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

transition functions paths and path integrals the main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which kolmogorov achieved via pde theory the theory of partial differential equations and his criterion for path continuity section 2 describes how ito calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process x by solving an sde stochastic differential equation driven by brownian motion of course sdes have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems though 2 begins heuristically the mathematics is then tightened up this paper is after all a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation once kolmogorov s precise language is available 2 then takes a quick sight seeing trip through some of the great developments by doob ito and their successors in an age in which so many do simulations of ito equations i have explained precisely in the briefest possible fashion what the exact theory is it is easy and usable you will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems and of what the malliavin calculus is about you will however be advised on guide books on these and other areas including physicists favourites large deviations measure valued diffusions etc so that you can later explore at your leisure confession the paper consists very largely of selected tracks remixed from the album rogers williams 1987 diffusions markov processes and martingales chichester wiley tributes to kolmogorov s work in probability and statistics have appeared in every book ever written on probability and in ann probability 17 1989 815 964 ann statist 18 1990 987 1031 bull lond math soc 22 1990 31 100 teor veroyatnost primenen 34 1989 no 1 usp mat nauk 43 1988 no 6 the official biography by shiryaev will be a wonderful volume for me this paper is a further expression of my thanks to kolmogorov and as he would have wished to levy doob and ito too

global optimization and simulated annealing in this paper we are concerned with global optimization which can be defined as the problem of finding points on a bounded subset of n in which some real valued functionf assumes its optimal maximal or minimal value we present a stochastic approach which is based on the simulated annealing algorithm the approach closely follows the formulation of the simulated annealing algorithm as originally given for discrete optimization problems the mathematical formulation is extended to continuous optimization problems and we prove asymptotic convergence to the set of global optima furthermore we discuss an implementation of the algorithm and compare its performance with other well known algorithms the performance evaluation is carried out for a standard set of test functions from the literature

a new method of constructing symmetry adapted linear combinations for a finite group ii correspondence theorem and basis operators abstract a correspondence theorem is introduced to describe the relation between an elementary basis function and a symmetry adapted linear combination belonging to the same irreducible representation of a point group by means of basis operators the general expressions are given for the symmetry adapted linear combinations the coupling coefficients the lattice harmonics and the hybrid atomic orbitals illustrative examples are discussed to demonstrate the effectiveness of the formalism

a two step approach to measurement error in time dependent covariates in nonlinear mixed effects models with application to igf i pharmacokinetics abstract the usual approach to the analysis of population pharmacokinetic studies is to represent the concentration time data by a nonlinear mixed effects model primary objectives are to characterize the pattern of drug disposition in the population and to identify individual specific covariates associated with pharmacokinetic behavior we consider data from a study of insulin like growth factor i igf i administered by intravenous infusion to patients with severe head trauma failure to maintain steady state levels of igf i was thought to be related to the temporal pattern of several covariates measured in the study and an analysis investigating this issue was of interest observations on these potentially relevant covariates for each subject were made at time points different from those at which igf i concentrations were determined moreover the covariates themselves were likely subject to measurement error the usual approach to time dependent covariates in population analysis is to invoke a simple

misspecified proportional hazard models summary the properties of an estimator based on a proportional hazards model are investigated when the model is incorrect the estimator from the partial likelihood is shown to be consistent for a parameter that is defined implicitly the results are used to investigate the effects on estimation if the true model is accelerated failure time or if covariates are omitted from the proportional hazards model

cure rate model with mismeasured covariates under transformation cure rate models explicitly account for the survival fraction in failure time data when the covariates are measured with errors naively treating mismeasured covariates as error free would cause estimation bias and thus lead to incorrect inference under the proportional hazards cure model we propose a corrected score approach as well as its generalization and implement a transformation on the mismeasured covariates toward error additivity and or normality the corrected score equations can be easily solved through the backfitting procedure and the biases in the parameter estimates are successfully eliminated we show that the proposed estimators for the regression coefficients are consistent and asymptotically normal we conduct simulation studies to examine the finite sample properties of the new method and apply it to a real data set for illustration

