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

decentralized delayed feedback control of a one way coupled ring map lattice abstract this paper presents a decentralized delayed feedback control for a one way coupled ring map lattice the delayed feedback controller is used at each local site this paper gives a simple sufficient condition for the control system to be stable this condition does not depend on the system size hence it allows us to obtain a systematic simple procedure for designing the controller in order to confirm the theoretical results we show some numerical examples

predictive accuracy and explained variation measures of the predictive accuracy of regression models quantify the extent to which covariates determine an individual outcome explained variation measures the relative gains in predictive accuracy when prediction based on covariates replaces unconditional prediction a unified concept of predictive accuracy and explained variation based on the absolute prediction error is presented for models with continuous binary polytomous and survival outcomes the measures are given both in a model based formulation and in a formulation directly contrasting observed and expected outcomes various aspects of application are demonstrated by examples from three forms of regression models it is emphasized that the likely degree of absolute or relative predictive accuracy often is low even if there are highly significant and relatively strong covariates

cross validation in survival analysis the predictive value of a statistical model is conceptually different from the explained variation in this paper we construct a measure of the predictive value of the cox proportional hazards model computed from the leave one out regression coefficients these coefficients can also be used to calculate a shrinkage factor which can be applied to improve the predictions and that can be used in r2 type measures of the proportion of explained variation our methods are illustrated by a study of chemotherapy for advanced ovarian cancer

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

explained variation for logistic regression different measures of the proportion of variation in a dependent variable explained by covariates are reported by different standard programs for logistic regression we review twelve measures that have been suggested or might be useful to measure explained variation in logistic regression models the definitions and properties of these measures are discussed and their performance is compared in an empirical study two of the measures squared pearson correlation between the binary outcome and the predictor and the proportional reduction of squared pearson residuals by the use of covariates give almost identical results agree very well with the multiple r2 of the general linear model have an intuitively clear interpretation and perform satisfactorily in our study for all measures the explained variation for the given sample and also the one expected in future samples can be obtained easily for small samples an adjustment analogous to radj2 in the general linear model is suggested we discuss some aspects of application and recommend the routine use of a suitable measure of explained variation for logistic models

controlling extended systems with spatially filtered time delayed feedback we investigate a control technique for spatially extended systems combining spatial filtering with a previously studied form of time delay feedback the scheme is naturally suited to real time control of optical systems we apply the control scheme to a model of a transversely extended semiconductor laser in which a desirable coherent traveling wave state exists but is a member of a nowhere stable family our scheme stabilizes this state and directs the system towards it from realistic distant and noisy initial conditions as confirmed by numerical simulation a linear stability analysis about the controlled state accurately predicts when the scheme is successful and illustrates some key features of the control including the individual merit of and interplay between the spatial and temporal degrees of freedom in the control

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

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

explained randomness in proportional hazards models a coefficient of explained randomness analogous to explained variation but for non linear models was presented by kent the construct hinges upon the notion of kullback leibler information gain kent and o quigley developed these ideas obtaining simple multiple and partial coefficients for the situation of proportional hazards regression their approach was based upon the idea of transforming a general proportional hazards model to a specific one of weibull form xu and o quigley developed a more direct approach more in harmony with the semi parametric nature of the proportional hazards model thereby simplifying inference and allowing for instance the use of time dependent covariates a potential drawback to the coefficient of xu and o quigley is its interpretation as explained randomness in the covariate given time an investigator might feel that the interpretation of the kent and o quigley coefficient as a proportion of explained randomness of time given the covariate is preferable one purpose of this note is to indicate that under an independent censoring assumption the two population coefficients coincide thus the simpler inferential setting for xu and o quigley can also be applied to the coefficient of kent and o quigley our second purpose is to point out that a sample based coefficient in common use in the sas statistical package can be interpreted as an estimate of explained randomness when there is no censoring when there is censoring the sas coefficient would not seem satisfactory in that its population counterpart depends on an independent censoring mechanism however there is a quick fix and we argue in favour of its use

feedback control of the kuramoto sivashinsky equation abstract this work focuses on linear finite dimensional output feedback control of the kuramoto sivashinsky equation kse with periodic boundary conditions under the assumption that the linearization of the kse around the zero solution is controllable and observable linear finite dimensional output feedback controllers are synthesized that achieve stabilization of the zero solution for any value of the instability parameter the controllers are synthesized on the basis of finite dimensional approximations of the kse which are obtained through galerkin s method the performance of the controllers is successfully tested through computer simulations

