multilevel time series models with applications to repeated measures data the analysis of repeated measures data can be conducted efficiently using a two level random coefficients model a standard assumption is that the within individual level 1 residuals are uncorrelated in some cases especially where measurements are made close together in time this may not be reasonable and this additional correlation structure should also be modelled a time series model for such data is proposed which consists of a standard multilevel model for repeated measures data augmented by an autocorrelation model for the level 1 residuals first and second order autoregressive models are considered in detail together with a seasonal component both discrete and continuous time are considered and it is shown how the autocorrelation parameters can themselves be structured in terms of further explanatory variables the models are fitted to a data set consisting of repeated height measurements on children

bayesian latent variable models for clustered mixed outcomes a general framework is proposed for modelling clustered mixed outcomes a mixture of generalized linear models is used to describe the joint distribution of a set of underlying variables and an arbitrary function relates the underlying variables to be observed outcomes the model accommodates multilevel data structures general covariate effects and distinct link functions and error distributions for each underlying variable within the framework proposed novel models are developed for clustered multiple binary unordered categorical and joint discrete and continuous outcomes a markov chain monte carlo sampling algorithm is described for estimating the posterior distributions of the parameters and latent variables because of the flexibility of the modelling framework and estimation procedure extensions to ordered categorical outcomes and more complex data structures are straightforward the methods are illustrated by using data from a reproductive toxicity study

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

estimation in covariance components models abstract estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes the estimation of fixed and random effects when the variances and covariances are known is presented in bayesian terms point estimates of the unknown variances and covariances are computed using the em algorithm for maximum likelihood estimation from incomplete data the techniques are illustrated with data on law schools field mice and professional football teams

a stochastic model for analysis of longitudinal aids data abstract in this paper we analyze serial cd4 t cell measurements from the los angeles portion of the multicenter aids cohort study our emphasis is on developing a plausible and parsimonious model to describe the stochastic process underlying the patterns of cd4 measurements the stochastic process that we use enables us to investigate the concept of derivative tracking for which it is assumed that the rank order of the individual s slopes is maintained over time a general model for the analysis of longitudinal repeated measures data is where yi tij is the measurement of subject i at time tij x tij represents fixed effect terms z tij bi represents random effect terms wi tij is a stochastic process allowing correlation between measurements and e ij is measurement error in the simplest case x tij and z tij contain the times of measurements for wi tij we use a two parameter integrated ornstein uhlenbeck ou process the ou process is the continuous mean zero gaussian markov proces

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

analysis of change in the presence of informative censoring application to a longitudinal clinical trial of progressive renal disease the rate of change in a continuous variable measured serially over time is often used as an outcome in longitudinal studies or clinical trials when patients terminate the study before the scheduled end of the study there is a potential for bias in estimation of rate of change using standard methods which ignore the missing data mechanism these methods include the use of unweighted generalized estimating equations methods and likelihood based methods assuming an ignorable missing data mechanism we present a model for analysis of informatively censored data based on an extension of the two stage linear random effects model where each subject s random intercept and slope are allowed to be associated with an underlying time to event the joint distribution of the continuous responses and the time to event variable are then estimated via maximum likelihood using the em algorithm and using the bootstrap to calculate standard errors we illustrate this methodology and compare it to simpler approaches and usual maximum likelihood using data from a multi centre study of the effects of diet and blood pressure control on progression of renal disease the modification of diet in renal disease mdrd study sensitivity analyses and simulations are used to evaluate the performance of this methodology in the context of the mdrd data under various scenarios where the drop out mechanism is ignorable as well as non ignorable

linear mixed models with flexible distributions of random effects for longitudinal data normality of random effects is a routine assumption for the linear mixed model but it may be unrealistic obscuring important features of among individual variation we relax this assumption by approximating the random effects density by the seminonparameteric snp representation of gallant and nychka 1987 econometrics 55 363 390 which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior controlled by a user chosen tuning parameter an advantage is that the marginal likelihood may be expressed in closed form so inference may be carried out using standard optimization techniques we demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality and we illustrate the approach via simulation and using longitudinal data from the framingham study

an approach to comparing treatments based on repeated measures summary a mixed effects model is assumed for comparing treatments based on repeated measurements the covariance structure of the data is interpreted as some systematic inhomogeneity of individual profiles along the time axis rather than as serial correlation then a method of comparing treatment effects is proposed as well as that of testing the homogeneity of individual profiles a follow up analysis of residuals for the resulting model is also mentioned

