regression models for relative survival four approaches to estimating a regression model for relative survival using the method of maximum likelihood are described and compared the underlying model is an additive hazards model where the total hazard is written as the sum of the known baseline hazard and the excess hazard associated with a diagnosis of cancer the excess hazards are assumed to be constant within pre specified bands of follow up the likelihood can be maximized directly or in the framework of generalized linear models minor differences exist due to for example the way the data are presented individual aggregated or grouped and in some assumptions e g distributional assumptions the four approaches are applied to two real data sets and produce very similar estimates even when the assumption of proportional excess hazards is violated the choice of approach to use in practice can therefore be guided by ease of use and availability of software we recommend using a generalized linear model with a poisson error structure based on collapsed data using exact survival times the model can be estimated in any software package that estimates glms with user defined link functions including sas stata s plus and r and utilizes the theory of generalized linear models for assessing goodness of fit and studying regression diagnostics

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

analysis of proportional mean residual life model with latent variables end stage renal disease esrd is one of the most serious diabetes complications numerous studies have been devoted to revealing the risk factors of the onset time of esrd in this article we propose a proportional mean residual life mrl model with latent variables to assess the effects of observed and latent risk factors on the mrl function of esrd in a cohort of chinese type 2 diabetic patients the proposed model generalizes the conventional proportional mrl model to accommodate the latent risk factor that cannot be measured by a single observed variable we employ a factor analysis model to characterize the latent risk factors via multiple observed variables we develop a borrow strength estimation procedure which incorporates the expectation maximization algorithm and an extended estimating equation approach the asymptotic properties of the proposed estimators are established simulation shows that the performance of the proposed methodology is satisfactory the application to the study of type 2 diabetes reveals insights into the prevention of esrd copyright 2016 john wiley sons ltd

on inconsistency of breslow s estimator as an estimator of the hazard rate in the cox model in the cox proportional hazards model breslow s estimator of the cumulative hazard function is well known this estimator is sometimes presented in such a way that it appears to offer a method of estimating the hazard rate function as well we state and prove an asymptotic result for this estimator of the hazard rate which shows that the estimator is inconsistent and that it is asymptotically unbiased

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

a note on eigenvalues of perturbed hermitian matrices abstract let a h 1 e e h 2 and a h 1 o o h 2 be hermitian matrices with eigenvalues 1 k and 1 k respectively denote by e the spectral norm of the matrix e and the spectral gap between the spectra of h 1 and h 2 it is shown that i i 2 e 2 2 4 e 2 which improves all the existing results similar bounds are obtained for singular values of matrices under block perturbations

omnibus tests for comparison of competing risks with adjustment for covariate effects summary this article develops omnibus tests for comparing cause specific hazard rates and cumulative incidence functions at specified covariate levels confidence bands for the difference and the ratio of two conditional cumulative incidence functions are also constructed the omnibus test is formulated in terms of a test process given by a weighted difference of estimates of cumulative cause specific hazard rates under cox proportional hazards models a simulation procedure is devised for sampling from the null distribution of the test process leading to graphical and numerical techniques for detecting significant differences in the risks the approach is applied to a cohort study of type specific hiv infection rates

simultaneous inferences on the contrast of two hazard functions with censored observations in survival analysis often times the pattern of instantaneous risk over time is more interesting than that of the cumulative risk for this case a nonparametric hazard function estimate is more appropriate for summarizing the risk experience of a group of patients than the corresponding kaplan meier estimate in comparing a new treatment with a standard therapy it is important to know if the treatment loses it potency during the follow up period and if it does one would like to know when it becomes ineffective unfortunately with a plot of the differences of two kaplan meier curves it is rather difficult to capture such temporal trends in this article we propose simple procedures for constructing confidence bands for the contrast of two hazard functions with censored data the simultaneous interval estimates are quite useful for identifying possible values of the contrast over time with a certain degree of confidence the new proposals are illustrated with an example and a small simulation study

