a proportional hazards model for interval censored failure time data this paper develops a method for fitting the proportional hazards regression model when the data contain left right or interval censored observations results given for testing the hypothesis of a zero regression coefficient lead to a generalization of the log rank test for comparison of several survival curves the method is used to analyze data from an animal tumorigenicity study and also a clinical trial

efficient estimation for the proportional hazards model with interval censoring the maximum likelihood estimator mle for the proportional hazards model with case 1 interval censored data is studied it is shown that the mle for the regression parameter is asymptotically normal with n convergence rate and achieves the information bound even though the mle for the baseline cumulative hazard function only converges at n 1 3 rate estimation of the asymptotic variance matrix for the mle of the regression parameter is also considered to prove our main results we also establish a general theorem showing that the mle of the finite dimensional parameter in a class of semiparametric models is asymptotically efficient even though the mle of the infinite dimensional parameter converges at a rate slower than n the results are iliustrated by applying them to a data set from a tumorigenicity study

a multiple imputation approach to cox regression with interval censored data we propose a general semiparametric method based on multiple imputation for cox regression with interval censored data the method consists of iterating the following two steps first from finite interval censored but not right censored data exact failure times are imputed using tanner and wei s poor man s or asymptotic normal data augmentation scheme based on the current estimates of the regression coefficient and the baseline survival curve second a standard statistical procedure for right censored data such as the cox partial likelihood method is applied to imputed data to update the estimates through simulation we demonstrate that the resulting estimate of the regression coefficient and its associated standard error provide a promising alternative to the nonparametric maximum likelihood estimate our proposal is easily implemented by taking advantage of existing computer programs for right censored data

smooth inference for survival functions with arbitrarily censored data we propose a procedure for estimating the survival function of a time to event random variable under arbitrary patterns of censoring the method is predicated on the mild assumption that the distribution of the random variable and hence the survival function has a density that lies in a class of smooth densities whose elements can be represented by an infinite hermite series truncation of the series yields a parametric expression that can well approximate any plausible survival density and hence survival function provided the degree of truncation is suitably chosen the representation admits a convenient expression for the likelihood for the parameters in the approximation under arbitrary censoring truncation that is straightforward to compute and maximize a test statistic for comparing two survival functions which is based on an integrated weighted difference of estimates of each under this representation is proposed via simulation studies and application to a number of data sets we demonstrate that the approach yields reliable inferences and can result in gains in efficiency over traditional nonparametric methods

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

inference based on imputed failure times for the proportional hazards model with interval censored data abstract we propose an approach to the proportional hazards model for interval censored data in which parameter estimates are obtained by solving estimating equations that are the partial likelihood score equations for the full data proportional hazards model averaged over all rankings of imputed failure times consistent with the observed censoring intervals imputed failure times are generated using a parametric estimate of the baseline distribution the parameters of the baseline distribution are estimated simultaneously with the proportional hazards regression parameters although a parametric form for the baseline distribution must be specified simulation studies show that the method performs well even when the baseline distribution is misspecified the estimating equations are solved using monte carlo techniques we present a recursive stochastic approximation scheme that converges to the zero of the estimating equations the solution has a random error that is asymptotically normally distributed w

a local likelihood proportional hazards model for interval censored data we discuss the use of local likelihood methods to fit proportional hazards regression models to right and interval censored data the assumed model allows for an arbitrary smoothed baseline hazard on which a vector of covariates operates in a proportional manner and thus produces an interpretable baseline hazard function along with estimates of global covariate effects for estimation we extend the modified em algorithm suggested by betensky lindsey ryan and wand we illustrate the method with data on times to deterioration of breast cosmeses and hiv 1 infection rates among haemophiliacs

functional anova modeling for proportional hazards regression the logarithm of the relative risk function in a proportional hazards model involving one or more possibly time dependent covariates is treated as a specified sum of a constant term main effects and selected interaction terms maximum partial likelihood estimation is used where the maximization is taken over a suitably chosen finite dimensional estimation space whose dimension increases with the sample size and which is constructed from linear spaces of functions of one covariate and their tensor products the l 2 rate of convergence for the estimate and its anova components is obtained an adaptive numerical implementation is discussed whose performance is compared to full likelihood hazard regression both with and without the restriction to proportional hazards

hazard regression with interval censored data in a recent paper kooperberg stone and truong 1995a introduced hazard regression hare in which linear splines and their tensor products are used to estimate the conditional log hazard function based on possibly censored positive response data and one or more covariates model selection is carried out in an adaptive fashion using maximum likelihood estimation of the unknown coefficients rao and wald statistics to carry out stepwise addition and deletion of basis functions and the bayesian information criterion bic to select the final model in the present paper the hare methodology is extended to accommodate interval censored data time dependent covariates and cubic splines the presence of interval censored data means that the log likelihood function may no longer be concave presenting additional numerical challenges the extended methodology is applied to a data set containing both interval censoring and time dependent covariates the new software will be available in a future release of s plus