the asymptotic properties of nonparametric tests for comparing survival distributions the asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary and possibly unequal right censoring the test statistics include equivalents to the log rank statistic the modified wilcoxon statistic and the class of rank invariant test procedures introduced by peto peto when there are equal censoring distributions and the hazard functions are proportional the sample size formula for the f test used to compare exponential samples is shown to be valid for the log rank test in certain situations the power of the log rank test falls as the amount of censoring decreases

on the treatment of thermal and quantum annealing by stochastic variational means we consider the control of a diffusion process represented by a stochastic differential equation of the ito type the diffusion process is one of a random walk on the configuration space of a multimodal function and the goal of the control is to determine the optimum variation of the temperature in order to anneal into the global minimum of the function such optimal controls are derived using stochastic variational means by the viscosity solution of a second order hamilton jacobi bellman equation or by minimizing a feynman kac wiener path integral the extension to quantized annealing and the possibilities of a differential game theoretic formulation for extra diffusive annealing are also considered

unobserved covariates in the two sample comparison of survival times a maximin efficiency robust test in analysing a clinical trial with the logrank test the hazards between the two groups are usually assumed to be proportional nevertheless this hypothesis is no longer valid with unobserved covariates as a consequence there is a loss of power of the logrank test for testing the null hypothesis h 0 of no treatment effect we propose a test suited for taking into account unobserved covariates the proposed approach is based on a proportional hazard frailty model whereby the omitted covariates are considered as an unobserved frailty variable the procedure is as follows in a first step the weighted logrank test optimal for testing h 0 against a general proportional hazard frailty model is obtained and its specialization for a gamma frailty variable is derived in a second step the proposed test is obtained by combining the maximin efficiency robustness principle and the gamma frailty distribution properties simulation studies investigate the power properties of the test for different frailty distributions a breast cancer clinical trial is analysed as an example the proposed test might be recommended rather than the logrank for practical situations in which one expects heterogeneity related to omitted covariates

failure time regression with continuous covariates measured with error summary we consider failure time regression analysis with an auxiliary variable in the presence of a validation sample we extend the nonparametric inference procedure of zhou and pepe to handle a continuous auxiliary or proxy covariate we estimate the induced relative risk function with a kernel smoother and allow the selection probability of the validation set to depend on the observed covariates we present some asymptotic properties for the kernel estimator and provide some simulation results the method proposed is illustrated with a data set from an on going epidemiologic study

the use of maximin efficiency robust tests in combining contingency tables and survival analysis abstract a test is the maximin efficiency robust test for a family of possible models underlying the data if no other test has higher minimum efficiency relative to the asymptotically optimum test for each model methods used to examine the efficiency robustness of rank tests for the two sample problem are adapted to obtain maximin efficiency robust procedures for testing the equality of proportions across several 2 2 tables for combining the results of tests for trend in several 2 x j tables in which the dose response function is one of a set of possible monotone functions and to analyze censored survival data when either the wilcoxon or log rank may be appropriate in the survival setting the robust test has maximin efficiency 93 3 relative to the wilcoxon or long rank when each is optimum in contrast to the 75 relative efficiency each statistic has when the other is optimum

on functional misspecification of covariates in the cox regression model model misspecification is discussed for the analysis of censored survival data with the cox regression model the maximum partial likelihood estimator for covariate effects has an asymptotic normal distribution under model misspecification lin wei 1989 situations where the assumed functional form for covariates is wrong are considered in a general framework in which we also derive the limit of breslow s 1972 estimator furthermore we give explicit expressions for the asymptotic variance of the maximum partial likelihood estimator numerical values for the errors of the maximum partial likelihood estimator and of the usual variance estimator are obtained for functional misspecification of a single relevant covariate and looking at the effect of a first factor representing treatment or something similar for functional wrong incorporation of auxiliary covariate measurements in the latter situations we analyse the impact of dependence between the covariates