semiparametric estimation of an accelerated failure time model with time dependent covariates summary a class of semiparametric accelerated failure time models is introduced that are useful for modelling the relationship of survival distributions to time dependent covariates a class of semiparametric rank estimators is derived for the parameters in this model when the survival data are right censored these estimates are shown to be consistent and asymptotically normal with variances that can be consistently estimated it is also shown that estimators within this class achieve the semiparametric efficiency bound we consider an accelerated failure time model for time to event data in which the survival distribution is scaled by a factor which may be a function of time dependent covariates these models directly model the effect of covariates on the length of survival this is in contrast to the more commonly used proportional hazards model which models the hazard rate itself in some applications it may be easier to visualize the concept that a treatment intervention or exposure to an environmental contaminant increases or decreases the length of survival by a certain proportion as compared to the concept that the hazard rate is changed for time independent covariates say z z1 zp the class of accelerated failure time models assumes that the survival distribution at time t given a set of covariates z z is given by f t i z f0 h z t the proportionality constant h z can be interpreted as the scale factor by which lifetime is decreased as a function of the covariates z often h z is taken to be log linear or h z exp b z in such cases if z 0 we can interpret fo t as the baseline survival distribution or the survival distribution for individuals with covariates all equal to zero a useful way of envisioning this model is to consider a hypothetical random variable u that corresponds to an individual s survival time if that individual had covariate values all equal to zero this baseline failure time u would be modified for values of the covariate different than zero for example if we denote the actual survival time by t then for an individual with covariate value z we have t e zu the parameters b in this model have a direct interpretation in terms of the increase or decrease in lifespan as a function of the covariates the assumption that u is independent of z in this hypothetical construct induces the probabilistic model

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

a two sample test sensitive to crossing hazards in uncensored and singly censored data savage score statistics are employed to develop a test for comparing survival distributions with right hand singly censored data the procedure is motivated by the interest in developing a powerful method for determining differences when true survival distributions cross examination of small sample characteristics under the null hypothesis indicate that asymptotic critical values yield a slightly conservative test power of the test compares favorably with other criteria including the modified smirnov procedure particularly if there is a single crossing of the survival curves

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

explained variation in survival analysis several measures of explained variation have been suggested for the cox proportional hazards regression model we have categorized these measures into three classes which correspond to three different definitions of multiple r2 of the general linear model in an empirical study we compared the performance of these measures and classified them by their adherence to a set of criteria which we think should be met by a measure of explained variation for survival data we suggest that currently there is no uniformly superior measure particularly as the concepts of either uncensored or censored populations may lead to different choices for uncensored populations a measure by kent and o quigley and the squared rank correlation between survival time and the predictor from a cox regression model appear recommendable choices for the latter censored survival times are terminated using a very recent data augmentation algorithm for multiple imputation under proportional hazards with censored populations schemper s measure v2 could be considered we give an introductory example discuss aspects of application and stress the desirability of routinely evaluating explained variation in studies of survival

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

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

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

ordered binary decision diagrams and minimal trellises ordered binary decision diagrams obdds are graph based data structures for representing boolean functions they have found widespread use in computer aided design and in formal verification of digital circuits minimal trellises are graphical representations of error correcting codes that play a prominent role in coding theory this paper establishes a close connection between these two graphical models as follows let spl cscr be a binary code of length n and let f sub c x sub 1 x sub n be the boolean function that takes the value 0 at x sub 1 x sub n if and only if x sub 1 x sub n spl isin spl cscr given this natural one to one correspondence between boolean functions and binary codes we prove that the minimal proper trellis for a code spl cscr with minimum distance d 1 is isomorphic to the single terminal obdd for its boolean indicator function f sub c x sub 1 x sub n prior to this result the extensive research during the past decade on binary decision diagrams in computer engineering and on minimal trellises in coding theory has been carried out independently as outlined in this work the realization that binary decision diagrams and minimal trellises are essentially the same data structure opens up a range of promising possibilities for transfer of ideas between these disciplines

a note on a general definition of the coefficient of determination summary a generalization of the coefficient of determination r2 to general regression models is discussed a modification of an earlier definition to allow for discrete models is proposed

stabilizing the absolutely or convectively unstable homogeneous solutions of reaction convection diffusion systems we study the problem of stabilization of a homogeneous solution in a two variable reaction convection diffusion one dimensional system with oscillatory kinetics in which moving or stationary patterns emerge in the absence of control we propose to use a formal spatially weighted feedback control to suppress patterns in an absolutely or convectively unstable system and pinning control for a convectively unstable system the latter approach is very effective and may require only one actuator to adjust feed conditions in the former approach the positive diagonal elements of the appropriate dynamics matrix are shifted to the left hand part of the complex plane to ensure linear asymptotic stability of the system according to gershgorin criterion moreover we construct a controller that with many actuators will approach the global stability of the solution according to liapunov s direct method we apply two alternative approaches to reveal the unstable modes an approximate one that is based on linear stability analysis of an unbounded system and an exact one that uses a traditional eigenstructure analysis of bounded systems the number of required actuators increases dramatically with system size and with the distance from the bifurcation point the methodology is developed for a system with learning cubic kinetics and is tested on a more realistic cross flow reactor model

