on perturbations of generalized landau lifshitz dynamics we consider deterministic and stochastic perturbations of dynamical systems with conservation laws in 3 the landau lifshitz equation for the magnetization dynamics in ferromagnetics is a special case of our system the averaging principle is a natural tool in such problems but bifurcations in the set of invariant measures lead to essential modification in classical averaging the limiting slow motion in this case in general is a stochastic process even if pure deterministic perturbations of a deterministic system are considered the stochasticity is a result of instabilities in the non perturbed system as well as of existence of ergodic sets of a positive measure we effectively describe the limiting slow motion

combined association and aggregation analysis of data from case control family studies genetic epidemiologists are increasingly interested in family studies using the case control family study design and this has motivated several recent developments in related statistical methodology by summarising and extending some of these developments this paper proposes an estimating equation based framework for analysing family data collected from case control studies motivated by the likelihood approach proposed by whittemore 1995 the proposed estimating equation approach requires modelling assumptions about means correlation coefficients and at most trivariate correlation coefficients and it can be used for analysing family data with arbitrary pedigree structures which would be prohibitive with the likelihood based approach further this new approach utilises all available data from case and control probands and from their relatives in assessing association between disease outcome and covariates and data from relatives in evaluating aggregation of disease phenotypes within families thus enhancing its efficiency the approach is illustrated by analysing data collected from case control studies on ovarian and colorectal cancers

the brownian snake and solutions of δu u2 in a domain we investigate the connections between the path valued process called the brownian snake and nonnegative solutions of the partial differential equation u u2 in a domain of d in particular we prove two conjectures recently formulated by dynkin the first one gives a complete characterization of the boundary polar sets which correspond to boundary removable singularities for the equation u u2 the second one establishes a one to one correspondence between nonnegative solutions that are bounded above by a harmonic function and finite measures on the boundary that do not charge polar sets this correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the brownian snake our proofs combine probabilistic and analytic arguments an important role is played by a new version of the special markov property which is of independent interest

random perturbations of dynamical systems and diffusion processes with conservation laws in this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral we calculate under some assumptions the limiting behavior of the slow component of the perturbed system in an appropriate time scale for a general class of perturbations the phase space of the slow motion is a graph defined by the first integral this is a natural generalization of the results concerning random perturbations of hamiltonian systems considering diffusion processes as the unperturbed system allows to study the multidimensional case and leads to a new effect the limiting slow motion can spend non zero time at some points of the graph in particular such delay at the vertices leads to more general gluing conditions our approach allows one to obtain new results on singular perturbations of pde s

the brownian cactus ii upcrossings and local times of super brownian motion we study properties of the random metric space called the brownian map for every r 0 we consider the connected components of the complement of the open ball of radius r centered at the root and we let mathbf n r varepsilon be the number of those connected components that intersect the complement of the ball of radius r varepsilon we then prove that varepsilon 3 mathbf n r varepsilon converges as varepsilon rightarrow 0 to a constant times the density at r of the profile of distances from the root in terms of the brownian cactus this gives asymptotics for the number of points at height r that have descendants at height r varepsilon our proofs are based on a similar approximation result for local times of super brownian motion by upcrossing numbers our arguments make a heavy use of the brownian snake and its special markov property

thermostat like perturbations of an oscillator we consider an oscillator with one degree of freedom perturbed by a deterministic thermostat like perturbation and another system in particular another oscillator coupled with the first one if the hamiltonian of the first system has saddle points the whole system has in a sense a stochastic behavior on long time intervals under certain conditions one can introduce the relative entropy and describe metastability and other large deviation effects in this deterministic system if the coupled system is also an oscillator the long time evolution of the energy of this oscillator has a diffusion approximation to get these results one has to regularize the system but the results are to some extent independent of the regularization the stochasticity is due to instabilities at saddle points of the original system

composite conditional likelihood for sparse clustered data summary sparse clustered data arise in finely stratified genetic and epidemiologic studies and pose at least two challenges to inference first it is difficult to model and interpret the full joint probability of dependent discrete data which limits the utility of full likelihood methods second standard methods for clustered data such as pairwise likelihood and the generalized estimating function approach are unsuitable when the data are sparse owing to the presence of many nuisance parameters we present a composite conditional likelihood for use with sparse clustered data that provides valid inferences about covariate effects on both the marginal response probabilities and the intracluster pairwise association our primary focus is on sparse clustered binary data in which case the method proposed utilizes doubly discordant quadruplets drawn from each stratum to conduct inference about the intracluster pairwise odds ratios

