survival analysis using a scale change random effects model abstract frailty models are effective in broadening the class of survival models and inducing dependence in multivariate survival distributions in proportional hazards the random effect multiplies the hazard function the scale change model incorporates unobserved heterogeneity through a random effect that enters the baseline hazard function to change the time scale we interpret this random effect as frailty or other unobserved risks that create heterogeneity in the population this model produces a wide range of shapes for univariate survival and hazard functions we extend this model to multivariate survival data by assuming that members of a group share a common random effect this structure induces association among the survival times in a group and provides alternative association structures to the proportional hazards frailty model we present parametric and semiparametric estimation techniques and illustrate these methods with an example

on the cox model with time varying regression coefficients in the analysis of censored failure time observations the standard cox proportional hazards model assumes that the regression coefficients are time invariant often these parameters vary over time and the temporal covariate effects on the failure time are of great interest in this article following previous work of cai and sun we propose a simple estimation procedure for the cox model with time varying coefficients based on a kernel weighted partial likelihood approach we construct pointwise and simultaneous confidence intervals for the regression parameters over a properly chosen time interval via a simple resampling technique we derive a prediction method for future patients survival with any specific set of covariates building on the estimates for the time varying coefficients we also consider the mixed case and present an estimation procedure for time independent parameters in the model furthermore we show how to use an integrated function of the estimate for a specific regression coeffic

estimating treatment effects with treatment switching via semicompeting risks models an application to a colorectal cancer study treatment switching is a frequent occurrence in clinical trials where during the course of the trial patients who fail on the control treatment may change to the experimental treatment analysing the data without accounting for switching yields highly biased and inefficient estimates of the treatment effect in this paper we propose a novel class of semiparametric semicompeting risks transition survival models to accommodate treatment switches theoretical properties of the proposed model are examined and an efficient expectation maximization algorithm is derived for obtaining the maximum likelihood estimates simulation studies are conducted to demonstrate the superiority of the model compared with the intent to treat analysis and other methods proposed in the literature the proposed method is applied to data from a colorectal cancer clinical trial

partially linear hazard regression with varying coefficients for multivariate survival data summary the paper studies estimation of partially linear hazard regression models with varying coefficients for multivariate survival data a profile pseudo partial likelihood estimation method is proposed the estimation of the parameters of the linear part is accomplished via maximization of the profile pseudo partial likelihood whereas the varying coefficient functions are considered as nuisance parameters that are profiled out of the likelihood it is shown that the estimators of the parameters are root n consistent and the estimators of the non parametric coefficient func tions achieve optimal convergence rates asymptotic normality is obtained for the estimators of the finite parameters and varying coefficient functions consistent estimators of the asymptotic variances are derived and empirically tested which facilitate inference for the model we prove that the varying coefficient functions can be estimated as well as if the parametric components were known and the failure times within each subject were independent simulations are con ducted to demonstrate the performance of the estimators proposed a real data set is analysed to illustrate the methodology proposed

the aalen additive gamma frailty hazards model in this paper we consider clustered right censored time to event data such data can be analysed either using a marginal model if one is interested in population effects or using so called frailty models if one is interested in covariate effects on the individual level and in estimation of correlation the cox frailty model has been studied extensively in the last decade or so and estimation techniques and large sample results are now available it is however difficult to deal with time changing covariate effects when using the cox model an appealing alternative model is the aalen additive hazards model in which it is easy to work with time dynamics in this paper we describe an innovative approach to estimation in the aalen additive gamma frailty hazards model we give the large sample properties of the estimators and investigate their small sample properties by monte carlo simulation a real example is provided for illustration copyright 2011 oxford university press

efficient estimation and inferences for varying coefficient models abstract this article deals with statistical inferences based on the varying coefficient models proposed by hastie and tibshirani local polynomial regression techniques are used to estimate coefficient functions and the asymptotic normality of the resulting estimators is established the standard error formulas for estimated coefficients are derived and are empirically tested a goodness of fit test technique based on a nonparametric maximum likelihood ratio type of test is also proposed to detect whether certain coefficient functions in a varying coefficient model are constant or whether any covariates are statistically significant in the model the null distribution of the test is estimated by a conditional bootstrap method our estimation techniques involve solving hundreds of local likelihood equations to reduce the computational burden a one step newton raphson estimator is proposed and implemented the resulting one step procedure is shown to save computational cost on an order of tens with no

semiparametric estimation of random effects using the cox model based on the em algorithm consider a survival experiment where individuals within a certain subset of the population share a common unobservable random frailty such a frailty could be an unobservable genetic or early environmental effect if individuals were in sibling groups or an environmental effect if individuals were grouped by households suppose that if the frailty omega is known the cox proportional hazards model for the observable covariates is valid with the consequence of the random effect being a multiplicative factor on the hazard rate assuming tht the random frailties follow a gamma distribution estimates of the fixed and random effects are obtained by using an em algorithm based on a profile likelihood construction the method developed is applied to the framingham heart study to examine the risks of smoking and cholesterol levels adjusting for potential random effects

