed methodology to real data from a t cell experiment reveals some interesting information including the dependency between repeated adhesion tests

consistent estimators in generalized linear mixed models abstract a simple method based on simulated moments is proposed for estimating the fixed effects and variance components in a generalized linear mixed model glmm it is shown that the method is not only computationally attractive but also leads to consistent estimators on the other hand simulation shows that the method can be quite inefficient

nonlinear component of variance models summary general aspects of nonlinearity in the context of component of variance models are discussed and two special topics are examined in detail firstly simple procedures both formal and informal are proposed for describing departures from normal theory linear models transformation models are shown to be a special case of a more general formulation and data on blood pressure are analyzed in illustration secondly an approximate likelihood is proposed and its accurate performance is examined numerically using examples of exponential regression and the analysis of several related 2 x 2 tables in the latter example the approximate score test has improved power over the mantelhaenszel test

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

a comparison of mixed effects logistic regression models for binary response data with two nested levels of clustering we compare mixed effects logistic regression models for binary response data with two nested levels of clustering the comparison of these models occurs in the context of developmental toxicity data sets for which multiple types of outcomes first level are measured on each rat pup second level nested within a litter third level because the nested nature of such data is occasionally accommodated by ignoring one level of clustering we consider three models i a three level model adjusting for clustering due to both pup and litter m1 ii a two level model adjusting for just pup m2 and iii another two level model adjusting for just litter m3 the three types of effects of interest are i differences among malformation types first level effects ii differences among groups of pups for example sex of pup second level effects and iii differences among groups of litters for example dose third level effects simulations and data analyses suggest that the m3 model leads to more bias than the m1 or m2 models for all three types of effects in terms of coverage of confidence intervals for fixed effects log odds ratio parameters the m1 model achieves nominal coverage whereas the m2 model reduces coverage for the third level effects and the m3 model obtains poor coverage for both first and second level effects these reductions in coverage for certain model parameter combinations worsen as baseline risk increases the data analyses support these simulation based conclusions to some extent copyright 1999 john wiley sons ltd

power analyses for longitudinal trials and other clustered designs summary existing methods for power and sample size estimation for longitudinal and other clustered study designs have limited applications in this paper we review and extend existing approaches to improve these limitations in particular we focus on power analysis for the two most popular approaches for clustered

maximizing generalized linear mixed model likelihoods with an automated monte carlo em algorithm summary two new implementations of the em algorithm are proposed for maximum likelihood fitting of generalized linear mixed models both methods use random independent and identically distributed sampling to construct monte carlo approximations at the e step one approach involves generating random samples from the exact conditional distribution of the random effects given the data by rejection sampling using the marginal distribution as a candidate the second method uses a multivariate t importance sampling approximation in many applications the two methods are complementary rejection sampling is more efficient when sample sizes are small whereas importance sampling is better with larger sample sizes monte carlo approximation using random samples allows the monte carlo error at each iteration to be assessed by using standard central limit theory combined with taylor series methods specifically we construct a sandwich variance estimate for the maximizer at each approximate e step this suggests a rule for automatically increasing the monte carlo sample size after iterations in which the true em step is swamped by monte carlo error in contrast techniques for assessing monte carlo error have not been developed for use with alternative implementations of monte carlo em algorithms utilizing markov chain monte carlo e step approximations three different data sets including the infamous salamander data of mccullagh and nelder are used to illustrate the techniques and to compare them with the alternatives the results show that the methods proposed can be considerably more efficient than those based on markov chain monte carlo algorithms however the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension

modelling spatial disease rates using maximum likelihood this paper concerns maximum likelihood estimation for a generalized linear mixed model glmm useful for modelling spatial disease rates the model allows for log linear covariate adjustment and local smoothing of rates through estimation of spatially correlated random effects the covariance structure of the random effects is based on a recently proposed model which parameterizes spatial dependence through the inverse covariance matrix a markov chain monte carlo algorithm for performing maximum likelihood estimation for this model is described results of a computer simulation study that compared maximum likelihood ml and penalized quasi likelihood pql estimators are presented compared with pql ml produced less biased estimates of the intercept but the ml estimates were slightly more variable estimates of the other regression coefficients were unbiased and nearly identical for the two methods ml estimators of the random effects standard deviation and spatial correlation were more biased than the corresponding pql estimators the conclusion is that ml estimators for glmms cannot be expected to perform better than pql for small samples copyright 2000 john wiley sons ltd

modeling the random effects covariance matrix for longitudinal data with covariates measurement error longitudinal data occur frequently in practice such as medical studies and life sciences generalized linear mixed models glmms are commonly used to analyze such data it is typically assumed that the random effects covariance matrix is constant among subjects in these models in many situations however the correlation structure may differ among subjects and ignoring this heterogeneity can lead to biases in model parameters estimate recently lee et al developed a heterogeneous random effects covariance matrix for glmms for error free covariates covariates measured with error also happen frequently in the longitudinal data set up eg blood pressure and cholesterol level ignoring this issue in the data may produce bias in model parameters estimate and lead to wrong conclusions in this paper we propose an approach to properly model the random effects covariance matrix based on covariates in the class of glmms where we also have covariates measured with error the resulting parameters from the decomposition of random effects covariance matrix have a sensible interpretation and can be easily modeled without the concern of positive definiteness of the resulting estimator the performance of the proposed approach is evaluated through simulation studies which show that the proposed method performs very well in terms of bias mean squared error and coverage rate an application of the proposed method is also provided using a longitudinal data from manitoba follow up study

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

interval oscillation criteria for nonlinear impulsive delay differential equations with damping term in this paper a class of second order nonlinear impulsive delay differential equations with damping term is studied three factors impulse delay and damping that affect the interval qualitative properties of solutions of equations are taken into account together several interval oscillation criteria which generalize some known ones such as in huang and feng 2010 xing and zheng 2008 and liu and xu 2007 are established moreover examples are also given to illustrate the effectiveness and non emptiness of our results

reml estimation with exact covariance in the logistic mixed model residual maximum likelihood reml estimation is adapted to certain logistic mixed models for which representation of the unconditional mean as a linear function of the fixed effects is possible only the first two moments of the unconditional distribution need be evaluated and except for the form of the covariance the maximization algorithm carries over directly from linear models the exact unconditional covariance is computed from the logistic normal mixture for input into the algorithm residual log likelihood plots provide a means of inference for the dispersion components the taylor series approximation to this covariance besides exhibiting insufficient accuracy fails to be positive definite for large values of the dispersion components as a consequence reml loglikelihood plots based on this approximate covariance attribute misleadingly high precision to the dispersion component estimates the method is presented in the context of a salamander mating experiment in which random effects corresponding to male and female animals occur in a crossed design analyses of these data by several methods are compared

robust estimation in generalised linear mixed models summary we propose robust methods for estimation of parameters of interest in an extended generalised linear mixed model in which only the conditional means of the responses given the random effects are specified a first step estimator 0 of the vector 0 of parameters is obtained by solving a system of estimating equations it is shown that 9 is consistent if furthermore the conditional variances are correctly specified a second step estimator 0 can be obtained by solving a system of optimal estimating equations whose coefficients a are estimated by 0 it is shown that 0 maintains the asymptotic optimality simulations a also indicate solid improvement of 0 over 0 two examples involving real data analysis are considered the methods are developed under a more general framework so that they may be applied to more general estimation problems

