modeling tumor onset and multiplicity using transition models with latent variables summary we describe a method for modeling carcinogenicity from animal studies where the data consist of counts of the number of tumors present over time the research is motivated by applications to transgenic rodent studies which have emerged as an alternative to chronic bioassays for screening possible carcinogens in transgenic mouse studies the endpoint of interest is frequently skin papilloma with weekly examinations determining how many papillomas each animal has at a particular point in time it is assumed that each animal has two unobservable latent variables at each time point the first indicates whether or not the tumors are in a multiplying state and the second is the potential number of additional tumors if the tumors are in a multiplying state the product of these variables follows a zero inflated poisson distribution and the em algorithm can be used to maximize the observed data pseudo likelihood based on the latent variables a generalized estimating equations robust variance estimator adjusts for dependency among outcomes within individual animals the method is applied to testing for a dose related trend in both tumor incidence and multiplicity in carcinogenicity studies

omitted variables in multilevel models statistical methodology for handling omitted variables is presented in a multilevel modeling framework in many nonexperimental studies the analyst may not have access to all requisite variables and this omission may lead to biased estimates of model parameters by exploiting the hierarchical nature of multilevel data a battery of statistical tools are developed to test various forms of model misspecification as well as to obtain estimators that are robust to the presence of omitted variables the methodology allows for tests of omitted effects at single and multiple levels the paper also introduces intermediate level tests these are tests for omitted effects at a single level regardless of the presence of omitted effects at a higher level a simulation study shows not surprisingly that the omission of variables yields bias in both regression coefficients and variance components it also suggests that omitted effects at lower levels may cause more severe bias than at higher levels important factors resulting in bias were found to be the level of an omitted variable its effect size and sample size a real data study illustrates that an omitted variable at one level may yield biased estimators at any level and in this study one cannot obtain reliable estimates for school level variables when omitted child effects exist however robust estimators may provide unbiased estimates for effects of interest even when the efficient estimators fail and the one degree of freedom test helps one to understand where the problem is located it is argued that multilevel data typically contain rich information to deal with omitted variables offering yet another appealing reason for the use of multilevel models in the social sciences

multivariate logistic models summary the multivariate logistic transform is a reparameterisation of cell probabilities in terms of marginal logistic contrasts it is known that an arbitrary set of logistic contrasts may not correspond to a valid joint distribution in this paper we present an efficient algorithm for detecting whether or not the inverse transform exists and for computing it if it does

model for the analysis of binary longitudinal pain data subject to informative dropout through remedication abstract we address the problem of accounting for informative dropout in the form of rescue medication when comparing pain relievers with respect to longitudinal binary pain relief outcomes we present a selection model approach for binary longitudinal data that accommodates informative dropout the relationship between dropout or remedication and the binary pain relief response is assumed to be characterized by a random effect that is conditional on this random effect response and dropout are independent unlike previous approaches to this problem which rely on numerical or approximation methods we obtain a closed form expression for the marginal log likelihood of response and dropout by specifying a complementary log log link function for both components and a conjugate log gamma random effect distribution a data analysis supported by simulation results suggest that the model fits reasonably well results are compared to those obtained from conventional but somewhat inappropriate analyses

on the determination of the basin of attraction of a periodic orbit in two dimensional systems we consider the general nonlinear differential equation x f x with x epsilon r 2 and develop a method to determine the basin of attraction of a periodic orbit borg s criterion provides a method to prove existence uniqueness and exponential stability of a periodic orbit and to determine a subset of its basin of attraction in order to use the criterion one has to find a function w epsilon c 1 r 2 r such that l w x w x l x is negative for all x epsilon k where k is a positively invariant set here l x is a given function and w x denotes the orbital derivative of w in this paper we prove the existence and smoothness of a function w such that l w x mu parallel to f x parallel to we approximate the function w which satisfies the linear partial differential equation w x mu parallel to f x parallel to l x using radial basis functions and obtain an approximation omega such that l omega x 0 using radial basis functions again we determine a positively invariant set k so that we can apply borg s criterion as an example we apply the method to the van der pol equation

empirical bayes estimation of rates in longitudinal studies abstract the usually irregular follow up intervals in epidemiologic studies preclude the use of classical growth curve analysis for short follow up intervals it is suggested that individual rates of change are useful for exploratory analysis empirical bayes estimates of these rates of change are recommended and developed an example of bone loss with age in women is also given