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

integrability by separation of variables abstract we study the integrability in the jacobi sense integrability by separation of variables of the hamiltonian differential systems using the levi civita theorem in particular we solve the stark problem for n 3

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

smooth estimation of the survival function for interval censored data interval censored data arise naturally in large scale panel studies where subjects can only be followed periodically and the event of interest can only be observed in some time intervals to estimate the survival function the non parametric maximum likelihood estimator npmle is commonly used which is a step function having some large jumps however in many applications the underlying survival function can be reasonably assumed to be smooth and then the npmle does not efficiently use this information two smooth estimators have been proposed in the literature one is based on the kernel smoothing of the npmle another is the logspline density model however to our knowledge there is no finite sample study yet to assess their performance in the literature in this paper we first show by simulation that both smooth estimators improve over the npmle for smooth survival functions we then apply these estimators to compare two survival curves the test statistics based on the maximum difference between two survival functions kolmogorov smirnov test and on the integrated weighted difference of two survival functions iwdb test are investigated via the bootstrap from our simulation studies the iwdb test seems particularly promising for some stochastically ordered survival functions that do not satisfy the proportional hazards model the methods are illustrated by reanalysing the breast cosmesis study data set copyright 2000 john wiley sons ltd

the problem of quantum integrability abstract the notion of integrability in quantum mechanics is investigated in order to prepare rigorous grounds for the study of regular and irregular behaviour of quantum systems its common sense definition turns out to have deficiencies which are illustrated by various explicit examples part of the ambiguities are shown to have their origin in the difficulty of transferring the concept of independent constants of motion into quantum mechanics due to a fundamental theorem on sets of commuting operators by von neumann taking into account the classical limit with coherent states does not resolve the problems as a result it is pointed out why the appealing phenomenological distinction between regular and chaotic quantum systems cannot be traced back to the present notion of quantum integrability in a mathematically rigorous way

flexible estimation of survival curves conditional on non linear and time dependent predictor effects prognostic studies often estimate survival curves for patients with different covariate vectors but the validity of their results depends largely on the accuracy of the estimated covariate effects to avoid conventional proportional hazards and linearity assumptions flexible extensions of cox s proportional hazards model incorporate non linear nl and or time dependent td covariate effects however their impact on survival curves estimation is unclear our primary goal is to develop and validate a flexible method for estimating individual patients survival curves conditional on multiple predictors with possibly nl and or td effects we first obtain maximum partial likelihood estimates of nl and td effects and use backward elimination to select statistically significant effects into a final multivariable model we then plug the selected nl and td estimates in the full likelihood function and estimate the baseline hazard function and the resulting survival curves conditional on individual covariate vectors the td and nl functions and the log hazard are modeled with unpenalized regression b splines in simulations our flexible survival curve estimates were unbiased and had much lower mean square errors than the conventional estimates in real life analyses of mortality after a septic shock our model improved significantly the deviance likelihood ratio test 84 8 df 20 p 0 0001 and changed substantially the predicted survival for several subjects

new existence and symmetry results for least energy positive solutions of schrodinger systems with mixed competition and cooperation terms abstract in this paper we focus on existence and symmetry properties of solutions to the cubic schrodinger system u i i u i j 1 d i j u j 2 u i in r n i 1 d where d 2 i i i 0 i j j i r for j i n 2 3 the underlying domain is either bounded or the whole space and u i h 0 1 or u i h rad 1 r n respectively we establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients as well as in the purely cooperative case

data generation for the cox proportional hazards model with time dependent covariates a method for medical researchers the proliferation of longitudinal studies has increased the importance of statistical methods for time to event data that can incorporate time dependent covariates the cox proportional hazards model is one such method that is widely used as more extensions of the cox model with time dependent covariates are developed simulations studies will grow in importance as well an essential starting point for simulation studies of time to event models is the ability to produce simulated survival times from a known data generating process this paper develops a method for the generation of survival times that follow a cox proportional hazards model with time dependent covariates the method presented relies on a simple transformation of random variables generated according to a truncated piecewise exponential distribution and allows practitioners great flexibility and control over both the number of time dependent covariates and the number of time periods in the duration of follow up measurement within this framework an additional argument is suggested that allows researchers to generate time to event data in which covariates change at integer valued steps of the time scale the purpose of this approach is to produce data for simulation experiments that mimic the types of data structures applied that researchers encounter when using longitudinal biomedical data validity is assessed in a set of simulation experiments and results indicate that the proposed procedure performs well in producing data that conform to the assumptions of the cox proportional hazards model

