the q laplace operator and q harmonic polynomials on the quantum vector space the aim of this paper is to study q harmonic polynomials on the quantum vector space generated by q commuting elements x1 x2 xn they are defined as solutions of the equation qp 0 where p is a polynomial in x1 x2 xn and the q laplace operator q is determined in terms of q derivatives the projector hm am hm is constructed where am and hm are the spaces of homogeneous of degree m polynomials and q harmonic polynomials respectively by using these projectors a q analog of classical associated spherical harmonics is constructed they constitute an orthonormal basis of hm a q analog of separation of variables is given representations of the nonstandard q deformed algebra uq son which plays the role of the rotation group so n in the case of classical harmonic polynomials on the spaces hm are explicitly constructed

nonparametric analysis of covariance for censored data the fully nonparametric model for nonlinear analysis of covariance proposed in akritas et al 2000 is considered in the context of censored observations under this model the distributions for each factor level combination and covariate value are not restricted to comply to any parametric or semiparametric model the data can be continuous or ordinal categorical the possibility of different shapes of covariate effect in different factor level combinations is also allowed this generality is useful whenever modelling assumptions such as additive risks proportional hazards or proportional odds appear suspect test statistics are obtained for the nonparametric hypotheses of no main effect and of no interaction effect which adjusts for the presence of a covariate they are quadratic forms based on averages over the covariate values of beran estimators of the conditional distribution of the survival time given each covariate value the derivation of the asymptotic chi 2 distribution of the test statistics uses a recently obtained asymptotic representation of the bekan estimator as average of independent random variables a real data set is analysed and results of simulation studies are reported

temporal process regression summary we consider regression for response and covariates which are temporal processes observed over intervals a functional generalised linear model is proposed which includes extensions of standard models in multi state survival analysis simple nonparametric esti mators of time indexed parameters are developed using working independence estimating equations and are shown to be uniformly consistent and to converge weakly to gaussian processes the procedure does not require smoothing or a markov assumption unlike approaches based on transition intensities the usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy the estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters providing novel goodness of fit tests the methodology s practical utility is illustrated in a data analysis

representations of the kauffman bracket skein algebra i invariants and miraculous cancellations we study finite dimensional representations of the kauffman bracket skein algebra of a surface s in particular we construct invariants of such irreducible representations when the underlying parameter q mathrm e 2 pi mathrm i hbar is a root of unity the main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group pi 1 s to mathrm sl 2 mathbb c or in a twisted version of this character variety the proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors these miraculous cancellations also play a fundamental role in subsequent work of the authors where novel examples of representations of the skein algebra are constructed

superposition rules for nonlinear coupled first order differential equations pairs of coupled nonlinear differential equations of the polynomial type have been studied no higher powers than quadratic were considered lie s theorem provides a superposition rule exists if certain operators generate a finite dimensional lie algebra the lie algebras possible were divided into twenty categories for each case the general form of the coupled differential equations is obtained the coupled differential equations are then separated where possible solutions are obtained expressed in terms of a finite number of particular solutions

statistical estimation in varying coefficient models varying coefficient models are a useful extension of classical linear models they arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age the appeal of these models is that the coef cient functions can easily be estimated via a simple local regression this yields a simple one step estimation procedure we show that such a one step method cannot be optimal when different coefficient functions admit different degrees of smoothness this drawback can be repaired by using our proposed two step estimation procedure the asymptotic mean squared error for the two step procedure is obtained and is shown to achieve the optimal rate of convergence a few simulation studies show that the gain by the two step procedure can be quite substantial the methodology is illustrated by an application to an environmental data set

censored quantile regression with partially functional effects quantile regression offers a flexible approach to analyzing survival data allowing each covariate effect to vary with quantiles in practice constancy is often found to be adequate for some covariates in this paper we study censored quantile regression tailored to the partially functional effect setting with a mixture of varying and constant effects such a model can offer a simpler view regarding covariate survival association and moreover can enable improvement in estimation efficiency we propose profile estimating equations and present an iterative algorithm that can be readily and stably implemented asymptotic properties of the resultant estimators are established a simple resampling based inference procedure is developed and justified extensive simulation studies demonstrate efficiency gains of the proposed method over a naive two stage procedure the proposed method is illustrated via an application to a recent renal dialysis study

center of quantum algebra uq so 3 we analyze the structure of the center of the quantum algebra uq so 3 this structure as expected depends substantially on the deformation parameter q when q is not a root of unity there exists only one casimir element which is the deformation of ordinary casimir element known from so3 the center has a structure of a ring of polynomials of one variable when qn 1 there are three more casimir elements of the form of polynomials in algebra generators uq so 3 can be seen as a finite dimensional module over the commutative ring of polynomials in these three casimir elements knowing three parametrical family of n dimensional irreducible representations we prove that the dimension of this module is n3 all four casimir elements are no longer algebraically independent we present the explicit description of the central variety that is polynomial dependence between these four casimir elements this dependence differs for n 2m 1 n 2 2m 1 and n 4m

