a unified bayesian semiparametric approach to assess discrimination ability in survival analysis the discriminatory ability of a marker for censored survival data is routinely assessed by the time dependent roc curve and the c index the time dependent roc curve evaluates the ability of a biomarker to predict whether a patient lives past a particular time t the c index measures the global concordance of the marker and the survival time regardless of the time point we propose a bayesian semiparametric approach to estimate these two measures the proposed estimators are based on the conditional distribution of the survival time given the biomarker and the empirical biomarker distribution the conditional distribution is estimated by a linear dependent dirichlet process mixture model the resulting roc curve is smooth as it is estimated by a mixture of parametric functions the proposed c index estimator is shown to be more efficient than the commonly used harrell s c index since it uses all pairs of data rather than only informative pairs the proposed estimators are evaluated through simulations and illustrated using a lung cancer dataset

american fractional lookback options valuation and premium decomposition this paper deals with valuation and premium decomposition of american fractional lookback options written on dividend paying assets for which exact formulas are unknown except for the perpetual case via a pde approach we derive laplace transforms of the values of lookback call and put options which can be decomposed into the associated european values plus the early exercise premiums using abelian theorems of laplace transforms we characterize asymptotic behaviors of the early exercise boundaries at a time to close to expiration and at infinite time to expiration based on the gaver stehfest inversion method combined with the newton method we develop a fast and accurate algorithm for computing both the option value and the early exercise boundary numerical analysis reveals some notable features of the american fractional lookback options

proportional hazards estimate of the conditional survival function summary we introduce a new estimator of the conditional survival function given some subset of the covariate values under a proportional hazards regression the new estimate does not require estimating the base line cumulative hazard function an estimate of the variance is given and is easy to compute involving only those quantities that are routinely calculated in a cox model analysis the asymptotic normality of the new estimate is shown by using a central limit theorem for kaplan meier integrals we indicate the straightforward extension of the estimation procedure under models with multiplicative relative risks including non proportional hazards and to stratified and frailty models the estimator is applied to a gastric cancer study where it is of interest to predict patients survival based only on measurements obtained before surgery the time at which the most important prognostic variable stage becomes known

valuing american floating strike lookback option and neumann problem for inhomogeneous black scholes equation abstract this paper presents our study of american floating strike lookback options written on dividend paying assets the valuation of these options can be mathematically formulated as a free boundary inhomogeneous black scholes pde with a neumann boundary condition which we by using a mellin transform convert into a relatively simple ordinary differential equation with dirichlet boundary conditions we then use these results to derive an integral equation that can be used to calculate the price of american floating strike lookback options in addition we also used mellin transforms to derive the closed form of the perpetual case

time dependent roc curves for censored survival data and a diagnostic marker roc curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker x for a binary disease variable d however many disease outcomes are time dependent d t and roc curves that vary as a function of time may be more appropriate a common example of a time dependent variable is vital status where d t 1 if a patient has died prior to time t and zero otherwise we propose summarizing the discrimination potential of a marker x measured at baseline t 0 by calculating roc curves for cumulative disease or death incidence by time t which we denote as roc t a typical complexity with survival data is that observations may be censored two roc curve estimators are proposed that can accommodate censored data a simple estimator is based on using the kaplan meier estimator for each possible subset x c however this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in x an alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of x t where t represents survival time akritas m j 1994 annals of statistics 22 1299 1327 we present an example where roc t is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the roc t curve displays the impact of modifying eligibility criteria for sample size and power in hiv prevention trials

characterizing and approximating eigenvalue sets of symmetric interval matrices we consider the eigenvalue problem for the case where the input matrix is symmetric and its entries are perturbed with perturbations belonging to some given intervals we present a characterization of some of the exact boundary points which allows us to introduce an inner approximation algorithm that in many case estimates exact bounds to our knowledge this is the first algorithm that is able to guarantee exactness we illustrate our approach by several examples and numerical experiments

the smallest eigenvalue of the signless laplacian recently the signless laplacian matrix of graphs has been intensively investigated while there are many results about the largest eigenvalue of the signless laplacian the properties of its smallest eigenvalue are less well studied the present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless laplacian

measures of explained variation for survival data the predictive power of a set of prognostic variables in a survival time model is a concept distinct from the statistical significance of the variables or the adequacy of the model fit in this paper we discuss the importance of quantifying the predictive power of a prognostic model and suggest measures of explained variation as a possible quantification the important features of our approach are that 1 the measures are completely model based 2 a specification of the time range of interest is easily incorporated and 3 the null models used for comparison are derived as mixtures of the predicted distributions