bias analysis and simex approach in generalized linear mixed measurement error models abstract we consider generalized linear mixed models glmms for clustered data when one of the predictors is measured with error when the measurement error is additive and normally distributed and the error prone predictor is itself normally distributed we show that the observed data also follow a glmm but with a different fixed effects structure from the original model a different and more complex random effects structure and restrictions on the parameters this characterization enables us to compute the biases that result in common glmms when one ignores measurement error for instance in one common situation the biases in parameter estimates become larger as the number of observations within a cluster increases both for regression coefficients and for variance components parameter estimation is described using the simex method a relatively new functional method that makes no assumptions about the structure of the unobservable predictors simulations and an example illustrate the results

properties of proportional hazards score tests under misspecified regression models the effects are investigated of misspecifying a proportional hazards regression model on the associated partial likelihood score test for comparing two randomized treatments in the presence of covariates the asymptotic efficiency of the proportional hazards score test relative to the optimal partial likelihood test declines slowly as the hazard functions for the two treatments deviate from proportionality the efficiency can be very low when the hazard functions cross or differ only at large survival times misspecification of the functional form of the regression portion of a proportional hazards model introduces a quantitative treatment covariate interaction in the situations that we examine based on a binary covariate this misspecification usually results in only a minor drop in efficiency the omission of a covariate that is balanced across treatments has a negligible effect on the size of the score test but can substantially reduce power when the covariate effect is strong the loss of power from mismodeling a balanced covariate is usually small

survival analysis with heterogeneous covariate measurement error this article is motivated by a time to event analysis where the covariate of interest was measured at the wrong time we show that the problem can be formulated as a special case of survival analysis with heterogeneous covariate measurement error and develop a general analytic framework we study the asymptotic behavior of the naive partial likelihood estimates and analytically demonstrate that under the heterogeneous measurement error structure and the assumption that all components of the covariate vector and the measurement error vector combined are mutually independent these naive estimates will shrink toward 0 and that the degree of attenuation increases as the measurement error increases we also give counterexamples for reverse attenuation when the independence conditions are violated we use our analytical results to derive a simple bias correcting estimator that performs well in simulations for small and moderate amounts of measurement error our framework can be used to provide insight into th

proportional hazards regression with missing covariates abstract nonparametric maximum likelihood npml is used to estimate regression parameters in a proportional hazards regression model with missing covariates the npml estimator is shown to be consistent and asymptotically normally distributed under some conditions em type algorithms are applied to solve the maximization problem variance estimates of the regression parameters are obtained by a profile likelihood approach that uses em aided numerical differentiation simulation results indicate that the npml estimates of the regression parameters are more efficient than the approximate partial likelihood estimates and estimates from complete case analysis when missing covariates are missing completely at random and that the proposed method corrects for bias when the missing covariates are missing at random

effect of frailty on marginal regression estimates in survival analysis unexplained heterogeneity in univariate survival data and association in multivariate survival can both be modelled by the inclusion of frailty effects this paper investigates the consequences of ignoring frailty in analysis fitting misspecified cox proportional hazards models to the marginal distributions regression coefficients are biased towards 0 by an amount which depends in magnitude on the variability of the frailty terms and the form of frailty distribution the bias is reduced when censoring is present fitted marginal survival curves can also differ substantially from the true marginals

likelihood based methods for missing covariates in the cox proportional hazards model problems associated with missing covariate data are well known but often ignored we present a method for estimating the parameters in the cox proportional hazards model when the missing data are missing at random mar and censoring is noninformative due to the computational burden of this method we introduce an approximation that allows us to use a weighted expectation maximization em algorithm to estimate the parameters more easily when the missing covariates are continuous rather than categorical we implement a monte carlo version of the em algorithm along with the gibbs sampler to obtain parameter estimates we also give the asymptotic distribution of these estimates the primary advantage of this method over complete case analysis is that it produces more efficient parameter estimates and corrects for bias in the mar setting to motivate the methodology we present an analysis of a phase iii melanoma clinical trial conducted by the eastern cooperative oncology group

approximate quasi likelihood estimation in models with surrogate predictors abstract we consider quasi likelihood estimation with estimated parameters in the variance function when some of the predictors are measured with error we review and extend four approaches to estimation in this problem all of them based on small measurement error approximations a taxonomy of the data sets likely to be available in measurement error studies is developed an asymptotic theory based on this taxonomy is obtained and includes measurement error and berkson error models as special cases

