time varying coefficients models for recurrent event data when different varying coefficients admit different degrees of smoothness application to heart disease modeling we consider a class of semiparametric marginal rate models for analyzing recurrent event data in these models both time varying and time free effects are present and the estimation of time varying effects may result in non smooth regression functions a typical approach for avoiding this problem and producing smooth functions is based on kernel methods the traditional kernel based approach however assumes a common degree of smoothness for all time varying regression functions which may result in suboptimal estimators if the functions have different levels of smoothness in this paper we extend the traditional approach by introducing different bandwidths for different regression functions first we establish the asymptotic properties of the suggested estimators next we demonstrate the superiority of our proposed method using two finite sample simulation studies finally we illustrate our methodology by analyzing a real world heart disease dataset copyright 2016 john wiley sons ltd

an extended hazard model with longitudinal covariates summary in clinical trials and other medical studies it has become increasingly common to observe an event time of interest and longitudinal covariates simultaneously in the literature joint modelling approaches have been employed to analyze both survival and longitudinal processes and to investigate their association early attention has mostly been placed on developing adaptive and flexible longitudinal processes based on a prespecified survival model most commonly the cox proportional model in this paper we propose a general class of semi parametric hazard regres

goodness of fit test for monotone proportional subdistribution hazards assumptions based on weighted residuals recently goodness of fit tests have been proposed for checking the proportional subdistribution hazards assumptions in the fine and gray regression model zhou fine and laird proposed weighted schoenfeld type residuals tests derived under an assumed model with specific form of time varying regression coefficients li sheike and zhang proposed an omnibus test based on cumulative sums of schoenfeld type residuals in this article we extend the class of weighted residuals tests by allowing random weights of schoenfeld type residuals at ordered event times in particular it is demonstrated that weighted residuals tests using monotone weight functions of time are consistent against monotone proportional subdistribution hazards assumptions extensive monte carlo studies were conducted to evaluate the finite sample performance of recent goodness of fit tests results from simulation studies show that weighted residuals tests using monotone random weight functions commonly used in non proportional hazards regression settings tend to be more powerful for detecting monotone departures than other goodness of fit tests assuming no specific time varying effect or misspecified time varying effects two examples using real data are provided for illustrations copyright 2016 john wiley sons ltd

joint modelling of accelerated failure time and longitudinal data summary the accelerated failure time model is an attractive alternative to the cox model when the proportionality assumption fails to capture the relationship between the survival time and longitudinal covariates several complications arise when the covariates are measured intermittently at different time points for different subjects possibly with measurement errors or measurements are not available after the failure time joint modelling of the failure time and longitudinal data offers a solution to such complications we explore the joint modelling approach under the accelerated failure time assumption when covariates are assumed to follow a linear mixed effects model with measurement errors the procedure is based on maximising the joint likelihood function with random effects treated as missing data a monte carlo em algorithm is used to estimate all the unknown parameters including the unknown baseline hazard function the performance of the proposed procedure is checked in simulation studies a case study of reproductive egg laying data for female mediterranean fruit flies and their relationship to longevity demonstrate the effectiveness of the new procedure

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

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

a regression survival model for testing the proportional hazards hypothesis a semi parametric generalization of the proportional hazards regression model is defined whereby the hazard functions can cross for different values of the covariates in the two sample comparison it includes in particular the case of two weibull distributions differing in scale and shape parameters a global test of the proportional hazards assumption is proposed against such defined alternatives its power in the two sample case is compared to that of previously described tests by using simulation experiments survival data of patients with breast carcinoma including several prognostic factors are presented as an illustration

multi parameter regression survival modeling an alternative to proportional hazards summary it is standard practice for covariates to enter a parametric model through a single distributional parameter of interest for example the scale parameter in many standard survival models indeed the well known proportional hazards model is of this kind in this article we discuss a more general approach whereby covariates enter the model through more than one distributional parameter simultaneously e g scale and shape parameters we refer to this practice as multi parameter regression mpr modeling and explore its use in a survival analysis context we find that multi parameter regression leads to more flexible models which can offer greater insight into the underlying data generating process to illustrate the concept we consider the two parameter weibull model which leads to time dependent hazard ratios thus relaxing the typical proportional hazards assumption and motivating a new test of proportionality a novel variable selection strategy is introduced for such multi parameter regression models it accounts for the correlation arising between the estimated regression coefficients in two or more linear predictors a feature which has not been considered by other authors in similar settings the methods discussed have been implemented in the mpr package in r

survival analysis with time varying regression effects using a tree based approach nonproportional hazards often arise in survival analysis as is evident in the data from the international non hodgkin s lymphoma prognostic factors project a tree based method to handle such survival data is developed for the assessment and estimation of time dependent regression effects under a cox type model the tree method approximates the time varying regression effects as piecewise constants and is designed to estimate change points in the regression parameters a fast algorithm that relies on maximized score statistics is used in recursive segmentation of the time axis following the segmentation a pruning algorithm with optimal properties similar to those of classification and regression trees cart is used to determine a sparse segmentation bootstrap resampling is used in correcting for overoptimism due to split point optimization the piecewise constant model is often more suitable for clinical interpretation of the regression parameters than the more flexible spline models the utility of the algorithm is shown on the lymphoma data where we further develop the published international risk index into a time varying risk index for non hodgkin s lymphoma