representations of the q deformed algebra uq so4 we study the nonstandard q deformation uq so4 of the universal enveloping algebra u so4 obtained by deforming the defining relations for skew symmetric generators of u so4 this algebra is used in quantum gravity and algebraic topology we construct a homomorphism of uq so4 to the certain nontrivial extension of the drinfeld jimbo quantum algebra uq sl2 2 and show that this homomorphism is an isomorphism by using this homomorphism we construct irreducible finite dimensional representations of the classical type and of the nonclassical type for the algebra uq so4 it is proved that for q not a root of unity each irreducible finite dimensional representation of uq so4 is equivalent to one of these representations we prove that every finite dimensional representation of uq so4 for q not a root of unity is completely reducible it is shown how to construct by using the homomorphism tensor products of irreducible representations of uq so4 note that no hopf algebra structure is known fo

universal formats for nonlinear ordinary differential systems it is shown that very general nonlinear ordinary differential systems embracing all that arise in practice may first be brought down to polynomial systems where the nonlinearities occur only as polynomials in the dependent variables by introducing suitable new variables into the original system second that polynomial systems are reducible to riccati systems where the nonlinearities are quadratic at most third that riccati systems may be brought to elemental universal formats containing purely quadratic terms with simple arrays of coefficients that are all zero or unity the elemental systems have representations as novel types of matrix riccati equations different starting systems and their associated riccati systems differ from one another at the final elemental level in order and in initial data but not in format

nonparametric survival analysis with time dependent covariate effects a penalized partial likelihood approach techniques are developed for nonparametric analysis of data under a cox regression like model permitting time dependent covariate effects determined by a regression function beta 0 t estimators resulting from maximization of an appropriate penalized partial likelihood are shown to exist and a computational approach is outlined weak uniform consistency with a rate of convergence and pointwise asymptotic normality of the estimators are established under regularity conditions a consistent estimator of a common baseline hazard function is presented and used to construct a consistent estimator of the asymptotic variance of the estimator of the regression function extensions to multiple covariates general relative risk functions and time dependent covariates are discussed

semiparametric analysis of the additive risk model in contrast to the proportional hazards model the additive risk model specifies that the hazard function associated with a set of possibly time varying covariates is the sum of rather than the product of the baseline hazard function and the regression function of covariates this formulation describes a different aspect of the association between covariates and the failure time than the proportional hazards model and is more plausible than the latter for many applications in the present paper simple procedures with high efficiencies are developed for making inference about the regression parameters under the additive risk model with an unspecified baseline hazard function

representations of the q deformed algebra uq iso2 an algebra homomorphism from the q deformed algebra uq iso2 with generating elements i t1 t2 and defining relations i t2 q t1 t1 i q t2 t2 t1 q 0 where a b q q1 2ab q 1 2ba to the extension uq m2 of the hopf algebra uq m2 is constructed the algebra uq iso2 at q 1 leads to the lie algebra iso2 m2 of the group iso 2 of motions of the euclidean plane the hopf algebra uq m2 which is not isomorphic to uq iso2 is treated as a hopf q deformation of the universal enveloping algebra of iso2 and is well known in the literature not all irreducible representations of uq m2 can be extended to representations of the extension uq m2 composing the homomorphism with irreducible representations of uq m2 we obtain representations of uq iso2 not all of these representations of uq iso2 are irreducible the reducible representations of uq iso2 are decomposed into irreducible components in this way we obtain all irreducible representations of uq iso2 when q is not a root of unity a part of these representations turns into irreducible representations of the lie algebra iso2 when q1 representations of the other part have no classical analogue

additive risk models for survival data with high dimensional covariates as a useful alternative to cox s proportional hazard model the additive risk model assumes that the hazard function is the sum of the baseline hazard function and the regression function of covariates this article is concerned with estimation and prediction for the additive risk models with right censored survival data especially when the dimension of the covariates is comparable to or larger than the sample size principal component regression is proposed to give unique and numerically stable estimators asymptotic properties of the proposed estimators component selection based on the weighted bootstrap and model evaluation techniques are discussed this approach is illustrated with analysis of the primary biliary cirrhosis clinical data and the diffuse large b cell lymphoma genomic data it is shown that this methodology is numerically stable and effective in dimension reduction while still being able to provide satisfactory prediction and classification results