smoothed and corrected score approach to censored quantile regression with measurement errors censored quantile regression is an important alternative to the cox proportional hazards model in survival analysis in contrast to the usual central covariate effects quantile regression can effectively characterize the covariate effects at different quantiles of the survival time when covariates are measured with errors it is known that naively treating mismeasured covariates as error free would result in estimation bias under censored quantile regression we propose smoothed and corrected estimating equations to obtain consistent estimators we establish consistency and asymptotic normality for the proposed estimators of quantile regression coefficients compared with the naive estimator the proposed method can eliminate the estimation bias under various measurement error distributions and model error distributions we conduct simulation studies to examine the finite sample properties of the new method and apply it to a lung cancer study supplementary materials for this article are available online

a semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error summary a common objective in longitudinal studies is to characterise the relationship between a failure time process and time independent and time dependent covariates timedependent covariates are generally available as longitudinal data collected periodically during the course of the study we assume that these data follow a linear mixed effects model with normal measurement error and that the hazard of failure depends both on the underlying random effects describing the covariate process and other time independent covariates through a proportional hazards relationship a routine assumption is that the random effects are normally distributed however this need not hold in practice within this framework we develop a simple method for estimating the proportional hazards model parameters that requires no assumption on the distribution of the random effects large sample properties are discussed and finite sample performance is assessed and compared to competing methods via simulation

double semiparametric method for missing covariates in cox regression models the problem of nuisance covariate model specification is considered in cox regression where the maximum semiparametric likelihood method is used to handle the missing covariates a component of the covariates is modeled nonparametrically to achieve robustness against covariate model misspecification and to reduce the number of possibly intractable integrations involved in the parametric modeling of the covariates the statistical properties of the proposed method are examined it is found that in some important situations the maximum semiparametric likelihood can be applied without making any additional parametric model assumptions on covariates the proposed method can yield a more efficient estimator than the nonparametric imputation methods and does not require specification of the missingness mechanism when compared with the inverse probability weighting method a real data example is analyzed to demonstrate use of the proposed method

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

a unified theory of point groups vi the projective corepresentations of the magnetic point groups of infinite order this paper provides all the p inequivalent projective irreducible unitary corepresentations of all the magnetic point groups of infinite order with full use of their isomorphisms

estimating equations with incomplete categorical covariates in the cox model incomplete covariate data is a common occurrence in many studies in which the outcome is survival time when a full likelihood is specified a useful technique for obtaining parameter estimates is the em algorithm we propose a set of estimating equations to estimate the parameters of cox s proportional hazards model when some covariate values are missing these estimating equations can be solved by an algorithm similar to the em algorithm because of the computational burden of finding a solution to these estimating equations we propose obtaining parameter estimates via monte carlo methods asymptotic variances of the parameter estimates are also derived we present a clinical trials example with three covariates two of which have some missing values

covariate measurement errors and parameter estimation in a failure time regression model summary estimation in cox s failure time regression model is considered when the regression vector is subject to measurement error a hazard function model is induced for the failure rate given the measured covariate and a partial likelihood function is derived for the relative risk parameters this partial likelihood function may involve the baseline hazard function as well as the regression parameter but useful inference techniques arise for testing whether the regression parameter equals zero and for more general inferences in important special cases explicit consideration is given to testing equality of survival curves when group membership is subject to misclassification and to relative risk estimation with normally distributed covariates approaches to regression estimation using the overall likelihood function and a marginal likelihood function based on failure time ranks are also indicated illustration of the possible effect of covariate errors on relative risk estimation is provided

a new method of constructing the symmetry coordinates of molecular vibrations based on the correspondence theorem a general method of constructing the symmetry coordinates of molecular vibrations is presented via the correspondence theorem which describes the parallelism between a salc symmetry adapted linear combination and an elementary basis function belonging to the same irreducible representation of the symmetry group of the molecule