goodness of fit test for proportional subdistribution hazards model this paper concerns using modified weighted schoenfeld residuals to test the proportionality of subdistribution hazards for the fine gray model similar to the tests proposed by grambsch and therneau for independently censored data we develop a score test for the time varying coefficients based on the modified schoenfeld residuals derived assuming a certain form of non proportionality the methods perform well in simulations and a real data analysis of breast cancer data where the treatment effect exhibits non proportional hazards copyright 2013 john wiley sons ltd

survival model predictive accuracy and roc curves summary the predictive accuracy of a survival model can be summarized using extensions of the proportion of variation explained by the model or r2 commonly used for continuous response models or using extensions of sensitivity and specificity which are commonly used for binary response models in this article we propose new time dependent accuracy summaries based on time specific versions of sensitivity and specificity calculated over risk sets we connect the accuracy summaries to a previously proposed global concordance measure which is a variant of kendall s tau in addition we show how standard cox regression output can be used to obtain estimates of time dependent sensitivity and specificity and time dependent receiver operating characteristic roc curves semiparametric estimation methods appropriate for both proportional and nonproportional hazards data are introduced evaluated in simulations and illustrated using two familiar survival data sets

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

on estimation of partially linear transformation models we study a general class of partially linear transformation models which extend linear transformation models by incorporating nonlinear covariate effects in survival data analysis a new martingale based estimating equation approach consisting of both global and kernel weighted local estimation equations is developed for estimating the parametric and nonparametric covariate effects in a unified manner we show that with a proper choice of the kernel bandwidth parameter one can obtain the consistent and asymptotically normal parameter estimates for the linear effects asymptotic properties of the estimated nonlinear effects are established as well we further suggest a simple resampling method to estimate the asymptotic variance of the linear estimates and show its effectiveness to facilitate the implementation of the new procedure an iterative algorithm is developed numerical examples are given to illustrate the finite sample performance of the procedure supplementary materials are available online

prediction of cumulative incidence function under the proportional hazards model in the presence of dependent competing risks in survival analysis the cox model can be utilized to examine the covariate effects on the cause specific hazard function for the failure type of interest for this situation the cumulative incidence function provides an intuitively appealing summary curve for marginal probabilities of this particular event in this paper we show how to construct confidence intervals and bands for such a function under the cox model for future patients with certain covariates our proposals are illustrated with data from a prostate cancer trial

direct regression models for longitudinal rates of change comparing rates of growth or rates of change across covariate defined subgroups is a primary objective for many longitudinal studies in the special case of a linear trend over time the interaction between a covariate and time will characterize differences in longitudinal rates of change however in the presence of a non linear longitudinal trajectory the standard mean regression approach does not permit parsimonious description or inference regarding differences in rates of change therefore we propose regression methodology for longitudinal data that allows a direct structured comparison of rates across subgroups even in the presence of a non linear trend over time our basic longitudinal rate regression method assumes a proportional difference across covariate groups in the rate of change across time but this assumption can be relaxed rates are compared relative to a generally specified time trend for which we discuss both parametric and non parametric estimating approaches we develop mixed model longitudinal methodology that explicitly characterizes subject to subject variation in rates as well as a marginal estimating equation based method in addition we detail a score test to detect violations of the proportionality assumption and we allow time varying rate effects as a natural generalization simulation results demonstrate potential gains in power for the longitudinal rate regression model relative to a linear mixed effects model in the presence of a non linear trend in time we apply our method to a study of growth among infants born to hiv infected mothers and conclude with a discussion of possible extensions for our methods

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

some diagnostic methods for cox regression models through hazard smoothing in this paper some graphical methods are proposed for evaluating the fit of cox regression models in survival analysis the simplest procedure proposed is to estimate the baseline hazard separately within subgroups by applying kernel based smoothing to standard cumulative hazard estimates the different estimates should be the same to within sampling variability when the model is correct a modification of this procedure gives a direct way of calculating smooth estimates of hazard ratios over time a further modification gives estimates of ratios as a function of a continuous covariate and time the methods are applied to data taken from a clinical trial of adjuvant therapy for breast cancer

reduced rank hazard regression for modelling non proportional hazards the cox proportional hazards model is the most common method to analyse survival data however the proportional hazards assumption might not hold the natural extension of the cox model is to introduce time varying effects of the covariates for some covariates such as surgical treatment non proportionality could be expected beforehand for some other covariates the non proportionality only becomes apparent if the follow up is long enough it is often observed that all covariates show similar decaying effects over time such behaviour could be explained by the popular gamma frailty model however the marginal effects of covariates in frailty models are not easy to interpret in this paper we propose the reduced rank model for time varying effects of covariates starting point is a cox model with p covariates and time varying effects modelled by q time functions constant included leading to a pxq structure matrix that contains the regression coefficients for all covariate by time function interactions by reducing the rank of this structure matrix a whole range of models is introduced from the very flexible full rank model identical to a cox model with time varying effects to the very rigid rank one model that mimics the structure of a gamma frailty model but is easier to interpret we illustrate these models with an application to ovarian cancer patients