marginally specified logistic normal models for longitudinal binary data summary likelihood based inference for longitudinal binary data can be obtained using a generalized linear mixed model breslow n and clayton d g 1993 journal of the american statistical association88 9 25 wolfinger r and o connell m 1993 journal of statistical computation and simulation48 233 243 given the recent improvements in computational approaches alternatively fitzmaurice and laird 1993 biometrika80 141 151 molenberghs and lesaffre 1994 journal of the american statistical association89 633 644 and heagerty and zeger 1996 journal of the american statistical association91 1024 1036 have developed a likelihood based inference that adopts a marginal mean regression parameter and completes full specification of the joint multivariate distribution through either canonical and or marginal higher moment assumptions each of these marginal approaches is computationally intense and currently limited to small cluster sizes in this manuscript an alternative parameterization of the logistic normal random effects model is adopted and both likelihood and estimating equation approaches to parameter estimation are studied a key feature of the proposed approach is that marginal regression parameters are adopted that still permit individual level predictions or contrasts an example is presented where scientific interest is in both the mean response and the covariance among repeated measurements

global cross ratio models for bivariate discrete ordered responses summary a family of statistical models is presented for bivariate discrete response to a regressor when both components of the response have ordered categories association between components is expressed in terms of global cross ratios cross product ratios of quadrant probabilities for each double dichotomy of the response table of probabilities into quadrants pearson and heron 1913 biometrika 9 159 315 these models are extensions to the work of plackett 1965 journal of the american statistical association 60 516 522 and mantel and brown 1973 biometrics 29 649 665 the marginal cumulative probabilities may satisfy linear logistic or other generalized linear models mccullagh 1980 journal of the royal statistical society series b 42 109 142 an analysis of patients postoperative pain level and medication frequency illustrates these methods

composite likelihood bayesian information criteria for model selection in high dimensional data for high dimensional data sets with complicated dependency structures the full likelihood approach often leads to intractable computational complexity this imposes difficulty on model selection given that most traditionally used information criteria require evaluation of the full likelihood we propose a composite likelihood version of the bayes information criterion bic and establish its consistency property for the selection of the true underlying marginal model our proposed bic is shown to be selection consistent under some mild regularity conditions where the number of potential model parameters is allowed to increase to infinity at a certain rate of the sample size simulation studies demonstrate the empirical performance of this new bic especially for the scenario where the number of parameters increases with sample size technical proofs of our theoretical results are provided in the online supplemental materials

inference for odds ratio regression models with sparse dependent data suppose the number of 2 x 2 tables is large relative to the average table size and the observations within a given table are dependent as occurs in longitudinal or family based case control studies we consider fitting regression models to the odds ratios using table level covariates the focus is on methods to obtain valid inferences for the regression parameters beta when the dependence structure is unknown in this setting liang 1985 biometrika 72 678 682 has shown that inference based on the noncentral hypergeometric likelihood is sensitive to misspecification of the dependence structure in contrast estimating functions based on the mantel haenszel method yield consistent estimators of beta we show here that under the estimating function approach wald s confidence interval for beta performs well in multiplicative regression models but unfortunately has poor coverage probabilities when an additive regression model is adopted as an alternative to wald inference we present a mantel haenszel quasi likelihood function based on integrating the mantel haenszel estimating function a simulation study demonstrates that in medium sized samples the mantel haenszel quasi likelihood approach yields better inferences than other methods under an additive regression model and inferences comparable to wald s method under a multiplicative model we illustrate the use of this quasi likelihood method in a study of the familial risk of schizophrenia

generalization of the mantel haenszel estimating function for sparse clustered binary data we extend the mantel haenszel estimating function to estimate both the intra cluster pairwise correlation and the main effects for sparse clustered binary data we propose both a composite likelihood approach and an estimating function approach for the analysis of such data the proposed estimators are consistent and asymptotically normally distributed simulation results demonstrate that the two approaches are comparable in terms of bias and efficiency however the estimating equation approach is computationally simpler analysis of the georgia high blood pressure survey is used for illustration

long time behavior of weakly coupled oscillators we consider small perturbations of a simple completely integrable system with many degrees of freedom a collection of independent one degree of freedom oscillators in the perturbed system the individual oscillators are no longer independent we show that the long time behavior of such a system even in the case of purely deterministic perturbations should in general be described as a stochastic process the limiting stochastic process is a markov process on an open book space corresponding to the collection of first integrals of the non perturbed system

a covariance estimator for gee with improved small sample properties summary in this paper we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations gee hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small resampling methods such as the jackknife and bootstrap have been suggested for covariance estimation when the number of clusters is small a drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near zero cell counts caused by resampling we propose a bias corrected covariance estimator that avoids this problem in a small simulation study we compare the bias corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10 40 subjects with equal and unequal cluster sizes of 16 64 observations the bias corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2 3 times the nominal level the methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease

approximate likelihood ratios for general estimating functions summary the method of estimating functions godambe 1991 is commonly used when one desires to conduct inference about some parameters of interest but the full distribution of the observations is unknown however this approach may have limited utility due to multiple roots for the estimating function a poorly behaved wald test or lack of a goodness of fit test this paper presents approximate likelihood ratios that can be used along with estimating functions when any of these three problems occurs we show that the approximate likelihood ratio provides correct large sample inference under very general circumstances including clustered data and misspecified weights in the estimating function two methods of constructing the approximate likelihood ratio one based on the quasi likelihood approach and the other based on the linear projection approach are compared and shown to be closely related in particular we show that quasi likelihood is the limit of the projection approach we illustrate the technique with two applications

a parametric model for cluster correlated categorical data summary a fully parametric copula model for symmetric dependent clustered categorical data is discussed the model accommodates any marginal regression models of interest and admits a broad range of within cluster association the form of the distribution is independent of cluster size and may be used to model data with varying cluster sizes the model contains an association parameter that is estimated from the data to give a measure of the strength of the within cluster association and also a test of independence two examples are given to illustrate the methods

logistic regression of family data from case control studies summary multivariate regression models are applied to binary disease data in families identified from case control studies attention is restricted to marginal or reproducible models i e those whose parameters have the same interpretations in the marginal distributions for all subsets of a family with a logistic specification of each individual s marginal disease probability for such models it is shown that the case control family data can be analysed as if they were obtained from a prospective study with the baseline disease probabilities of case and control probands differing from that of their relatives this result extends that of anderson 1972 and prentice pyke 1979 for the probands data to include disease outcomes and covariates for their families it contrasts with inconsistent estimates of parameters in nonreproducible models that result when the case control sampling design is ignored tosteson rosner redline 1991 the contrast underscores the need to check the plausibility of the reproducibility assumption which requires that the covariates be independent of any unmeasured factors responsible for the correlation of familial disease occurrence before analysing case control data prospectively

tree based models for random distribution of mass a mathematical model for distribution of mass ind dimensional space based upon randomly embedding random trees into space is introduced and studied the model is a variant of thesuper brownian motion process studied by mathematicians we present calculations relating to i the distribution of position of a typical mass element ii moments of the center of mass iii large deviation behavior and iv a recursive self similarity property

extended mantel haenszel estimating procedure for multivariate logistic regression models a class of estimating functions is proposed for the estimation of multivariate relative risk in stratified case control studies it reduces to the well known mantel haenszel estimator when there is a single binary risk factor large sample properties of the solutions to the proposed estimating equations are established for two distinct situations efficiency calculations suggest that the proposed estimators are nearly fully efficient relative to the conditional maximum likelihood estimator for the parameters considered application of the proposed method to family data and longitudinal data where the conditional likelihood approach fails is discussed two examples from case control studies and one example from a study on familial aggregation are presented

pairwise likelihood inference for nested hidden markov chain models for multilevel longitudinal data in the context of multilevel longitudinal data where sample units are collected in clusters an important aspect that should be accounted for is the unobserved heterogeneity between sample units and between clusters for this aim we propose an approach based on nested hidden latent markov chains which are associated with every sample unit and with every cluster the approach allows us to account for the previously mentioned forms of unobserved heterogeneity in a dynamic fashion it also allows us to account for the correlation that may arise between the responses provided by the units belonging to the same cluster under the assumed model computing the manifest distribution of these response variables is infeasible even with a few units per cluster therefore we make inference on this model through a composite likelihood function based on all the possible pairs of subjects within each cluster properties of the composite likelihood estimator are assessed by simulation the proposed approach is illu

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

geodesics in large planar maps and in the brownian map we study geodesics in the random metric space called the brownian map which appears as the scaling limit of large planar maps in particular we completely describe geodesics starting from the distinguished point called the root and we characterize the set s of all points that are connected to the root by more than one geodesic the set s is dense in the brownian map and homeomorphic to a non compact real tree furthermore for every x in s the number of distinct geodesics from x to the root is equal to the number of connected components of s x in particular points of the brownian map can be connected to the root by at most three distinct geodesics our results have applications to the behavior of geodesics in large planar maps

averaging of incompressible flows on two dimensional surfaces we consider a process on a compact two dimensional surface which consists of the fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise it gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow it has been shown by freidlin and wentzell that if the unperturbed motion is periodic for almost all the initial points then the corresponding process on the graph weakly converges to a markov process we consider the situation when the unperturbed motion is not periodic and the flow has ergodic components of positive measure we show that the process on the graph still converges to a markov process which spends a positive proportion of time in the vertices corresponding to the ergodic components of the flow as shown in a companion paper 5 these results allow one to describe the viscosity regularization of dissipative deterministic perturbations of incompressible flows in the case of surfaces of higher genus the limiting process may exhibit intermittent behavior in the sense that the time axis can be divided into the intervals of random lengths where the particle stays inside one ergodic component separated by the intevals where the particle moves deterministically between the components