the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

markov and semi markov switching linear mixed models used to identify forest tree growth components tree growth is assumed to be mainly the result of three components i an endogenous component assumed to be structured as a succession of roughly stationary phases separated by marked change points that are asynchronous among individuals ii a time varying environmental component assumed to take the form of synchronous fluctuations among individuals and iii an individual component corresponding mainly to the local environment of each tree to identify and characterize these three components we propose to use semi markov switching linear mixed models i e models that combine linear mixed models in a semi markovian manner the underlying semi markov chain represents the succession of growth phases and their lengths endogenous component whereas the linear mixed models attached to each state of the underlying semi markov chain represent in the corresponding growth phase both the influence of time varying climatic covariates environmental component as fixed effects and interindividual heterogeneity individual component as random effects in this article we address the estimation of markov and semi markov switching linear mixed models in a general framework we propose a monte carlo expectation maximization like algorithm whose iterations decompose into three steps i sampling of state sequences given random effects ii prediction of random effects given state sequences and iii maximization the proposed statistical modeling approach is illustrated by the analysis of successive annual shoots along corsican pine trunks influenced by climatic covariates resume d auteur

approaches for empirical bayes confidence intervals abstract parametric empirical bayes eb methods of point estimation date to the landmark paper by james and stein 1961 interval estimation through parametric empirical bayes techniques has a somewhat shorter history which was summarized by laird and louis 1987 in the exchangeable case one obtains a naive eb confidence interval by simply taking appropriate percentiles of the estimated posterior distribution of the parameter where the estimation of the prior parameters hyperparameters is accomplished through the marginal distribution of the data unfortunately these naive intervals tend to be too short since they fail to account for the variability in the estimation of the hyperparameters that is they do not attain the desired coverage probability in the eb sense defined by morris 1983a b they also provide no statement of conditional calibration rubin 1984 in this article we propose a conditional bias correction method for developing em intervals that corrects these deficiencies

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

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

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

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

standard errors for em estimation log likelihood the well known sem algorithm uses the second approach we consider three additional algorithms one that uses the first approach and two that use the second we evaluate the complexity and precision of these three and the sem algorithm in seven examples the first is a single parameter example used to give insight the others are three examples in each of two areas of em application poisson mixture models and the estimation of covariance from incomplete data the examples show that there are algorithms that are much simpler and more accurate than the sem algorithm hopefully their simplicity will increase the availability of standard error estimates in em applications it is shown that as previously conjectured a symmetry diagnostic can accurately estimate errors arising from numerical differentiation some issues related to the speed of the em algorithm and algorithms that differentiate the em operator are identified

stable estimates for source solution of critical fractal burgers equation abstract in this paper we provide two sided estimates for the source solution of d dimensional critical fractal burgers equation u t 2 b u u q 0 q 1 d 1 2 b r d by the density function of the isotropic stable process

censored observations repeated measures and mixed effects models an approach using the em algorithm and normal errors summary maximum likelihood estimation of a vector regression parameter and variance com ponents is considered for the mixed effects model when observations are right censored a general scheme of estimation is given using the em algorithm and detailed results found for the model with between and within block variation this model is applied to the logarithms of survival times from a repeated measures design in this paper we consider the estimation of the parameters when some of the observa tions are right censored this can occur with repeated measurements on subjects where the response is a waiting time and the variability in log times is modelled by a normal distribution alternative models for this situation are given by clayton cuzick 1985 and their methods involve partially parametric techniques using ranks and are based on models involving extreme value log gamma and logistic distributions we are concerned with use of the em algorithm dempster laird rubin 1977 to estimate the parameters of the mixed model for uncensored data hartley rao 1967 have considered this solution the analysis leads to a method which is straightforward to implement for the repeated measures model dempster et al 1984 give an account of the method with uncensored observations in 2 we consider a general implementation of the em algorithm for the pure random effects model with censored data and then in 3 we give explicit results for the simple random effects model section 4 uses these results for the analysis of repeated measures and 5 discusses some miscellaneous points arising finally in 6 we give an example involving skin graft data where there has been extensive analysis using matched pairs techniques

hierarchical models a current computational perspective 2000 hierarchical models a current computational perspective journal of the american statistical association vol 95 no 452 pp 1312 1316