testing model fit in longitudinal data analysis against alternatives with omitted covariates several types of common model misspecifications can be re formulated as problems of omitted covariates these include situations with unmeasured confounders measurement errors in observed covariates and informative censoring longitudinal data present special opportunities for detecting omitted covariates that are related to the observed ones differently across time than across individuals this situation arises with period and cohort effects as well as with usual formulations of classical measurement error in observed covariates in this article we focus on testing for the existence of omitted covariates in longitudinal data analysis when models are fit by generalized estimation equations when omitted covariates are present specification of the correct link function conditionally on only observed covariates under the alternative usually involves complicated numerical integration we propose a quasi score test statistic that avoids the need to fit such alternative models the statistic is asymptotically chi square distributed under the null hypothesis of no omitted covariates with degrees of freedom determined by the assumed alternative structure we study the significance level and the power of the quasi score test in linear and logistic regression models the test is then applied to an analysis of excessive daytime sleepiness

on the determination of the basin of attraction of periodic orbits in three and higher dimensional systems the determination of the basin of attraction of a periodic orbit can be achieved using a lyapunov function a lyapunov function can be constructed by approximation of a first order linear pde for the orbital derivative via meshless collocation however if the periodic orbit is only accessible numerically a different method has to be used near the periodic orbit borg s criterion provides a method to obtain information about the basin of attraction by measuring whether adjacent solutions approach each other with respect to a riemannian metric using a numerical approximation of the periodic orbit and its first variation equation a suitable riemannian metric is constructed

estimating vaccine efficacy from secondary attack rates epidemiologists have used secondary attack rates sars to estimate the protective effects of vaccination since the 1930s sars can also be used to estimate the effect of vaccination on reducing infectiousness in breakthrough cases the conventional sar approach has been to pool the denominators and numerators across transmission units then to use a confidence interval for a simple relative risk we demonstrate appropriate model based methods to estimate vaccine efficacy ve from sars using generalized estimating equations taking correlation within transmission units into account the model based procedures require transformation of the parameter estimates to the sar scale to obtain vaccine efficacy estimates appropriate confidence intervals are then based on the bootstrap with resampling done by transmission unit we show that the usual confidence intervals are too narrow we estimated the effect of pertussis vaccination on person to person transmission the results show that pertussis vaccination re

improving generalised estimating equations using quadratic inference functions summary generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified however under such misspecification the estimator of the regression parameter can be inefficient in this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter and that remains optimal even if the working correlation structure is misspecified the idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices a representation that is valid for the working correlations most commonly used both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations this approach also provides a chi squared inference function for testing nested models and a chi squared regression misspecification test furthermore the test statistic follows a chi squared distribution asymptotically whether or not the working correlation structure is correctly specified

nonparametric empirical bayes growth curve analysis abstract nonparametric and semiparametric regression have been suggested as alternatives to parametric models for growth curve analysis in this article we demonstrate that empirical bayes estimation can be used to improve linear smoothing estimates when multiple curves are available

some considerations in the analysis of rates of change in longitudinal studies this paper discusses and compares several estimators of mean rate of change in unbalanced longitudinal data based on a model with randomly distributed regression coefficients across individuals the estimators are unweighted and weighted means of these coefficients the paper also evaluates commonly used variance estimates corresponding to the estimators results show that in situations of very slight imbalance the choice of method is not critical when imbalance is substantial however one should weight the regression coefficients by their estimated precision an example using data from a nutritional study on premature neonates illustrates some issues encountered in the analysis of longitudinal clinical data sets

general methods for analysing repeated measures we present an overview of issues and methods for analysing repeated measures of a continuous random variable we discuss modelling the mean vector and covariance structure statistical efficiency regression diagnostics and discrepancies between longitudinal and cross sectional methods we illustrate key points with examples and discuss areas requiring further development

comparisons of predictive values of binary medical diagnostic tests for paired designs positive and negative predictive values of a diagnostic test are key clinically relevant measures of test accuracy surprisingly statistical methods for comparing tests with regard to these parameters have not been available for the most common study design in which each test is applied to each study individual in this paper we propose a statistic for comparing the predictive values of two diagnostic tests using this paired study design the proposed statistic is a score statistic derived from a marginal regression model and bears some relation to mcnemar s statistic as mcnemar s statistic can be used to compare sensitivities and specificities of diagnostic tests parameters that condition on disease status our statistic can be considered as an analog of mcnemar s test for the problem of comparing predictive values parameters that condition on test outcome we report on the results of a simulation study designed to examine the properties of this test under a variety of conditions the method is illustrated with data from a study of methods for diagnosis of coronary artery disease