confidence bands for cumulative incidence curves under the additive risk model in the context of competing risks the cumulative incidence function is often used to summarize the cause specific failure time data as an alternative to the proportional hazards model the additive risk model is used to investigate covariate effects by specifying that the subject specific hazard function is the sum of a baseline hazard function and a regression function of covariates based on such a formulation we present an approach to constructing simultaneous confidence intervals for the cause specific cumulative incidence function of patients with given risk factors a melanoma data set is used for the purpose of illustration

quantum deformations of su 2 explicit representations of two deformations of su 2 one in the cartan basis and one in the cartesian basis are constructed the cartan deformation subsumes previously known cases while the cartesian deformation appears to be new in the latter case explicit reversible mappings in terms of representations of su 2 and a casimir for the algebra are constructed

representations of the cyclically symmetric q deformed algebra soq 3 an algebra homomorphism from the nonstandard q deformed cyclically symmetric algebra uq so3 to the extension uq sl2 of the hopf algebra uq sl2 is constructed not all irreducible representations of uq sl2 can be extended to representations of uq sl2 composing the homomorphism with irreducible representations of uq sl2 we obtain representations of uq so3 not all of these representations of uq so3 are irreducible reducible representations of uq so3 are decomposed into irreducible components in this way we obtain all irreducible representations of uq so3 when q is not a root of unity a part of these representations turns into irreducible representations of the lie algebra so3 when q 1 representations of the other part have no classical analog using the homomorphism it is shown how to construct tensor products of finite dimensional representations of uq so3 irreducible representations of uq so3 when q is a root of unity are constructed some of them are obtained from irreducible re

on the classification of irreducible finite dimensional representations of uq so3 algebra in an earlier work m havlicek et al j math phys 40 2135 1999 we defined for any finite dimension five nonequivalent irreducible representations of the nonstandard deformation uq so3 of the lie algebra so3 where q is not a root of unity for each dimension only one of them called classical admits limit q 1 in the first part of this paper we show that any finite dimensional irreducible representation is equivalent to some of these representations in the case qn 1 we derive new casimir elements of uq so3 and show that a dimension of any irreducible representation is not higher than n these elements are casimir elements of uq som for all m and even of uq isom 1 due to inonu wigner contraction according to the spectrum of one of the generators the representations are found to belong to two main disjoint sets we give full classification and explicit formulas for all representations from the first set we call them nonsingular representations if n is odd we have full classificatio

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography

modeling partly conditional means with longitudinal data abstract we propose a general modeling approach to longitudinal data that is a hybrid of the marginal regression models of zeger and liang and of the classical transition models such as used in time series analyses rather than conditioning at time t only on covariate values as is typical with the marginal approach or on the entire history of the process up to t as is typical with the transition model approach we suggest models that condition on a subset of the process history estimation proceeds using generalized estimating equation methodology but with the restriction that the working covariance matrix is diagonal the proposed regression models share common features with cox regression models for failure time data in that they are composed of a nuisance baseline function of time and a simple parametric function of the covariates two illustrative examples are presented

comments on superposition rules for nonlinear coupled first order differential equations some comments are made on the classification of finite dimensional subalgebras of the lie algebra of vector fields in n variables and of the related nonlinear ordinary differential equations with superposition principles in particular for n 2 a very natural requirement of indecomposability implies that only two types of equations need be considered

invariants of 2 1 gravity in 1 2 we established and dicussed the algebra of observables for 2 1 gravity at both the classical and quantum level here our treatment broadens and extends previous results to any genusg with a systematic discussion of the centre of the algebra the reduction of the number of independent observables to 6g 6 g 1 is treated in detail with a precise classification forg 1 andg 2

survival analysis with temporal covariate effects summary we propose a natural generalization of the cox regression model in which the regression coefficients have direct interpretations as temporal covariate effects on the survival function under the conditionally independent censoring mechanism we develop a smoothing free estimation procedure with a set of martingale based equations our estimator is shown to be uniformly consistent and to converge weakly to a gaussian process a simple resampling method is proposed for approximating the limiting distribution of the estimated coefficients second stage inferences with time varying coefficients are developed accordingly simulations and a real example illustrate the practical utility of the proposed method finally we extend this proposal of temporal covariate effects to the general class of linear transformation models and also establish a connection with the additive hazards model

analysis of additive risk model with high dimensional covariates using partial least squares in this paper we construct a partial additive regression par model to predict the survival times of cancer patients based on microarray gene expression data with right censoring the area under time dependent receiver operating characteristic curve is used as a model evaluation criterion we conduct a simulation study to compare the proposed method with other methods i e partial cox regression and supervised principal component analysis two data sets of breast cancer and diffuse large b cell lymphoma are analyzed to illustrate our procedure the outcome indicates great predictive performance on both dimension reduction and predictive performance of the proposed par model