regression analysis of additive hazards model with latent variables we propose an additive hazards model with latent variables to investigate the observed and latent risk factors of the failure time of interest each latent risk factor is characterized by correlated observed variables through a confirmatory factor analysis model we develop a hybrid procedure that combines the expectation maximization em algorithm and the borrow strength estimation approach to estimate the model parameters we establish the consistency and asymptotic normality of the parameter estimators various nice features including finite sample performance of the proposed methodology are demonstrated by simulation studies our model is applied to a study concerning the risk factors of chronic kidney disease for type 2 diabetic patients supplementary materials for this article are available online

nonparametric specification tests for conditional duration models this paper deals with the testing of autoregressive conditional duration acd models by gauging the distance between the parametric density and hazard rate functions implied by the duration process and their non parametric estimates we derive the asymptotic justification using the functional delta method for fixed and gamma kernels and then investigate the finite sample properties through monte carlo simulations although our tests display some size distortion bootstrapping suffices to correct the size without compromising their excellent power we show the practical usefulness of such testing procedures for the estimation of intraday volatility patterns

superfast divide and conquer method and perturbation analysis for structured eigenvalue solutions we present a superfast divide and conquer method for finding all the eigenvalues as well as all the eigenvectors in a structured form of a class of symmetric matrices with off diagonal ranks or n

improved random perturbation intervals of symmetric eigenvalue problem abstract in this paper by transforming the symmetric random perturbed matrix into a special form and using the result from li and li 2005 7 we first present an improved random perturbation interval under the random perturbation in the interval the simple eigenvalue of the original matrix is still simple then we extend the result to the case of several eigenvalues finally numerical experiments are given to illustrate the obtained results

two sample goodness of fit tests for additive risk models with censored observations summary the additive risk model assumes that the hazard function associated with a set of covariates is the sum of the baseline hazard function and the regression function of covariates we propose two different test procedures for checking the adequacy of two sample additive risk models for randomly censored observations one is based on the martingale residuals and the other on the difference between weighted estimators of the excess risk the test statistics are shown to be asymptotically normal under appropriate regularity conditions and consistent under any model misspecifications finally two real examples are provided along with results of a simulation study

perturbation of multiple eigenvalues of hermitian matrices this paper is concerned with the perturbation of a multiple eigenvalue mu of the hermitian matrix a mbox diag mu i a 22 when it undergoes an off diagonal perturbation e whose columns have widely varying magnitudes when some of e s columns are much smaller than the others some copies of mu are much less sensitive than any existing bound suggests we explain this phenomenon by establishing individual perturbation bounds for different copies of mu they show that when a 22 mu i is definite the i th bound scales quadratically with the norm of the i th column and in the indefinite case the bound is necessarily proportional to the product of e s i th column norm and e s norm an extension to the generalized hermitian eigenvalue problem is also presented

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

locally adaptive hazard smoothing we consider nonparametric estimation of hazard functions and their derivatives under random censorship based on kernel smoothing of the nelson 1972 estimator one critically important ingredient for smoothing methods is the choice of an appropriate bandwidth since local variance of these estimates depends on the point where the hazard function is estimated and the bandwidth determines the trade off between local variance and local bias data based local bandwidth choice is proposed a general principle for obtaining asymptotically efficient data based local bandwiths is obtained by means of weak convergence of a local bandwidth process to a gaussian limit process several specific asymptotically efficient bandwidth estimators are discussed we propose in particular an asymptotically efficient method derived from direct pilot estimators of the hazard function and of the local mean squared error this bandwidth choice method has practical advantages and is also of interest in the uncensored case as well as for density estimation

nonsmooth backfitting for the excess risk additive regression model with two survival time scales we consider an extension of aalen s additive regression model allowing covariates to have effects that vary on two different time scales the two time scales considered are equal up to a constant that varies for each individual such as for example follow up time and age in medical studies or calendar time and age in longitudinal studies the model has been introduced in scheike 2001 where it was solved via smoothing techniques we present a new backfitting 20 algorithm estimating the structured model without having to use smoothing estimators of the cumulative regression functions on the two time scales are suggested by solving local estimating equations jointly on the two time scales we provide large sample properties and simultaneous confidence bands the model is applied to data on myocardial infarction providing a separation of the two effects stemming from time since diagnosis and age

estimation for a semimartingale regression model using the method of sieves estimation by the method of sieves for a semimartingale regression model introduced by aalen 1980 is studied it is of interest to estimate functions which describe the influence of the covariates over time an estimator for these functions is introduced and conditions which ensure consistency of the estimator in l 2 norm are given applications to diffusion processes and point processes with censored data are also discussed