conditional generalized estimating equations for the analysis of clustered and longitudinal data a common and important problem in clustered sampling designs is that the effect of within cluster exposures i e exposures that vary within clusters on outcome may be confounded by both measured and unmeasured cluster level factors i e measurements that do not vary within clusters when some of these are ill not accounted for estimation of this effect through population averaged models or random effects models may introduce bias we accommodate this by developing a general theory for the analysis of clustered data which enables consistent and asymptotically normal estimation of the effects of within cluster exposures in the presence of cluster level confounders semiparametric efficient estimators are obtained by solving so called conditional generalized estimating equations we compare this approach with a popular proposal by neuhaus and kalbfleisch 1998 biometrics 54 638 645 who separate the exposure effect into a within and a between cluster component within a random intercept model we find that the latter approach yields consistent and efficient estimators when the model is linear but is less flexible in terms of model specification under nonlinear models this approach may yield inconsistent and inefficient estimators though with little bias in most practical settings

semiparametric regression analysis for recurrent event interval counts this paper deals with analysis of data from longitudinal studies where the rate of a recurrent event characterizing morbidity is the primary criterion for treatment evaluation we consider clinical trials which require patients to visit their clinical center at successive scheduled times as part of follow up at each visit the patient reports the number of events that occurred since the previous visit or an examination reveals the number of accumulated events such as skin cancers the exact occurrence times of the events are unavailable and the actual patient visit times typically vary randomly about the scheduled follow up times each patient s record thus consists of a sequence of clinic visit dates event counts corresponding to the successive time intervals between clinic visits and baseline covariates we propose a semiparametric regression model extending the fully parametric model of thall 1988 biometrics 44 197 209 to estimate and test for covariate effects on the rate of events over time while also accounting for the possibly time varying nature of the underlying event rate covariate effects enter the model parametrically while the underlying time varying event rate is modelled nonparametrically the method of severini and wong 1992 annals of statistics 20 1768 1802 is used to construct asymptotically efficient estimators of the parametric component and to specify their asymptotic distribution a simulation study and application to a data set are provided

ranking and estimation of related means in the presence of a covariate a bayesian approach abstract choosing the largest of several means can be a demanding problem especially in the presence of a covariate a hierarchical bayesian approach to ranking and selection as well as estimation of related means in the presence of a covariate is considered for the multiple slopes model we compute in addition to the posterior means and standard deviations of the parameters the posterior probabilities that each mean at a given value of the covariate is the largest the vector of posterior probabilities thus obtained provides an easily understandable answer to the selection problem although calculation of the posterior probabilities may involve four dimensional numerical integration in the difficult unbalanced design and unknown variance case an efficient monte carlo method of evaluation has been developed and is given in the article by reanalyzing a well known data set on the breaking strength and thickness of starch films we illustrate how our bayesian approach produces meaningful conclusions

converse theorems on contraction metrics for an equilibrium the stability and basin of attraction of an equilibrium can be determined by a contraction metric a contraction metric is a riemannian metric with respect to which the distance between adjacent trajectories decreases the advantage of a contraction metric over e g a lyapunov function is that the contraction condition is robust under perturbations of the system while the sufficiency of a contraction metric for the existence stability and basin of attraction of an equilibrium has been extensively studied in this paper we will prove converse theorems showing the existence of several different contraction metrics this will be useful to develop algorithms for the construction of contraction metrics

efficiency of estimating equations and the use of pivots summary methods are proposed for deriving unbiased estimating equations for the parameters of interest in a statistical model which includes further incidental or nuisance parameters we start with pivot like quantities which eliminate the dependence on the incidental parameters and derive functions of these which give estimating equations which are most efficient in a certain sense these equations involve the incidental parameters which are then replaced by estimators the methods overcome some of the difficulties encountered in likelihood and least squares estimation when there are many incidental parameters and our theory reduces to likelihood and least squares estimation when there are none thus the optimality results apply to these methods too some examples are discussed

an intrinsic observer for a class of lagrangian systems we propose a new design method of asymptotic observers for a class of nonlinear mechanical systems lagrangian systems with configuration position measurements our main contribution is to introduce a state position and velocity observer that is invariant under any changes of the configuration coordinates the observer dynamics equations as the euler lagrange equations are intrinsic the design method uses the riemannian structure defined by the kinetic energy on the configuration manifold the local convergence is proved by showing that the jacobian of the observer dynamics is negative definite contraction for a particular metric defined on the state space a metric derived from the kinetic energy and the observer gains from a practical point of view such intrinsic observers can be approximated when the estimated configuration is close to the true one by an explicit set of differential equations involving the riemannian curvature tensor these equations can be automatically generated via symbolic differentiations of the metric and potential up to order two numerical simulations for the ball and beam system an example where the scalar curvature is always negative show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics

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

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

brief paper stability and robustness analysis of nonlinear systems via contraction metrics and sos programming a wide variety of stability and performance questions about linear dynamical systems can be reformulated as convex optimization problems involving linear matrix inequalities lmis these techniques have been recently extended to nonlinear systems with polynomial or rational dynamics through the use of sum of squares sos programming in this paper we further extend the class of systems that can be analyzed with convexity based methods we show how to analyze the robust stability properties of uncertain nonlinear systems with polynomial or rational dynamics via contraction analysis and sos programming since the existence of a global contraction metric is a sufficient condition for global stability of an autonomous system we develop an algorithm for finding such contraction metrics using sos programming the search process is made computationally tractable by relaxing matrix definiteness constraints the feasibility of which indicates the existence of a contraction metric to sos constraints on polynomial matrices we illustrate our results through examples from the literature and show how our contraction based approach offers advantages when compared with traditional lyapunov analysis

the analysis of longitudinal polytomous data generalized estimating equations and connections with weighted least squares in recent years methods have been developed for modelling repeated observations of a categorical response obtained over time on the same individual although situations in which the repeated response is binary or poisson have been studied extensively relatively little attention has been given to polytomous categorical response variable in this paper we extend the estimating equations initially developed for clustered discrete data by liang and zeger 1986 biometrika 73 13 22 and subsequently extended by prentice 1988 biometrics 44 1033 1048 to polytomous response variables under certain assumptions we illustrate that these estimating equations simplify to the weighted least squares wls equations formalized by koch et al 1977 biometrics 33 133 158 this connection provides a formal framework for obtaining iterated weighted least squares model parameter estimates cumulative logit models are developed and applied to a representative longitudinal data set simulation results comparing wls an iterative form of wls and independence estimating equations using a robust estimate of the variance are presented

asymptotics for generalized estimating equations with large cluster sizes generalized estimating equations are used in regression analysis of longitudinal data where observations on each subject are correlated statistical analysis using such methods is based on the asymptotic properties of regression parameter estimators this paper presents asymptotic results when either the number of independent subjects or the cluster sizes the number of observations on each subject or both go to infinity a set of information matrix based general conditions is developed which leads to the weak and strong consistency as well as the asymptotic normality of the estimators most of the results are parallel to the elegant work of fahrmeir and kaufmann on maximum likelihood estimators related to the generalized linear models the conditions for weak consistency and asymptotic normality are verified for several examples of general interest

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

necessary conditions for a limit cycle and its basin of attraction in this paper we consider a general differential equation of the form x f x with f epsilon c l r n r n and n greater than or equal to 2 borg hartman leonov and others have studied sufficient conditions for the existence uniqueness and exponential stability of a periodic orbit and for a set to belong to its basin of attraction they used a certain contraction property of the flow with respect to the euclidian or a riemannian metric in this paper we also prove sufficient conditions including upper bounds for the floquet exponents of the periodic orbit moreover we show the necessity of these conditions using floquet theory and a lyapunov function

iterative weighted least squares estimation in heteroscedastic linear models this article addresses the problem of choosing weights for iterative weighted least squares estimation in heteroscedastic linear models an asymptotically optimal method for determining weights at each iteration is derived under a bayesian model for the variances the method uses a compromise between model based and model free variance estimates consider a heteroscedastic linear regression model in which responses are grouped so that the variance is constant within each group let denote the vector of regression parameters and let denote a vector of parameters determining a prior distribution for the variances iterative weighted least squares estimators are defined as follows given estimates and calculate a weight for the ith group as a function of the values in the ith group of the predictor variables and the average of the squared residuals from the estimated mean responses in the ith group given weights calculate the weighted least squares estimate and a new estimate continue until conv

prediction using partly conditional time varying coefficients regression models summary data collected longitudinally in time provide the opportunity to develop predictive models of future observations given current data for an individual such models may be of particular value in defining individuals at high risk and thereby in suggesting subgroups for targeting of prevention intervention research efforts in this paper we propose a method for estimating predictive functions the method uses an extension of the marginal regression analysis methods of liang and zeger 1986 biometrika73 13 22 and is implemented using simple estimating equations a key feature of the models is that regression coefficients are modelled as smooth functions of the times both at and for prediction data from a study of obesity in childhood and early adulthood is used to demonstrate the methodology criteria for defining individuals to be at high risk can be defined on the basis of estimated predictive functions we suggest methods for evaluating the diagnostic accuracy sensitivity and specificity of such rules using cross validation the method holds promise as a robust and technically easy way of evaluating information about future prognosis that may be gleaned from a patient s current and past clinical status