a multivariate reduced rank growth curve model with unbalanced data a multivariate reduced rank growth curve model is proposed that extends the univariate reducedrank growth curve model to the multivariate case in which several response variables are measured over multiple time points the proposed model allows us to investigate the relationships among a number of response variables in a more parsimonious way than the traditional growth curve model in addition the method is more flexible than the traditional growth curve model for example response variables do not have to be measured at the same time points nor the same number of time points it is also possible to apply various kinds of basis function matrices with different ranks across response variables it is not necessary to specify an entire set of basis functions in advance examples are given for illustration

nonparametric regression analysis of longitudinal data abstract nonparametric methods are developed for estimating the dose effect when a response consists of correlated observations over time measured in a dose response experiment the methods can also be applied to data collected from a completely randomized design experiment methods are developed for the detection and description of the effects of dose time and their interaction the methods allow for individual variation in the timing and number of observations a generalization allowing baseline covariates to be incorporated is addressed these results may be used in an exploratory fashion in the process of building a random effects model for longitudinal data

a general framework for random effects survival analysis in the cox proportional hazards setting summary the use of random effects modeling in statistics has increased greatly in recent years the introduction of such modeling into event time analysis has proceeded more slowly however previously random effects models for survival data have either required assumptions regarding the form of the baseline hazard function or restrictions on the classes of models that can be fit in this paper we develop a method of random effect analysis of survival data the hierarchical cox model that is an extension of cox s original formulation in that the baseline hazard function remains unspecified this method also allows an arbitrary distribution for the random effects we accomplish this using markov chain monte carlo methods in a bayesian setting the method is illustrated with three models for a dataset with times to multiple occurrences of mammory tumors for 48 rats treated with a carcinogen and then randomized to either treatment or control this analysis is more satisfying than standard approaches such as studying the first event for each subject which does not fully use the data or assuming independence which in this case would overestimate the precision

permutation tests for random effects in linear mixed models summary inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis as a result the asymptotic null distribution of the wald score and likelihood ratio tests will not have the typical 2 distribution although it has been proved that the correct asymptotic distribution is a mixture of 2 distributions the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect as alternatives we present two permutation tests one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic both methods involve weighted residuals with the weights determined by the among and within subject variance components the null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples we examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients

functional linear discriminant analysis for irregularly sampled curves we introduce a technique for extending the classical method of linear discriminant analysis lda to data sets where the predictor variables are curves or functions this procedure which we call functional linear discriminant analysis flda is particularly useful when only fragments of the curves are observed all the techniques associated with lda can be extended for use with flda in particular flda can be used to produce classifications on new test curves give an estimate of the discriminant function between classes and provide a one or two dimensional pictorial representation of a set of curves we also extend this procedure to provide generalizations of quadratic and regularized discriminant analysis

mean squared error of empirical predictor the term empirical predictor refers to a two stage predictor of a linear combination of fixed and random effects in the first stage a predictor is obtained but it involves unknown parameters thus in the second stage the unknown parameters are replaced by their estimators in this paper we consider mean squared errors mse of empirical predictors under a general setup where ml or reml estimators are used for the second stage we obtain second order approximation to the mse as well as an estimator of the mse correct to the same order the general results are applied to mixed linear models to obtain a second order approximation to the mse of the empirical best linear unbiased predictor eblup of a linear mixed effect and an estimator of the mse of eblup whose bias is correct to second order the general mixed linear model includes the mixed anova model and the longitudinal model as special cases

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

sequential monitoring for comparison of changes in a response variable in clinical studies the spending function approach proposed by lan and demets 1983 biometrika 70 659 663 for sequential monitoring of clinical trials is applied to situations where comparison of changes in a continuous response variable between two groups is the primary concern death loss to follow up and missed visits could cause follow up measurements to be right censored or missing for some participants furthermore the probability of being censored may be dependent on the parameter value of the response variable informative censoring we propose to compare treatment effects by comparing areas under the expected response change curves between the two groups when the response curves are linear as a function of time in both groups this comparison is equivalent to comparing the rates of change in the response variable covariances of the sequential test statistics are derived conditions for having independent increments are presented for studies designed to evaluate long term treatment effects spending functions obtained by shifting the usual spending functions kim and demets 1987 biometrika 74 149 154 to the right and then rescaling to the remaining interval are also proposed such a shifted spending function is applied to the monitoring plan for the lung health study anthonisen 1989 american review of respiratory diseases 140 871 872