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

estimation of the survival function for gray s piecewise constant time varying coefficients model gray s extension of cox s proportional hazards ph model for right censored survival data allows for a departure from the ph assumption via introduction of time varying regression coefficients tvc using penalized splines gray s work focused on estimation inference and residual analyses but no estimator for the survival function has been proposed we derive a survival function estimator for one important member of the class of tvc models a piecewise constant time varying coefficients pc tvc model we also derive an estimate for the confidence limits of the survival function accuracy in estimating underlying survival times and survival quantiles is assessed for both cox s and gray s pc tvc model using a simulation study featuring scenarios violating the ph assumption finally an example of the estimated survival functions and the corresponding confidence limits derived from cox s ph and gray s pc tvc model respectively is presented for a liver transplant data set in the second part of the thesis we examine the effect of model misspecification for two classes of regression models for right censored survival data additive and multiplicative models for the conditional hazard rate a particular attention is given to data exhibiting time varying regression coefficients the class of multiplicative models is represented by cox ph model and gray s tvc model respectively and for additive models we use aalen s linear model both gray s tvc model and aalen s linear model incorporate time varying coefficients a simulation study is performed to cross analyze survival data which follows either a multiplicative or an additive model for the conditional hazard rate the effect of misspecifying the true model for the conditional hazard rate is assessed by looking at the power of the individual models to detect an existing effect bias and mean square error observed for each conditional model based estimator of survival we also show that aalen s model formulae is a first order taylor series approximation of that of gray s model which explains the comparably higher flexibility on part of the aalen s model as compared to the cox ph when the gray s tvc model for the data is misspecified

some tests for comparing cumulative incidence functions and cause specific hazard rates abstract we consider the competing risks problem with the available data in the form of times and causes of failure in many practical situations e g in reliability testing it is important to know whether two risks are equal or whether one is more serious than the other we propose some distribution free tests for comparing cumulative incidence functions and cause specific hazard rates against ordered alternatives without making any assumptions on the nature of dependence between the risks both the censored and the uncensored cases are studied the performance of the proposed tests is assessed in a simulation study as an illustration we compare the risks of two types of cancer mortality thymic lymphoma and reticulum cell carcinoma in a strain of laboratory mice

bilinear approach to delay painleve equations we present the list of all known to date delay painleve equations together with their hirota bilinear forms

estimation of excess risk from case control data using aalen s linear regression model we introduce methods for statistical inference in aalen s non parametric linear regression model of disease incidence aalen 1989 statistics in medicine 8 907 925 from nested case control data these methods provide the basis for estimation of excess risk as a linear function of dose and absolute risk for a given exposure history the methods are illustrated by estimating excess and absolute risks associated with radon exposure and smoking from nested case control samples from the colorado plateau uranium miners cohort language en

estimating net survival the importance of allowing for informative censoring net survival the one that would be observed if cancer were the only cause of death is the most appropriate indicator to compare cancer mortality between areas or countries several parametric and non parametric methods have been developed to estimate net survival particularly when the cause of death is unknown these methods are based either on the relative survival ratio or on the additive excess hazard model the latter using the general population mortality hazard to estimate the excess mortality hazard the hazard related to net survival the present work used simulations to compare estimator abilities to estimate net survival in different settings such as the presence absence of an age effect on the excess mortality hazard or on the potential time of follow up knowing that this covariate has an effect on the general population mortality hazard too it showed that when age affected the excess mortality hazard most estimators including specific survival were biased only two estimators were appropriate to estimate net survival the first is based on a multivariable excess hazard model that includes age as covariate the second is non parametric and is based on the inverse probability weighting these estimators take differently into account the informative censoring induced by the expected mortality process the former offers great flexibility whereas the latter requires neither the assumption of a specific distribution nor a model building strategy because of its simplicity and availability in commonly used software the nonparametric estimator should be considered by cancer registries for population based studies