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

the survival of nonattractive interacting particle systems on z we consider interacting particle systems on z which allow five types of pairwise interaction annihilation birth coalescence death and exclusion with corresponding rates a b c d e we show that whatever the values of a c d e if the birthrate is high enough there is a positive probability the particle system will survive starting from any finite occupied set in particular an ips with rates a b c d e has a positive probability of survival if b 4d 6a c a d e or b 7d 3a 3c 3e c a d e we create a suitable supermartingale by extending the method used by holley and liggett in their treatment of the contact process

conditional inference methods for incomplete poisson data with endogenous time varying covariates we investigate the effect of protease inhibitors pis on the rate of emergency room er visits among hiv infected women from a longitudinal cohort study one strategy for accounting for serial correlation in longitudinal studies is to assume that observations are independent conditional on unit specific nuisance parameters it is possible to estimate these models using unconditional maximum likelihood where the nuisance parameters are assigned a parametric distribution and integrated out of the likelihood alternatively we can proceed using conditional inference where we eliminate the nuisance parameters from the likelihood by conditioning on a sufficient statistic for these parameters an advantage of conditional inference methods over parametric random effects models is that all patient level time invariant factors both measured and unmeasured are accounted for in the analysis a limitation is that standard conditional inference methods assume that missing data are missing completely at random a

construction of a cpa contraction metric for periodic orbits using semidefinite optimization abstract a riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction while the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit the explicit construction of the metric is a difficult problem in this paper the construction of such a contraction metric is achieved by formulating it as an equivalent problem namely a feasibility problem in semidefinite optimization the contraction metric a matrix valued function is constructed as a continuous piecewise affine cpa function which is affine on each simplex of a triangulation of the phase space the contraction conditions are formulated as conditions on the values at the vertices the paper states a semidefinite optimization problem we prove on the one hand that a feasible solution of the optimization problem determines a cpa contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough an objective function can be used to obtain a bound on the largest floquet exponent of the periodic orbit

time period effects in longitudinal studies measuring average rates of change random time period effects are unexplained increases or decreases in the observed value for all individuals measured at a particular time point in a longitudinal study they can be caused by learning effects changes in equipment personnel and overall subject co operation we investigate the consequences of time period effects in random coefficient regression models where interest is in the average rate of change slope of a continuous outcome in a study with a single group of subjects they can lead to conditionally biased estimates of the mean slope and its variance conditional on the time period effects calculations suggest that the increase in sample size required to maintain a specified precision of the mean slope estimate over repeated studies may be substantial in a study with a concurrent control group however time period effects do not distort the expectation estimated variance or the distribution of the difference between the mean slopes with missing data in addition to time period effects an unbiased estimate of a single mean slope remains problematic but one can use standard maximum likelihood techniques to obtain consistent estimators of the difference in mean slopes and its variance this suggests the importance of a concurrent control group when potential time period effects are of concern

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

convergence properties for discrete time nonlinear systems three similar convergence notions are considered two of them are the long established notions of convergent dynamics and incremental stability the other is the more recent notion of contraction analysis all three convergence notions require that all solutions of a system converge to each other in this note we investigate the differences between these convergence properties for discrete time and time varying nonlinear systems by comparing the properties in pairs and using examples we also demonstrate a time varying smooth lyapunov function characterization for each of these convergence notions and with appropriate assumptions we provide several sufficient conditions to establish relationships between these properties in terms of lyapunov functions

maximum likelihood estimation and large sample inference for generalized linear and nonlinear regression models summary the class of generalized linear models is extended to allow for correlated observations nonlinear models and error distributions not of the exponential family form the extended class of models include a number of important examples particularly of the composite transformational type large sample inference and maximum likelihood estimation for the extended class of generalized linear models are discussed and the analysis of deviance is generalized to the extended class of models calculation of the maximum likelihood estimate for a general likelihood by fisher s scoring method and a related method is considered and the relation with the gauss newton method is discussed

effect of omitted confounders on the analysis of correlated binary data summary marginal analysis using the generalized estimating equation approach is widely applied to correlated observations as occur in studies with clusters and in longitudinal follow up of individuals in this article we investigate the effect of confounding in such models we assume that a risk factor x and a confounder z are related by a generalized linear model to the outcome y which can be binary or ordinal in order to investigate confounding arising from the omission of z a joint structure for x and z must be specified modeling normally distributed x z as sums of between and within individual or cluster components allows us to incorporate different degrees of betweenand within individual correlation such a structure includes as special cases cohort and period effects in longitudinal settings and random intercept models the latter situation corresponds to allowing z to vary only on the between individual or cluster level and to be uncorrelated with x and leads to attenuation of the coefficient of x in marginal models with the logit and probit links more complex situations occur when z is allowed to also vary on the within individual or cluster level and when z is correlated with x we examine the model specification and the expected bias when fitting a marginal model in the presence of the omitted confounder z we derive general formulas and interpret the parameters and results in an ongoing cohort study testing for omitted covariates is also discussed