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

ascent based monte carlo expectation maximization summary the expectation maximization em algorithm is a popular tool for maximizing likelihood functions in the presence of missing data unfortunately em often requires the evaluation of analytically intractable and high dimensional integrals the monte carlo em mcem algorithm is the natural extension of em that employs monte carlo methods to estimate the relevant integrals typically a very large monte carlo sample size is required to estimate these integrals within an acceptable tolerance when the algorithm is near convergence even if this sample size were known at the onset of implementation of mcem its use throughout all iterations is wasteful especially when accurate starting values are not available we propose a data driven strategy for controlling monte carlo resources in mcem the algorithm proposed improves on similar existing methods by recovering em s ascent i e likelihood increasing property with high probability being more robust to the effect of user defined inputs and handling classical monte carlo and markov chain monte carlo methods within a common framework because of the first of these properties we refer to the algorithm as ascent based mcem we apply ascent based mcem to a variety of examples including one where it is used to accelerate the convergence of deterministic em dramatically

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

sampling bias and logistic models summary in a regression model the joint distribution for each finite sample of units is determined by a function px y depending only on the list of covariate values x x u1 x un on the sampled units no random sampling of units is involved in biological work random sampling is frequently unavoidable in which case the joint distribution p y x depends on the sampling scheme regression models can be used for the study of dependence provided that the conditional distribution p y x for random samples agrees with px y as determined by the regression model for a fixed sample having a non random configuration x the paper develops a model that avoids the concept of a fixed population of units thereby forcing the sampling plan to be incorporated into the sampling distribution for a quota sample having a predetermined covariate configuration x the sampling distribution agrees with the standard logistic regression model with correlated components for most natural sampling plans such as sequential or simple random sampling the conditional distribution p y x is not the same as the regression distribution unless px y has independent components in this sense most natural sampling schemes involving binary random effects models are biased the implications of this formulation for subject specific and population averaged procedures are explored

the hierarchical logistic regression model for multilevel analysis abstract a hierarchical logistic regression model is proposed for studying data with group structure and a binary response variable the group structure is defined by the presence of micro observations embedded within contexts macro observations and the specification is at both of these levels at the first micro level the usual logistic regression model is defined for each context the same regressors are used in each context but the micro regression coefficients are free to vary over contexts at the second level the micro coefficients are treated as functions of macro regressors an empirical bayes estimation procedure is proposed for estimating the micro and macro coefficients explicit formulas are provided that are computationally feasible for large scale data analyses these include an algorithm for finding the maximum likelihood estimates of the covariance components representing within and between macro equation error variability the methodology is applied to world fertility survey data

bias reduction of approximate maximum likelihood estimates for heritability in threshold models approximate maximum likelihood estimates such as penalized quasi likelihood and iterated reweighted reml estimates for heritability or intraclass correlation in threshold models for binary data can be seriously biased two approaches to correct for this bias are studied with emphasis on animal breeding models for binary data scope for reduction of bias of heritability estimates is found to be slim because of the commonly large number of fixed effects in animal breeding models minimal dimensions for the data are identified such that bias and root mean squared error are of modest size and useful inference on heritability or intraclass correlation is feasible

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

analysis of repeated ordered categorical outcomes with possibly missing observations and time dependent covariates abstract this article describes a method for comparing responses in two groups of subjects observed repeatedly at a common set of observation times when the response is an ordered categorical outcome the method which allows both time dependent covariates and missing observations consists of two analytic steps in the first step the data are analyzed separately at each occasion using a regression model chosen from the class of models for ordinal data proposed by mccullagh 1980 the joint asymptotic distribution of the estimates of these occasion specific regression coefficients and a consistent estimator of their asymptotic covariance matrix are obtained without imposing any parametric model of dependence on the repeated observations in the second step this asymptotic distribution together with appropriate simultaneous inference procedures is used to characterize the overall difference between groups and the variation in group differences over time the missing data process may differ between gro

a hybrid pairwise likelihood method a modification to the pairwise likelihood method is proposed which aims to improve the estimation of the marginal distribution parameters this is achieved by replacing the pairwise likelihood score equations for estimating such parameters by the optimal linear combinations of the marginal score functions a further advantage of the proposed estimator of marginal parameters over pairwise likelihood is that it is robust to misspecification of the bivariate distributions as long as the univariate marginal distributions are correctly specified while alternating logistic regression can be seen as a special case of the proposed method it is shown that an existing generalization of alternating logistic regression applicable to ordinal data is not the same as and is inferior to the proposed method because it replaces certain conditional densities by pseudodensities that assume working independence the fitting of the multivariate negative binomial distribution is another scenario involving intractable likelihood that calls for the use of pairwise likelihood methods and the superiority of the modified method is demonstrated in a simulation study two examples based on the analyses of salamander mating and patient controlled analgesia data demonstrate the usefulness of the proposed method the possibility of combining optimally the pairwise rather than marginal scores is also considered and its difficulty and potential are discussed