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

goodness of fit of relative survival models additive regression models are preferred over multiplicative models in the analysis of relative survival data such preferences are mainly grounded in practical experience with mostly cancer registries data where the basic assumption of the additivity of hazards is more likely to be met also the interpretation of coefficients is more meaningful in additive than in multiplicative models nonetheless the question of goodness of fit of the assumed model must still be addressed and while there is an abundance of methods to check the goodness of fit of multiplicative models the respective arsenal for additive models is almost empty we propose here a variety of procedures for testing the null hypothesis of a good fit these are based on partial residuals defined similarly to schoenfeld residuals familiar for cox model diagnostics the tests have appropriate sizes under the null hypothesis and good power under different alternatives we investigate their performance through simulations and apply the methods to data from a study into survival of colon cancer patients

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

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

semiparametric and nonparametric regression analysis of longitudinal data this article deals with the regression analysis of repeated measurements taken at irregular and possibly subject specific time points the proposed semiparametric and nonparametric models postulate that the marginal distribution for the repeatedly measured response variable y at time t is related to the vector of possibly time varying covariates x through the equations e y t x t 0 t 0x t and e y t x t 0 t 0 t x t where 0 t is an arbitrary function of t 0 is a vector of constant regression coefficients and 0 t is a vector of time varying regression coefficients the stochastic structure of the process y is completely unspecified we develop a class of least squares type estimators for 0 which is proven to be n consistent and asymptotically normal with simple variance estimators furthermore we develop a closed form estimator for a cumulative function of 0 t which is shown to be n consistent and on proper normalization converges weakly to a zero mean gaussian pr

painleve analysis and singularity confinement the ultimate conjecture the authors propose a new integrability criterion based on the combination of two powerful integrability detectors the painleve method for continuous systems and the singularity confinement for discrete systems with this new criterion they can treat a large variety of systems that combine discrete and continuous characteristics differential difference systems ordinary and partial differential delay problems and even a class of integro differential equations

goodness of fit tests in models for life history data based on cumulative hazard rates to check the validity of an assumed parametric model for survival data one may compare t the nonparametric nelson aalen plot of the cumulative hazard rate with a t the estimated parametric cumulative hazard rate being for example the maximum likelihood estimator convergence in distribution of n t a t and more general processes is studied employing the general framework of counting processes which allows for quite general models for life history data and for quite general censoring schemes the results are applied to the construction of 2 type statistics for goodness of fit cramer von mises and kolmogorov smirnov type tests are presented in the case where the unknown parameter is one dimensional

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

discrete painleve equations from singularity patterns the asymmetric trihomographic case we derive the discrete painleve equations associated with the affine weyl group e8 1 that can be represented by an in the quispel roberts thompson sense asymmetric trihomographic system the method used in this paper is based on singularity confinement we start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns that cannot lead to confined singularities working with the remaining ones we implement the confinement conditions and derive the corresponding discrete painleve equations which involve two variables by eliminating either of these variables we obtain a symmetric equation examining all these equations of a single variable we find that they coincide exactly with those derived in previous works of ours thereby establishing the completeness of our results

sharpness in rates of convergence for the symmetric lanczos method the lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem there is a well established convergence theory that produces bounds to predict the rates of convergence good for a few extreme eigenpairs these bounds suggest at least linear convergence in terms of the number of lanczos steps assuming there are gaps between individual eigenvalues in practice often superlinear convergence is observed the question is do the existing bounds tell the correct convergence rate in general an affirmative answer is given here for the two extreme eigenvalues by examples whose lanczos approximations have errors comparable to the error bounds for all lanczos steps

methods for the analysis of sampled cohort data in the cox proportional hazards model methods are provided for regression parameter and cumulative baseline hazard estimation in the cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law the key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes leading to estimation methods parallel to those for full cohort data these techniques are illustrated for a number of sampling designs including three novel designs counter matching with additional randomly sampled controls quota sampling of controls and nested case control sampling with number of controls dependent on the failure s exposure status general asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic distributions for estimators from a large class of designs

