models for longitudinal data with random effects and ar 1 errors abstract for longitudinal data on several individuals linear models that contain both random effects across individuals and autocorrelation in the within individual errors are studied a score test for autocorrelation in the within individual errors for the conditional independence random effects model is first developed an explicit maximum likelihood estimation procedure using the scoring method for the model with random effects and autoregressive ar 1 errors is then derived empirical bayes estimation of the random effects and prediction of future responses of an individual based on this random effects with ar 1 errors model are also considered a numerical example is presented to illustrate these methods

linear and nonlinear mixed effects models for censored hiv viral loads using normal independent distributions summary hiv rna viral load measures are often subjected to some upper and lower detection limits depending on the quantification assays hence the responses are either left or right censored linear and nonlinear mixed effects models with modifications to accommodate censoring are routinely used to analyze this type of data and are based on normality assumptions for the random terms however those analyses might not provide robust inference when the normality assumptions are questionable in this article we develop a bayesian framework for censored linear and nonlinear models replacing the gaussian assumptions for the random terms with normal independent ni distributions the ni is an attractive class of symmetric heavy tailed densities that includes the normal student s t slash and the contaminated normal distributions as special cases the marginal likelihood is tractable using approximations for nonlinear models and can be used to develop bayesian case deletion influence diagnostics based on the kullback leibler divergence the newly developed procedures are illustrated with two hiv aids studies on viral loads that were initially analyzed using normal censored mixed effects models as well as simulations

a hybrid estimator in nonlinear and generalised linear mixed effects models summary a hybrid method that combines laplace s approximation and monte carlo simulations to evaluate integrals in the likelihood function is proposed for estimation of the parameters in nonlinear mixed effects models that assume a normal parametric family for the random effects simulations show that these parametric estimates of fixed effects are close to the nonparametric estimates even though the mixing distribution is far from the assumed normal parametric family an asymptotic theory of this hybrid method for parametric estimation without requiring the true mixing distribution to belong to the assumed parametric family is developed to explain these results this hybrid method and its asymptotic theory are also extended to generalised linear mixed effects models

smoothing spline nonlinear nonparametric regression models almost all of the current nonparametric regression methods such as smoothing splines generalized additive models and varyingcoefficients models assume a linear relationship when nonparametric functions are regarded as parameters in this article we propose a general class of smoothing spline nonlinear nonparametric models that allow nonparametric functions to act nonlinearly they arise in many fields as either theoretical or empirical models our new estimation methods are based on an extension of the gauss newton method to infinite dimensional spaces and the backfitting procedure we extend the generalized cross validation and generalized maximum likelihood methods to estimate smoothing parameters we establish connections between some nonlinear nonparametric models and nonlinear mixed effects models we derive approximate bayesian confidence intervals for inference we illustrate the methods with an application to term structure of interest rates and conduct simulations to evaluate the finite sampl

simultaneous inference for semiparametric nonlinear mixed effects models with covariate measurement errors and missing responses semiparametric nonlinear mixed effects nlme models are flexible for modeling complex longitudinal data covariates are usually introduced in the models to partially explain interindividual variations some covariates however may be measured with substantial errors moreover the responses may be missing and the missingness may be nonignorable we propose two approximate likelihood methods for semiparametric nlme models with covariate measurement errors and nonignorable missing responses the methods are illustrated in a real data example simulation results show that both methods perform well and are much better than the commonly used naive method

mixed effects nonlinear regression for unbalanced repeated measures repeated measures data such as clinical pharmacokinetic data growth data and dose response data are often inherently nonlinear with respect to a given response function and are frequently incomplete and or unbalanced nonlinear random effects models together with a variety of estimation procedures have been proposed for the analysis of such data this paper is concerned with a straightforward procedure for estimating and comparing the parameters of a generalized mixed effects nonlinear regression model the asymptotic properties of the proposed estimators are given and large sample tests of hypothesis provided the results are applied to in vitro data on the water transport kinetics of hemodialyzers used in the treatment of patients with chronic renal failure

a population pharmacokinetic model with time dependent covariates measured with errors we propose a population pharmacokinetic pk model with time dependent covariates measured with errors this model is used to model s oxybutynin s kinetics following an oral administration of ditropan and allows the distribution rate to depend on time dependent covariates blood pressure and heart rate which are measured with errors we propose two two step estimation methods the second order two step method with numerical solutions of differential equations 2ordernd and the second order two step method with closed form approximate solutions of differential equations 2orderad the proposed methods are computationally easy and require fitting a linear mixed model at the first step and a nonlinear mixed model at the second step we apply the proposed methods to the analysis of the ditropan data and evaluate their performance using a simulation study our results show that the 2ordernd method performs well while the 2orderad method can yield pk parameter estimators that are subject to considerable biases