functional association models for multivariate survival processes we consider multivariate temporal processes that are continuously observed within overlapping time windows the intended application is in censored multistate and multivariate survival settings where point processes are continuously observed these data differ from other functional data like longitudinal data which are discretely observed at irregular times existing functional approaches to survival processes use intensity models which require smoothing and depend critically on the choice of smoothing parameters similarly to discretely observed data in this article we study functional mean and association regression models for the point processes with unspecified time varying coefficients the continuous observation scheme is exploited the coefficients may be estimated nonparametrically by extending generalized estimating equations to continuously observed data the estimators automatically converge at the parametric rate without smoothing unlike with discretely observed data uniform consistenc

time series modelling with semiparametric factor dynamics high dimensional regression problems which reveal dynamic behavior are typically analyzed by time propagation of a few number of factors the inference on the whole system is then based on the low dimensional time series analysis such high dimensional problems occur frequently in many different fields of science in this article we address the problem of inference when the factors and factor loadings are estimated by semiparametric methods this more flexible modeling approach poses an important question is it justified from an inferential point of view to base statistical inference on the estimated times series factors we show that the difference of the inference based on the estimated time series and true unobserved time series is asymptotically negligible our results justify fitting vector autoregressive processes to the estimated factors which allows one to study the dynamics of the whole high dimensional system with a low dimensional representation we illustrate the theory with a simulati

efficient estimation for the cox model with varying coefficients summary a proportional hazards model with varying coefficients allows one to examine the extent to which covariates interact nonlinearly with an exposure variable a global partial likelihood method in contrast with the local partial likelihood method of fan et al 2006 is proposed for estimation of varying coefficient functions the proposed estimators are proved to be consistent and asymptotically normal semiparametric efficiency of the estimators is demonstrated in terms of their linear functionals evidence in support of the superiority of the method is presented in numerical studies and real examples

an estimating equation for parametric shared frailty models with marginal additive hazards summary multivariate failure time data arise when data consist of clusters in which the failure times may be dependent a popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption in some contexts however it may be more reasonable to use the marginal additive hazards model we derive asymptotic properties of the lin and ying estimators for the marginal additive hazards model for multivariate failure time data furthermore we suggest estimating equations for the regression parameters and association parameters in parametric shared frailty models with marginal additive hazards by using the lin and ying estimators we give the large sample properties of the estimators arising from these estimating equations and investigate their small sample properties by monte carlo simulation a real example is provided for illustration

partly functional temporal process regression with semiparametric profile estimating functions summary marginal mean models of temporal processes in event time data analysis are gaining more attention for their milder assumptions than the traditional intensity models recent work on fully functional temporal process regression tpr offers great flexibility by allowing all the regression coefficients to be nonparametrically time varying the existing esti mation procedure however prevents successive goodness of fit test for covariate coefficients in comparing a sequence of nested models this article proposes a partly functional tpr model in the line of marginal mean models some covari ate effects are time independent while others are completely unspecified in time this class of models is very rich includ ing the fully functional model and the semiparametric model as special cases to estimate the parameters we propose semiparametric profile estimating equations which are solved via an iterative algorithm starting at a consistent estimate from a fully functional model in the existing work no smoothing is needed in contrast to other varying coefficient meth ods the weak convergence of the resultant estimators are developed using the empirical process theory successive tests of time varying effects and backward model selection procedure can then be carried out the practical usefulness of the methodology is demonstrated through a simulation study and a real example of recurrent exacerbation among cystic fibrosis patients

partially linear additive hazards regression with varying coefficients please see the supplementary materials for a correction to this article to explore the nonlinear interactions between some covariates and an exposure variable we propose the partially linear additive hazards model for survival data in a semiparametric setting we construct a local pseudoscore function to estimate the varying and constant coefficients and establish the asymptotic normality of the proposed estimators moreover we develop the weak convergence property for the local estimator of the baseline cumulative hazard function we conduct simulation studies to empirically examine the finite sample performance of the proposed methods and use real data from a breast cancer study for illustration

predicting cumulative incidence probability by direct binomial regression we suggest a new simple approach for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model we consider a semiparametric regression model where some effects may be time varying and some may be constant over time our estimator can be implemented by standard software our simulation study shows that the estimator works well and has finite sample properties comparable with the subdistribution approach we apply the method to bone marrow transplant data and estimate the cumulative incidence of death in complete remission following a bone marrow transplantation here death in complete remission and relapse are two competing events

dynamical correlation for multivariate longitudinal data nonparametric methodology for longitudinal data analysis is becoming increasingly popular the analysis of multivariate longitudinal data where data on several time courses are recorded for each subject has received considerably less attention despite its importance for practical data analysis in particular there is a need for measures and estimates to capture dependency between the components of vector valued longitudinal data we propose and analyze a simple and effective nonparametric method to quantify the covariation of components of multivariate longitudinal observations which are viewed as realizations of a random process this includes the notion of a correlation between derivatives and time shifted versions the concept of dynamical correlation is based on a scalar product obtained from pairs of standardized smoothed curves the proposed method can be used when observation times are irregular and not matching between subjects or between responses within a subject for higher dimensional dat