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

comparing two correlated c indices with right censored survival outcome a one shot nonparametric approach the area under the receiver operating characteristic curve is often used as a summary index of the diagnostic ability in evaluating biomarkers when the clinical outcome truth is binary when the clinical outcome is right censored survival time the c index motivated as an extension of area under the receiver operating characteristic curve has been proposed by harrell as a measure of concordance between a predictive biomarker and the right censored survival outcome in this work we investigate methods for statistical comparison of two diagnostic or predictive systems of which they could either be two biomarkers or two fixed algorithms in terms of their c indices we adopt a u statistics based c estimator that is asymptotically normal and develop a nonparametric analytical approach to estimate the variance of the c estimator and the covariance of two c estimators a z score test is then constructed to compare the two c indices we validate our one shot nonparametric method via simulation studies in terms of the type i error rate and power we also compare our one shot method with resampling methods including the jackknife and the bootstrap simulation results show that the proposed one shot method provides almost unbiased variance estimations and has satisfactory type i error control and power finally we illustrate the use of the proposed method with an example from the framingham heart study

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

choice of prognostic estimators in joint models by estimating differences of expected conditional kullback leibler risks prognostic estimators for a clinical event may use repeated measurements of markers in addition to fixed covariates these measurements can be linked to the clinical event by joint models that involve latent features when the objective is to choose between different prognosis estimators based on joint models the conventional akaike information criterion is not well adapted and decision should be based on predictive accuracy we define an adapted risk function called expected prognostic cross entropy we define another risk function for the case of right censored observations the expected prognostic observed cross entropy epoce these risks can be estimated by leave one out cross validation for which we give approximate formulas and asymptotic distributions the approximated cross validated estimator cvpola of epoce is studied in simulation and applied to the comparison of several joint latent class models for prognosis of recurrence of prostate cancer using prostate specific antigen measurements

novel metrics for evaluating improvement in discrimination net reclassification and integrated discrimination improvement for normal variables and nested models net reclassification and integrated discrimination improvements have been proposed as alternatives to the increase in the auc for evaluating improvement in the performance of risk assessment algorithms introduced by the addition of new phenotypic or genetic markers in this paper we demonstrate that in the setting of linear discriminant analysis under the assumptions of multivariate normality all three measures can be presented as functions of the squared mahalanobis distance this relationship affords an interpretation of the magnitude of these measures in the familiar language of effect size for uncorrelated variables furthermore it allows us to conclude that net reclassification improvement can be viewed as a universal measure of effect size our theoretical developments are illustrated with an example based on the framingham heart study risk assessment model for high risk men in primary prevention of cardiovascular disease

bias and variance reduction in nonparametric estimation of time dependent accuracy measures a new nonparametric approach is developed to estimate the time dependent accuracy measure curves which are defined on the cumulative cases and dynamic controls for censored survival data based on an estimable survival process the main intention of this study is to reduce the finite sample biases of nearest neighbor estimators the asymptotic variances of some retrospective accuracy measure estimators are further reduced by applying a smoothing technique to the underlying process of a marker meanwhile practically feasible and theoretically valid procedures are proposed for bandwidth selection in the presented estimators in addition the proposed methodology can be reasonably extended to accommodate stratified survival data and survival data with multiple markers as shown in the simulations our new estimators outperform the nearest neighbor and inverse censoring weighted estimators data from the aids clinical trials group study 175 and an angiographic coronary artery disease study are also used to illustrate the proposed methodology copyright 2016 john wiley sons ltd