testing for omitted variables and non linearity in regression models for longitudinal data when fitting regression models to investigate the relationship between an outcome variable and independent variables of primary interest there is often concern whether omitted variables or assuming a different functional relationship could have changed the conclusion or interpretation of the results in longitudinal studies of aging the concern with omitted variables is well known in the context of cohort and period effects which refer to unmeasured variables systematically related to the individual s year of birth and secular trends in outcome respectively we present and compare three approaches to detecting omitted confounders and non linearity in the random effects model for longitudinal data laird and ware 1982 with random slope and intercept across individuals the first approach compares simple unweighted within and between regression coefficients the second is the hausman specification test for regression models and the third approach involves testing directly the significance of functions of individual specific covariate means means i in the random effects regression model this last approach is motivated by the models that arise when cohort or period effects are ignored we compare the three approaches and illustrate their application

semiparametric gee analysis in partially linear single index models for longitudinal data in this article we study a partially linear single index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases a semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations gee is introduced to estimate the two parameter vectors as well as the unknown link function under some mild conditions we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data we also discuss the estimation of the covariance or weight matrices involved in the semiparametric gee method furthermore we provide some numerical studies including monte carlo simulation and an empirical application to illustrate our methodology and theory

the power to detect differences in average rates of change in longitudinal studies with considerable current interest in longitudinal epidemiologic studies little is available regarding sample size requirements this paper considers a method for analysis of longitudinal data where one compares the mean rates of change for two or more groups and proposes a statistic for use in determining sample size requirements one calculates individual rates of change with least squares estimates of slopes of individuals responses regressed over time the assumption of linear change over time while clearly not applicable for some data applies to many biological measurements either as recorded or with some transformation the variances of these estimated slopes have two components within individual variability based on measurement error and length and frequency of follow up and true between individual slope variability it is assumed that measurement error is the same for all subjects so that the total variances differ due to differences in follow up the question addressed is when can one use the usual anova f statistic to compare group means of estimated slopes expected mean squares demonstrate that this f is appropriate when either each group has the same number of subjects or when each subject has the same length and frequency of follow up a procedure for computing power and sample size is presented where one can specify the maximum detectable difference in any two average slopes moment estimation and maximum likelihood estimation of variance components from prior data are discussed

projected partial likelihood and its application to longitudinal data summary an estimating equation which we call the projected partial score is introduced for longitudinal data analysis the estimating equation is obtained by projecting the partial likelihood score function onto the vector space spanned by a class of conditionally linear estimating equations we demonstrate that removing certain terms from the projection of the full likelihood score does not alter important inferential properties of the estimating equation and doing so is advantageous in handling missing data and time varying covariates within a prequential frame of reference it is shown that the estimating equation is optimal among the largest collection of estimating equations determined by the conditional moments furthermore the method possesses similar properties to generalised estimating equations in particular the correct conditional variance specification is necessary for efficiency but not for asymptotic consistency and distribution theory

graphs and stochastic relaxation for hierarchical bayes modelling this expository paper describes two useful tools for the statistical analysis of processes that generate repeated measures and longitudinal data the first tool is a graph for a visual description of dependency structures the second tool is a stochastic relaxation method gibbs sampling for fitting hierarchical bayes models graphs are concise and accessible summaries of stochastic models graphs aid communications between statistical and subject matter scientists during which formulations of scientific questions are modified an uncluttered picture of the dependency structure of a model augments effectively its corresponding formulaic description stochastic relaxation is a computationally intense method that allows experimentation with broader classes of models than were previously thought feasible because of analytic intractability stochastic relaxation is intuitive and easily described to non statisticians several sample graphs show how hierarchical bayes models can use stochastic relaxation to obtain their fits an example based on estimating drug shelf life demonstrates some uses of graphs and stochastic relaxation compared with several frequentist growth curve analyses that use restricted maximum likelihood and generalized estimating equations approaches

borg s criterion for almost periodic differential equations borg s criterion is used to prove the existence of an exponentially asymptotically stable periodic orbit of an autonomous differential equation and to determine its domain of attraction in this article this method is generalized to almost periodic differential equations both sufficient and necessary conditions are obtained for the existence of an exponentially stable almost periodic solution the condition uses a riemannian metric and an example for the explicit construction of such a metric is presented

the analysis of binary longitudinal data with time independent covariates summary this paper considers extensions of logistic regression to the case where the binary outcome variable is observed repeatedly for each subject we propose two working models that lead to consistent estimates of the regression parameters and of their variances under mild assumptions about the time dependence within each subject s data the efficiency of the proposed estimators is examined an analysis of stress in mothers with infants is presented to illustrate the proposed method

tree structured logistic model for over dispersed binomial data with application to modeling developmental effects this article proposes tree structured logistic regression modeling for over dispersed binomial data recursive partitioning is performed using a combination of statistical tests and residual analysis the splitting criterion in cross validation is based on the deviance function a nested grid algorithm to estimate the bootstrap parameters is developed the regression tree procedure provides a new approach for exploring in detail the relationship between the binomial response and explanatory variables the proposed procedure is used to model the relationship between the incidence of malformation and dose and fetal weight using data from a developmental experiment conducted at the national center for toxicological research a conditional gaussian chain model is used to account for the effect of fetal weight by dose