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

the 32 minimal general generator sets of 230 double space groups the 32 minimal general generator sets mggs for the three dimensional double space groups are provided each mggs has a very limited number of parameters that describe the translational parts of the space groups belonging to a crystal class the method of construction is based on the algebraic defining relations of the abstract generators of the point groups the equivalence or inequivalence criteria for the space groups with respect to lattice transformations are established by introducing the definite set u of the unimodular matrices that leaves a lattice type l of each crystal class g based on the set u it will be shown that mere shifts of the lattice origin are necessary and sufficient to determine the inequivalence of almost all the space groups belonging to a crystal class of high symmetry it is simpler to construct mggs of higher symmetry than those of lower symmetry this contrasts with the existing methods which are based on the solvability of the space groups

a new method of constructing the symmetry adapted linear combinations based on the correspondence theorem and induced representations the theory of induced representations is incorporated into the method of constructing symmetry adapted linear combinations for a symmetry group based on the correspondence theorem developed previously the theory is effective for a highly symmetric molecule with a set of equivalent radicals each of which again consists of a number of equivalent atoms an illustrative example is given to construct the internal symmetry coordinates of molecular vibrations of c ch3 4 with the frozen configuration belonging to the point group td

a unified theory of the point groups iii classification and basis functions of improper point groups this paper introduces a new system of classification of improper point groups which is most effective for describing their general irreducible representations the complete set of the general angular momentum eigenfunctions is classified according to the general irreducible representations of the seven sets of the improper point groups corresponding to cn c and dn d for an aribtrary n

on using the cox proportional hazards model with missing covariates summary we propose two methods for handling missing covariates in using the cox proportional hazards model the maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information also it is biased when the probability of missingness depends on the failure or censoring time our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information the proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates also the proposed estimator is more efficient than the estimator suggested previously by lin ying 1993 when data are missing completely at random

a new method of constructing symmetry adapted linear combinations for a finite group i salc s of equivalent invariant functions abstract a theory of representation for a finite group g has been presented based on the basis functions defined on a set of equivalent points s n with respect to the symmetry operations of g the general expressions are given for the symmetry adapted linear combinations of equivalent invariant functions and the bonding hybrid atomic orbitals

sample size formula for the proportional hazards regression model a formula is derived for determining the number of observations necessary to test the equality of two survival distributions when concomitant information is incorporated this formula should be useful in designing clinical trials with a heterogeneous patient population schoenfeld 1981 biometrika 68 316 319 derived the asymptotic power of a class of statistics used to test the equality of two survival distributions that result is extended to the case where concomitant information is available for each individual and where the proportional hazards model holds the loss of efficiency caused by ignoring concomitant variables is also computed

a unified theory of the point groups iv the general corepresentations of the crystallographic and noncrystallographic shubnikov point groups a new system of classification for all the shubnikov point groups gs is presented which is best suited for describing their isomorphisms as well as their construction by the new classification and the modified form of the corepresentations there are presented the simple general expressions of the irreducible corepresentations for the 10 minimum sets of proper gs from which all coirreps of the remaining gs follow icosahedral groups are excluded

decomposition in general mathematical programming in this paper a unifying framework is presented for the generalization of the decomposition methods originally developed by benders 1962 and dantzig and wolfe 1960 these generalizations calledvariable decomposition andconstraint decomposition respectively are based on the general duality theory developed by tind and wolsey the framework presented is of a general nature since there are no restrictive conditions imposed on problem structure moreover inaccuracies and duality gaps that are encountered during computations are accounted for the two decomposition methods are proven not to cycle if certain fairly general conditions are met furthermore finite convergence can be ensured under the traditional finiteness conditions and asymptotic convergence can be guaranteed once certain continuity conditions are met the obvious symmetry between both types of decomposition methods is explained by establishing a duality relation between the two which extends a similar result in linear programming a remaining asymmetry in the asymptotic convergence results is argued to be a direct consequence of a fundamental asymmetry that resides in the tind wolsey duality theory it can be shown that in case the latter asymmetry disappears the former does too other decomposition techniques such as lagrangean decomposition and cross decomposition turn out to be captured by the general framework presented here as well