a parametric model for long term follow up data from phase iii breast cancer clinical trials we propose a parametric version of a univariate gamma frailty model the proposed model is shown to be flexible enough to model long term follow up survival data from breast cancer clinical trials when the treatment effect diminishes as time progresses a case for which neither the proportional hazards nor proportional odds assumptions are satisfied the observed information matrix is computed to evaluate the variances of parameter estimates a simple parametric test statistic to test proportional odds assumption is also constructed the model is applied to a data set from a phase iii clinical trial on breast cancer

a note on linear rank tests and gill and schumacher s tests of proportionality summary gill schumacher 1987 assess proportionality of hazards in the two sample case by comparing two generalized linear rank tests with different weights they consider the difference of the two linear rank statistics under proportionality the expected value of this difference is zero and its variance may be estimated the purpose of this note is to show that one can perform a similar procedure within the traditional regression setting the procedure has the advantage of generalizability to multi sample or continuous data and is computable using existing regression programs

a nonparametric test to compare survival distributions with covariate adjustment summary the analysis of covariance is a technique that is used to improve the power of a k sample test by adjusting for concomitant variables if the end point is the time of survival and some observations are right censored the score statistic from the cox proportional hazards model is the method that is most commonly used to test the equality of conditional hazard functions in many situations however the proportional hazards model assumptions are not satisfied specifically the relative risk function is not time invariant or represented as a log linear function of the covariates we propose an asymptotically valid k sample test statistic to compare conditional hazard functions which does not require the assumption of proportional hazards a parametric specification of the relative risk function or randomization of group assignment simulation results indicate that the performance of this statistic is satisfactory the methodology is demonstrated on a data set in prostate cancer

diagnostic plots to reveal functional form for covariates in multiplicative intensity models abstract we show how plots based on the residuals from a proportional hazards model may be used to reveal the correct functional form for covariates in the model a smoothed plot of the martingale residues was suggested for this purpose by therneau grambsch and fleming 1990 biometrika 77 147 160 however its consistency required that the covariates be independent they also noted that the plot could be biased for large covariate effects we introduce two refinements which overcome these difficulties the first is based on a ratio of scatter plot smooths where the numerator is the smooth of the observed count plotted against the covariate and the denominator is a smooth of the expected count this is related to the arjas goodness of fit plot 1988 journal of the american statistical association 83 204 212 the second technique smooths the martingale residuals divided by the expected count using expected count as a weight this latter approach is related to a glm partial residual plot as well as to the iterative methods of hastie and tibshirani 1990 biometrics 46 1005 1016 and gentleman and crowley 1991 biometrics 47 1283 1296 applications to survival data sets are given

the construction of solvable lie algebras from equidimensional nilpotent algebras abstract the study of gradings of solvable lie algebras l of finite dimensionover a field f of zero characteristic led the authors to the discovery of equidimensional nilpotent algebras l uniquely determined by l up to f isomorphy conversely for any finite dimensional lie algebra l over f an algorithm is developed which yields in parametric form all solvable lie algebras l determining l as the corresponding nilpotent algebra the exposition is independent of lie grading theory it organizes in a novel way the classification of solvable lie algebras of given dimension around the same task for nilpotent algebras every isomorphy class of solvable f algebras is obtained in this way

multiple imputation in cox regression when there are time varying effects of covariates in cox regression it is important to test the proportional hazards assumption and sometimes of interest in itself to study time varying effects tves of covariates tves can be investigated with log hazard ratios modelled as a function of time missing data on covariates are common and multiple imputation is a popular approach to handling this to avoid the potential bias and efficiency loss resulting from a complete case analysis two multiple imputation methods have been proposed for when the substantive model is a cox proportional hazards regression an approximate method imputing missing covariate values for the cox model in statistics in medicine 2009 by white and royston and a substantive model compatible method multiple imputation of covariates by fully conditional specification accommodating the substantive model in statistical methods in medical research 2015 by bartlett et al at present neither accommodates tves of covariates we extend them to do so for a general form for the tves and give specific details for tves modelled using restricted cubic splines simulation studies assess the performance of the methods under several underlying shapes for tves our proposed methods give approximately unbiased tve estimates for binary covariates with missing data but for continuous covariates the substantive model compatible method performs better the methods also give approximately correct type i errors in the test for proportional hazards when there is no tve and gain power to detect tves relative to complete case analysis ignoring tves at the imputation stage results in biased tve estimates incorrect type i errors and substantial loss of power in detecting tves we also propose a multivariable tve model selection algorithm the methods are illustrated using data from the rotterdam breast cancer study r code is provided

iterated residuals and time varying covariate effects in cox regression summary the cox proportional hazards model which is widely used for the analysis of treatment and prognostic effects with censored survival data makes the assumption that the hazard ratio is constant over time nonparametric estimators have been developed for an extended model in which the hazard ratio is allowed to change over time estimators based on residuals are appealing as they are easy to use and relate in a simple way to the more restricted cox model estimator after fitting a cox model and calculating the residuals one can obtain a crude estimate of the time varying coefficients by adding a smooth of the residuals to the initial constant estimate treating the crude estimate as the fit one can re estimate the residuals iteration leads to consistent estimation of the nonparametric time varying coefficients this approach leads to clear guidelines for residual analysis in applications the results are illustrated by an analysis of the medical research council s myeloma trials and by simulation