analysis of clustered binary outcomes using within cluster paired resampling summary conditional logistic regression clr is useful for analyzing clustered binary outcome data when interest lies in estimating a cluster specific exposure parameter while treating the dependency arising from random cluster effects as a nuisance clr aggregates unmeasured cluster specific factors into a cluster specific baseline risk and is invalid in the presence of unmodeled heterogeneous covariate effects or within cluster dependency we propose an alternative resampling based method for analyzing clustered binary outcome data within cluster paired resampling wcpr which allows for within cluster dependency not solely due to baseline heterogeneity for example dependency may be in part caused by heterogeneity in response to an exposure across clusters due to unmeasured cofactors when both clr and wcpr are valid our simulations suggest that the two methods perform comparably when clr is invalid wcpr continues to have good operating characteristics for illustration we apply both wcpr and clr to a periodontal data set where there is heterogeneity in response to exposure across clusters

tweedie family of generalized linear models with distribution free random effects for skewed longitudinal data generalized linear mixed models have played an important role in the analysis of longitudinal data however traditional approaches have limited flexibility in accommodating skewness and complex correlation structures in addition the existing estimation approaches generally rely heavily on the specifications of random effects distributions therefore the corresponding inferences are sometimes sensitive to the choice of random effect distributions under certain circumstance in this paper we incorporate serially dependent distribution free random effects into tweedie generalized linear models to accommodate a wide range of skewness and covariance structures for discrete and continuous longitudinal data an optimal estimation of our model has been developed using the orthodox best linear unbiased predictors of random effects our approach unifies population averaged and subject specific inferences our method is illustrated through the analyses of patient controlled analgesia data and framingham cholesterol data

additive multilevel item structure models with random residuals item modeling for explanation and item generation an additive multilevel item structure amis model with random residuals is proposed the model includes multilevel latent regressions of item discrimination and item difficulty parameters on covariates at both item and item category levels with random residuals at both levels the amis model is useful for explanation purposes and also for prediction purposes as in an item generation context the parameters can be estimated with an alternating imputation posterior algorithm that makes use of adaptive quadrature and the performance of this algorithm is evaluated in a simulation study

generalized linear mixed models with varying coefficients for longitudinal data the routinely assumed parametric functional form in the linear predictor of a generalized linear mixed model for longitudinal data may be too restrictive to represent true underlying covariate effects we relax this assumption by representing these covariate effects by smooth but otherwise arbitrary functions of time with random effects used to model the correlation induced by among subject and within subject variation due to the usually intractable integration involved in evaluating the quasi likelihood function the double penalized quasi likelihood dpql approach of lin and zhang 1999 journal of the royal statistical society series b61 381 400 is used to estimate the varying coefficients and the variance components simultaneously by representing a nonparametric function by a linear combination of fixed effects and random effects a scaled chi squared test based on the mixed model representation of the proposed model is developed to test whether an underlying varying coefficient is a polynomial of certain degree we evaluate the performance of the procedures through simulation studies and illustrate their application with indonesian children infectious disease data

a semiparametric recurrent events model with time varying coefficients we consider a recurrent events model with time varying coefficients motivated by two clinical applications a random effects gaussian frailty model is used to describe the intensity of recurrent events the model can accommodate both time varying and time constant coefficients the penalized spline method is used to estimate the time varying coefficients laplace approximation is used to evaluate the penalized likelihood without a closed form the smoothing parameters are estimated in a similar way to variance components we conduct simulations to evaluate the performance of the estimates for both time varying and time independent coefficients we apply this method to analyze two data sets a stroke study and a child wheeze study

inference in generalized additive mixed modelsby using smoothing splines generalized additive mixed models are proposed for overdispersed and correlated data which arise frequently in studies involving clustered hierarchical and spatial designs this class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression while accounting for correlation between observations by using random effects we estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi likelihood because numerical integration is often required by maximizing the objective functions double penalized quasi likelihood is proposed to make approximate inference frequentist and bayesian inferences are compared a key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software a bias correction procedure is also proposed to improve the performance of double penalized quasi likelihood for sparse data we illustrate the method with an application to infectious disease data and we evaluate its performance through simulation