bootstrap based methods for estimating standard errors in cox s regression analyses of clustered event times we propose two bootstrap based methods to correct the standard errors ses from cox s model for within cluster correlation of right censored event times the cluster bootstrap method resamples with replacement only the clusters whereas the two step bootstrap method resamples i the clusters and ii individuals within each selected cluster with replacement in simulations we evaluate both methods and compare them with the existing robust variance estimator and the shared gamma frailty model which are available in statistical software packages we simulate clustered event time data with latent cluster level random effects which are ignored in the conventional cox s model for cluster level covariates both proposed bootstrap methods yield accurate ses and type i error rates and acceptable coverage rates regardless of the true random effects distribution and avoid serious variance under estimation by conventional cox based standard errors however the two step bootstrap method over estimates the variance for individual level covariates we also apply the proposed bootstrap methods to obtain confidence bands around flexible estimates of time dependent effects in a real life analysis of cluster event times

estimating the parameters in the cox model when covariate variables are measured with error the cox proportional hazards model is commonly used to model survival data as a function of covariates because of the measuring mechanism or the nature of the environment covariates are often measured with error and are not directly observable a naive approach is to use the observed values of the covariates in the cox model which usually produces biased estimates of the true association of interest an alternative strategy is to take into account the error in measurement which may be carried out for the cox model in a number of ways we examine several such approaches and compare and contrast them through several simulation studies we introduce a likelihood based approach which we refer to as the semiparametric method and show that this method is an appealing alternative the methods are applied to analyze the relationship between survival and cd4 count in patients with aids

a risk set calibration method for failure time regression by using a covariate reliability sample regression parameter estimation in the cox failure time model is considered when regression variables are subject to measurement error assuming that repeat regression vector measurements adhere to a classical measurement model we can consider an ordinary regression calibration approach in which the unobserved covariates are replaced by an estimate of their conditional expectation given available covariate measurements however since the rate of withdrawal from the risk set across the time axis due to failure or censoring will typically depend on covariates we may improve the regression parameter estimator by recalibrating within each risk set the asymptotic and small sample properties of such a risk set regression calibration estimator are studied a simple estimator based on a least squares calibration in each risk set appears able to eliminate much of the bias that attends the ordinary regression calibration estimator under extreme measurement error circumstances corresponding asymptotic distribution theory is developed small sample properties are studied using computer simulations and an illustration is provided

the loss in efficiency from misspecifying covariates in proportional hazards regression models summary this note considers partial likelihood score tests for association between a covariate process and survival time based on a proportional hazards regression model we derive the asymptotic efficiency of the test based on a misspecified form of the covariate process relative to the test based on the correct form the results can be used to assess the consequences of various types of misspecification

model misspecification in proportional hazards regression the proportional hazards model is frequently used to evaluate the effect of treatment on failure time events in randomised clinical trials concomitant variables are usually available and may be considered for use in the primary analyses under the assumption that incorporating them may reduce bias or improve efficiency in this paper we consider two approaches to including covariate information regression modelling and stratification we focus on the setting where covariate effects are nonproportional and we compare the bias efficiency and coverage properties of these approaches these results indicate that our intuition based on linear model analysis of covariance is misleading covariate adjustment in proportional hazards models has little effect on the variance but may significantly improve the accuracy of the treatment effect estimator

a unified theory of the point groups v the general projective corepresentations of the magnetic point groups and their applications to magnetic space groups this paper presents the general expressions of the projective irreducible corepresentations of the 20 homologous sets of the magnetic double point groups from which follow all those of the remaining groups of finite order through isomorphisms the icosahedral group is excluded these are explicitly given in terms of the irreducible representations of the unitary double point groups their special cases provide all the irreducible corepresentations of all magnetic space groups of wave vector through simple gauge transformations a method of determining the gauge transformations is discussed through typical examples