model based analyses of bioequivalence crossover trials using the stochastic approximation expectation maximisation algorithm in this work we develop a bioequivalence analysis using nonlinear mixed effects models nlmem that mimics the standard noncompartmental analysis nca we estimate nlmem parameters including between subject and within subject variability and treatment period and sequence effects we explain how to perform a wald test on a secondary parameter and we propose an extension of the likelihood ratio test for bioequivalence we compare these nlmem based bioequivalence tests with standard nca based tests we evaluate by simulation the nca and nlmem estimates and the type i error of the bioequivalence tests for nlmem we use the stochastic approximation expectation maximisation saem algorithm implemented in monolix we simulate crossover trials under h 0 using different numbers of subjects and of samples per subject we simulate with different settings for between subject and within subject variability and for the residual error variance the simulation study illustrates the accuracy of nlmem based geometric means estimated with the saem algorithm whereas the nca estimates are biased for sparse design nca based bioequivalence tests show good type i error except for high variability for a rich design type i errors of nlmem based bioequivalence tests wald test and likelihood ratio test do not differ from the nominal level of 5 type i errors are inflated for sparse design we apply the bioequivalence wald test based on nca and nlmem estimates to a three way crossover trial showing that omnitrope sandoz gmbh kundl austria powder and solution are bioequivalent to genotropin pfizer pharma gmbh karlsruhe germany nlmem based bioequivalence tests are an alternative to standard nca based tests however caution is needed for small sample size and highly variable drug

regular subalgebras and nilpotent orbits of real graded lie algebras abstract for a semisimple lie algebra over the complex numbers dynkin 1952 developed an algorithm to classify the regular semisimple subalgebras up to conjugacy by the inner automorphism group for a graded semisimple lie algebra over the complex numbers vinberg 1979 showed that a classification of a certain type of regular subalgebras called carrier algebras yields a classification of the nilpotent orbits in a homogeneous component of that lie algebra here we consider these problems for graded semisimple lie algebras over the real numbers first we describe an algorithm to classify the regular semisimple subalgebras of a real semisimple lie algebra this also yields an algorithm for listing up to conjugacy the carrier algebras in a real graded semisimple real algebra we then discuss what needs to be done to obtain a classification of the nilpotent orbits from that such classifications have applications in differential geometry and theoretical physics our algorithms are implemented in the language of the computer algebra system gap using our package corelg we report on example computations

a bayesian population model with hierarchical mixture priors applied to blood count data abstract population pharmacokinetic and pharmacodynamic studies require analyzing nonlinear growth curves fit to multiple measurements from study subjects we propose a class of nonlinear population models with nonparametric second stage priors for analyzing such data the proposed models apply a flexible class of mixtures to implement the nonparametric second stage the discussion is based on a pharmacodynamic study involving longitudinal data consisting of hematologic profiles i e blood counts measured over time of cancer patients undergoing chemotherapy we describe a full posterior analysis in a bayesian framework this includes prediction of future observations profiles and end points for new patients estimation of the mean response function for observed individuals and inference on population characteristics the mixture model is specified and given a hyperprior distribution by means of a dirichlet processes prior on the mixing measure estimation is implemented by a combination of various m

bootstrap approach for constructing confidence intervals for population pharmacokinetic parameters i a use of bootstrap standard error in population pharmacokinetic studies one of the main objectives is to estimate population pharmacokinetic parameters specifying the population distributions of pharmacokinetic parameters confidence intervals for population pharmacokinetic parameters are generally estimated by assuming the asymptotic normality which is a large sample property that is a property which holds for the cases where sample sizes are large enough in actual clinical trials however sample sizes are limited and not so large in general likelihood functions in population pharmacokinetic modelling include a multiple integral and are quite complicated we hence suspect that the sample sizes of actual trials are often not large enough for assuming the asymptotic normality and that the asymptotic confidence intervals underestimate the uncertainties of the estimates of population pharmacokinetic parameters as an alternative to the asymptotic normality approach we can employ a bootstrap approach this paper proposes a bootstrap standard error approach for constructing confidence intervals for population pharmacokinetic parameters comparisons between the asymptotic and bootstrap confidence intervals are made through applications to a simulated data set and an actual phase i trial

high dimensional maximum marginal likelihood item factor analysis by adaptive quadrature although the bock aitkin likelihood based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations a serious limitation of the method is its reliance on fixed point gauss hermite g h quadrature in the solution of the likelihood equations and likelihood ratio tests when the number of latent dimensions is large computational considerations require that the number of quadrature points per dimension be few but with large numbers of items the dispersion of the likelihood given the response pattern becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space in this paper we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data in particular we show that adaptive g h quadrature combined with mean and covariance adjustments at each iteration of an em algorithm produces an accurate fast converging solution with as few as two points per dimension evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions unlike an earlier application of adaptive gibbs sampling to this problem by meng and schilling the simulations also confirm the validity of the present method in calculating likelihood ratio chi square statistics for determining the number of factors required in the model finally we apply the method to a sample of real data from a test of teacher qualifications