joint estimation of time dependent and non linear effects of continuous covariates on survival in order to yield more flexible models the cox regression model lambda t x lambda 0 t exp betax has been generalized using different non parametric model estimation techniques one generalization is the relaxation of log linearity in x lambda t x lambda 0 t exp r x another is the relaxation of the proportional hazards assumption lambda t x lambda 0 t exp beta t x these generalizations are typically considered independently of each other we propose the product model lambda t x lambda 0 t exp beta t r x which allows for joint estimation of both effects and investigate its properties the functions describing the time dependent beta t and non linear r x effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood likelihood ratio tests are proposed to compare alternative models simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model by contrast type i error rates may be highly inflated and the estimates considerably biased if the model is misspecified applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors

smooth semiparametric regression analysis for arbitrarily censored time to event data a general framework for regression analysis of time to event data subject to arbitrary patterns of censoring is proposed the approach is relevant when the analyst is willing to assume that distributions governing model components that are ordinarily left unspecified in popular semiparametric regression models such as the baseline hazard function in the proportional hazards model have densities satisfying mild smoothness conditions densities are approximated by a truncated series expansion that for fixed degree of truncation results in a parametric representation which makes likelihood based inference coupled with adaptive choice of the degree of truncation and hence flexibility of the model computationally and conceptually straightforward with data subject to any pattern of censoring the formulation allows popular models such as the proportional hazards proportional odds and accelerated failure time models to be placed in a common framework provides a principled basis for choosing among them and renders useful extensions of the models straightforward the utility and performance of the methods are demonstrated via simulations and by application to data from time to event studies

a general framework for parametric survival analysis parametric survival models are being increasingly used as an alternative to the cox model in biomedical research through direct modelling of the baseline hazard function we can gain greater understanding of the risk profile of patients over time obtaining absolute measures of risk commonly used parametric survival models such as the weibull make restrictive assumptions of the baseline hazard function such as monotonicity which is often violated in clinical datasets in this article we extend the general framework of parametric survival models proposed by crowther and lambert journal of statistical software 53 12 2013 to incorporate relative survival and robust and cluster robust standard errors we describe the general framework through three applications to clinical datasets in particular illustrating the use of restricted cubic splines modelled on the log hazard scale to provide a highly flexible survival modelling framework through the use of restricted cubic splines we can derive the cumulative hazard function analytically beyond the boundary knots resulting in a combined analytic numerical approach which substantially improves the estimation process compared with only using numerical integration user friendly stata software is provided which significantly extends parametric survival models available in standard software

a multiple imputation approach to regression analysis for doubly censored data with application to aids studies sun liao and pagano 1999 proposed an interesting estimating equation approach to cox regression with doubly censored data here we point out that a modification of their proposal leads to a multiple imputation approach where the double censoring is reduced to single censoring by imputing for the censored initiating times for each imputed data set one can take advantage of many existing techniques and software for singly censored data under the general framework of multiple imputation the proposed method is simple to implement and can accommodate modeling issues such as model checking which has not been adequately discussed previously in the literature for doubly censored data here we illustrate our method with an application to a formal goodness of fit test and a graphical check for the proportional hazards model for doubly censored data we reanalyze a well known aids data set

semiparametric regression analysis of interval censored data summary we propose a semiparametric approach to the proportional hazards regression analysis of interval censored data an em algorithm based on an approximate likelihood leads to an m step that involves maximizing a standard cox partial likelihood to estimate regression coefficients and then using the breslow estimator for the unknown baseline hazards the e step takes a particularly simple form because all incomplete data appear as linear terms in the complete data log likelihood the algorithm of turnbull 1976 journal of the royal statistical society series b38 290 295 is used to determine times at which the hazard can take positive mass we found multiple imputation to yield an easily computed variance estimate that appears to be more reliable than asymptotic methods with small to moderately sized data sets in the right censored survival setting the approach reduces to the standard cox proportional hazards analysis while the algorithm reduces to the one suggested by clayton and cuzick 1985 applied statistics34 148 156 the method is illustrated on data from the breast cancer cosmetics trial previously analyzed by finkelstein 1986 biometrics42 845 854 and several subsequent authors