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

coexistence does not imply joint measurability one of the hallmarks of quantum theory is the realization that distinct measurements cannot in general be performed simultaneously in stark contrast to classical physics in this context the notions of coexistence and joint measurability are employed to analyze the possibility of measuring together two general quantum observables characterizing different degrees of compatibility between measurements it is known that two jointly measurable observables are always coexistent and that the converse holds for various classes of observables including the case of observables with two outcomes here we resolve in the negative the open question of whether this equivalence holds in general our resolution strengthens the notions of coexistence and joint measurability by showing that both are robust against small imperfections in the measurement setups

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

finite monoids and the fine structure of nc 1 recently a new connection was discovered between the parallel complexity class nc 1 and the theory of finite automata in the work of barrington on bounded width branching programs there nonuniform nc 1 was characterized as those languages recognized by a certain nonuniform version of a dfa here we extend this characterization to show that the internal structures of nc 1 and the class of automata are closely related in particular using therien s classification of finite monoids we give new characterizations of the classes ac 0 depth k ac 0 and acc the last being the ac 0 closure of the mod q functions for all constant q we settle some of the open questions in 3 give a new proof that the dot depth hierarchy of algebraic automata theory is infinite 8 and offer a new framework for understanding the internal structure of nc 1

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

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

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

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

on a conjecture concerning dot depth two languages abstract in this paper we study the second level of the dot depth hierarchy for star free regular languages we investigate a necessary condition stated by straubing for a language to have dot depth two and prove that it is sufficient for languages whose syntactic monoid is inverse with three inverse generators also we disprove a conjecture according to which straubing s condition would be equivalent to both dot depth two and another condition expressed in terms of two sided semidirect product

going higher in first order quantifier alternation hierarchies on words we investigate quantifier alternation hierarchies in first order logic on finite words levels in these hierarchies are defined by counting the number of quantifier alternations in formulas we prove that one can decide membership of a regular language in the levels b 2 finite boolean combinations of formulas having only one alternation and 3 formulas having only two alternations and beginning with an existential block our proofs work by considering a deeper problem called separation which once solved for lower levels allows us to solve membership for higher levels

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

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

semigroups and languages of dot depth two this paper is a contribution to the problem of effectively determining the dot depth of a star free language a problem in the theory of automata and formal languages with close connections to algebra and formal logic we conjecture an effective criterion for determining whether a given language has dot depth 2 the condition is shown to be necessary in general and sufficient for languages over a two letter alphabet the condition involves a novel use of categories in the study of semigroup theoretic problems

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

on coexistence and joint measurability of rank 1 quantum observables we show that a rank 1 quantum observable povm is jointly measurable with a quantum observable exactly when is a smearing of if is extreme rank 1 and discrete then and are coexistent if and only if they are jointly measurable

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

scaling regression inputs by dividing by two standard deviations interpretation of regression coefficients is sensitive to the scale of the inputs one method often used to place input variables on a common scale is to divide each numeric variable by its standard deviation here we propose dividing each numeric variable by two times its standard deviation so that the generic comparison is with inputs equal to the mean 1 standard deviation the resulting coefficients are then directly comparable for untransformed binary predictors we have implemented the procedure as a function in r we illustrate the method with two simple analyses that are typical of applied modeling a linear regression of data from the national election study and a multilevel logistic regression of data on the prevalence of rodents in new york city apartments we recommend our rescaling as a default option an improvement upon the usual approach of including variables in whatever way they are coded in the data file so that the magnitudes of coefficients can be directly compared as a matter of routine statistical practice copyright 2007 john wiley sons ltd

nondisturbing quantum measurements we consider pairs of discrete quantum observables povms and analyze the relation between the notions of nondisturbance joint measurability and commutativity we specify conditions under which these properties coincide or differ depending for instance on the interplay between the number of outcomes and the hilbert space dimension or on algebraic properties of the effect operators we also show that non disturbance is in general not a symmetric relation and that it can be decided and quantified by means of a semidefinite program

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 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

algebraic tools for the concatenation product this paper is a contribution to the algebraic study of the concatenation product in the first part of the paper we extend to the ordered case standard algebraic tools related to the concatenation product like the schutzenberger product and the relational morphisms we show in a precise way how the ordered schutzenberger product corresponds to polynomial operations on languages in the second part of the paper we apply these results to establish a bridge between the three standard concatenation hierarchies namely the straubing therien hierarchy the brzozowski s or dot depth hierarchy and the group hierarchy

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

inverse monoids of dot depth two resume nous etudions une condition necessaire enoncee par straubing pour qu un langage rationnel sans etoile appartienne au niveau deux de la hierarchie de concatenation de straubing nous prouvons que la condition est suffisante pour les langages dont le monoide syntaxique est un monoide inversif a deux generateurs inversifs

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

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

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

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

an empirical comparison of statistical tests for assessing the proportional hazards assumption of cox s model in the analysis of survival data using the cox proportional hazard ph model it is important to verify that the explanatory variables analysed satisfy the proportional hazard assumption of the model this paper presents results of a simulation study that compares five test statistics to check the proportional hazard assumption of cox s model the test statistics were evaluated under proportional hazards and the following types of departures from the proportional hazard assumption increasing relative hazards decreasing relative hazards crossing hazards diverging hazards and non monotonic hazards the test statistics compared include those based on partitioning of failure time and those that do not require partitioning of failure time the simulation results demonstrate that the time dependent covariate test the weighted residuals score test and the linear correlation test have equally good power for detection of non proportionality in the varieties of non proportional hazards studied using illustrative data from the literature these test statistics performed similarly

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

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

a graphical method for assessing goodness of fit in cox s proportional hazards model abstract suggested here is a simple graphical method for studying the goodness of fit in cox s regression model for survival data the method is easy to use as it does not require the estimation of alternative models and only involves quantities similar to those already appearing in the partial likelihood expression that is needed in the parameter estimation the rationale behind the graphs is very intuitive they make a direct comparison between observed and expected failure frequencies as estimated from the model in a correctly specified model one anticipates an approximate balance between such frequencies otherwise there will typically be groups of individuals for which the expected frequencies are systematically too high or too low to match with the data and this shows in the graphs introduced here in the concrete applications of the method the individuals are stratified in a way that depends on what aspect of the model is being checked against data there is always one graph for each stratum s

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

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

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

complete characterization of extreme quantum observables in infinite dimensions we give a complete characterization for extreme quantum observables i e for normalized positive operator valued measures povms which are extremals in the convex set of all povms the characterization is valid both in discrete and continuous cases and also in the case of an infinite dimensional hilbert space we show that sharp povms are pre processing clean i e they cannot be irreversibly connected to other povms via quantum channels