aalen additive hazards change point model we study a test comparing the full aalen additive hazards model and the change point model and suggest how to estimate the parameters of the change point model we also study a test for no change point effect both tests are provided with large sample properties and a resampling method is applied to obtain p values the finite sample properties of the proposed inference procedures and estimators are assessed through a simulation study the methods are further applied to a dataset concerning myocardial infarction

bayesian estimators for conditional hazard functions summary this article introduces a new bayesian approach to the analysis of right censored survival data the hazard rate of interest is modeled as a product of conditionally independent stochastic processes corresponding to 1 a baseline hazard function and 2 a regression function representing the temporal influence of the covariates these processes jump at times that form a time homogeneous poisson process and have a pairwise dependency structure for adjacent values the two processes are assumed to be conditionally independent given their jump times features of the posterior distribution such as the mean covariate effects and survival probabilities conditional on the covariate are evaluated using the metropolis hastings green algorithm we illustrate our methodology by an application to nasopharynx cancer survival data

additive hazards model with multivariate failure time data summary marginal additive hazards models are considered for multivariate survival data in which individuals may experience events of several types and there may also be correlation between individuals estimators are proposed for the parameters of such models and for the baseline hazard functions the estimators of the regression coefficients are shown asymptotically to follow a multivariate normal distribution with a sandwich type covariance matrix that can be consistently estimated the estimated baseline and subject specific cumulative hazard processes are shown to converge weakly to a zero mean gaussian random field the weak convergence properties for the corresponding survival processes are established a resampling technique is proposed for constructing simultaneous confidence bands for the survival curve of a specific subject the method ology is extended to a multivariate version of a class of partly parametric additive hazards model simulation studies are conducted to assess finite sample properties and the method is illustrated with an application to development of coronary heart diseases and cardiovascular accidents in the framingham heart study

finite dimensional irreducible modules of the universal askey wilson algebra since the introduction of askey wilson algebras by zhedanov in 1991 the classification of the finite dimensional irreducible modules of askey wilson algebras remains open a universal analog triangle q of the askey wilson algebras was recently studied in this paper we consider a family of infinite dimensional triangle q modules by the universal property of these triangle q modules we classify the finite dimensional irreducible triangle q modules when q is not a root of unity

nonparametric smoothing estimates of time varying coefficient models with longitudinal data this paper considers nonparametric estimation in a varying coefficient model with repeated measurements y ij x ij t ij for i 1 n and j 1 n i where x ij x ij x ijk t and y ij x ij t ij denote the jth outcome covariate and time design points respectively of the ith subject the model considered here is y ij x ij t t ij e i t ij where t 0 t k t t for k 0 are smooth nonparametric functions of interest and e i t is a zero mean stochastic process the measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject two nonparametric estimators of t namely a smoothing spline and a locally weighted polynomial are derived for such repeatedly measured data a crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters asymptotic properties such as consistency rates of convergence and asymptotic mean squared errors are established for kernel estimators a special case of the local polynomials these asymptotic results give useful insights into the reliability of our general estimation methods an example of predicting the growth of children born to hiv infected mothers based on gender hiv status and maternal vitamin a levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies

estimating the effect of treatment in a proportional hazards model in the presence of non compliance and contamination summary methods for adjusting for non compliance and contamination which respect the randomization are extended from binary outcomes to time to event analyses by using a proportional hazards model a simple non iterative method is developed when there are no covariates which is a generalization of the mantel haenszel estimator more generally a partial likelihood is developed which accommodates covariates under the assumption that they are independent of compliance a key feature is that the proportion of contaminators and non compliers in the risk set is updated at each failure time when covariates are not independent of compliance a full likelihood is developed and explored but this leads to a complex estimator estimating equations and information matrices are derived for these estimators and they are evaluated by simulation studies

regression analysis with multiplicative and time varying additive regression coefficients with examples from breast and colon cancer regression analysis may be used to simplify the representation of mortality rates when there are many significant prognostic covariates or to adjust for confounding effects the principal request of the regression model in this range of use is to have unbiased parameter estimates a model with constant multiplicative and time varying additive regression coefficients is discussed the model allows some covariate effects to be multiplicative while allowing others to have a time varying additive effect thus it is a mix of classical cox regression and aalen s additive risk model a major characteristic of cancer mortality rates in contrast to general mortality rates is that hazard rates after a potentially initial increase decrease although not always tending to zero cancer diseases like breast and colon cancer have significantly increased cause specific mortality rates even 20 years after diagnosis another major feature in cancer survival analysis is that many covariate effects are time varying some covariate effects like age at diagnosis may only be significant for a limited time after diagnosis furthermore some treatment procedures may initially decrease the mortality while the long term effect may be opposite a third issue is that average covariate effects are very often not multiplicative estimation is carried out iteratively the cumulative additive regression functions are estimated non parametrically using a least squares method and the multiplicative parameters are estimated from the partial likelihood the method is applied on 3201 female breast cancer and 1372 male colon cancer patients