an em algorithm for nonlinear random effects models the pharmaceutical industry is currently interested in the population approach and population models also known as mixed effects models and random effects models depending on the precise form population models are useful in that they can account for both withinand between individual sources of variability and serial correlation within individual observations when analyzing unbalanced repeated measures data the modelling of population pharmacodynamic or pharmacokinetic profiles typically involves nonlinear random effects models each individual s observations are modelled by identical up to unknown parameter values nonlinear regression models with the distribution of the observations or a transformation of the observations about expected responses taken to be normal with the degree of variability described by a variance model between individual variability is modelled by a population distribution for the individual regression parameter values random effects in a parametric analysis the population distribution is taken to be normal the parameters of which along with the parameters of the variance model are known as the population parameters maximum likelihood estimation of the population parameters for nonlinear random effects models was pioneered by beal and sheiner 1979 and since then a number of algorithms have appeared for approximate maximum likelihood including steimer et al 1984 lindstrom and bates 1990 beal and sheiner 1992 and mentre and gomeni 1995 all of these algorithms are approximate in some way for a summary see beal and sheiner 1992 wolfinger 1993 pinheiro and bates 1994 and davidian and giltinan 1995 in this paper an em algorithm for exact maximum likelihood estimation is introduced an em algorithm obtaining maximum likelihood estimates for linear random effects models was introduced by dempster laird and rubin 1977 laird and ware 1982 lindstrom and bates 1988 jennrich and schluchter 1986 and liu and rubin 1994 all describe hybrid em algorithms for the linear random effects model a true em algorithm for the linear model is described by jamshidian and jennrich 1993 mentre and gomeni 1995 describe an approximate em algorithm for nonlinear random effects models and from the algorithm given in this paper it can be seen clearly how their approximations arise the present algorithm uses monte carlo methods to perform the e step a strategy previously adopted in an altogether different model by guo and thompson 1994 guo and thompson require a gibbs sampler that is a markov chain monte carlo method for their e step but the present algorithm uses independent samples in section 2 of this paper the nonlinear random effects model is described section 3 gives the em algorithm without random effect covariates while section 4 gives the modified algorithm in the

shape invariant modeling of circadian rhythms with random effects and smoothing spline anova decompositions medical studies often collect physiological and or psychological measurements over time from multiple subjects to study dynamics such as circadian rhythms under the assumption that the expected response functions of all subjects are the same after shift and scale transformations shape invariant models have been applied to analyze this kind of data the shift and scale parameters provide efficient and interpretable data summaries while the common shape function is usually modeled nonparametrically to provide flexibility however due to the deterministic nature of the shift and scale parameters potential correlations within a subject are ignored furthermore the shape of the common function may depend on other factors such as disease in this article we propose shape invariant mixed effects models a second stage model with fixed and random effects is used to model individual shift and scale parameters a second stage smoothing spline anova model is used to study potential covariate effects on the common shape function we apply our methods to a real data set to investigate disease effects on circadian rhythms of cortisol a hormone that is affected by stress we find that patients with cushing s syndrome lost circadian rhythms and their 24 hour means were elevated to very high levels patients with major depression had the same circadian shape and phases as normal subjects however their 24 hour mean levels were elevated and amplitudes were dampened for some patients

semi parametric latent process model for longitudinal ordinal data application to cognitive decline ordinal and quantitative discrete data are frequent in biomedical and neuropsychological studies we propose a semi parametric model for the analysis of the change over time of such data in longitudinal studies a threshold model is defined where the outcome value depends on the current value of an underlying gaussian latent process the latent process model is a gaussian linear mixed model with a non parametric function of time f t to model the expected change over time this model includes random effects and a stochastic error process to flexibly handle correlation between repeated measures the function f t and all the model parameters are estimated by penalized likelihood using a cubic spline approximation for f t the smoothing parameter is estimated by an approximate cross validation criterion confidence bands may be computed for the estimated curves for the latent process and using a monte carlo approach for the outcome in its natural scale the method is applied to the paquid cohort data to compare the time course over 14 years of two cognitive scores in a sample of 350 future alzheimer patients and in a matched sample of healthy subjects

linear discriminant models for unbalanced longitudinal data this paper discusses statistical methods for the classification of observations into one of two or more groups based on longitudinal observations measurements on subjects in longitudinal medical studies are often collected at different times and on a different number of occasions classical multivariate methods for linear discriminant analysis are difficult to apply to repeated measurements due to the highly unbalanced structure observed in these data linear models for the analysis of longitudinal data proposed by laird and ware and non linear models proposed by lindstrom and bates can be used to estimate population parameters for a discriminant model that classifies individuals into distinct predefined groups or populations an example is presented using data from a study in 150 pregnant women in santiago chile in order to predict normal versus abnormal pregnancy outcomes

comparing linear and nonlinear mixed model approaches to cosinor analysis the cosinor model used for variables governed by circadian and other biological rhythms is a nonlinear model in the amplitude and acrophase parameters that has a linear representation upon transformation with linear cosinor analysis amplitude and acrophase for each harmonic can be computed as nonlinear functions of the estimated linear regression coefficients here a flexible mixed model approach to cosinor analysis is considered where the fixed effect parameters may enter nonlinearly as acrophase and amplitude for each harmonic or linearly after transformation to regression coefficients in addition the random effects may enter nonlinearly as subject specific deviations from the acrophases and amplitudes or linearly as subject specific deviations from the regression coefficients it is also possible for the fixed effects to enter nonlinearly while the random effects enter linearly additionally we evaluate whether including higher order linear harmonic terms as random effects that is rao khatri covariance adjustment improves precision applying the delta method to nonlinear functions of the parameters from linear mixed cosinor models to obtain approximate variances produces results that are often identical to results from nonlinear mixed models consequently traditional linear cosinor analysis can often be used to estimate and compare the nonlinear parameters of interest that is amplitudes and acrophases via the delta method this is advantageous since the nonlinear mixed model may have convergence difficulties for more complex models however for some multiple group analyses the linear cosinor transformation should not be used and we clarify when the two methods are equivalent and when they differ