the statistical analysis of failure time data preface 1 introduction 1 1 failure time data 1 2 failure time distributions 1 3 time origins censoring and truncation 1 4 estimation of the survivor function 1 5 comparison of survival curves 1 6 generalizations to accommodate delayed entry 1 7 counting process notation bibliographic notes exercises and complements 2 failure time models 2 1 introduction 2 2 some continuous parametric failure time models 2 3 regression models 2 4 discrete failure time models bibliographic notes exercises and complements 3 inference in parametric models and related topics 3 1 introduction 3 2 censoring mechanisms 3 3 censored samples from an exponential distribution 3 4 large sample likelihood theory 3 5 exponential regression 3 6 estimation in log linear regression models 3 7 illustrations in more complex data sets 3 8 discrimination among parametric models 3 9 inference with interval censoring 3 10 discussion bibliographic notes exercises and complements 4 relative risk cox regression models 4 1 introduction 4 2 estimation of beta 4 3 estimation of the baseline hazard or survivor function 4 4 inclusion of strata 4 5 illustrations 4 6 counting process formulas 4 7 related topics on the cox model 4 8 sampling from discrete models bibliographic notes exercises and complements 5 counting processes and asymptotic theory 5 1 introduction 5 2 counting processes and intensity functions 5 3 martingales 5 4 vector valued martingales 5 5 martingale central limit theorem 5 6 asymptotics associated with chapter 1 5 7 asymptotic results for the cox model 5 8 asymptotic results for parametric models 5 9 efficiency of the cox model estimator 5 10 partial likelihood filtration bibliographic notes exercises and complements 6 likelihood construction and further results 6 1 introduction 6 2 likelihood construction in parametric models 6 3 time dependent covariates and further remarks on likelihood construction 6 4 time dependence in the relative risk model 6 5 nonnested conditioning events 6 6 residuals and model checking for the cox model bibliographic notes exercises and complements 7 rank regression and the accelerated failure time model 7 1 introduction 7 2 linear rank tests 7 3 development and properties of linear rank tests 7 4 estimation in the accelerated failure time model 7 5 some related regression models bibliographic notes exercises and complements 8 competing risks and multistate models 8 1 introduction 8 2 competing risks 8 3 life history processes bibliographic notes exercises and complements 9 modeling and analysis of recurrent event data 9 1 introduction 9 2 intensity processes for recurrent events 9 3 overall intensity process modeling and estimation 9 4 mean process modeling and estimation 9 5 conditioning on aspects of the counting process history bibliographic notes exercises and complements 10 analysis of correlated failure time data 10 1 introduction 10 2 regression models for correlated failure time data 10 3 representation and estimation of the bivariate survivor function 10 4 pairwise dependency estimation 10 5 illustration australian twin data 10 6 approaches to nonparametric estimation of the bivariate survivor function 10 7 survivor function estimation in higher dimensions bibliographic notes exercises and complements 11 additional failure time data topics 11 1 introduction 11 2 stratified bivariate failure time analysis 11 3 fixed study period survival studies 11 4 cohort sampling and case control studies 11 5 missing covariate data 11 6 mismeasured covariate data 11 7 sequential testing with failure time endpoints 11 8 bayesian analysis of the proportional hazards model 11 9 some analyses of a particular data set bibliographic notes exercises and complements glossary of notation appendix a some sets of data appendix b supporting technical material bibliography author index subject index

varying coefficient models we explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables general algorithms are presented for estimating the models flexibly and some examples are given this class of models ties together generalized additive models and dynamic generalized linear models into one common framework when applied to the proportional hazards model for survival data this approach provides a new way of modelling departures from the proportional hazards assumption

estimating a time dependent concordance index for survival prediction models with covariate dependent censoring given a predictive marker and a time to event response variable the proportion of concordant pairs in a data set is called concordance index a specifically useful marker is the risk predicted by a survival regression model this article extends the existing methodology for applications where the length of the follow up period depends on the predictor variables a class of inverse probability of censoring weighted estimators is discussed in which the estimates rely on a working model for the conditional censoring distribution the estimators are consistent for a truncated concordance index if the working model is correctly specified and if the probability of being uncensored at the truncation time is positive in this framework all kinds of prediction models can be assessed and time trends in the discrimination ability of a model can be captured by varying the truncation time point for illustration we re analyze a study on risk prediction for prostate cancer patients the effects of misspecification of the censoring model are studied in simulated data

commutativity and spectra of hermitian matrices abstract if two hermitian matrices commute then the eigenvalues of their sum are just the sums of the eigenvalues of the two matrices in a suitable order examples show that the converse is not true in general in this paper partial converses are obtained the technique involves a characterization of the equality cases for weyl s inequalities moreover a new proof on the commutativity of two hermitian matrices with property l and analogous results for the product of two positive definite hermitian matrices are included

addressing issues associated with evaluating prediction models for survival endpoints based on the concordance statistic prediction models for disease risk and prognosis play an important role in biomedical research and evaluating their predictive accuracy in the presence of censored data is of substantial interest the standard concordance c statistic has been extended to provide a summary measure of predictive accuracy for survival models motivated by a prostate cancer study we address several issues associated with evaluating survival prediction models based on c statistic with a focus on estimators using the technique of inverse probability of censoring weighting ipcw compared to the existing work we provide complete results on the asymptotic properties of the ipcw estimators under the assumption of coarsening at random car and propose a sensitivity analysis under the mechanism of noncoarsening at random ncar in addition we extend the ipcw approach as well as the sensitivity analysis to high dimensional settings the predictive accuracy of prediction models for cancer recurrence after prostatectomy is assessed by applying the proposed approaches we find that the estimated predictive accuracy for the models in consideration is sensitive to ncar assumption and thus identify the best predictive model finally we further evaluate the performance of the proposed methods in both settings of low dimensional and high dimensional data under car and ncar through simulations