generalised structured models summary we present a general class of nonlinear regression and time series models that we call generalised structured models the class is a natural generalisation of generalised additive models and it includes generalised interaction models structured volatility models visual garch generalised autoregressive conditional heteroscedasticity models and varying coefficient models we discuss estimation principles including smoothing splines and a generalisation of the projection approach of mammen et al 1999 we finish the paper with some theoretical considerations about the asymptotic performance of the estimator for the general class of generalised structured models

residuals for relative risk regression summary several possible definitions of residuals are given for relative risk regression with time varying covariates each such residual has a representation as an estimator of a stochastic integral with respect to the martingale arising from a subject s failure time counting process previously proposed residuals for individual study subjects and for specific time points are shown to be special cases of this definition as are previously derived regression diagnostics an illustration and various generalizations are also given able methods are required to detect various departures from modelling assumptions suitably defined residuals may play an important role in such identification however the nonparametric aspect of the model the possibility that modelled regression variables may be varying with follow up time and most importantly the usual presence of right censorship implies that specialized residual definitions are required the class of residuals considered here is most easily formulated using counting process notation for the failure time data let ni t yi t and zi t represent respectively for the ith subject the values of counting censoring and covariate processes at follow up time t i 1 n while ni u yi u zi u 0c u t specifies the corresponding histories for the ith subject prior to time t thus in a typical univariate failure time application ni with right continuous sample paths will take value zero prior to the time of failure on the ith subject and value one thereafter while yi with left continuous sample paths will take value one at times at which the ith subject is at risk for an observed failure and value zero otherwise the counting process ni can be uniquely decomposed so that for all t i

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

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

the partly aalen s model for recurrent event data with a dependent terminal event recurrent event data are commonly observed in biomedical longitudinal studies in many instances there exists a terminal event which precludes the occurrence of additional repeated events and usually there is also a nonignorable correlation between the terminal event and recurrent events in this article we propose a partly aalen s additive model with a multiplicative frailty for the rate function of recurrent event process and assume a cox frailty model for terminal event time a shared gamma frailty is used to describe the correlation between the two types of events consequently this joint model can provide the information of temporal influence of absolute covariate effects on the rate of recurrent event process which is usually helpful in the decision making process for physicians an estimating equation approach is developed to estimate marginal and association parameters in the joint model the consistency of the proposed estimator is established simulation studies demonstrate that the proposed approach is appropriate for practical use we apply the proposed method to a peritonitis cohort data set for illustration

dynamic regression hazards models for relative survival a natural way of modelling relative survival through regression analysis is to assume an additive form between the expected population hazard and the excess hazard due to the presence of an additional cause of mortality within this context the existing approaches in the parametric semiparametric and non parametric setting are compared and discussed we study the additive excess hazards models where the excess hazard is on additive form this makes it possible to assess the importance of time varying effects for regression models in the relative survival framework we show how recent developments can be used to make inferential statements about the non parametric version of the model this makes it possible to test the key hypothesis that an excess risk effect is time varying in contrast to being constant over time in case some covariate effects are constant we show how the semiparametric additive risk model can be considered in the excess risk setting providing a better and more useful summary of the data estimators have explicit form and inference based on a resampling scheme is presented for both the non parametric and semiparametric models we also describe a new suggestion for goodness of fit of relative survival models which consists on statistical and graphical tests based on cumulative martingale residuals this is illustrated on the semiparametric model with proportional excess hazards we analyze data from the trace study using different approaches and show the need for more flexible models in relative survival

age dependent u shaped risk functions and aalen s additive risk model epidemiologic studies of time to some failure often show a quadratic relation between the risk of failure and covariate s we study the nadir for a given covariate i e the value of the covariate associated with the lowest risk supposing a u shape within aalen s additive risk model this model was applied since the effect of the covariate s is allowed to vary over time and as a consequence a given nadir can vary over time we propose a test for the null hypothesis that the nadir is time independent and if this is the case an estimate of the nadir large sample properties of the test statistic and estimator are derived the methods are illustrated with data where time to death is related to body mass index