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

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

maximum likelihood computations with repeated measures application of the em algorithm abstract the purpose of this article is to consider the use of the em algorithm dempster laird and rubin 1977 for both maximum likelihood ml and restricted maximum likelihood reml estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure anderson 1973 several models and methods of analysis have been proposed in recent years for repeated measures data ware 1985 presented an overview because the em algorithm is a general purpose iterative method for computing ml estimates with incomplete data it has often been used in this particular setting dempster et al 1977 andrade and helms 1984 jennrich and schluchter 1985 there are two apparently different approaches to using the em algorithm in this setting in one application each experimental unit is observed under a standard protocol specifying measurements at each of n occasions or under n conditions and incompleteness implies that the number of measurements actua

a general approach for two stage analysis of multilevel clustered non gaussian data in this article we propose a two stage approach to modeling multilevel clustered non gaussian data with sufficiently large numbers of continuous measures per cluster such data are common in biological and medical studies utilizing monitoring or image processing equipment we consider a general class of hierarchical models that generalizes the model in the global two stage gts method for nonlinear mixed effects models by using any square root n consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model and by extending the stage 2 model to accommodate random effects from multiple levels of clustering the second stage model is a standard linear mixed effects model with normal random effects but the cluster specific distributions conditional on random effects can be non gaussian this methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest for estimation of the population parameters we propose a conditional restricted maximum likelihood creml approach and establish the asymptotic properties of the creml estimators the proposed general approach is illustrated using quartiles as cluster specific parameters estimated in the first stage and applied to the data example from a collagen fibril development study we demonstrate using simulations that in samples with small numbers of independent clusters the creml estimators may perform better than conditional maximum likelihood estimators which are a direct extension of the estimators from the gts method

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

self modeling for two dimensional response curves two dimensional response curves are an important experimental outcome in speech kinematics and other areas of research these parameterized curves are usually obtained by recording the two dimensional location of an object over time in this setting time is the independent variable and the x and y locations on specified coordinate axes define the multivariate response collections of such parameterized curves can be obtained either from one subject or from a number of different subjects each producing one or several realizations of the response curve when only one dependent variable is observed over time and no parametric model is specified self modeling regression semor is an attractive modeling approach semor assumes that each of a collection of curves differs from a smooth average shape function by some simple parametric transformation of the coordinate axes usually linear we will describe the extension of semor to two dimensional parameterized curves using affine transformations of a two dimensional time parameterized shape function

influence of confounding factors on designs for dose effect relationship estimates three types of designs can be used to estimate the drug dose effect relationship during phase ii clinical trials parallel dose designs parallel cross over designs x and dose escalation designs symbol see text despite the use of non linear mixed effect models the potential influence of confounding factors on symbol see text designs has not been previously fully elucidated we undertook simulations to investigate this for all three experimental designs we found that i monotonic spontaneous evolution of the effect ev did not affect the maximum effect estimation emax and the dose giving 50 per cent of this ed50 ii ev similar to a regression to the mean gave rise to biases for symbol see text designs iii the introduction of a pharmacodynamic carry over generates important biases and imprecision for symbol see text designs even when the carry over is adjusted for iv the introduction of non responders resulted in bias and imprecision for both emax and ed50 in all three designs

a non linear mixed effect modelling approach for metabolite correction of the arterial input function in pet studies abstract quantitative pet studies with arterial blood sampling usually require the correction of the measured total plasma activity for the presence of metabolites in particular if labelled metabolites are found in the plasma in significant amounts their presence has to be accounted for because it is the concentration of the parent tracer which is required for data quantification this is achieved by fitting a parent plasma fraction ppf model to discrete metabolite measurements the commonly used method is based on an individual approach i e for each subject the ppf model parameters are estimated from its own metabolite samples which are in general sparse and noisy this fact can compromise the quality of the reconstructed arterial input functions and consequently affect the quantification of tissue kinetic parameters in this study we proposed a non linear mixed effect modelling nlmem approach to describe metabolite kinetics since nlmem has been developed to provide robust parameter estimates in the case of sparse and or noisy data it has the potential to be a reliable method for plasma metabolite correction three different pet datasets were considered 11c phno 54 scans 11c pib 22 scans and 11c dasb 30 scans for each tracer both simulated and measured data were considered and nlmem performance was compared with that provided by individual analysis results showed that nlmem provided improved estimates of the plasma parent input function over the individual approach when the metabolite data were sparse or contained outliers

a case study in exploratory functional data analysis geometrical features of the internal carotid artery this pilot study is a product of the aneurisk project a scientific program that aims at evaluating the role of vascular geometry and hemodynamics in the pathogenesis of cerebral aneurysms by means of functional data analyses we explore the aneurisk dataset to highlight the relations between the geometric features of the internal carotid artery expressed by its radius profile and centerline curvature and the aneurysm location after introducing a new similarity index for functional data we eliminate ancillary variability of vessel radius and curvature profiles through an iterative registration procedure we then reduce data dimension by means of functional principal components analysis last a quadratic discriminant analysis of functional principal components scores allows us to discriminate patients with aneurysms in different districts

the matrix logarithmic covariance model abstract a flexible method is introduced to model the structure of a covariance matrix c and study the dependence of the covariances on explanatory variables by observing that for any real symmetric matrix a the matrix exponential transformation c exp a is a positive definite matrix because there is no constraint on the possible values of the upper triangular elements on a any possible structure of interest can be imposed on them the method presented here is not intended to replace the existing special models available for a covariance matrix but rather to provide a broad range of further structures that supplements existing methodology maximum likelihood estimation procedures are used to estimate the parameters and the large sample asymptotic properties are obtained a simulation study and two real life examples are given to illustrate the method introduced