estimation of covariate specific time dependent roc curves in the presence of missing biomarkers summary covariate specific time dependent roc curves are often used to evaluate the diagnostic accuracy of a biomarker with time to event outcomes when certain covariates have an impact on the test accuracy in many medical studies measurements of biomarkers are subject to missingness due to high cost or limitation of technology this article considers estimation of covariate specific time dependent roc curves in the presence of missing biomarkers to incorporate the covariate effect we assume a proportional hazards model for the failure time given the biomarker and the covariates and a semiparametric location model for the biomarker given the covariates in the presence of missing biomarkers we propose a simple weighted estimator for the roc curves where the weights are inversely proportional to the selection probability we also propose an augmented weighted estimator which utilizes information from the subjects with missing biomarkers the augmented weighted estimator enjoys the double robustness property in the sense that the estimator remains consistent if either the missing data process or the conditional distribution of the missing data given the observed data is correctly specified we derive the large sample properties of the proposed estimators and evaluate their finite sample performance using numerical studies the proposed approaches are illustrated using the us alzheimer s disease neuroimaging initiative adni dataset

predictive accuracy and explained variation in cox regression summary we suggest a new measure of the proportion of the variation of possibly censored survival times explained by a given proportional hazards model the proposed measure termed v shares several favorable properties with an earlier v1 but also improves the handling of censoring the statistic contrasts distance measures between individual 1 0 survival processes and fitted survival curves with and without covariate information these distance measures dx and d respectively are themselves informative as summaries of absolute rather than relative predictive accuracy we recommend graphical comparisons of survival curves for prognostic index groups to improve the understanding of obtained values for v dx and d their use and interpretation is exemplified for a yorkshire lung cancer study on survival from this and an overview for several well known clinical data sets we show that the likely amount of relative or absolute predictive accuracy is often low even if there are highly significant and relatively strong prognostic factors

estimating time dependent roc curves using data under prevalent sampling prevalent sampling is frequently a convenient and economical sampling technique for the collection of time to event data and thus is commonly used in studies of the natural history of a disease however it is biased by design because it tends to recruit individuals with longer survival times this paper considers estimation of time dependent receiver operating characteristic curves when data are collected under prevalent sampling to correct the sampling bias we develop both nonparametric and semiparametric estimators using extended risk sets and the inverse probability weighting techniques the proposed estimators are consistent and converge to gaussian processes while substantial bias may arise if standard estimators for right censored data are used to illustrate our method we analyze data from an ovarian cancer study and estimate receiver operating characteristic curves that assess the accuracy of the composite markers in distinguishing subjects who died within 3 5 years from subjects who remained alive copyright 2016 john wiley sons ltd

predictive accuracy of covariates for event times we propose a graphical measure the generalized negative predictive function to quantify the predictive accuracy of covariates for survival time or recurrent event times this new measure characterizes the event free probabilities over time conditional on a thresholded linear combination of covariates and has direct clinical utility we show that this function is maximized at the set of covariates truly related to event times and thus can be used to compare the predictive accuracy of different sets of covariates we construct nonparametric estimators for this function under right censoring and prove that the proposed estimators upon proper normalization converge weakly to zero mean gaussian processes to bypass the estimation of complex density functions involved in the asymptotic variances we adopt the bootstrap approach and establish its validity simulation studies demonstrate that the proposed methods perform well in practical situations two clinical studies are presented

performance of binary markers for censored failure time outcome nonparametric approach based on proportions this work focuses on the assessment of the discrimination ability of a binary marker to identify patients that will relapse in time we consider the cumulative definition of sensitivity and dynamic definition of specificity at a time horizon that is the probability of a positive marker in the population that will relapse cases and that will not relapse controls in the presence of censoring sensitivity and specificity cannot be estimated by proportions because it is not known whether censored subjects should be considered as cases or controls the solutions proposed do not enable to obtain asymptotic confidence intervals we explore the use of inverse probability of censoring weighting imputation borrowed from the methodology used to correct for verification bias to adjust the classification matrix for the presence of censoring the adjustment based on weights estimated conditional on the marker turned to be equivalent to the adjustment based on imputation these approaches which address for the presence of marker dependent censoring showed a better performance than the adjustment based on weights estimated on the entire sample even in the case of marker independent censoring we derived single intervals and confidence region for sensitivity and 1 specificity using the delta method the confidence region is particularly useful for a binary marker because the marker has some ability to discriminate among cases and controls only if the region does not intersect the first quadrant bisector