mendelian randomization using semiparametric linear transformation models mendelian randomization mr uses genetic information as an instrumental variable iv to estimate the causal effect of an exposure of interest on an outcome in the presence of unknown confounding we are interested in the causal effect of cigarette smoking on lung cancer survival which is subject to confounding by underlying pulmonary functions despite the well developed iv analyses for the continuous and binary outcomes the scarcity of methodology for the survival outcome limits its utility for the time to event data collected in many observational studies we propose an iv analysis method in the survival context estimating causal effects on a transformed survival time and survival probabilities using semiparametric linear transformation models we study the conditions under which hazard ratio and the effect on survival probability can be approximated for statistical inference we construct estimating equations to circumvent the difficulty in deriving joint likelihood of the exposure and the outcome due to the unknown confounding asymptotic properties of the proposed estimators are established without parametric assumptions about confounders we study the finite sample performance in extensive simulation studies the mr analysis of a lung cancer study suggests a harmful prognostic effect of smoking pack years that would have been missed by the crude association

avoiding infinite estimates of time dependent effects in small sample survival studies we address the phenomenon of monotone likelihood in cox regression with time dependent effects monotone likelihood occurs in the fitting process of a cox model if at least one parameter estimate diverges to infinity we show that the probability of monotone likelihood is increased by the inclusion of time dependent effects particularly in small samples with several unbalanced and highly predictive covariates and with a high percentage of censoring firth s bias reduction procedure was shown to provide an ideal solution to monotone likelihood here we extend his idea to cox regression with time dependent effects by penalized maximum likelihood estimation finite hazard ratio estimates of constant and time dependent effects can be obtained penalized likelihood ratio tests and profile penalized likelihood confidence intervals are proposed as tools for inference a monte carlo study of cox regression with time dependent effects confirms advantages of firth corrected fc over standard cox analysis in terms of average bias and median absolute deviation we also compare the fc and standard cox approaches by means of analyses of two studies with time dependent effects an sas macro and an r package for fc cox regression with time varying covariates and time dependent effects are available at http www muw ac at msi biometrie programs copyright 2008 john wiley sons ltd

solvable lie algebras of dimension 4 over perfect fields abstract the solvable lie algebras of dimension not greater than four over a perfect field of reference are described in terms of their nilpotent frame counted over finite fields of q elements and classified over the real and complex number fields the number dq n of nonisomorphic solvable lie algebras of dimension n over f q roughly speaking grows as a polynomial in n with qn 2 as highest term but there are smaller additional terms depending on the residue class of q modulo n 1 if n 4

flcrm functional linear cox regression model ummary we consider a functional linear cox regression model for characterizing the association between time to event data and a set of functional and scalar predictors the functional linear cox regression model incorporates a functional principal component analysis for modeling the functional predictors and a high dimensional cox regression model to characterize the joint effects of both functional and scalar predictors on the time to event data we develop an algorithm to calculate the maximum approximate partial likelihood estimates of unknown finite and infinite dimensional parameters we also systematically investigate the rate of convergence of the maximum approximate partial likelihood estimates and a score test statistic for testing the nullity of the slope function associated with the functional predictors we demonstrate our estimation and testing procedures by using simulations and the analysis of the alzheimer s disease neuroimaging initiative adni data our real data analyses show that high dimensional hippocampus surface data may be an important marker for predicting time to conversion to alzheimer s disease data used in the preparation of this article were obtained from the adni database adni loni usc edu

graphical methods for assessing violations of the proportional hazards assumption in cox regression a major assumption of the cox proportional hazards model is that the effect of a given covariate does not change over time if this assumption is violated the simple cox model is invalid and more sophisticated analyses are required this paper describes eight graphical methods for detecting violations of the proportional hazards assumption and demonstrates each on three published datasets with a single binary covariate i discuss the relative merits of these methods smoothed plots of the scaled schoenfeld residuals are recommended for assessing ph violations because they provide precise usable information about the time dependence of the covariate effects

score tests for homogeneity of regression effect in the proportional hazards model a simple model containing the proportional hazards regression model as a special case is presented the purpose of the model is to provide a framework in which specific alternatives to the proportional hazards assumption may be tested rank invariant score tests for linear quadratic or exponential trends can for instance all be undertaken within this framework in the case of the two sample problem the required calculations are shown to take a particularly simple form special consideration is given to the two sample case in which there is an inversion of the regression effect i e where the hazard functions cross at some given point both of the motivating examples are concerned with this problem computational aspects are relatively straightforward and some discussion on this is provided

the estimation of average hazard ratios by weighted cox regression often the effect of at least one of the prognostic factors in a cox regression model changes over time which violates the proportional hazards assumption of this model as a consequence the average hazard ratio for such a prognostic factor is under or overestimated while there are several methods to appropriately cope with non proportional hazards in particular by including parameters for time dependent effects weighted estimation in cox regression is a parsimonious alternative without additional parameters the methodology which extends the weighted k sample logrank tests of the tarone ware scheme to models with multiple binary and continuous covariates has been introduced in the nineties of the last century and is further developed and re evaluated in this contribution the notion of an average hazard ratio is defined and its connection to the effect size measure p x y is emphasized the suggested approach accomplishes estimation of intuitively interpretable average hazard ratios and provides tools for inference a monte carlo study confirms the satisfactory performance advantages of the approach are exemplified by comparing standard and weighted analyses of an international lung cancer study sas and r programs facilitate application