equivalent theorems of the convergence between proximal type algorithms abstract the main aim of this work is to give the equivalence of the convergence between proximal type algorithms our convergence analysis covers rockafellar type proximal point algorithm with a weak contraction for finding zeros of accretive operators as well as browder and halpern type iterations and viscosity approximation methods for finding fixed points of nonexpansive type mappings in banach spaces we also provide simple proofs of some results of wong sahu and yao n c wong d r sahu j c yao solving variational inequalities involving nonexpansive type mappings nonlinear anal 69 12 2008 4732 4753

unbalanced repeated measures models with structured covariance matrices the question of how to analyze unbalanced or incomplete repeated measures data is a common problem facing analysts we address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within subject covariances models that can be fit include standard univariate and multivariate models with incomplete data random effects models and models with time series and factor analytic error structures we describe newton raphson and fisher scoring algorithms for computing maximum likelihood estimates and generalized em algorithms for computing restricted and unrestricted maximum likelihood estimates an example fitting several models to a set of growth data is included

some simple methods for estimating intraindividual variability in nonlinear mixed effects models methods are proposed to improve both population and individual inference within the nonlinear random coefficient regression framework by incorporating possible heterogeneity in the intraindividual variance structure these methods extend existing variance function estimation techniques by pooling information across individuals the methods are appropriate when it is reasonable to assume that there exists a common intraindividual variance structure

the nonlinear mixed effects model with a smooth random effects density summary the fixed parameters of the nonlinear mixed effects model and the density of the random effects are estimated jointly by maximum likelihood the density of the random effects is assumed to be smooth but is otherwise unrestricted the method uses a series expansion that follows from the smoothness assumption to represent the density and quadrature to compute the likelihood standard algorithms are used for optimization empirical bayes estimates of random coefficients are obtained by computing posterior modes the method is applied to data from pharmacokinetics and properties of the method are investigated by application to simulated data

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

random effects and variances as a synthesis of nonlinear regression analyses of mitochondrial electron transport summary this cross disciplinary paper develops random effects and variances methods especially for small samples within effects and applies them to mitochondrial enzyme kinetics a new force flux kinetic model is introduced for the alternative oxidase of plant mitochondria the results of 42 experiments are published and the force flux function is fitted by nonlinear regression for each experiment the results of the regression analyses are synthesized by a random effects model specifying random nonlinear regression coefficients and random variances within experiments distributed in an inverse gamma distribution the paper shows how random nonlinear regression coefficients analysis can summarize the results of a long series of similar biological experiments and sharpen the analysis of the results of each a large data set often consists of a family of components each of which can be fitted with the same model there is a within component variation for each compo nent and random effects variation between components the interpretation of the data requires not only regression analysis of each component but also a subsequent synthesis of the results of such analysis into an overall summary achieved by specifying random regression coefficients and random error variances in the example which follows the individual samples are so small that random variances are specified and contracted predictors of them are computed which are more accurate than the individual estimates they are required for weights in the analysis the residual maximum likelihood estimates of the random effects variance matrices are sometimes singular a lagrange multiplier method is introduced to compute them the strength of the integrated analysis comes however at the price of a complex set of decisions as to how the different components of the computation are to be fitted together the paper shows how this is done by a flow diagram for the computations as well as describing each individual component

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

fieller s theorem scheffe simultaneous confidence intervals and ratios of parameters of linear and nonlinear mixed effects models the issue of joint confidence region and simultaneous confidence interval estimation for ratios of the parameters of a nonlinear mixed effects model is addressed using a fieller s theorem approach the method presented is similarly applicable to linear mixed effects models in addition previous work on linear fixed effects models is demonstrated to be a special case of the present method the methodology is applied to the ratios of slopes in a study of gestational maturation of placental glucose transfer capacity in sheep

accelerated degradation tests modeling and analysis high reliability systems generally require individual system components having extremely high reliability over long periods of time short product development times require reliability tests to be conducted with severe time constraints frequently few or no failures occur during such tests even with acceleration thus it is difficult to assess reliability with traditional life tests that record only failure times for some components degradation measures can be taken over time a relationship between component failure and amount of degradation makes it possible to use degradation models and data to make inferences and predictions about a failure time distribution this article describes degradation reliability models that correspond to physical failure mechanisms we explain the connection between degradation reliability models and failure time reliability models acceleration is modeled by having an acceleration model that describes the effect that temperature or another accelerating variable has on

design in nonlinear mixed effects models optimization using the fedorov wynn algorithm and power of the wald test for binary covariates we extend the methodology for designs evaluation and optimization in nonlinear mixed effects models with an illustration of the decrease of human immunodeficiency virus viral load after antiretroviral treatment initiation described by a bi exponential model we first show the relevance of the predicted standard errors ses given by the computation of the population fisher information matrix using the r function pfim in comparison to those computed with the stochastic approximation expectation maximization algorithm implemented in the monolix software we then highlight the usefulness of the fedorov wynn fw algorithm for designs optimization compared to the simplex algorithm from the predicted se of pfim we compute the predicted power of the wald test to detect a treatment effect as well as the number of subjects needed to achieve a given power using the fw algorithm we investigate the influence of the design on the power and show that for optimized designs with the same total number of samples the power increases when the number of subjects increases and the number of samples per subject decreases a simulation study is also performed with the nlme function of r to confirm this result and show the relevance of the predicted powers compared to those observed by simulation

regional optimization of fertilization using a hierarchical linear model utilization of regional data for optimizing fertilizer strategy is considered it is argued that a hierarchical mixed effects model is a reasonable parsimonious way to model the data such models take into account the correlations between yield responses at the same site also they conveniently let one model the differences between years or between sites as a function of year and site characteristics we specifically develop the case of a linear in the parameters response model but hierarchical models could also be based on nonlinear response functions given the estimated parameters the calculation of an optimal fertilizer strategy which depends on the site and year characteristics is usually straightforward a number of measures of the quality of this strategy are proposed among these are the risk which measures yield loss caused by using estimated rather than true parameter values another measure of interest is the potential gain from having a model that explains all the between year or all the between site variability the effect of potassium on sown prairies in france is treated as an example