estimating and comparing time dependent areas under receiver operating characteristic curves for censored event times with competing risks the area under the time dependent roc curve auc may be used to quantify the ability of a marker to predict the onset of a clinical outcome in the future for survival analysis with competing risks two alternative definitions of the specificity may be proposed depending of the way to deal with subjects who undergo the competing events in this work we propose nonparametric inverse probability of censoring weighting estimators of the auc corresponding to these two definitions and we study their asymptotic properties we derive confidence intervals and test statistics for the equality of the aucs obtained with two markers measured on the same subjects a simulation study is performed to investigate the finite sample behaviour of the test and the confidence intervals the method is applied to the french cohort paquid to compare the abilities of two psychometric tests to predict dementia onset in the elderly accounting for death without dementia competing risk the timeroc r package is provided to make the methodology easily usable

an application of harrell s c index to ph frailty models frailty models are encountered in many medical applications yet little research has been devoted to develop measures that quantify the predictive ability of these models in this paper we elaborate on the concept of the concordance probability to clustered data resulting in an overall conditional c index or bfc o c and an overall marginal c index or c o m both overall c indices can be split up into a between conditional or c b c and a between marginal c index or c b m and into a within conditional or c w c and a within marginal c index or c w m for ph frailty models of the power variance family c w c and c w m are equivalent resulting in one within c index c w we propose an application of harrell s c index to estimate the proposed indices within a likelihood and a bayesian context and the performances of their point estimates and confidence credible intervals are compared in an extensive simulation study this simulation study shows that the point estimates of c w and c b m perform good within both a likelihood and bayesian context but that the point estimates of c b c show less bias for the bayesian approach than for the likelihood approach the 95 per cent confidence credible intervals also possess good coverage properties given that the point estimates perform good the performance of the c indices is evaluated on a real data set

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

robust risk prediction with biomarkers under two phase stratified cohort design identification of novel biomarkers for risk prediction is important for disease prevention and optimal treatment selection however studies aiming to discover which biomarkers are useful for risk prediction often require the use of stored biological samples from large assembled cohorts and thus the depletion of a finite and precious resource to make efficient use of such stored samples two phase sampling designs are often adopted as resource efficient sampling strategies especially when the outcome of interest is rare existing methods for analyzing data from two phase studies focus primarily on single marker analysis or fitting the cox regression model to combine information from multiple markers however the cox model may not fit the data well under model misspecification the composite score derived from the cox model may not perform well in predicting the outcome under a general two phase stratified cohort sampling design we present a novel approach to combining multiple markers to optimize prediction by fitting a flexible nonparametric transformation model using inverse probability weighting to account for the outcome dependent sampling we propose to estimate the model parameters by maximizing an objective function which can be interpreted as a weighted c statistic for survival outcomes regardless of model adequacy the proposed procedure yields a sensible composite risk score for prediction a major obstacle for making inference under two phase studies is due to the correlation induced by the finite population sampling which prevents standard inference procedures such as the bootstrap from being used for variance estimation we propose a resampling procedure to derive valid confidence intervals for the model parameters and the c statistic accuracy measure we illustrate the new methods with simulation studies and an analysis of a two phase study of high density lipoprotein cholesterol hdl c subtypes for predicting the risk of coronary heart disease

bivariate marker measurements and roc analysis this paper considers receiver operating characteristic roc analysis for bivariate marker measurements the research interest is to extend tools and rules from univariate marker to bivariate marker setting for evaluating predictive accuracy of markers using a tree based classification rule using an and or classifier an roc function together with a weighted roc function wroc and their conjugate counterparts are proposed for examining the performance of bivariate markers the proposed functions evaluate the performance of and or classifiers among all possible combinations of marker values and are ideal measures for understanding the predictability of biomarkers in target population specific features of roc and wroc functions and other related statistics are discussed in comparison with those familiar properties for univariate marker nonparametric methods are developed for estimating roc related functions partial area under curve and concordance probability with emphasis on average performance of markers the proposed procedures and inferential results are useful for evaluating marker predictability based on a single or bivariate marker or test measurements with different choices of markers and for evaluating different and or combinations in classifiers the inferential results developed in this paper also extend to multivariate markers with a sequence of arbitrarily combined and or classifier

on the invariance of the sign pattern of matrix eigenvectors under perturbation abstract we study how much perturbation a in a real matrix a is allowed for the i th real eigenvector not to change sign