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

inverses of infinite sign regular matrices let a be an infinite sign regular sr matrix which can be viewed as a bounded linear operator from lx to itself it is proved here that if the range of a contains the sequence 1 1 1 1 then a is onto if a exists then da d is also sr where d is the diagonal matrix with diagonal entries alternately 1 and 1 in case a is totally positive tp then da d is also tp under additional assumptions on a 0 introduction if the problem of spline interpolation is expressed in terms of b splines then the question of existence of a bounded spline interpolant to bounded data is seen to be equivalent to the question of whether a certain bounded band matrix has all bounded sequences in its range in 5 c a micchelli conjectured that there exists a unique bounded spline interpolant of a given order and a given knot sequence to any data sequence t y z in the plane with t strictly increasing and y bounded provided only that it is possible to interpolate the particular data sequence t l ez by sucn a sp ine there is apparently nothing special about the particular spline problem other than that it leads to a banded totally positive matrix therefore one of us quoted this conjecture in 3 p 319 prob lem 4 as a bi infinite banded totally positive matrix a is boundedly invert ible if and only if the linear system ax 1 has a bounded solution 2 micchelli gave a simple argument for the case when a is a toeplitz matrix cavaretta dahmen micchelli and smith 2 recently proved the conjecture in case a is a block toeplitz matrix this is all the more remarkable since it is easy to see in hindsight that the conjecture is faulty even in the original context of spline interpolation for example interpolation by bounded broken lines with breakpoint sequence z at the sequence t z 0 is possible to any bounded ordinate sequence y but not uniquely so since the value of the interpolant at 0 is freely choosable in matrix terms this corresponds to the matrix obtained from the bi infinite identity matrix by dropping one row but with the condition changed to has a unique bounded solution the conjecture was proved in 1

surjectivity and invertibility properties of totally positive matrices abstract a new finite section type theorem is used to show that the members of an interesting class of bounded totally positive matrices map l onto l if and only if their range contains a vector which alternates in sign and has coordinates bounded away from zero the class of matrices studied contains all banded totally positive matrices and thus all infinite spline collocation matrices connections to related work and extension to matrices which are not sign regular are indicated

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

structure of invertible bi infinite totally positive matrices an invertible nonfinite totally positive matrix a is shown to have one and only one main diagonal this means that exactly one diagonal of a has the property that all finite sections of a principal with respect to this diagonal are invertible and their inverses converge boundedly and entrywise to a i this is shown to imply restrictions on the possible shapes of such a matrix in the proof such a matrix is also shown to have an la invertible ldu factorization in addition decay of the entries of such a matrix away from the main diagonal is demonstrated it is also shown that a bounded sign regular matrix carrying some bounded sequence to a uniformly alternating sequence must have all its columns in co

some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates abstract recurrent event time data are common in medical research examples include infections in aids patients and seizures in epilepsy patients in this context as well as in the more usual context of a single failure time variable time dependent covariates are frequently of interest we suggest some rate functions that might be displayed when analyzing recurrent failure time data or when the effect of a categorical time dependent covariate is of interest estimators of these functions are provided along with two sample test statistics a new approach to regression modeling of these data is suggested and contrasted with existing methods our methods do not require that an explicit model be formulated for the probabilistic association between failure times within an individual this is in line with the currently popular generalized estimating equation approach to longitudinal data if the nature of such associations is known or is of particular interest then alternative methods may be appropriate

variable selection for cox s proportional hazards model and frailty model a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in fan and li 2001a it has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance such a property is called an oracle property the proposed procedures were illustrated in the context of linear regression robust linear regression and generalized linear models in this paper the nonconcave penalized likelihood approach is extended further to the cox proportional hazards model and the cox proportional hazards frailty model two commonly used semi parametric models in survival analysis as a result new variable selection procedures for these two commonly used models are proposed it is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function further with a proper choice of the regularization parameter and the penalty function the proposed estimators possess an oracle property standard error formulae are derived and their accuracies are empirically tested simulation studies show that the proposed procedures are more stable in prediction and more effective in computation than the best subset variable selection and they reduce model complexity as effectively as the best subset variable selection compared with the lasso which is the penalized likelihood method with the l 1 penalty proposed by tibshirani the newly proposed approaches have better theoretic properties and finite sample performance