the mizon richard encompassing test for the cox and aalen additive hazards models summary the cox hazards model cox 1972 journal of the royal statistical society series b 34 187 220 for survival data is routinely used in many applied fields sometimes however with too little emphasis on the fit of the model a useful alternative to the cox model is the aalen additive hazards model aalen 1980 in lecture notes in statistics 2 1 25 that can easily accommodate time changing covariate effects it is of interest to decide which of the two models that are most appropriate to apply in a given application this is a nontrivial problem as these two classes of models are nonnested except only for special cases in this article we explore the mizon richard encompassing test for this particular problem it turns out that it corresponds to fitting of the aalen model to the martingale residuals obtained from the cox regression analysis we also consider a variant of this method which relates to the proportional excess model martinussen and scheike 2002 biometrika 89 283 298 large sample properties of the suggested methods under the two rival models are derived the finite sample properties of the proposed procedures are assessed through a simulation study the methods are further applied to the well known primary biliary cirrhosis data set

visualizing covariates in proportional hazards model we present a graphical method called the rank hazard plot that visualizes the relative importance of covariates in a proportional hazards model the key idea is to rank the covariate values and plot the relative hazard as a function of ranks scaled to interval 0 1 the relative hazard is plotted with respect to the reference hazard which can be for example the hazard related to the median of the covariate transformation to scaled ranks allows plotting of covariates measured in different units in the same graph which helps in the interpretation of the epidemiological relevance of the covariates rank hazard plots show the difference of hazards between the extremes of the covariate values present in the data and can be used as a tool to check if the proportional hazards assumption leads to reasonable estimates for individuals with extreme covariate values alternative covariate definitions or different transformations applied to covariates can be also compared using rank hazard plots we apply rank hazard plots to the data from the finrisk study where population based cohorts have been followed up for events of cardiovascular diseases and compare the relative importance of the covariates cholesterol smoking blood pressure and body mass index the data from the study to understand prognoses preferences outcomes and risks of treatment support are used to visualize nonlinear covariate effects the proposed graphics work in other regression models with different interpretations of the y axis

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

asymptotic properties of a general class of nonparametric tests for survival analysis many of the popular nonparametric test statistics for censored survival data used in two sample s sample trend and single continuous covariate situations are special cases of a general statistic differing only in the choice of covariate based label and weight function formulated in terms of counting processes and martingales this general statistic standardized by the square root of its consistent variance estimator is shown to be asymptotically normal under the null hypothesis and under a sequence of contiguous hazard alternatives that includes both relative and excess risk models

a comparison of estimators to evaluate the discriminatory power of time to event models discrimination measures for continuous time to event outcomes have become an important tool in medical decision making the idea behind discrimination measures is to evaluate the performance of a prediction model by measuring its ability to distinguish between observations having an event and those having no event researchers proposed a variety of approaches to estimate discrimination measures from a set of right censored data these approaches rely on different regularity assumptions that are needed to ensure consistency of the respective estimators typical examples of regularity assumptions include the proportional hazards assumption in cox regression and the random censoring assumption because regularity assumptions are often violated in practice conducting a sensitivity analysis of the estimators is of considerable interest the aim of the paper is to analyze and to compare the most popular estimators of discrimination measures for event time outcomes on the basis of the results of an extensive simulation study and the analysis of molecular data we investigate the behavior of the estimators in situations where the underlying regularity assumptions do not hold we show that violations of the regularity assumptions may induce a nonignorable bias and may therefore result in biased medical decision making

testing causal effects in observational survival data using propensity score matching design time to event data are very common in observational studies unlike randomized experiments observational studies suffer from both observed and unobserved confounding biases to adjust for observed confounding in survival analysis the commonly used methods are the cox proportional hazards ph model the weighted logrank test and the inverse probability of treatment weighted cox ph model these methods do not rely on fully parametric models but their practical performances are highly influenced by the validity of the ph assumption also there are few methods addressing the hidden bias in causal survival analysis we propose a strategy to test for survival function differences based on the matching design and explore sensitivity of the p values to assumptions about unmeasured confounding specifically we apply the paired prentice wilcoxon ppw test or the modified ppw test to the propensity score matched data simulation studies show that the ppw type test has higher power in situations when the ph assumption fails for potential hidden bias we develop a sensitivity analysis based on the matched pairs to assess the robustness of our finding following rosenbaum s idea for nonsurvival data for a real data illustration we apply our method to an observational cohort of chronic liver disease patients from a mayo clinic study the ppw test based on observed data initially shows evidence of a significant treatment effect but this finding is not robust as the sensitivity analysis reveals that the p value becomes nonsignificant if there exists an unmeasured confounder with a small impact

dynamic cox modelling based on fractional polynomials time variations in gastric cancer prognosis the most popular model used for survival analysis is the proportional hazards regression model proposed by cox this is mainly due to its exceptional simplicity nevertheless the fundamental assumption of the cox model is the proportionality of the hazards for many applications however this assumption is doubtful proposals to extend the cox model for non proportional hazards to allow for dynamic effect structures usually either depend on prespecifications or require non standard estimation techniques and are thus not favoured in application moreover tests to verify the dynamic effect structures are not straightforward or lack omnibus power in this paper we propose a flexible method for modelling dynamic effects in survival data within the cox framework the method is based on fractional polynomials as introduced by royston and altman this allows for a transformation of the dynamic predictor which leads back to the conventional cox model and hence fitting is straightforward using standard estimation techniques in addition it offers the possibility of easily verifying the existence of time variation we describe a model selection algorithm which selects time varying effects only when evidence is given in the data we illustrate the properties of the approach in a simulation study and compare it with other methods in a survival study of gastric cancer patients we apply the approach to analyse dynamic structures in the effects of prognostic factors