instrumental variable estimation in semi parametric additive hazards models instrumental variable methods allow unbiased estimation in the presence of unmeasured confounders when an appropriate instrumental variable is available two stage least squares and residual inclusion methods have recently been adapted to additive hazard models for censored survival data the semi parametric additive hazard model which can include time independent and time dependent covariate effects is particularly suited for the two stage residual inclusion method since it allows direct estimation of time independent covariate effects without restricting the effect of the residual on the hazard in this article we prove asymptotic normality of two stage residual inclusion estimators of regression coefficients in a semi parametric additive hazard model with time independent and time dependent covariate effects we consider the cases of continuous and binary exposure estimation of the conditional survival function given observed covariates is discussed and a resampling scheme is proposed to obtain simultaneous confidence bands the new methods are compared to existing ones in a simulation study and are applied to a real data set the proposed methods perform favourably especially in cases with exposure dependent censoring

a nonparametric dynamic additive regression model for longitudinal data in this work we study additive dynamic regression models for longitudinal data these models provide a flexible and nonparametric method for investigating the time dynamics of longitudinal data the methodology is aimed at data where measurements are recorded at random time points we model the conditional mean of responses given the full internal history and possibly time varying covariates we derive the asymptotic distribution for a new nonparametric least squares estimator of the cumulative time varying regression functions based on the asymptotic results confidence bands may be computed and inference about time varying coefficients may be drawn we propose two estimators of the cumulative regression function one estimator that involves smoothing and one that does not the latter however has twice the variance as the smoothing based estimator goodness of fit of the model is considered using martingale residuals finally we also discuss how partly conditional mean models in which the mean of the response is regressed onto selected time varying covariates may be analysed in the same framework we apply the methods to longitudinal data on height development for cystic fibrosis patients

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

a linear regression model for the analysis of life times a linear model is suggested for the influence of covariates on the intensity function this approach is less vulnerable than the cox model to problems of inconsistency when covariates are deleted or the precision of covariate measurements is changed a method of non parametric estimation of regression functions is presented this results in plots that may give information on the change over time in the influence of covariates a test method and two goodness of fit plots are also given the approach is illustrated by simulation as well as by data from a clinical trial of treatment of carcinoma of the oropharynx

totally bipartite leonard pairs and totally bipartite leonard triples of q racah type abstract let k denote an algebraically closed field of characteristic zero let v denote a vector space over k with finite positive dimension a leonard pair on v is an ordered pair of linear transformations in end v such that for each of these transformations there exists a basis for v with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal whenever these tridiagonal matrices are bipartite the leonard pair is said to be totally bipartite the notion of a leonard triple and the corresponding notion of totally bipartite are similarly defined in this paper we first classify up to isomorphism the totally bipartite leonard pairs the classification reveals that these leonard pairs are of the q racah krawtchouk or bannai ito type then we determine all leonard triples extended from given totally bipartite leonard pairs of q racah type finally we classify up to isomorphism the totally bipartite leonard triples of q racah type

leonard triples of q racah type and their pseudo intertwiners abstract let f denote a field and let v denote a vector space over f with finite positive dimension pick a nonzero q f such that q 4 1 and let a b c denote a leonard triple on v that has q racah type we show that there exist invertible w w w in end v such that i a commutes with w and w 1 b w c ii b commutes with w and w 1 c w a iii c commutes with w and w 1 a w b moreover each of w w w is unique up to multiplication by a nonzero scalar in f we show that the three elements w w w w w w mutually commute and their product is a scalar multiple of the identity a number of related results are obtained we call w w w the pseudo intertwiners for a b c

further results on the non parametric linear regression model in survival analysis this paper gives further developments of a non parametric linear regression model in survival analysis three subjects are studied first martingale residuals originally developed for the cox model are introduced for our linear model their theory is developed and they are shown to be useful for judging goodness of fit the second focus of the paper is on the use of bootstrap replications to judge which features of the cumulative regression plots are likely to reflect real phenomena and not merely random variation in particular this is applied to judging whether the effect of a covariate disappears over time a problem for which no formal test exists the third subject is density type or kernel estimation of the regression functions themselves this might give more direct information than the cumulative plots the approaches are illustrated by data from a clinical trial of carcinoma of the oropharynx and by survival times of grafts in renal patients

a partly parametric additive risk model summary aalen s additive risk model allows the influence of each covariate to vary separately over time although allowing greater flexibility of temporal structure than a cox model aalen s model is more limited in the number of covariates it can handle we introduce a partly parametric version of aalen s model in which the influence of only a few covariates varies nonparametrically over time and that of the remaining covariates is constant efficient procedures for fitting this new model are developed and studied the approach is applied to data from the medical research council s myelomatosis trials it is the first step of a taylor series expansion of a general hazard function about the zero of the covariate vector however in estimating the unknown functions in such a general model there is a variance bias trade off that may be critical in small and medium samples also after fitting the model one does not have parameters or formulae that are easily reported we propose a model that takes the additive structure of aalen s model and imposes parametric constraints to obtain a semiparametric submodel which may be more appropriate in some applications the model will be illustrated with data from clinical trials on myelomatosis covariates include treatment sex and four age strata which will be treated parametrically together with serum levels of haemoglobin and f32 microglobulin whose effects will be investigated nonparametrically the additive form can be interpreted loosely in terms of unobserved competing risks since the hazard function for the minimum of independent random vari ables is the sum of the hazard functions for the individual variables microglobulin levels are related to kidney function and tumour mass whereas haemoglobin is unaffected by kidney function hence one might anticipate that the hazard function associated with each covariate represents a different cause of death