asymptotic theory for the frailty model the frailty model is a generalization of cox s proportional hazards model which includes a random effect nielsen gill andersen and sorensen 1992 proposed an em algorithm to estimate the cumulative baseline hazard and the variance of the random effect here the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance

the inverse of a totally positive biinfinite band matrix it is shown that a bounded bi infinite banded totally positive matrix a is boundedly invertible iff there is one and only one bounded sequence mapped by a to the sequence the argument shows that such a matrix has a main diagonal i e the inverse of a is the bounded pointwise limit of inverses of finite sections of a principal with respect to a particular diagonal hence 4 j 7 or its inverse is again totally positive 0 introduction this paper is a further step in a continuing effort to understand certain linear spline approximation schemes convergence of such processes is intimately tied to their stability i e to their boundedness as maps on c say use of the b spline basis shows this question to be equivalent to bounding the inverse of certain totally positive band matrices the calculation of bounds on the inverse of a given matrix is in general a difficult task it is hoped that the present investigation into the consequences of bandedness and total positivity for the structure of the inverse may ultimately prove helpful in obtaining such bounds the results in this paper were obtained in the study of a conjecture due to c a micchelli 7 in connection with his work on the specific approximation scheme of interpolation at a strictly increasing point sequence t by elements of sm t i e by splines of some order m with some knot sequence t micchelli became convinced that every bounded function has one and only one bounded spline interpolant iff the particular function which satisfies t all has a bounded spline interpolant in sm if ni ni mt denotes the corresponding b spline basis for sm then micchelli s conjecture can be phrased thus the matrix

consistency in a proportional hazards model incorporating a random effect supported in part by the north atlantic treaty organization under a grant awarded in 1990 conducted at the mathematical institute of the university of utrecht the netherlands

semiparametric transformation models with random effects for recurrent events in this article we study a class of semiparametric transformation models with random effects for the intensity function of the counting process these models provide considerable flexibility in formulating the effects of possibly time dependent covariates on the developments of recurrent events while accounting for the dependence of the recurrent event times within the same subject we show that the nonparametric maximum likelihood estimators npmles for the parameters of these models are consistent and asymptotically normal the limiting covariance matrices for the estimators of the regression parameters achieve the semiparametric efficiency bounds and can be consistently estimated the limiting covariance function for the estimator of any smooth functional of the cumulative intensity function also can be consistently estimated we develop a simple and stable em algorithm to compute the npmles as well as the variance and covariance estimators simulation studies demonstrate that the proposed methods per

estimating equations for hazard ratio parameters based on correlated failure time data summary weighted partial likelihood estimating equations are proposed for the estimation of marginal hazard ratio parameters based on correlated failure time data asymptotic distribution theory is derived for the solution to such equations using martingale convergence results and inverse function theory simulation studies and theoretical efficiency calculations indicate that the inclusion of weights in the estimating equation produces important efficiency gains only if the dependencies among the failure times are strong

hazard models with varying coefficients for multivariate failure time data statistical estimation and inference for marginal hazard models with varying coefficients for multivariate failure time data are important subjects in survival analysis a local pseudo partial likelihood procedure is proposed for estimating the unknown coefficient functions a weighted average estimator is also proposed in an attempt to improve the efficiency of the estimator the consistency and asymptotic normality of the proposed estimators are established and standard error formulas for the estimated coefficients are derived and empirically tested to reduce the computational burden of the maximum local pseudo partial likelihood estimator a simple and useful one step estimator is proposed statistical properties of the one step estimator are established and simulation studies are conducted to compare the performance of the one step estimator to that of the maximum local pseudo partial likelihood estimator the results show that the one step estimator can save computational cost without compromising performance both asymptotically and empirically and that an optimal weighted average estimator is more efficient than the maximum local pseudo partial likelihood estimator a data set from the busselton population health surveys is analyzed to illustrate our proposed methodology

estimating time varying effects for overdispersed recurrent events data with treatment switching in the analysis of multivariate event times frailty models assuming time independent regression coefficients are often considered mainly due to their mathematical convenience in practice regression coefficients are often time dependent and the temporal effects are of clinical interest motivated by a phase iii clinical trial in multiple sclerosis we develop a semiparametric frailty modelling approach to estimate time varying effects for overdispersed recurrent events data with treatment switching the proposed model incorporates the treatment switching time in the time varying coefficients theoretical properties of the proposed model are established and an efficient expectation maximization algorithm is derived to obtain the maximum likelihood estimates simulation studies evaluate the numerical performance of the proposed model under various temporal treatment effect curves the ideas in this paper can also be used for time varying coefficient frailty models without treatment switching as well as for alternative models when the proportional hazard assumption is violated a multiple sclerosis dataset is analysed to illustrate our methodology