a global goodness of fit statistic for cox regression models in this paper a global goodness of fit test statistic for a cox regression model which has an approximate chi squared distribution when the model has been correctly specified is proposed our goodness of fit statistic is global and has power to detect if interactions or higher order powers of covariates in the model are needed the proposed statistic is similar to the hosmer and lemeshow 1980 communications in statistics a10 1043 1069 goodness of fit statistic for binary data as well as schoenfeld s 1980 biometrika 67 145 153 statistic for the cox model the methods are illustrated using data from a mayo clinic trial in primary billiary cirrhosis of the liver fleming and harrington 1991 counting processes and survival analysis in which the outcome is the time until liver transplantation or death the are 17 possible covariates two cox proportional hazards models are fit to the data and the proposed goodness of fit statistic is applied to the fitted models

regression by local fitting methods properties and computational algorithms abstract local regression is a procedure for estimating regression surfaces by the local fitting of linear or quadratic functions of the independent variables in a moving fashion that is analogous to how a moving average is computed for a time series the advantage of the methodology over the global fitting of parametric functions of the independent variables by least squares the current paradigm in regression studies is that a much wider class of regression functions can be estimated without distortion in this paper we discuss the methods their statistical properties and computational algorithms

smoothed rank regression with censored data a weighted rank estimating function is proposed to estimate the regression parameter vector in an accelerated failure time model with right censored data in general rank estimating functions are discontinuous in the regression parameter creating difficulties in determining the asymptotic distribution of the estimator a local distribution function is used to create a rank based estimating function that is continuous and monotone in the regression parameter vector a weight is included in the estimating function to produce a bounded influence estimate the asymptotic distribution of the regression estimator is developed and simulations are performed to examine its finite sample properties a lung cancer dataset is used to illustrate the methodology

regression on quantile residual life a time specific log linear regression method on quantile residual lifetime is proposed under the proposed regression model any quantile of a time to event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring consistency and asymptotic normality of the regression estimator are established an asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point evaluation of the test statistic does not require estimation of the variance covariance matrix of the regression estimators which involves the probability density function of the survival distribution with censoring simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic the new regression method is applied to a breast cancer data set with long term follow up to estimate the patients median residual lifetimes adjusting for important prognostic factors

between within models for survival analysis a popular way to control for confounding in observational studies is to identify clusters of individuals e g twin pairs such that a large set of potential confounders are constant shared wit

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

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

goodness of fit analysis for the cox regression model based on a class of parameter estimators abstract in this article we propose a class of estimation functions for the vector of regression parameters in the cox proportional hazards model with possibly time dependent covariates by incorporating the weight functions commonly used in weighted log rank tests into the partial likelihood score function the resulting estimators behave much like the conventional maximum partial likelihood estimator in that they are consistent and asymptotically normal when the cox model is inappropriate however the estimators with different weight functions generally converge to nonidentical constant vectors for example the magnitude of the parameter estimator using the kaplan meier survival estimator as the weight function will be stochastically larger than that of the maximum partial likelihood estimator if covariate effects diminish over time such facts motivate us to develop goodness of fit methods for the cox regression model by comparing parameter estimators with different weight functions under the assume

a class of transformed mean residual life models with censored survival data this article features online supplementary material the mean residual life function is an attractive alternative to the survival function or the hazard function of a survival time in practice it provides the remaining life expectancy of a subject surviving up to time t in this study we propose a class of transformed mean residual life models for fitting survival data under right censoring to estimate the model parameters we make use of the inverse probability of censoring weighting approach and develop a system of estimating equations efficiency and robustness of the estimators are also studied both asymptotic and finite sample properties of the proposed estimators are established and the approach is applied to two real life datasets collected from clinical trials

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

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

a simple test for goodness of fit of cox s proportional hazards model cox s proportional hazards model for survival analysis 1972 has gained widespread attention from both applied and theoretical statisticians and is central to most work on survival analysis to verify the assumption of proportional hazards underlying the model several methods have recently been put forward cox 1979 schoenfeld 1980 andersen 1982 kalbfleisch and prentice 1980 these methods consist either of graphical techniques designed so as to visualize departures from the proportional hazards ph assumption or of formal tests based on parameterizing the interaction between covariates and time in this paper a simple test statistic is proposed for detecting departures from the ph assumption the proposed test is computationally and conceptually straightforward expressions for the first two moments of the test statistic are presented

adjusted restricted mean survival times in observational studies in observational studies with censored data exposure outcome associations are commonly measured with adjusted hazard ratios from multivariable cox proportional hazards models the difference in restricted mean survival times rmsts up to a pre specified time point is an alternative measure that offers a clinically meaningful interpretation several regression based methods exist to estimate an adjusted difference in rmsts but they digress from the model free method of taking the area under the survival function we derive the adjusted rmst by integrating an adjusted kaplan meier estimator with inverse probability weighting ipw the adjusted difference in rmsts is the area between the two ipw adjusted survival functions in a monte carlo type simulation study we demonstrate that the proposed estimator performs as well as two regression based approaches the ancova type method of tian et al and the pseudo observation method of andersen et al we illustrate the methods by reexamining the association between total cholesterol and the 10 year risk of coronary heart disease in the framingham heart study