semiparametric frailty models for clustered failure time data summary we consider frailty models with additive semiparametric covariate effects for clustered failure time data we propose a doubly penalized partial likelihood dppl procedure to estimate the nonparametric functions using smoothing splines we show that the dppl estimators could be obtained from fitting an augmented working frailty model with parametric covariate effects whereas the nonparametric functions being estimated as linear combinations of fixed and random effects and the smoothing parameters being estimated as extra variance components this approach allows us to conveniently estimate all model components within a unified frailty model framework we evaluate the finite sample performance of the proposed method via a simulation study and apply the method to analyze data from a study of sexually transmitted infections sti

fence methods for mixed model selection many model search strategies involve trading off model fit with model complexity in a penalized goodness of fit measure asymptotic properties for these types of procedures in settings like linear regression and arma time series have been studied but these do not naturally extend to nonstandard situations such as mixed effects models where simple definition of the sample size is not meaningful this paper introduces a new class of strategies known as fence methods for mixed model selection which includes linear and generalized linear mixed models the idea involves a procedure to isolate a subgroup of what are known as correct models of which the optimal model is a member this is accomplished by constructing a statistical fence or barrier to carefully eliminate incorrect models once the fence is constructed the optimal model is selected from among those within the fence according to a criterion which can be made flexible in addition we propose two variations of the fence the first is a stepwise procedure to handle situations of many predictors the second is an adaptive approach for choosing a tuning constant we give sufficient conditions for consistency of fence and its variations a desirable property for a good model selection procedure the methods are illustrated through simulation studies and real data analysis

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

empirical bayes estimation of random effects parameters in mixed effects logistic regression models we extend an approach for estimating random effects parameters under a random intercept and slope logistic regression model to include standard errors thereby including confidence intervals the procedure entails numerical integration to yield posterior empirical bayes eb estimates of random effects parameters and their corresponding posterior standard errors we incorporate an adjustment of the standard error due to kass and steffey ks 1989 journal of the american statistical association 84 717 726 to account for the variability in estimating the variance component of the random effects distribution in assessing health care providers with respect to adult pneumonia mortality comparisons are made with the penalized quasi likelihood pql approximation approach of breslow and clayton 1993 journal of the american statistical association 88 9 25 and a bayesian approach to make comparisons with an eb method previously reported in the literature we apply these approaches to crossover trials data previously analyzed with the estimating equations eb approach of waclawiw and liang 1994 statistics in medicine 13 541 551 we also perform simulations to compare the proposed ks and pql approaches these two approaches lead to eb estimates of random effects parameters with similar asymptotic bias however for many clusters with small cluster size the proposed ks approach does better than the pql procedures in terms of coverage of nominal 95 confidence intervals for random effects estimates for large cluster sizes and a few clusters the pql approach performs better than the ks adjustment these simulation results agree somewhat with those of the data analyses

a two part random effects model for semicontinuous longitudinal data a semicontinuous variable has a portion of responses equal to a single value typically 0 and a continuous often skewed distribution among the remaining values in cross sectional analyses variables of this type may be described by a pair of regression models for example a logistic model for the probability of nonzero response and a conditional linear model for the mean response given that it is nonzero we extend this two part regression approach to longitudinal settings by introducing random coefficients into both the logistic and the linear stages fitting a two part random effects model poses computational challenges similar to those found with generalized linear mixed models we obtain maximum likelihood estimates for the fixed coefficients and variance components by an approximate fisher scoring procedure based on high order laplace approximations to illustrate we apply the technique to data from the adolescent alcohol prevention trial examining reported recent alcohol use for students in g

approximate estimation in generalized linear mixed models with applications to the rasch model this article discusses two different approaches to estimate the difficulty parameters fixed effects parameters and the variance of latent traits variance components in the mixed rasch model the first one is the generalized estimating equations gee2 which uses an approximation of the marginal likelihood to derive the joint moments whilst the second approach uses the maximum of the approximate likelihood we illustrate these methods with a simulation study and with an analysis of real data from a quality of life