asymptotic theory for the correlated gamma frailty model the frailty model is a generalization of cox s proportional hazard model where a shared unobserved quantity in the intensity induces a positive correlation among the survival times murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator npmle for the shared gamma frailty model without covariates in this paper we extend this result to the correlated gamma frailty model and we allow for covariates we discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator which goes back to wald our proof of the consistency for the npmle is essentially the same as the classical proof for the maximum likelihood estimator a new central limit theorem for processes of bounded variation is given furthermore we prove that a consistent estimator for the asymptotic variance of the npmle is given by the inverse of a discrete observed information matrix

marginal regression models for multivariate failure time data abstract in this article we propose a general cox type regression model to formulate the marginal distributions of multivariate failure time data this model has a nested structure in that it allows different baseline hazard functions among distinct failure types and imposes a common baseline hazard function on the failure times of the same type we prove that the maximum quasi partial likelihood estimator for the vector of regression parameters under the independence working assumption is consistent and asymptotically normal with a covariance matrix for which a consistent estimator is provided furthermore we establish the uniform consistency and joint weak convergence of the aalen breslow type estimators for the cumulative baseline hazard functions and develop a resampling technique to approximate the joint distribution of these processes which enables one to make simultaneous inference about the survival functions over the time axis and across failure types finally we assess the small sample pro

l boundedness of l 2 projections on splines for a multiple geometric mesh this paper concerns the l2 projectors from loo to the normed linear space of polynomial splines it is shown that for the multiple geometric meshes the loo norms of the corresponding l2 projectors are bounded independently of the mesh ratio 0 introduction let us begin with some notation let k be a positive integer and x x i z a real nondecreasing knot sequence with xi xi k all i set x 00 lim xi i x o lim xji i x oo x00 i 0 i o00o fori e z and x e i define mi k x k xi xi k x i ni k x xi k xi mi k x k where po pr f denotes the rth divided difference of the function f at the points po 0 pr then ni k is called the i th b spline of order k on the knot sequence x for a e rz the rule f x a i niak x defines a function on i if we take the sum to be pointwise every such function is called a polynomial spline of order k with the knot sequence x and their collection is denoted by skx further let denote the normed linear space of bounded polynomial splines of order k with the knot sequence x and norm fll ii sup i f x ix xei we shall be concerned with p the orthogonal projector onto with respect to the ordinary inner product f g ff x g x dx but restricted to l i we want to bound its norm 11 p ii sup iipsf ii 1 iii o fel j received march 26 1984 revised december 16 1985 1980 mathematics subject classification primary 41a15 15a60 this is a part of the author s thesis which was submitted to the graduate school university of wisconsin madison in partial fullfillment of the requirement for ph d degree present address department of mathematics zhejiang university hangzhou zhejiang the people s republic of china 1987 american mathematical society 0025 5718 87 1 00 25 per page 675 this content downloaded from 207 46 13 178 on wed 15 jun 2016 07 00 20 utc all use subject to http about jstor org terms

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

semiparametric regression for the mean and rate functions of recurrent events the counting process with the cox type intensity function has been commonly used to analyse recurrent event data this model essentially assumes that the underlying counting process is a time transformed poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process recently pepe and cai and lawless and co workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the poisson type assumption in this paper we provide a rigorous justification of such robust procedures through modern empirical process theory furthermore we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean rate model the advantages of the robust procedures are demonstrated through simulation studies an illustration with multiple infection data taken from a clinical study on chronic granulomatous disease is also provided

on the cohomology of joins of operator algebras abstract by analogy with the join in topology the join a b for operator algebras a and b acting on hilbert spaces h and k respectively was defined by gilfeather and smith amer j math 116 1994 541 561 assuming that k is finite dimensional they calculated the hochschild cohomology groups for a b with coefficients in l k h we assume that a is a maximal abelian von neumann algebra acting on h a is a subalgebra of a l k and b is an ultraweakly closed subalgebra of m n a containing a 1 n we show that b may be decomposed into a finite sum of free modules in this context we redefine the join of a and b generalize the calculations of gilfeather and smith and calculate h m a b a l c n k for all m 0