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

some alternatives to asymptotic tests for the analysis of pharmacogenetic data using nonlinear mixed effects models nonlinear mixed effects models allow investigating individual differences in drug concentration profiles pharmacokinetics and responses pharmacogenetics focuses on the genetic component of this variability two tests often used to detect a gene effect on a pharmacokinetic parameter are 1 the wald test assessing whether estimates for the gene effect are significantly different from 0 and 2 the likelihood ratio test comparing models with and without the genetic effect because those asymptotic tests show inflated type i error on small sample size and or with unevenly distributed genotypes we develop two alternatives and evaluate them by means of a simulation study first we assess the performance of the permutation test using the wald and the likelihood ratio statistics second for the wald test we propose the use of the f distribution with four different values for the denominator degrees of freedom we also explore the influence of the estimation algorithm using both the first order conditional estimation with interaction linearization based algorithm and the stochastic approximation expectation maximization algorithm we apply these methods to the analysis of the pharmacogenetics of indinavir in hiv patients recruited in the cophar2 anrs 111 trial results of the simulation study show that the permutation test seems appropriate but at the cost of an additional computational burden one of the four f distribution based approaches provides a correct type i error estimate for the wald test and should be further investigated

discriminant analysis for longitudinal data with multiple continuous responses and possibly missing data multiple outcomes are often used to properly characterize an effect of interest this article discusses model based statistical methods for the classification of units into one of two or more groups where for each unit repeated measurements over time are obtained on each outcome we relate the observed outcomes using multivariate nonlinear mixed effects models to describe evolutions in different groups due to its flexibility the random effects approach for the joint modeling of multiple outcomes can be used to estimate population parameters for a discriminant model that classifies units into distinct predefined groups or populations parameter estimation is done via the expectation maximization algorithm with a linear approximation step we conduct a simulation study that sheds light on the effect that the linear approximation has on classification results we present an example using data from a study in 161 pregnant women in santiago chile where the main interest is to predict normal versus abnormal pregnancy outcomes

fitting a regression model for genotype by environment data on heading dates in grasses by methods for nonlinear mixed models summary the analysis of agricultural crop variety trials is usually complicated by the presence of genotype by environment interaction a number of methods and models have been proposed to tackle this problem one of the most common methods is the regression approach due to yates and cochran 1938 journal of agricultural science28 556 580 in which performances of genotypes in the environments are regressed onto environmental means the underlying regression model contains a multiplicative term with two unknown parameters one for genotypes and one for environments in the present paper the model is modified by exchanging the role of genotypes and environments various diagnostic plots show that this modified model is adequate for a data set on heading dates in the grass species dactylis glomerata if environments are considered as a random factor while genotypes are taken as fixed the model falls into the class of nonlinear mixed models recently a number of procedures have been suggested for this class of models which are based on first order taylor series expansion alternatively the model can be estimated by maximum likelihood this paper discusses the application of these methods for estimating parameters of the model

on analyzing circadian rhythms data using nonlinear mixed models with harmonic terms summarywang ke and brown 2003 biometrics59 804 812 developed a smoothing based approach for modeling circadian rhythms with random effects their approach is flexible in that fixed and random covariates can affect both the amplitude and phase shift of a nonparametrically smoothed periodic function in motivating their approach wang et al stated that a simple sinusoidal function is too restrictive in addition they stated that although adding harmonics can improve the fit it is difficult to decide how many harmonics to include in the model and the results are difficult to interpret we disagree with the notion that harmonic models cannot be a useful tool in modeling longitudinal circadian rhythm data in this note we show how nonlinear mixed models with harmonic terms allow for a simple and flexible alternative to wang et al s approach we show how to choose the number of harmonics using penalized likelihood to flexibly model circadian rhythms and to estimate the effect of covariates on the rhythms we fit harmonic models to the cortisol circadian rhythm data presented by wang et al to illustrate our approach furthermore we evaluate the properties of our procedure with a small simulation study the proposed parametric approach provides an alternative to wang et al s semiparametric approach and has the added advantage of being easy to implement in most statistical software packages