evaluating prediction rules for t year survivors with censored regression models suppose that we are interested in establishing simple but reliable rules for predicting future t year survivors through censored regression models in this article we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate sensitivity and specificity and positive and negative predictive values specifically under various working models we derive consistent estimators for the above measures through substitution and cross validation estimation procedures furthermore we provide large sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified confidence intervals for example for the difference of the precision measures between two competing rules can then be constructed all of the proposals are illustrated with real examples and their finite sample properties are evaluated through a simulation study

nearest neighbor estimation of a bivariate distribution under random censoring we consider the problem of estimating the bivariate distribution of the random vector x y when y may be subject to random censoring the censoring variable c is allowed to depend on x but it is assumed that y and c are conditionally independent given x x the estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of y given x x over a range of values of x the weak convergence of the centered estimator multiplied by n 1 2 is obtained and a closed form expression for the covariance function of the limiting process is given it is shown that the proposed estimator is optimal in the beran sense this is similar to an optimality property the product limit estimator enjoys using the proposed estimator of the bivariate distribution an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established

a robust alternative to the schemper henderson estimator of prediction error in clinical applications the prediction error of survival models has to be taken into consideration to assess the practical suitability of conclusions drawn from these models different approaches to evaluate the predictive performance of survival models have been suggested in the literature in this article we analyze the properties of the estimator of prediction error developed by schemper and henderson 2000 biometrics 56 249 255 which quantifies the absolute distance between predicted and observed survival functions we provide a formal proof that the estimator proposed by schemper and henderson is not robust against misspecification of the survival model that is the estimator will only be meaningful if the model family used for deriving predictions has been specified correctly to remedy this problem we construct a new estimator of the absolute distance between predicted and observed survival functions we show that this modified schemper henderson estimator is robust against model misspecification allowing its practical application to a wide class of survival models the properties of the schemper henderson estimator and its new modification are illustrated by means of a simulation study and the analysis of two clinical data sets

recursively imputed survival trees we propose recursively imputed survival tree rist regression for right censored data this new nonparametric regression procedure uses a novel recursive imputation approach combined with extremely randomized trees that allows significantly better use of censored data than previous tree based methods yielding improved model fit and reduced prediction error the proposed method can also be viewed as a type of monte carlo em algorithm which generates extra diversity in the tree based fitting process simulation studies and data analyses demonstrate the superior performance of rist compared with previous methods

a measure of explained variation for event history data there is no shortage of proposed measures of prognostic value of survival models in the statistical literature they come under different names including explained variation correlation explained randomness and information gain but their goal is common to define something analogous to the coefficient of determination r 2 in linear regression none however have been uniformly accepted none have been extended to general event history data including recurrent events and many cannot incorporate time varying effects or covariates we present here a measure specifically tailored for use with general dynamic event history regression models the measure is applicable and interpretable in discrete or continuous time with tied data or otherwise with time varying time fixed or dynamic covariates with time varying or time constant effects with single or multiple event times with parametric or semiparametric models and under general independent censoring observation for single event survival data with neither censoring nor time dependency it reduces to the concordance index we give expressions for its population value and the variance of the estimator and explore its use in simulations and applications a web link to r software is provided

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

time dependent predictive accuracy in the presence of competing risks competing risks arise naturally in time to event studies in this article we propose time dependent accuracy measures for a marker when we have censored survival times and competing risks time dependent versions of sensitivity or true positive tp fraction naturally correspond to consideration of either cumulative or prevalent cases that accrue over a fixed time period or alternatively to incident cases that are observed among event free subjects at any select time time dependent dynamic specificity 1 false positive fp can be based on the marker distribution among event free subjects we extend these definitions to incorporate cause of failure for competing risks outcomes the proposed estimation for cause specific cumulative tp dynamic fp is based on the nearest neighbor estimation of bivariate distribution function of the marker and the event time on the other hand incident tp dynamic fp can be estimated using a possibly nonproportional hazards cox model for the cause specific hazards and riskset reweighting of the marker distribution the proposed methods extend the time dependent predictive accuracy measures of heagerty lumley and pepe 2000 biometrics 56 337 344 and heagerty and zheng 2005 biometrics 61 92 105

valuing vulnerable geometric asian options an option on a defaultable instrument is prone to default risk arising from the probability that the instrument s issuer will not honor its contractual obligations under this financial circumstance we consider vulnerable asian options in this paper we obtain parabolic partial differential equations with time dependent coefficients on the vulnerable asian option model and derive a closed formula of the pricing of the vulnerable geometric asian option using the pricing formula of the vulnerable european option with time dependent coefficients