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

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

analysis of longitudinal data with unmeasured confounders confounding in longitudinal or clustered data creates special problems and opportunities because the relationship between the confounder and covariate of interest may differ across and within individuals or clusters a well known example of such confounding in longitudinal data is the presence of cohort and period effects in models of aging in epidemiologic research we first formulate a data generating model with confounding and derive the distribution of the response variable unconditional on the confounder we then examine the properties of the regression coefficient for some analytic approaches when the confounder is omitted from the fitted model the expected value of the regression coefficient differs in across and within individual regression in the multivariate case within and between individual information is combined and weighted according to the assumed covariance structure we assume compound symmetry in the fitted covariance matrix and derive the variance bias and mean squared error of the slope estimate as a function of the fitted within individual correlation we find that even in this simplest multivariate case the trade off between bias and variance depends on a large number of parameters it is generally preferable to fit correlations somewhat above the true correlation to minimize the effect of between individual confounders or cohort effects period effects can lead to situations where it is advantageous to fit correlations that are below the true correlation the results highlight the trade offs inherent in the choice of method for analysis of longitudinal data and show that an appropriate choice can be made only after determining whether within or between individual confounding is the major concern

statistical analysis of aberrant crypt assays for colon cancer promotion studies summary the size of aberrant crypt foci in rodent colons is a short term assay for the promotion of colon carcinogenesis an approximate model for the number of aberrant crypts in a focus is derived by assuming a branching process with a gamma lifetime maximum likelihood estimation can be implemented using symbolic manipulation packages although in most cases the model provides a reasonable fit for the distribution of the number of aberrant crypts per focus there is some evidence of lack of fit a more robust analysis can be based on summarizing the data for each animal by their mean number of aberrant crypts per focus the branching process with gamma lifetime is used to obtain expressions for the functional relation between the variance of the number of aberrant crypts per focus and their mean it is shown that in most cases this relation is quadratic the quadratic relation also holds for other lifetime distributions the parameters of the quadratic relation can be estimated by pseudolikelihood an experiment for assessing the promotional effect of the compound hmf is used for illustration

a differential lyapunov framework for contraction analysis lyapunov s second theorem is an essential tool for stability analysis of differential equations the paper provides an analog theorem for incremental stability analysis by lifting the lyapunov function to the tangent bundle the lyapunov function endows the state space with a finsler structure incremental stability is inferred from infinitesimal contraction of the finsler metrics through integration along solutions curves

using empirical bayes methods in biopharmaceutical research a compound sampling model where a unit specific parameter is sampled from a prior distribution and then observed are generated by a sampling distribution depending on the parameter underlies a wide variety of biopharmaceutical data for example in a multi centre clinical trial the true treatment effect varies from centre to centre observed treatment effects deviate from these true effects through sampling variation knowledge of the prior distribution allows use of bayesian analysis to compute the posterior distribution of clinic specific treatment effects frequently summarized by the posterior mean and variance more commonly with the prior not completely specified observed data can be used to estimate the prior and use it to produce the posterior distribution an empirical bayes or variance component analysis in the empirical bayes model the estimated prior mean gives the typical treatment effect and the estimated prior standard deviation indicates the heterogeneity of treatment effects in both the bayes and empirical bayes approaches estimated clinic effects are shrunken towards a common value from estimates based on single clinics this shrinkage produces more efficient estimates in addition the compound model helps structure approaches to ranking and selection provides adjustments for multiplicity allows estimation of the histogram of clinic specific effects and structures incorporation of external information this paper outlines the empirical bayes approach coverage will include development and comparison of approaches based on parametric priors for example a gaussian prior with unknown mean and variance and non parametric priors discussion of the importance of accounting for uncertainty in the estimated prior comparison of the output and interpretation of fixed and random effects approaches to estimating population values estimating histograms and identification of key considerations in the use and interpretation of empirical bayes methods

improved upper bounds for the contact process critical value the best known upper bound for the critical value c of the basic one dimensional contact process is 2 most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds this seemed not to be the case for the approach which yielded c 2 for the basic contact process in this paper we propose a technique for generating better bounds in this context to illustrate its use we carry out the full program in one case with the conclusion that c 1 942