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

a new concordance measure for risk prediction models in external validation settings concordance measures are frequently used for assessing the discriminative ability of risk prediction models the interpretation of estimated concordance at external validation is difficult if the case mix differs from the model development setting we aimed to develop a concordance measure that provides insight into the influence of case mix heterogeneity and is robust to censoring of time to event data we first derived a model based concordance mbc measure that allows for quantification of the influence of case mix heterogeneity on discriminative ability of proportional hazards and logistic regression models this mbc can also be calculated including a regression slope that calibrates the predictions at external validation c mbc hence assessing the influence of overall regression coefficient validity on discriminative ability we derived variance formulas for both mbc and c mbc we compared the mbc and the c mbc with commonly used concordance measures in a simulation study and in two external validation settings the mbc was asymptotically equivalent to a previously proposed resampling based case mix corrected c index the c mbc remained stable at the true value with increasing proportions of censoring while harrell s c index and to a lesser extent uno s concordance measure increased unfavorably variance estimates of mbc and c mbc were well in agreement with the simulated empirical variances we conclude that the mbc is an attractive closed form measure that allows for a straightforward quantification of the expected change in a model s discriminative ability due to case mix heterogeneity the c mbc also reflects regression coefficient validity and is a censoring robust alternative for the c index when the proportional hazards assumption holds copyright 2016 john wiley sons ltd

additive expectancy regression regression is a useful tool for studying association between an outcome variable and its covariates in a classical linear regression model the outcome s expectation is usually a function of the covariates and some regression parameters and the underlying distribution is often normal in this article we propose a new class of additive expectancy regression models in which the outcomes tend to be positively skewed in the new regression models regression parameters are practically meaningful and also useful with inferences on the means of skewed outcomes parametric and semiparametric methods are developed for model estimation and inferences model based prediction and model adequacy assessment are discussed the proposed methodologies are demonstrated by two real data analyses

spatial extended hazard model with application to prostate cancer survival summary this article develops a bayesian semiparametric approach to the extended hazard model with generalization to high dimensional spatially grouped data county level spatial correlation is accommodated marginally through the normal transformation model of li and lin 2006 journal of the american statistical association 101 591 603 using a correlation structure implied by an intrinsic conditionally autoregressive prior efficient markov chain monte carlo algorithms are developed especially applicable to fitting very large highly censored areal survival data sets per variable tests for proportional hazards accelerated failure time and accelerated hazards are efficiently carried out with and without spatial correlation through bayes factors the resulting reduced interpretable spatial models can fit significantly better than a standard additive cox model with spatial frailties

on dependence estimation using correlated failure time data from case control family studies correlated failure time data arise routinely in case control family studies to measure the dependence between bivariate failure times a cross ratio i e conditional hazard ratio function has been used many approaches ranging from estimating equations to likelihood score equations have been proposed for the estimation of parameters in the cross ratio function from prospective family data however the adaptation of these approaches to case control family data appears too complicated and computationally intensive to be practical in this paper a connection is established between the cross ratio function and the relative risk function in a stratified proportional hazards model leading to the development of a set of partial likelihood based estimating equations a simulation study is conducted to evaluate the performance of this estimation procedure and two datasets are analysed to illustrate this method

a semi parametric threshold regression analysis of sexually transmitted infections in adolescent women time to event analysis of sexually transmitted infection data is often complicated by the existence of nonproportional hazards and nonlinear independent variable effects methods without the proportional hazards assumption such as threshold regression models have been successfully used in many applications this paper seeks to extend the existing threshold regression models to accommodate the nonlinear independent variable effects specifically we incorporated penalized and regression splines to the threshold regression models for added modeling flexibility cross validation methods were used for the selection of the number of knots and for the determination of smoothing parameters variance estimates were proposed for inference purposes simulation results showed that the proposed methods were able to achieve nonparametric function and parametric coefficient estimates that are close to their true values simulation also demonstrated satisfactory performance of variance estimates using the proposed methods we analyzed time from sexual debut to the first infection with chlamydia trachomatis infection in a group of young women analysis shows that the lifetime number of sexual partners has a nonlinear effect on the risk of c trachomatis infection and the infection risks were differential by ethnicity and age of sexual debut

testing goodness of fit for proportional hazards model with censored observations abstract a numerical omnibus test of fit for the two sample proportional hazards model is proposed for randomly censored observations the test is derived from cox s partial likelihood and does not need a dummy time dependent covariate or any partition of the time axis consistency of the test is established examples are provided for illustration