a covariate specific time dependent receiver operating characteristic curve for correlated survival data several studies for the clinical validity of circulating tumor cells ctcs in metastatic breast cancer were conducted showing that it is a prognostic biomarker of overall survival in this work we consider an individual patient data meta analysis for nonmetastatic breast cancer to assess the discrimination of ctcs regarding the risk of death data are collected in several centers and present correlated failure times for subjects of the same center however although the covariate specific time dependent receiver operating characteristic roc curve has been widely used for assessing the performance of a biomarker there is no methodology yet that can handle this specific setting with clustered censored failure times we propose an estimator for the covariate specific time dependent roc curves and area under the roc curve when clustered failure times are detected we discuss the assumptions under which the estimators are consistent and their interpretations we assume a shared frailty model for modeling the effect of the covariates and the biomarker on the outcome in order to account for the cluster effect a simulation study was conducted and it shows negligible bias for the proposed estimator and a nonparametric one based on inverse probability censoring weighting while a semiparametric estimator ignoring the clustering is markedly biased finally in our application to breast cancer data the estimation of the covariate specific area under the curves illustrates that the ctcs discriminate better patients with inflammatory tumor than patients with noninflammatory tumor with respect to their risk of death

three measures of explained variation for correlated survival data under the proportional hazards mixed effects model measures of explained variation are useful in scientific research as they quantify the amount of variation in an outcome variable of interest that is explained by one or more other variables we develop such measures for correlated survival data under the proportional hazards mixed effects model because different approaches have been studied in the literature outside the classical linear regression model we investigate three measures r 2 rres2 and 2 that quantify three different population coefficients we show that although the three population measures are not the same they reflect similar amounts of variation explained by the predictors among the three measures we show that r 2 which is the simplest to compute is also consistent for the first population measure under the usual asymptotic scenario when the number of clusters tends to infinity the other two measures on the other hand all require that in addition the cluster sizes be large we study the properties of the measures both analytically and through simulation studies we illustrate their different usage on a multi center clinical trial and a recurrent events data set copyright 2016 john wiley sons ltd

an algorithm for addressing the real interval eigenvalue problem in this paper we present an algorithm for approximating the range of the real eigenvalues of interval matrices such matrices could be used to model real life problems where data sets suffer from bounded variations such as uncertainties e g tolerances on parameters measurement errors or to study problems for given states the algorithm that we propose is a subdivision algorithm that exploits sophisticated techniques from interval analysis the quality of the computed approximation and the running time of the algorithm depend on a given input accuracy we also present an efficient c implementation and illustrate its efficiency on various data sets in most of the cases we manage to compute efficiently the exact boundary points limited by floating point representation

on modified mellin transforms gauss laguerre quadrature and the valuation of american call options we extend a framework based on mellin transforms and show how to modify the approach to value american call options on dividend paying stocks we present a new integral equation to determine the price of an american call option and its free boundary using modified mellin transforms we also show how to derive the pricing formula for perpetual american call options using the new framework a result due to kim 1990 24 regarding the optimal exercise price at expiry is also recovered finally we apply gauss laguerre quadrature for the purpose of an efficient and accurate numerical valuation

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

time dependent tree structured survival analysis with unbiased variable selection through permutation tests incorporating time dependent covariates into tree structured survival analysis tssa may result in more accurate prognostic models than if only baseline values are used available time dependent tssa methods exhaustively test every binary split on every covariate however this approach may result in selection bias towards covariates with more observed values we present a method that uses unbiased significance levels from newly proposed permutation tests to select the time dependent or baseline covariate with the strongest relationship with the survival outcome the specific splitting value is identified using only the selected covariate simulation results show that the proposed time dependent tssa method produces tree models of equal or greater accuracy as compared to baseline tssa models even with high censoring rates and large within subject variability in the time dependent covariate to illustrate the proposed method is applied to data from a cohort of bipolar youth to identify subgroups at risk for self injurious behavior

real eigenvalue bounds of standard and generalized real interval eigenvalue problems abstract in this paper we study the real eigenvalue bounds of standard and generalized real interval eigenvalue problems based on some known sufficient conditions for the regularity of interval matrices a new algorithm for computing real eigenvalue bounds for real interval matrices is proposed it can easily be extended for applying to generalized real interval eigenvalue problems one advantage of the method is that the computation procedure takes less time than former methods we have proposed therefore it can be applied for solving larger interval eigenvalue problems finally some numerical examples are presented which show the effectiveness of the method