kaplan meier marginal or conditional probability curves in summarizing competing risks failure time data in the context of competing risks the kaplan meier estimator is often unsuitable for summarizing failure time data we discuss some alternative descriptive methods including marginal probability and conditional probability estimators two sample test statistics are also presented

a note on the variable kernel estimator of the hazard function from randomly censored data in a recent paper tanner and wong 1983b a family of data based nonparametric hazard estimators was introduced several of these estimators were studied in an extensive simulation experiment the estimator which allows for variable bandwidth was found to have a superior performance in this note sufficient conditions for the variable kernel estimator to be strongly consistent are presented

additive and multiplicative models for relative survival rates the relative survival rate is the ratio of the overall survival rate to that expected for demographically similar individuals in a reference population it is commonly used to estimate the effect of a particular disease on mortality when the cause of death is not reliably known the effect of the disease on the underlying hazards may be multiplicative or additive the former case has been considered elsewhere but seems biologically less plausible than the latter both models are examined in this paper the effect of the disease is assumed to be constant throughout the follow up period or to be piecewise constant within k follow up intervals maximum likelihood estimates and related statistics are presented as are simpler statistics based on moment estimators of the disease effect the moment based statistic for testing the homogeneity of r groups may be expressed as a sum of individual scores which are shown to be closely related to the logrank scores when follow up intervals are made arbitrarily small

additive hazards regression with current status data current status data arise when the only knowledge about the failure time of interest is whether the failure occurs before or after a random monitoring time we propose to analyse such data by the semiparametric additive hazards model which specifies that the hazard function for the failure time associated with a set of possibly time dependent covariates is the sum of an arbitrary baseline hazard function and a regression function of covariates under certain conditions on the monitoring time one can make inferences about the regression parameters of the additive hazards model by using the familiar asymptotic theory and software for the proportional hazards model with right censored data an application to a carcinogenicity experiment is provided

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

lehmann bounds and eigenvalue error estimation the paper investigates the properties of lehmann s optimal bounds for eigenvalues of hermitian problems in order to find a way to efficiently use them for eigenvalue error estimation a practical error estimation scheme is described that can be employed in the framework of a subspace iteration algorithm and is actually implemented by the hsl ea19 software package from the hsl mathematical software library of rutherford appleton laboratory

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

a semiparametric additive regression model for longitudinal data in previous work we have studied a nonparametric additive time varying regression model for longitudinal data recorded at irregular intervals the model allows the influence of each covariate to vary separately with time for small datasets however only a limited number of covariates may be handled in this way in this paper we introduce a semiparametric regression model for longitudinal data the influence of some of the covariates varies nonparametrically with time while the effect of the remaining covariates are constant no smoothing is necessary in the estimation of the parametric terms of the model asymptotics are derived using martingale techniques for the cumulative regression functions which are much easier to estimate and study than the regression functions themselves the approach is applied to longitudinal data from the copenhagen study group for liver diseases schlichting et al 1983

integrability criteria for differential difference systems a comparison of singularity confinement and low growth requirements a new approach for the study of the integrability of differential difference systems is introduced given a differential difference system of such a form that it can be iterated without necessitating the integration of a differential equation at any step a two stage strategy is used first the singularity confinement necessary integrability criterion is used in order to limit the possible choices once the system is sufficiently reduced the much stronger requirement of nonexponential growth of the degree of the iterates of some initial condition is implemented this method turns out to be powerful and practical the investigation of a given class of differential difference systems has resulted in some well known integrable systems but also to two promising integrability candidates one of which is reduced to a known integrable case

local convergence of empirical measures in the random censorship situation with application to density and rate estimators on etudie les deviations locales de la mesure empirique definie par l estimateur de kaplan meier 1958 pour la fonction de survie

delay differential equations and the painleve transcendents abstract we apply the recently proposed integrability criterion for differential difference systems that blends the classical painleve analysis with singularity confinement for discrete systems to a class of first order differential delay equations our analysis singles out the family of bi riccati equations as integrability candidates among these equations that pass the test some are integrable in a straightforward way usually by reduction to a standard riccati equation for some transformed variable while the remaining ones define new hysterodifferential forms of the painleve transcendental equations

analysis of oldest old mortality lifetables revisited ing the hazard function we provide rigorous asymptotic results for the smoothing of lifetables and show that the transformation approach is supported by both asymptotic and simulation results we also demonstrate how the information contained in many samples of lifetables as typically obtained in aging experiments can be summarized in a twodimensional hazard surface