evaluation by simulation of tests based on non linear mixed effects models in pharmacokinetic interaction and bioequivalence cross over trials we propose tests based on non linear mixed effects models nlmem in pharmacokinetic interaction and bioequivalence cross over trials comparing two treatments or two formulations to compare the logarithm of the area under the curve auc using these models two approaches are studied in the first one concentration data are analysed globally with and without the estimation of a treatment effect and in the second one they are analysed separately in each treatment group with the estimation of the individual parameters four tests for comparison of the logarithm auc between two treatment arms are studied a likelihood ratio test lrt a wald test and two tests parametric and non parametric comparing the individual empirical bayes eb estimates these tests are adapted to the case of equivalence except the lrt which does not have any simple extension we evaluate by simulation of the type i error and the power for both comparison and equivalence tests they are compared to the standard tests recommended by the fda and the emea based on non compartmental nc auc trials for a usual pk model are simulated under h0 and several h1 using s plus software and analysed with the nlme function different configurations of the number of subjects n 12 24 and 40 and of the number of samples per subject j 10 5 and 3 are studied the type i error of lrt and wald comparison test in the 5000 replications of interaction cross over trials is found to be 20 9 per cent and 21 7 per cent respectively in the original design n 12 j 10 which is far superior to 5 per cent and decreases when n increases when n is fixed is found to increase with j power is satisfactory for both tests after correction of the significance threshold results of eb and nc tests are similar with satisfactory powers and a type i error close to 5 per cent except when j 3 for eb tests similar results are obtained for equivalence tests except for eb and nc student tests which are not of a great interest nc tests keep their place when the number of samples per subject j is large but nlmem seem useful for cross over studies performed in special populations where j limited the evaluation by monte carlo simulations of empirical threshold seems however necessary because of the inflation of the type i error copyright 2005 john wiley sons ltd

cluster analysis using multivariate mixed effects models a common situation in the biological and social sciences is to have data on one or more variables measured longitudinally on a sample of individuals a problem of growing interest in these areas is the grouping of individuals into one of two or more clusters according to their longitudinal behavior recently methods have been proposed to deal with cases where individuals are classified into clusters through a linear model of mixed univariate effects deriving from a longitudinally measured variable the method proposed in the current work deals with the case of clustering and then classification based on two or more variables measured longitudinally through the fitting of non linear multivariate mixed effect models and with consideration given to parameter estimation for balanced and unbalanced data using an em algorithm the application of the method is illustrated with an example in which the clusters are identified and the classification into clusters is compared with the true membership of individuals in one of two groups which is known at the end of the follow up period

approximating the marginal likelihood estimate for models with random parameters often a model for the mean and variance of a measurement set is naturally expressed in terms of both deterministic and random parameters each of the deterministic parameters has one fixed value while the random parameters come from a distribution of values we restrict our attention to the case where the random parameters and the measurement error have a gaussian distribution in this case the joint likelihood of the data and random parameters is an extended least squares function the likelihood of the data alone is the integral of this extended least squares function with respect to the random parameters this is the likelihood that we would like to optimize but we do not have a closed form expression for the integral we use laplace s method to obtain an approximation for the likelihood of the data alone maximizing this approximation is less computationally demanding than maximizing the integral expression but this yields a different estimator in addition evaluation of the approximation requires second derivatives of the original model functions if we were to use this approximation as our objective function evaluation of the derivative of the objective would require third derivatives of the original model functions we present modified approximations that are expressed using only values of the original model functions evaluation of the derivative of the modified approximations only requires first derivatives of the original model functions we use monte carlo techniques to approximate the difference between an arbitrary estimator and the estimator that maximizes the likelihood of the data alone in addition we approximate the information matrix corresponding to the estimator that maximizes the likelihood of the data alone

reml estimation in unbalanced multivariate variance components models using an em algorithm estimation of covariance matrices when data are missing is a difficult problem in the multivariate mixed model setting problems can arise either because the data are unbalanced or because the response vectors are incomplete jennrich and schluchter 1986 biometrics 42 805 820 and laird lange and stram 1987 journal of the american statistical association 82 97 105 have proposed em algorithms for estimating the parameters associated with longitudinal data models in this paper a new em algorithm is proposed for the general multivariate mixed model the algorithm is derivativefree and based on an algorithm for balanced data proposed by calvin and dykstra 1991 annals qf statistics 19 850 869 once the algorithm is described it is used to produce estimates of the variance components matrices associated with a completely random two factor nested model for a bivariate response vector the data for this example come from calvin and sedransk 1991 journal of the american statistical association 86 36 48 who analyze a study of the quality of care received by cancer patients receiving radiation therapy

a smooth nonparametric estimate of a mixing distribution using mixtures of gaussians abstract we propose a method of estimating mixing distributions using maximum likelihood over the class of arbitrary mixtures of gaussians subject to the constraint that the component variances be greater than or equal to some minimum value h this approach can lead to estimates of many shapes with smoothness controlled by parameter h we show that the resulting estimate will always be a finite mixture of gaussians each having variance h the nonparametric maximum likelihood estimate can be viewed as a special case with h 0 the method can be extended to estimate multivariate mixing distributions examples and the results of a simulation study are presented

a bayesian approach to nonlinear random effects models nonlinear random effects models are considered from the bayesian point of view the method of analysis follows closely that of lindley and smith 1972 journal of the royal statistical society series b 34 1 42 the numerical method is related to the em algorithm

optimal design in random effects regression models summary an approach is proposed to optimal design of experiments for estimating randomeffects regression models the population designs are defined by the number of subjects and the individual designs to be performed cost functions associated with individual designs are incorporated for a given maximal cost an algorithm is proposed for finding the statistical population design that maximises the determinant of the fisher information matrix of the population parameters the fisher information matrix is formulated for linear models and normal distributions the approach is applied to the design of an optimal experiment in toxicokinetics using a first order linearisation of the model several cost functions and designs of various orders are studied an example illustrates the optimal population designs and the increased efficiency of some optimal designs over more standard designs