the pricing of vulnerable options with double mellin transforms abstract many options traded in the over the counter markets are subject to default risks resulting from the probability that the option writer could not honor its contractual obligations there have been growing concerns about financial derivatives subject to default risks in particular since the global financial crisis and eurozone crisis this paper uses double mellin transforms to study european vulnerable options under constant as well as stochastic the hull white interest rates we obtain explicitly an analytic closed form pricing formula in each interest rate case so that the pricing of the options can be computed both accurately and efficiently

time dependent summary receiver operating characteristics for meta analysis of prognostic studies prognostic studies are widely conducted to examine whether biomarkers are associated with patient s prognoses and play important roles in medical decisions because findings from one prognostic study may be very limited meta analyses may be useful to obtain sound evidence however prognostic studies are often analyzed by relying on a study specific cut off value which can lead to difficulty in applying the standard meta analysis techniques in this paper we propose two methods to estimate a time dependent version of the summary receiver operating characteristics curve for meta analyses of prognostic studies with a right censored time to event outcome we introduce a bivariate normal model for the pair of time dependent sensitivity and specificity and propose a method to form inferences based on summary statistics reported in published papers this method provides a valid inference asymptotically in addition we consider a bivariate binomial model to draw inferences from this bivariate binomial model we introduce a multiple imputation method the multiple imputation is found to be approximately proper multiple imputation and thus the standard rubin s variance formula is justified from a bayesian view point our simulation study and application to a real dataset revealed that both methods work well with a moderate or large number of studies and the bivariate binomial model coupled with the multiple imputation outperforms the bivariate normal model with a small number of studies copyright 2016 john wiley sons ltd

a risk based measure of time varying prognostic discrimination for survival models prognostic survival models are commonly evaluated in terms of both their calibration and their discrimination comparing observed and predicted survival curves can assess calibration while discrimination is typically summarized through comparison of the properties of cases or subjects who experience an event and the properties of controls represented by event free individuals for binary data discrimination is characterized either by using the relative ranks of cases and controls and a receiver operating characteristic roc curve or by summarizing the magnitude of risk placed on cases and controls through calculation of the discrimination slope ds in this article we propose a risk based measure of time varying discrimination that generalizes the discrimination slope to allow use with incident events and hazard models we refer to the new measure as the hazard discrimination summary hds since it compares the relative risk among incident cases to their associated dynamic risk set controls we introduce both a model based estimation procedure that adopts the cox model and an alternative approach that locally relaxes the proportional hazards assumption we illustrate the proposed methods using both a benchmark survival data set and an oncology study where primary interest is in the time varying performance of candidate biomarkers

lower bounds for the minimum eigenvalue of hadamard product of an m matrix and its inverse for the hadamard product a a 1 of an m matrix a and its inverse a 1 we give new lower bounds for the minimum eigenvalue of a a 1 these bounds are strong enough to prove the conjecture of fiedler and markham an inequality for the hadamard product of an m matrix and inverse m matrix linear algebra appl 101 1988 1 8

on the sign patterns of the smallest signless laplacian eigenvector abstract let h be a connected bipartite graph whose signless laplacian matrix is q h suppose that the bipartition of h is s t and that x is the eigenvector of the smallest eigenvalue of q h it is well known that x is positive and constant on s and negative and constant on t the resilience of the sign pattern of x under addition of edges into the subgraph induced by either s or t is investigated and a number of cases in which the sign pattern of x persists are described

several methods to assess improvement in risk prediction models extension to survival analysis risk prediction models have been widely applied for the prediction of long term incidence of disease several parameters have been identified and estimators developed to quantify the predictive ability of models and to compare new models with traditional models these estimators have not generally accounted for censoring in the survival data normally available for fitting the models this paper remedies that problem the primary parameters considered are net reclassification improvement nri and integrated discrimination improvement idi we have previously similarly considered a primary measure of concordance area under the roc curve auc also called the c statistic we also include here consideration of population attributable risk par and ratio of predicted risk in the top quintile of risk to that in the bottom quintile we evaluated estimators of these various parameters both with simulation studies and also as applied to a prospective study of coronary heart disease chd our simulation studies showed that in general our estimators had little bias and less bias and smaller variances than the traditional estimators we have applied our methods to assessing improvement in risk prediction for each traditional chd risk factor compared to a model without that factor these traditional risk factors are considered valuable yet when adding any of them to a risk prediction model that has omitted the one factor the improvement is generally small for any of the parameters this experience should prepare us to not expect large values of the risk prediction improvement evaluation parameters for any new risk factor to be discovered