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

analysis of covariance with incomplete data via semiparametric model transformations we propose a method for fitting semiparametric models such as the proportional hazards ph additive risks ar and proportional odds po models each of these semiparametric models implies that some transformation of the conditional cumulative hazard function at each t depends linearly on the covariates the proposed method is based on nonparametric estimation of the conditional cumulative hazard function forming a weighted average over a range of t values and subsequent use of least squares to estimate the parameters suggested by each model an approximation to the optimal weight function is given this allows semiparametric models to be fitted even in incomplete data cases where the partial likelihood fails e g left censoring right truncation however the main advantage of this method rests in the fact that neither the interpretation of the parameters nor the validity of the analysis depend on the appropriateness of the ph or any of the other semiparametric models in fact we propose an integrated method for data analysis where the role of the various semiparametric models is to suggest the best fitting transformation a single continuous covariate and several categorical covariates factors are allowed simulation studies indicate that the test statistics and confidence intervals have good small sample performance a real data set is analyzed

nonparametric kernel estimation in counting processes with explanatory variables summary cox s proportional hazards model assumes that the duration hazard rate factors into a product of a baseline hazard rate and a nonnegative function of explanatory variables an estimate of the baseline hazard rate hence also of overall hazard rate is proposed based on a smoothing procedure using a kernel function it is shown that the proposed estimator is uniformly consistent and converges weakly to a gaussian process bandwidth selection issues are also discussed

additive and multiplicative covariate regression models for relative survival incorporating fractional polynomials for time dependent effects relative survival is used to estimate patient survival excluding causes of death not related to the disease of interest rather than using cause of death information from death certificates which is often poorly recorded relative survival compares the observed survival to that expected in a matched group from the general population models for relative survival can be expressed on the hazard mortality rate scale as the sum of two components where the total mortality rate is the sum of the underlying baseline mortality rate and the excess mortality rate due to the disease of interest previous models for relative survival have assumed that covariate effects act multiplicatively and have thus provided relative effects of differences between groups using excess mortality rate ratios in this paper we consider i the use of an additive covariate model which provides estimates of the absolute difference in the excess mortality rate and ii the use of fractional polynomials in relative survival models for the baseline excess mortality rate and time dependent effects the approaches are illustrated using data on 115 331 female breast cancer patients diagnosed between 1 january 1986 and 31 december 1990 the use of additive covariate relative survival models can be useful in situations when the excess mortality rate is zero or slightly less than zero and can provide useful information from a public health perspective the use of fractional polynomials has advantages over the usual piecewise estimation by providing smooth estimates of the baseline excess mortality rate and time dependent effects for both the multiplicative and additive covariate models all models presented in this paper can be estimated within a generalized linear models framework and thus can be implemented using standard software copyright 2005 john wiley sons ltd

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

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

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

estimation of population size based on additive hazards models for continuous time recapture experiments summary we use the semiparametric additive hazards model to formulate the effects of individual covariates on the capture rates in the continuous time capture recapture experiment and then construct a horvitz thompson type estimator for the unknown population size the resulting estimator is consistent and asymptotically normal with an easily estimated variance simulation studies show that the asymptotic approximations are adequate for practical use when the average capture probabilities exceed 5 ignoring covariates would underestimate the population size and the coverage probability is poor a wildlife example is provided