on doubly robust estimation of the hazard difference the estimation of conditional treatment effects in an observational study with a survival outcome typically involves fitting a hazards regression model adjusted for a high dimensional covariate standard estimation of the treatment effect is then not entirely satisfactory as the misspecification of the effect of this covariate may induce a large bias such misspecification is a particular concern when inferring the hazard difference because it is difficult to postulate additive hazards models that guarantee non negative hazards over the entire observed covariate range we therefore consider a novel class of semiparametric additive hazards models which leave the effects of covariates unspecified the efficient score under this model is derived we then propose two different estimation approaches for the hazard difference and hence also the relative chance of survival both of which yield estimators that are doubly robust the approaches are illustrated using simulation studies and data on right heart catheterization and mortality from the support study

spectra eigenvectors and overlap functions for representation operators of q deformed algebras operators of representations corresponding to symmetric elements of theq deformed algebrasuq su1 1 uq so2 1 uq so3 1 uq son and representable by jacobi matrices are studied closures of unbounded symmetric operators of representations of the algebrasuq su1 1 anduq so2 1 are not selfadjoint operators for representations of the discrete series their deficiency indices are 1 1 bounded symmetric operators of these representations are trace class operators or have continuous simple spectra eigenvectors of some operators of representations are evaluated explicitly coefficients of transition to eigenvectors overlap coefficients are given in terms ofq orthogonal polynomials it is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq deformed algebras

additive hazards markov regression models illustrated with bone marrow transplant data summary when there are covariate effects to be considered multi state survival analysis is dominated either by parametric markov regression models or by semiparametric markov regression models using cox s 1972 proportional hazards models for transition intensities between the states the purpose of this research work is to study alternatives to cox s model in a general finite state markov process setting we shall look at two alternative models aalen s 1989 nonparametric additive hazards model and lin ying s 1994 semiparametric additive hazards model the former allows the effects of covariates to vary freely over time while the latter assumes that the regression coefficients are constant over time with the basic tools of the product integral and the functional delta method we present an estimator of the transition probability matrix and develop the large sample theory for the estimator under each of these two models data on 1459 hla identical sibling transplants for acute leukaemia from the international bone marrow transplant registry serve as illustration

2 1 gravity for genus 1 in 1 we analysed the algebra of observables for the simple case of a genus 1 initial data surface 2 for 2 1 de sitter gravity here we extend the analysis to higher genus we construct for genus 2 the group of automorphismsh of the homotopy group 1 induced by the mapping class group the grouph induces a groupd of canonical transformations on the algebra of observables which is related to the braid group for 6 threads

weighted least squares estimation for aalen s additive risk model abstract cox s proportional hazards model has so far been the most popular model for the regression analysis of censored survival data however the additive risk model of aalen can provide a useful and biologically more plausible alternative aalen s model stipulates that the conditional hazard function for a subject whose covariates are y y 1 yp has the form h t y y t where 1 p is an unknown vector of hazard functions this article discusses inference for based on a weighted least squares wls estimator of the vector of cumulative hazard functions the asymptotic distribution of the wls estimator is derived and used to obtain confidence intervals and bands for the cumulative hazard functions both a grouped data and a continuous data version of the estimator are examined an extensive simulation study is carried out the method is applied to grouped data on the incidence of cancer mortality among japanese atomic bomb survivors

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

assessing cumulative incidence functions under the semiparametric additive risk model in analyzing competing risks data a quantity of considerable interest is the cumulative incidence function often the effect of covariates on the cumulative incidence function is modeled via the proportional hazards model for the cause specific hazard function as the proportionality assumption may be too restrictive in practice we consider an alternative more flexible semiparametric additive hazards model of biometrika 1994 81 501 514 for the cause specific hazard this model specifies the effect of covariates on the cause specific hazard to be additive as well as allows the effect of some covariates to be fixed and that of others to be time varying we present an approach for constructing confidence intervals as well as confidence bands for the cause specific cumulative incidence function of subjects with given values of the covariates furthermore we also present an approach for constructing confidence intervals and confidence bands for comparing two cumulative incidence functions given values of the covariates the finite sample property of the proposed estimators is investigated through simulations we conclude our paper with an analysis of the well known malignant melanoma data using our method