comparison of case cohort estimators based on data on premature death of adult adoptees a case cohort sample of adoptees was collected to investigate genetic and environmental influences on premature death which motivated us to supplement existing simulation results to explore the performance of various estimators proposed for case cohort samples of survival data we studied six regression coefficients estimators which differ with regard to the weighting scheme used in a pseudo likelihood function and two different estimators of their variances compared to earlier simulation studies we changed the following conditions type of explanatory variable the distribution of lifetimes and the percentage of deaths in the full cohort the latter condition affected the performance of the estimated variances of the regression coefficients where we found a systematic bias of the estimator proposed by self and prentice dependent on the percentages of deaths this dependence of percentages of death was different for different sizes of case cohort studies a robust variance estimator showed a better overall performance the estimators of regression coefficients compared did not differ much the estimators proposed by kalbfleisch and lawless and by prentice performing very well results of the case cohort data of adoptees were not in conflict with earlier findings of a moderate genetic influence on premature death in adulthood copyright 2003 john wiley sons ltd

adaptive regression splines in the cox model we develop a method for constructing adaptive regression spline models for the exploration of survival data the method combines cox s 1972 journal of the royal statistical society series b 34 187 200 regression model with a weighted least squares version of the multivariate adaptive regressi on spline mars technique of friedman 1991 annals of statistics 19 1 141 to adaptively select the knots and covariates the new technique can automatically fit models with terms that represent nonlinear effects and interactions among covariates applications based on simulated data and data from a clinical trial for myeloma are presented results from the myeloma application identified several important prognostic variables including a possible nonmonotone relationship with survival in one laboratory variable results are compared to those from the adaptive hazard regression hare method of kooperberg stone and truong 1995 journal of the american statistical association 90 78 94

martingale difference residuals as a diagnostic tool for the cox model the proportional hazards model makes two major assumptions the hazard ratio is constant over time and the relationship between the hazard and continuous covariates is log linear methods exist for checking and relaxing each of these assumptions but in both cases the methods rely on the other assumption being true problems can occur if neither of the assumptions is appropriate or even if only one of the assumptions is appropriate but it is not known which we propose a new kind of residual for checking the two assumptions simultaneously the smoothed residuals provide a flexible estimate of the hazard ratio which may deviate from the standard proportional hazards model by having a time dependent hazard ratio transformed covariates or both the methods are illustrated using data from the medical research council s myeloma trials

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

a special case of reduced rank models for identification and modelling of time varying effects in survival analysis flexible survival models are in need when modelling data from long term follow up studies in many cases the assumption of proportionality imposed by a cox model will not be valid instead a model that can identify time varying effects of fixed covariates can be used although there are several approaches that deal with this problem it is not always straightforward how to choose which covariates should be modelled having time varying effects and which not at the same time it is up to the researcher to define appropriate time functions that describe the dynamic pattern of the effects in this work we suggest a model that can deal with both fixed and time varying effects and uses simple hypotheses tests to distinguish which covariates do have dynamic effects the model is an extension of the parsimonious reduced rank model of rank 1 as such the number of parameters is kept low and thus a flexible set of time functions such as b splines can be used the basic theory is illustrated along with an efficient fitting algorithm the proposed method is applied to a dataset of breast cancer patients and compared with a multivariate fractional polynomials approach for modelling time varying effects copyright 2016 john wiley sons ltd

regression calibration in semiparametric accelerated failure time models summary in large cohort studies it often happens that some covariates are expensive to measure and hence only measured on a validation set on the other hand relatively cheap but error prone measurements of the covariates are available for all subjects regression calibration rc estimation method prentice 1982 biometrika 69 331 342 is a popular method for analyzing such data and has been applied to the cox model by wang et al 1997 biometrics 53 131 145 under normal measurement error and rare disease assumptions in this article we consider the rc estimation method for the semiparametric accelerated failure time model with covariates subject to measurement error asymptotic properties of the proposed method are investigated under a two phase sampling scheme for validation data that are selected via stratified random sampling resulting in neither independent nor identically distributed observations we show that the estimates converge to some well defined parameters in particular unbiased estimation is feasible under additive normal measurement error models for normal covariates and under berkson error models the proposed method performs well in finite sample simulation studies we also apply the proposed method to a depression mortality study

a nonparametric multiplicative hazard model for event history analysis a major issue in exploring and analyzing life history data with multiple states and events is the development and availability of flexible methods that allow simultaneous incorporation and estimation of baseline hazards detection and modelling of nonlinear functional forms of covariates and time varying effects and the possibility to include time dependent covariates in this paper we consider a nonparametric multiplicative hazard model that takes into account these aspects embedded in the counting process approach estimation is based on penalized likelihoods and splines the methods are illustrated by two real data applications one to a more conventional survival data set with two absorbing states and one to more complex sleep electroencephalography data with multiple recurrent states of sleep

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

a score test on measurement errors in rating transition times abstract we model credit rating histories as continuous time discrete state markov processes infrequent monitoring of the debtors solvency will result in erroneous observations of the rating transition times and consequently in biased parameter estimates we develop a score test against such measurement errors in the transition data that is independent of the error distribution we derive the asymptotic 2 distribution for the test statistic under the null by stochastic limit theory the test is applied to an international corporate portfolio while accounting for economic and debtor specific covariates the test indicates that measurement errors in the transition times are a real problem in practice

the four sets of additive quantum numbers of su 3 the four maximal sets of additive quantum numbers and related fine gradings of the lie algebra sl 3 c are described in detail the quantum numbers are determined by a grading of the lie algebra fine gradings providing maximal sets of them two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the lie algebra