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

a generalized additive regression model for survival times we present a non parametric survival model withtwo time scales the time scales are equivalent up to a constant that varies over the subjects covariate effects are modelled linearly on eachtime scale by additive aalen models estimators of the cumulative intensities on the two time scales are suggested by solving approximate local maximum likelihood estimating equations the local estimating equations necessitate only the choice of one bandwidth the estimators are provided with large sample properties the model is applied to data on patients with myocardial infarction and used to describe the prognostic effect of covariates on the two time scales time since myocardial infarction and age 1 introduction in many bio medical applications in survival analysis it is of interest to study the effect of covariates on various time scales we consider the situation where multiple time scales are involved and focus on the specific situation withtwo time scales that are equivalent up to a constant for eachindividual suchas for example follow up time and age a class of models where it is relevant to consider multiple time scales is the the three state model known as the illness death model or the disability model where the additional time scale may be duration in the illness state of the model see keiding 1991 for a general discussion of these models oakes 1995 discussed how multiple time scales may be combined into a single scale previous work has considered semi parametric survival models where one time scale is modelled parametrically and the other time scale is considered non parametric an example of this type of analysis with three time scales applied to diabetes patients can be found in ramlau hansen et al 1987 in th is paper we present a non parametric regression approachwithtwo time scales where each time scale contribute additively to the mortality the effect of covariates are modelled by additive aalen models on eachtime scale aalen 1980 1989 1993 mckeague 1988 huffer and mckeague 1991 this allows covariates to have effects that vary on two different time scales in a motivating example we consider patients that experience myocardial infarction and aim at predicting the intensity considering the two time scales age and time since myocardial infarction as an example one of the covariates describes the heart function and is allowed to have an effect that varies

a test of fit for a semiparametric additive risk model summary kolmogorov smirnov and cramer von mises type test statistics based on the standardised cumulative hazard process are proposed it is very difficult to evaluate their asymptotic distributions but they can be approximated by the use of the bootstrap the advantages of the goodness of fit test are that arbitrary partitions of the time axis and covariate spaces are not needed for evaluating test statistics and that it has excellent consistency properties the test is applied to data from the mayo clinic trial in primary biliary cirrhosis of the liver a simulation study indicates that the proposed test is suitable for practical use

computing the common zeros of two bivariate functions via bezout resultants the common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor chebyshev basis from here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and bezout matrices with polynomial entries using techniques including domain subdivision bezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree ge 1 000 1 000 we analyze the resultant method and its conditioning by noting that the bezout matrices are matrix polynomials two implementations are available one on the matlab central file exchange and another in the roots command in chebfun2 that is adapted to suit chebfun s methodology

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

hazard function estimators a simulation study kernel based methods for the smooth non parametric estimation of the hazard function have received considerable attention in the statistical literature although the mathematical properties of the kernel based hazard estimators have been carefully studied their statistical properties have not we reviewed various kernel based methods for hazard function estimation from right censored data and compared the statistical properties of these estimators through computer simulations our simulations covered seven distributions three levels of random censoring four types of bandwidth functions two sample sizes and three types of boundary correction we conducted a total of 504 simulation experiments with 500 independent samples each our results confirmed the advantages of two recent innovations in kernel estimation boundary correction and locally optimal bandwidths the median relative improvement decrease in mean square error over fixed bandwidth estimators without boundary correction was 3 per cent for fixed bandwidth estimators with left boundary correction 52 per cent locally optimal bandwidths without boundary correction and 66 per cent for locally optimal bandwidths with left boundary correction the locally optimal bandwidth estimators with left boundary correction also outperformed three previously published and publicly available algorithms with median relative improvements in mean square error of 31 per cent 77 per cent and 80 per cent

simple parametric and nonparametric models for excess and relative mortality this paper studies two classes of hazard rate based models for the mortality in a group of individuals taking normal life expectancy into account in a multiplicative hazard model the estimate for the relative mortality generalises the standardised mortality ratio and the adequacy of a model with constant relative mortality can be tested using a type of total time on test statistic in an additive hazard model continuous time generalisations of a corrected survival curve and a normal survival curve are obtained and the adequacy of a model with constant excess mortality can again be tested using a type of total time on test statistic a model including both the multiplicative hazard model and the additive hazard model is briefly considered the use of the models is illustrated on a set of data concerning survival after operation for malignant melanoma