dynamic conditionally linear mixed models for longitudinal data we develop a new class of models dynamic conditionally linear mixed models for longitudinal data by decomposing the within subject covariance matrix using a special cholesky decomposition here dynamic means using past responses as covariates and conditional linearity means that parameters entering the model linearly may be random but nonlinear parameters are nonrandom this setup offers several advantages and is surprisingly similar to models obtained from the first order linearization method applied to nonlinear mixed models first it allows for flexible and computationally tractable models that include a wide array of covariance structures these structures may depend on covariates and hence may differ across subjects this class of models includes e g all standard linear mixed models antedependence models and vonesh carter models second it guarantees the fitted marginal covariance matrix of the data is positive definite we develop methods for bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited

sensitivity analysis for nonrandom dropout a local influence approach diggle and kenward 1994 applied statistics 43 49 93 proposed a selection model for continuous longitudinal data subject to nonrandom dropout it has provoked a large debate about the role for such models the original enthusiasm was followed by skepticism about the strong but untestable assumptions on which this type of model invariably rests since then the view has emerged that these models should ideally be made part of a sensitivity analysis this paper presents a formal and flexible approach to such a sensitivity assessment based on local influence cook 1986 journal of the royal statistical society series b 48 133 169 the influence of perturbing a missing at random dropout model in the direction of nonrandom dropout is explored the method is applied to data from a randomized experiment on the inhibition of testosterone production in rats

modeling subject specific phase dependent effects and variations in longitudinal responses via a geometric brownian motion process we address statistical issues regarding modeling a collection of longitudinal response trajectories characterized by the presence of subject specific phase dependent effects and variation to accommodate these two time varying individual characteristics we employ a geometric stochastic differential equation for modeling based on a brownian motion process and develop a two step paradigm for statistical analysis this paradigm reverses the order of statistical inference in random effects model we first extract individual information about phase dependent treatment effects and volatility parameters for all subjects then we derive the association relationship between the parameters characterizing the individual longitudinal trajectories and the corresponding covariates by means of multiple regression analysis the stochastic differential equation model and the two step paradigm together provide significant advantages both in modeling flexibility and in computational efficiency the modeling flexibility is due to the easy adaptation of temporal change points for subject specific phase transition in treatment effects whereas the computational efficiency benefits in part from the independent increment property of brownian motion that avoids high dimensional integration we demonstrate our modeling approach and statistical analysis on a real data set of longitudinal measurements of disease activity scores from a rheumatoid arthritis study copyright 2011 john wiley sons ltd

multivariate linear mixed models for multiple outcomes we propose a multivariate linear mixed mlmm for the analysis of multiple outcomes which generalizes the latent variable model of sammel and ryan the proposed model assumes a flexible correlation structure among the multiple outcomes and allows a global test of the impact of exposure across outcomes in contrast to the sammel ryan model the mlmm separates the mean and correlation parameters so that the mean estimation will remain reasonably robust even if the correlation is misspecified the model is applied to birth defects data where continuous data on the size of infants who were exposed to anticonvulsant medications in utero are compared to controls

comments on multiphase systolic algorithms for spectral decomposition and reply comments that the paper by liu and yao see ibid vol 40 no 1 p 190 1992 presented an efficient algorithm for spectral decomposition but the parallel algorithm and architecture for the hessenberg reduction has a problem the matrix obtained from the unitary similarity transformation is not necessarily a hessenberg matrix the authors reply that the hessenberg reduction described is not in the hessenberg form only the first column is in its proper form there are many ways to overcome such a problem one simple way is to use the multiphase rectangular systolic array

influence diagnostics for linear longitudinal models abstract influence diagnostics are important for analyzing cross sectional regression studies because they allow the analyst to understand the impact of individual observations on the estimated regression model in this article we consider the role of influence diagnostics in subject specific longitudinal models diagnostics are proposed under both fixed and random subject effects our approach is based on subject deletion which in this setting involves deleting a group of correlated observations we develop partial influence statistics to understand the combined impact of observations from a subject on population parameters simple computational formulas make the procedures feasible finally we illustrate the use of our new influence statistics by examining a dataset to model a taxpayer s charitable givings

application of nonlinear models with random coefficients to growth data summary the application of nonlinear random coefficient models to the analysis of growth curve data is described the approach is further developed for the estimation of mean growth curves and their variability from mark recapture data when the age of an animal at first capture is unknown but the time between successive captures is known these methods are of wide applicability as shown by the analysis of data on the growth of emus dromaius novaehollandiae noisy scrub birds atrichornis clamosus and whelks dicathais aegaota simulations of a mark recapture experiment on the rock spiny lobster panulirus argus in mexico using a known growth curve showed the effectiveness of the approach in estimating both the population s growth curve and the variability in individual growth

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