newton raphson and em algorithms for linear mixed effects models for repeated measures data abstract we develop an efficient and effective implementation of the newton raphson nr algorithm for estimating the parameters in mixed effects models for repeated measures data we formulate the derivatives for both maximum likelihood and restricted maximum likelihood estimation and propose improvements to the algorithm discussed by jennrich and schluchter 1986 to speed convergence and ensure a positive definite covariance matrix for the random effects at each iteration we use matrix decompositions to develop efficient and computationally stable implementations of both the nr algorithm and an em algorithm laird and ware 1982 for this model we compare the two methods em vs nr in terms of computational order and performance on two sample data sets and conclude that in most situations a well implemented nr algorithm is preferable to the em algorithm or em algorithm with aitken s acceleration the term repeated measures refers to experimental designs where there are several individuals and several

bayesian approach for a nonlinear growth model nonlinear least squares methods are currently used for fitting a well known growth model namely the jenss model to the length measurements of a child followed throughout the first six years of life an empirical bayes approach is developed for fitting the model and the prior distribution of the growth model parameters is estimated from a large sample of least squares parameters an expression which is proportional to the posterior distribution is derived so that the posterior mode can be estimated given the observations on a child this posterior mode provides bayes estimates of the jenss curve parameters for the child

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

estimating data transformations in nonlinear mixed effects models a routine practice in the analysis of repeated measurement data is to represent individual responses by a mixed effects model on some transformed scale for example for pharmacokinetic growth and other data both the response and the regression model are typically transformed to achieve approximate within individual normality and constant variance on the new scale however the choice of transformation is often made subjectively or by default with adoption of a standard choice such as the log we propose a mixed effects framework based on the transform both sides model where the transformation is represented by a monotone parametric function and is estimated from the data for this model we describe a practical fitting strategy based on approximation of the marginal likelihood inference is complicated by the fact that estimation of the transformation requires modification of the usual standard errors for estimators of fixed effects however we show that under conditions relevant to common applications this complication is asymptotically negligible allowing straightforward implementation via standard software

analysis of multi arm tumor growth trials in xenograft animals using phase change adaptive piecewise quadratic models xenograft trials allow tumor growth in human cell lines to be monitored over time in a mouse model we consider the problem of inferring the effect of treatment combinations on tumor growth a piecewise quadratic model with flexible phase change locations is proposed to model the effect of change in therapy over time each piece represents a growth phase with phase changes in response to change in treatment piecewise slopes represent phase specific log linear growth rates and curvature parameters represent departure from linear growth trial data are analyzed in two stages i subject specific curve fitting ii analysis of slope and curvature estimates across subjects a least squares approach with penalty for phase change point location is proposed for curve fitting in simulation studies the method is shown to give consistent estimates of slope and curvature parameters under independent and ar 1 measurement error the piecewise quadratic model is shown to give excellent fit median r 2 0 98 to growth data from a six armed xenograft trial on a lung carcinoma cell line

estimation in forest yield models using composite link functions with random effects a sigmoidal forest yield model is estimated from longitudinal measurements of sample plots where measurement times and intervals vary the projection form of the yield model is fitted as a generalized linear model glm with composite link functions and an offset this allows age and additional covariates to be easily incorporated in a more flexible nonlinear structure than that provided by single link glms also existing algorithms for fitting generalized linear mixed models glmms are easily adapted to fit the projection model thereby allowing incorporation of random plot effects estimation for the composite link glmm is described as well as an alternative to the glmm in which random effects are incorporated as linear components on the same scale as the response this model is referred to as an additive generalized linear mixed model aglmm unlike the glmm the fixed effect parameter estimates in the aglmm are population average pa the aglmm is fitted using a modification of a subject specific ss algorithm using marginal expectation the random effects in the aglmm may have a natural interpretation or may provide a means of generating a marginal covariance structure as is the case with linear mixed models a simple example of an aglmm for binomial data and logit link is given in both this and the forest yield example a single artificial random effect is constructed to give exchangeable correlation for observations within subjects and a comparison of fit of the aglmm and glmm is made using a conditional quasi deviance if the fit is similar and pa parameter estimates are required the aglmm is preferred

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

testing departure from additivity in tukey s model using shrinkage application to a longitudinal setting while there has been extensive research developing gene environment interaction gei methods in case control studies little attention has been given to sparse and efficient modeling of gei in longitudinal studies in a two way table for gei with rows and columns as categorical variables a conventional saturated interaction model involves estimation of a specific parameter for each cell with constraints ensuring identifiability the estimates are unbiased but are potentially inefficient because the number of parameters to be estimated can grow quickly with increasing categories of row column factors on the other hand tukey s one degree of freedom model for non additivity treats the interaction term as a scaled product of row and column main effects because of the parsimonious form of interaction the interaction estimate leads to enhanced efficiency and the corresponding test could lead to increased power unfortunately tukey s model gives biased estimates and low power if the model is misspecified when screening multiple geis where each genetic and environmental marker may exhibit a distinct interaction pattern a robust estimator for interaction is important for gei detection we propose a shrinkage estimator for interaction effects that combines estimates from both tukey s and saturated interaction models and use the corresponding wald test for testing interaction in a longitudinal setting the proposed estimator is robust to misspecification of interaction structure we illustrate the proposed methods using two longitudinal studies the normative aging study and the multi ethnic study of atherosclerosis

a relative off set orthogonality convergence criterion for nonlinear least squares an orthogonality convergence criterion using relative offset is proposed this criterion is compared to currently used criteria and its advantages are discussed