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

a method for finding bounds on critical values for non attractive interacting particle systems the method is to use a computer to find a function for the process that always has negative expectation by considering all relevant configurations of 0 s and 1 s at the boundary of a finite process in one dimension it is shown that a branching annihilating random walk will die out if the diffusion parameter is greater than 0 176 the method may also be applied to attractive processes and is used for the contact process in one dimension obtaining the same values as ziezold and grillenberger 1988 for up to 10 places in from the boundaries

simple and trustworthy cluster robust gmm inference abstract this paper develops a new asymptotic theory for gmm estimation and inference in the presence of clustered dependence the key feature of our alternative asymptotics is that the number of clusters g is regarded as fixed as the sample size increases under the fixed g asymptotics we show that the wald and t tests in two step gmm are asymptotically pivotal only if we recenter the estimated moment process in the clustered covariance estimator cce also the j statistic the trinity of two step gmm statistics qlr lm and wald and the t statistic can be modified to have an asymptotic standard f distribution or t distribution we suggest a finite sample variance correction to further improve the accuracy of the f and t approximations the proposed tests are very appealing to practitioners because the test statistics are simple modifications of conventional gmm test statistics and critical values are readily available from f and t tables no further simulations or resampling methods are needed a monte carlo study shows that our proposed tests are more accurate than the conventional large g asymptotic inferences

application of empirical bayes inference to estimation of rate of change in the presence of informative right censoring we apply parametric empirical bayes inference of morris7 to the estimation of rate of change from incomplete longitudinal studies where the right censoring process is considered informative that is the length of time the subjects participate in the study is associated with level of the study variable ignoring such an association can result in a biased estimate of rate of change the proposed method provides estimates of rate of change for individual subjects as well as for the entire group adjusted for informative right censoring the method is considered more robust than those based on a specific parametric model for the censoring distribution under non informative right censoring these estimators of slopes are equivalent to the bayes estimators derived by fearn 11 we illustrate the method with an example involving renal transplant data we evaluate the method s performance through a simulation study

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

using the general linear mixed model to analyse unbalanced repeated measures and longitudinal data the general linear mixed model provides a useful approach for analysing a wide variety of data structures which practising statisticians often encounter two such data structures which can be problematic to analyse are unbalanced repeated measures data and longitudinal data owing to recent advances in methods and software the mixed model analysis is now readily available to data analysts the model is similar in many respects to ordinary multiple regression but because it allows correlation between the observations it requires additional work to specify models and to assess goodness of fit the extra complexity involved is compensated for by the additional flexibility it provides in model fitting the purpose of this tutorial is to provide readers with a sufficient introduction to the theory to understand the method and a more extensive discussion of model fitting and checking in order to provide guidelines for its use we provide two detailed case studies one a clinical trial with repeated measures and dropouts and one an epidemiological survey with longitudinal follow up

selecting the number of classes under latent class regression a factor analytic analogue recently the regression extension of latent class analysis rlca model has received much attention in the field of medical research the basic rlca model summarizes shared features of measured multiple indicators as an underlying categorical variable and incorporates the covariate information in modeling both latent class membership and multiple indicators themselves to reduce complexity and enhance interpretability one usually fixes the number of classes in a given rlca often goodness of fit methods comparing various estimated models are used as a criterion to select the number of classes in this paper we propose a new method that is based on an analogous method used in factor analysis and does not require repeated fitting two ideas with application to many settings other than ours are synthesized in deriving the method a connection between latent class models and factor analysis and techniques of covariate marginalization and elimination a monte carlo simulation study is presented to evaluate the behavior of the selection procedure and compare to alternative approaches data from a study of how measured visual impairments affect older persons functioning are used for illustration

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

a simulation study of estimators for rates of change in longitudinal studies with attrition many longitudinal studies and clinical trials are designed to compare rates of change over time in one or more outcome variables in several groups most such studies have incomplete data because some patients drop out before completing the study the missing data may induce bias and inefficiency in naive estimates of important parameters this paper uses monte carlo methods to compare the bias and efficiency of several two stage estimators of the effect of treatment on the mean rate of change when the missing data arise from one of four processes we also study the validity of confidence intervals and the power of hypothesis tests based on these estimates and their standard errors in general the weighted least squares estimator does relatively well as does an analysis of covariance type estimator proposed by wu et al the best estimates of variance components are based on complete cases or maximum likelihood

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

marginal models for longitudinal continuous proportional data continuous proportional data arise when the response of interest is a percentage between zero and one e g the percentage of decrease in renal function at different follow up times from the baseline in this paper we propose methods to directly model the marginal means of the longitudinal proportional responses using the simplex distribution of barndorff nielsen and jorgensen that takes into account the fact that such responses are percentages restricted between zero and one and may as well have large dispersion parameters in such a marginal model are estimated using an extended version of the generalized estimating equations where the score vector is a nonlinear function of the observed response the method is illustrated with an ophthalmology study on the use of intraocular gas in retinal repair surgeries