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

publication bias and meta analysis for 2x2 tables an average markov chain monte carlo em algorithm summary a major difficulty in meta analysis is publication bias studies with positive outcomes are more likely to be published than studies reporting negative or inconclusive results correcting for this bias is not possible without making untestable assumptions in this paper a sensitivity analysis is discussed for the meta analysis of 2 2 tables using exact conditional distributions a markov chain monte carlo em algorithm is used to calculate maximum likelihood estimates a rule for increasing the accuracy of estimation and automating the choice of the number of iterations is suggested

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

existence and uniqueness of nonnegative solution of m gb 1 queueing model abstract in this paper we set up m g b 1 queueing model as an abstract cauchy problem and prove the existence of a unique positive solution using the theory of strongly continuous semigroups of operators

using em to obtain asymptotic variance covariance matrices the sem algorithm abstract the expectation maximization em algorithm is a popular and often remarkably simple method for maximum likelihood estimation in incomplete data problems one criticism of em in practice is that asymptotic variance covariance matrices for parameters e g standard errors are not automatic byproducts as they are when using some other methods such as newton raphson in this article we define and illustrate a procedure that obtains numerically stable asymptotic variance covariance matrices using only the code for computing the complete data variance covariance matrix the code for em itself and code for standard matrix operations the basic idea is to use the fact that the rate of convergence of em is governed by the fractions of missing information to find the increased variability due to missing information to add to the complete data variance covariance matrix we call this supplemented em algorithm the sem algorithm theory and particular examples reinforce the conclusion that the sem alg

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

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

the analysis of binomial data by a generalized linear mixed model summary methods for generalized linear models are extended to provide estimates of location and variance parameters for mixed models fitted to binomial data formed by classifying samples from an underlying normal distribution the method estimates the parameters directly on the underlying scale for a balanced one way random effects model the variance estimator simplifies to the usual analysis of variance one the estimation of variances and the prediction of random effects for binomial traits is required by animal breeders the predictors given are analogous to best linear unbiased predictors henderson 1973 but differ from those presented by harville mee 1984

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

maximum likelihood estimation for probit linear mixed models with correlated random effects the probit normal model for binary data mcculloch 1994 journal of the american statistical association 89 330 335 is extended to allow correlated random effects to obtain maximum likelihood estimates we use the em algorithm with its m step greatly simplified under the assumption of a probit link and its e step made feasible by gibbs sampling standard errors are calculated by inverting a monte carlo approximation of the information matrix rather than via the sem algorithm a method is also suggested that accounts for the monte carlo variation explicitly as an illustration we present a new analysis of the famous salamander mating data unlike previous analyses we find it necessary to introduce different variance components for different species of animals finally we consider models with correlated errors as well as correlated random effects

bayesian analysis of constrained parameter and truncated data problems using gibbs sampling abstract constrained parameter problems arise in a wide variety of applications including bioassay actuarial graduation ordinal categorical data response surfaces reliability development testing and variance component models truncated data problems arise naturally in survival and failure time studies ordinal data models and categorical data studies aimed at uncovering underlying continuous distributions in many applications both parameter constraints and data truncation are present the statistical literature on such problems is very extensive reflecting both the problems widespread occurrence in applications and the methodological challenges that they pose however it is striking that so little of this applied and theoretical literature involves a parametric bayesian perspective from a technical viewpoint this perhaps is not difficult to understand the fundamental tool for bayesian calculations in typical realistic models is multidimensional numerical integration which often is problem

a mixed model procedure for analyzing ordered categorical data a mixed model procedure is developed for predicting the value of an ordered categorical response from knowledge of various predictor variables this procedure resembles the best linear unbiased procedure of henderson 1975 biometrics 31 423 447 for predicting the value of a quantitative response the approach is based on a mixed model version of the threshold model in which it is assumed that the observed category is determined by the value of an underlying unobservable continuous response that follows a mixed linear model the results are illustrated by an application to the problem of predicting the degree of difficulty that will be experienced by a dairy cow in the birth of her calf

likelihood inference in a correlated probit regression model correlated binary observations arise in a variety of applications for example in animal studies the term litter effect is used to describe the greater alikeness of responses within a litter as compared to that between litters at a given set of experimental conditions in a setting that motivated this work cells on individual atomic bomb survivors were scored as to the presence or absence of chromosomal aberrations the number of aberrant cells varied among subjects at a specified radiation exposure estimate age at exposure city of exposure and sex in a manner that substantially exceeds that consistent with a binomial error structure otake prentice 1984 this extra binomial variation or overdispersion reflects the fact that the binary response of cells from a given survivor tend to be more alike than are the responses of cells from distinct survivors having the same estimated radiation exposure level and other covariate values such overdispersion may result from individual differences in susceptibility to radiation damage from omitted covariates or most plausibly in this setting from substantial random errors in the estimated radiation exposure levels failure to acknowledge overdispersion may lead to serious underestimation of the standard errors associated with regression parameter estimates and to unduly precise inferences

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

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

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