on casting random effects models in a survival framework summary logistic random effects models are often employed in the analysis of correlated binary data however fitting these models is challenging since the marginal distribution of the response variables is analytically intractable often the random effects are treated as missing data for constructing traditional data augmentation algorithms we create a novel alternative data augmentation scheme that simplifies the likelihood based inference for logistic random effects models we cast the random effects model in a survival framework where each binary response is the censoring indicator for a survival time that is treated as additional missing data under this augmentation framework the conditional expectations are free of unknown regression parameters such a construction has a particular advantage that in the case of discrete covariates the score equations for regression parameters have analytical solutions consequently one does not need to resort to a search algorithm in estimating the regression parameters we further create a parameter expansion scheme for logistic random effects models under this survival data augmentation framework the proposed data augmentation is illustrated when the random effects distribution follows a multivariate gaussian and multivariate t distribution the performance of the method is assessed through simulation studies and a real data analysis

bias correction in generalized linear mixed models with multiple components of dispersion abstract general formulas are derived for the asymptotic bias in regression coefficients and variance components estimated by penalized quasi likelihood pql in generalized linear mixed models with canonical link function and multiple sets of independent random effects easily computed correction matrices result in variance component estimates that have satisfactory asymptotic behavior for small values of the variance components and significantly reduce bias for larger values both first order and second order correction procedures are developed for regression coefficients estimated by pql the methods are illustrated through an analysis of an experiment on salamander matings involving crossed male and female random effects and their properties are evaluated in a simulation study

heterogeneity in mantel haenszel type models summary the mantel haenszel problem involves inferences about a common odds ratio in a set of 2 x 2 tables although it is a fairly standard practice to test whether the odds ratios are indeed constant there is remarkably little methodology available for proceeding when there is evidence of some heterogeneity our interest is in models where the log odds ratios ok for tables k 1 2 k are thought of as a sample from a population with mean 0 and standard deviation a and inferences are desired regarding the parameters 0 a by mantel haenszel type models we mean to include the generalization involving pairs of poisson observations rather than 2 x 2 tables where the ok are logarithms of ratios of the poisson means within pairs direct computation of the likelihood function for 0 a in these settings involves numerical integration and the main point here is a simple approximation to this likelihood the approximation is based on laplace s method and is very accurate for practical applications inference regarding the parameter 0 of primary interest may be made from the profile likelihood function an alternative approach to likelihood methods based on approximations to the marginal means and variances is also considered the methods explored here can be readily generalized to settings where the parameters ok depend on covariables as well

concordance correlation coefficient applied to discrete data in any field in which decisions are subject to measurements interchangeability between the methods used to obtain these measurements is essential to consider methods as interchangeable a certain degree of agreement is needed between the measurements they provide the concordance correlation coefficient is an index that assesses the strength of agreement and it has been widely applied in situations in which measurements are made on a continuous scale recently the concordance correlation coefficient has been defined as a specific intraclass correlation coefficient estimated by the variance components of a normal normal mixed linear model although this coefficient was defined for the continuous scale case it may also be used with a discrete scale in this case the data are often transformed and normalized and the concordance correlation is applied this study discusses the expression of the concordance correlation coefficient for discrete poisson data by means of the poisson normal generalized linear mixed model the behaviour of the concordance correlation coefficient estimate is assessed by means of a simulation study in which the estimates were compared using four models three normal normal mixed models with raw data log transformed data and square root transformed data and the poisson normal generalized linear mixed model an example is provided in which two different methods are used to measure cd34 cells

large deviations for markov processes with discontinuous statistics ii random walks let 1 and 2 be borel probability measures on d with finite moment generating functions the main theorem in this paper proves the large deviation principle for a random walk whose transition mechanism is governed by 1 when the walk is in the left halfspace 1 x d x 1 0 and whose transition mechanism is governed by 2 when the walk is in the right halfspace 2 x d x 1 0 when the measures 1 and 2 are equal the main theorem reduces to cramer s theorem