classification of configurations of systems of identical particles on finite sets all possible configurations of a system of n identical particles are classified for the case when the configuration space q of a single particle reduces to a finite set the classification is achieved by means of a combinatorial analogue of the duality of weyl two groups the pauli group permuting identical particles and the symmetric group on q act on the nth cartesian power of q these actions mutually commute and their orbit structure is classified by appropriate epikernels moreover each stratum of each group is invariant under the dual action which yields a coarsening of permutation representations acting on appropriate sets of orbits it is shown that such a coarsening recovers the geometric symmetry for the case when particles are indistinguishable

bias correction for score tests arising from misspecified proportional hazards regression models when censoring depends on both treatment group and covariates tests for a randomised treatment effect arising from cox s proportional hazards model are not in general centred at zero under the null and can be seriously distorted we propose a corrected test that modifies the at risk indicator in a way that corrects for this bias the resulting test is asymptotically valid regardless of whether or not the fitted model is correctly specified implementation of the corrected test requires that the dependence of censoring on treatment group and covariates be modelled simulations indicate that the bias corrected test performs well even when the dependence of censoring on treatment group and covariates is not modelled exactly and maintains high efficiency relative to the uncorrected test when the latter is valid

cox regression with dependent error in covariates many survival studies have error contaminated covariates due to the lack of a gold standard of measurement furthermore the error distribution can depend on the true covariates but the structure may be difficult to characterize heteroscedasticity is a common manifestation we suggest a novel dependent measurement error model with minimal assumptions on the dependence structure and propose a new functional modeling method for cox regression when an instrumental variable is available this proposal accommodates much more general error contamination than existing approaches including nonparametric correction methods of huang and wang 2000 journal of the american statistical association 95 1209 1219 2006 statistica sinica 16 861 881 the estimated regression coefficients are consistent and asymptotically normal and a consistent variance estimate is provided for inference simulations demonstrate that the procedure performs well even under substantial error contamination illustration with a clinical study is provided

robust covariate adjusted logrank tests summary in testing for significance of treatment effect within two sample censored survival data with measured covariates it is known tsiatis rosner tritchler 1985 that adjusting the logrank test statistic using a proportional hazards model for covariate effect can substantially increase efficiency extending the robust score statistics described by lin wei 1989 we show how to estimate and optimise the relative efficiencies of such score statistics based on various possible working models providing a flexible and statistically valid two sample testing methodology which improves upon the purely nonparametric logrank without sacrificing model robustness numerical calculations indicate some nonproportional hazard settings within which the power of the logrank is erratic while the tests described here have reliably good power

cox regression with accurate covariates unascertainable a nonparametric correction approach abstract many survival studies involve covariates that are not accurately ascertainable cd4 lymphocyte count in hiv aids research is a typical example for which the gold standard of the measurement is not available this article proposes a consistent estimation procedure for cox regression under the additive measurement error model distinct from existing methods the proposed estimation for regression coefficients does not require any additional assumptions we establish the normalized partial score function as a functional of empirical processes which facilitates the construction of an estimating function with the same limit using replicated mismeasured covariates the resulting regression coefficient estimators are shown to be consistent and asymptotically normal a consistent sandwich variance estimate is presented we also suggest estimators for the baseline cumulative hazard function numerical studies demonstrate that the procedure performs well under practical sample sizes application to an aid

auxiliary covariate data in failure time regression summary we consider the problem of missing covariate data in the context of censored failure time relative risk regression auxiliary covariate data which are considered informative about the missing data but which are not explicitly part of the relative risk regression model may be available full covariate information is available for a validation set an estimated partial likelihood method is proposed for estimating relative risk parameters this method is an extension of the estimated likelihood regression analysis method for uncensored data pepe 1992 pepe fleming 1991 a key feature of the method is that it is nonparametric with respect to the association between the missing and observed including auxiliary covariate components asymptotic distribution theory is derived for the proposed estimated partial likelihood estimator in the case where the auxiliary or mismeasured covariates are categorical asymptotic efficiencies are calculated for exponential failure times using an exponential relative risk model the estimated partial likelihood estimator compares favourably with a fully parametric maximum likelihood analysis comparisons are also made with a standard partial likelihood analysis which ignores the incomplete observations important efficiency gains can be made with the estimated partial likelihood method small sample properties are investigated through simulation studies