nonparametric analysis of changes in hazard rates for censored survival data an alternative to change point models summary as a nonparametric estimator for the point of the most rapid change of a hazard rate we propose the location of an extremum of a nonparametric estimate of the derivative or equivalently of a zero of a nonparametric estimate of the second derivative using the kernel method for the nonparametric estimation of derivatives of the hazard rate the asymptotic local limiting distribution and uniform consistency are applied to prove consistency and to find the limiting distribution of these estimators under random censoring and to construct confidence intervals both for the derivatives of the hazard rate and for the point of most rapid change an application to leukaemia data illustrates this concept and we discuss its relations to change point modelling the monte carlo method is used to assess the reliability of finite sample analyses and in particular of the given example

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

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

estimations in homoscedastic linear regression models with censored data an empirical process approach pertaining to the estimating equations proposed by tsiatis based on the linear rank test we show the existence of local confounding between the baseline hazard function and the covariates due to the local confounding an estimating equation in tsiatis family with a larger time point of truncation could contain less information about the regression parameter than the estimating equation with a smaller time point of truncation this phenomenon further indicates significant loss of efficiency of tsiatis estimating equations as well as the power loss of log rank type tests when the baseline hazard function goes up and down along the time scale to take care of this local confounding without using nonparametric estimates of the derivative of the baseline hazard function we propose the empirical process approach epa based on an empirical process constructed from tsiatis log rank estimating equation by varying its truncating time point the epa will provide very tractable estimations of the regression parameters as well as pearson s chi squared statistics for testing the model s assumptions specifically the performance of the epa estimator is shown to be very close to the best estimator in tsiatis family

stochastic calculus as a tool in survival analysis a review this paper gives a brief survey of the uses of stochastic calculus in survival analysis the role played by martingale central limit theory in deriving asymptotic distributions of estimators and test statistics is described the nelson aalen estimator the kaplan meier estimator cox s proportional hazards model aalen s additive risk model and a goodness of fit test for cox s model are discussed sketches of the proofs of the main results are included

a geometric approach to singularity confinement and algebraic entropy a geometric approach to the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but exhibits chaotic behaviour is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be the action of an extended weyl group of indefinite type a method to calculate its algebraic entropy by using the theory of intersection numbers is presented

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

cluster robust accuracy bounds for ritz subspaces abstract given an approximating subspace for a hermitian matrix a the rayleigh ritz procedure is commonly used to compute a few approximate eigenvalues called ritz values and corresponding approximate eigenvectors called ritz vectors in this paper new bounds on the canonical angles between the invariant subspace of a associated with its few extreme smallest or largest eigenvalues and its approximating ritz subspace in terms of the differences between ritz values and the targeted eigenvalues are obtained from this result various bounds are readily available to estimate how accurate the ritz vectors computed from the approximating subspace may be based on information on approximation accuracies in the ritz values the result is helpful in understanding how ritz vectors move towards eigenvectors while ritz values are made to move towards eigenvalues

linear regression analysis for multivariate failure time observations abstract in this article we consider the case that each patient in a longitudinal study may experience two or more distinct failures the corresponding failure times which are possible censored are recorded for each patient the logarithm of each marginal failure time is assumed to be linearly related to its covariates however the distributional form of the error term in the model does not have to be specified in the analysis furthermore no specific structure of dependence among the distinct failure times on each subject has to be imposed various linear regression methods for analyzing multivariate failure time observations are proposed our procedures do not involve the unstable nonparametric hazard function estimation extensive numerical studies are conducted to evaluate the new proposals recommendations are also made for their practical usage

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure

dynamic modelling and penalized likelihood estimation for discrete time survival data summmary this paper describes a dynamic or state space approach for analyzing discrete time or grouped survival data simultaneous estimation of baseline hazard functions and of timevarying covariate effects is based on maximization of posterior densities or equivalently a penalized likelihood leading to kalman type smoothing algorithms data driven choice of unknown smoothing parameters is possible via an em type procedure the methods are illustrated by applications to real data