a unified theory of the point groups ii the general projective representations and their application to space groups the general expressions of all inequivalent irreducible projective representations of crystallographic and noncrystallographic point groups except for the icosahedral group are given in terms of the vector representations of the proper double point groups its application to the representations of the space group of wave vector is discussed construction of the basis sets is discussed

a unified theory of the point groups and their general irreducible representations point groups and their general irreducible vector and projective representations are characterized by the subgroup conditions for su 2 in a unified scheme these conditions are given by simple polynomial equations imposed on the matrix elements of the 2 2 unitary matrices of su 2 the general irreducible representations for the point groups d dn with arbitrary integer n 2 o and t are given by four simple and effective tables

biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates summary certain important nonlinear regression models lead to biased estimates of treatment effect even in randomized experiments if needed covariates are omitted the asymptotic bias is determined both for estimates based on the method of moments and for maximum likelihood estimates the asymptotic bias from omitting covariates is shown to be zero if the regression of the response variable on treatment and covariates is linear or exponential and in regular cases this is a necessary condition for zero bias many commonly used models do have such exponential regressions thus randomization ensures unbiased treatment estimates in a large number of important nonlinear models for moderately censored exponential survival data analysis with the exponential survival model yields less biased estimates of treatment effect than analysis with the proportional hazards model of cox if needed covariates are omitted simulations confirm that calculations of asymptotic bias are in excellent agreement with the bias observed in experiments of modest size

effects of covariate omission and categorization when analysing randomized trials with the cox model when analysing the survival of patients in comparative randomized clinical trials using the cox proportional hazards model important prognostic factors may be included for the adjustment of the treatment effect in this paper we examine two of the most common misspecifications of the model i an important prognostic factor is omitted from the analysis and ii an important prognostic factor originally present on continuous scale is included in categorized form both situations may emerge from the occurrence of missing values we investigate the properties of the maximum partial likelihood estimator of the treatment effect under this kind of misspecification the estimate of the treatment effect is found to be asymptotically biased toward zero for its asymptotic variance we obtain a quantity with the so called sandwich structure thus variance estimation by the inverse of the second order derivative of the likelihood is not consistent the magnitude of overestimation or underestimation is evaluated numerically for specific settings the precision of the treatment effect estimate under covariate omission or categorization is compared with the precision of the estimate in the correct and not misspecified model it turns out that correct adjustment does not lead to a higher precision of the treatment effect estimate but due to the resulting underestimation covariate omission or categorization lead to loss of power of the test of no treatment effect 1997 by john wiley sons ltd

regression dilution in the proportional hazards model the problem of regression dilution arising from covariate measurement error is investigated for survival data using the proportional hazards model the naive approach to parameter estimation is considered whereby observed covariate values are used inappropriately in the usual analysis instead of the underlying covariate values a relationship between the estimated parameter in large samples and the true parameter is obtained showing that the bias does not depend on the form of the baseline hazard function when the errors are normally distributed with high censorship adjustment of the naive estimate by the factor 1 lambda where lambda is the ratio of within person variability about an underlying mean level to the variability of these levels in the population sampled removes the bias as censorship increases the adjustment required increases and when there is no censorship is markedly higher than 1 lambda and depends also on the true risk relationship

cox regression with incomplete covariate measurements abstract this article provides a general solution to the problem of missing covariate data under the cox regression model the estimating function for the vector of regression parameters is an approximation to the partial likelihood score function with full covariate measurements and reduces to the pseudolikelihood score function of self and prentice in the special setting of case cohort designs the resulting parameter estimator is consistent and asymptotically normal with a covariance matrix for which a simple and consistent estimator is provided extensive simulation studies show that the large sample approximations are adequate for practical use the proposed approach tends to be more efficient than the complete case analysis especially for large cohorts with infrequent failures for case cohort designs the new methodology offers a variance covariance estimator that is much easier to calculate than the existing ones and allows multiple subcohort augmentations to improve efficiency real data taken f