pattern formation capacity of spatially extended systems we analyze a class of spatially extended systems which are capable of generating many complicated patterns these systems are given by the ginzburg landau equation coupled with a system of two linear equations and describe nonlinear media with localized defects we find a connection between these systems and spin glass systems we show that the system is capable to produce many patterns and describe patterning algorithms

quantification of the prentice criteria for surrogate endpoints in a recent discussion on the prentice criteria for surrogate endpoints 1989 and the role of meta analyses in this context alonso et al expand on a number of points among these is a suggestion to base a measure of the degree at which an endpoint meets the prentice criteria on a likelihood reduction factor although new in this context the suggestion has been studied elsewhere and the measure of alonso et al as presented requires some modification if it is to be useful in general situations as it stands it depends upon the censoring mechanism even when independent of the failure mechanism this is however easily fixed the measure suggested by alonso et al can be seen to fit in with a well established theory so that once corrected for the presence of censoring it can be employed with confidence of particular interest is the wide availability of a standard sas program producing a coefficient which can be immediately transformed to provide an estimate of the population quantity behind the alonso et al proposal

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

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

solving boolean equations using rosop forms boolean equations are important tools in digital logic previous algorithms for solving boolean equations are based on the boolean algebra of disjoint sop forms in this paper we develop a new boolean algebra with more efficient boolean operation algorithms called the reduced ordered sop rosop forms which are canonical representations rosops are closely related to the well known obdd data structure the results here also show the algebraic structure of obdds

proportional hazards models with frailties and random effects we discuss some of the fundamental concepts underlying the development of frailty and random effects models in survival one of these fundamental concepts was the idea of a frailty model where each subject has his or her own disposition to failure their so called frailty additional to any effects we wish to quantify via regression although the concept of individual frailty can be of value when thinking about how data arise or when interpreting parameter estimates in the context of a fitted model we argue that the concept is of limited practical value individual random effects frailties whenever detected can be made to disappear by elementary model transformation in consequence unless we are to take some model form as unassailable beyond challenge and carved in stone and if we are to understand the term frailty as referring to individual random effects then frailty models have no value random effects models on the other hand in which groups of individuals share some common effect can be used to advantage even in this case however if we are prepared to sacrifice some efficiency we can avoid complex modelling by using the considerable power already provided by the stratified proportional hazards model stratified models and random effects models can both be seen to be particular cases of partially proportional hazards models a view that gives further insight the added structure of a random effects model viewed as a stratified proportional hazards model with some added distributional constraints will for group sizes of five or more provide no more than modest efficiency gains even when the additional assumptions are exactly true on the other hand for moderate to large numbers of very small groups of sizes two or three the study of twins being a well known example the efficiency gains of the random effects model can be far from negligible for such applications the case for using random effects models rather than the stratified model is strong this is especially so in view of the good robustness properties of random effects models nonetheless the simpler analysis based upon the stratified model remains valid albeit making a less efficient use of resources

least upper bounds on obdd sizes this paper derives exact equations for the maximum number of nonterminal vertexes in reduced and quasi reduced ordered binary decision diagrams obdd s a reduced obdd is reduced by both merging and deleting vertices and a quasi reduced obdd is reduced only by merging these formulas are used to tighten lee s original bounds and to correct the bounds recently reported by h t liaw and c s lin 1992

concordance probability and discriminatory power in proportional hazards regression summary the concordance probability is used to evaluate the discriminatory power and the predictive accuracy of nonlinear statistical models we derive an analytical expression for the concordance probability in the cox proportional hazards model the proposed estimator is a function of the regression parameters and the covariate distribution only and does not use the observed event and censoring times for this reason it is asymptotically unbiased unlike harrell s c index based on informative pairs the asymptotic distribution of the concordance probability estimate is derived using u statistic theory and the methodology is applied to a predictive model in lung cancer

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

the size of reduced obdd s and optimal read once branching programs for almost all boolean functions boolean functions are often represented by ordered binary decision diagrams obdds introduced by bryant 1986 liaw and lin 1992 have proved upper and lower bounds on the minimal obdd size of almost all boolean functions now tight bounds are proved for the minimal obdd size for arbitrary or optimal variable orderings and for the minimal read once branching program size of almost all functions almost all boolean functions have a sensitivity of almost 1 i e the minimal obdd size for an optimal variable ordering differs from the minimal obdd size for a worst variable ordering by a factor of at most 1 spl epsi n where spl epsi n converges exponentially fast to 0

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

on comparing 2 correlated c indices with censored survival data as new biomarkers and risk prediction procedures are in rapid development it is of great interest to develop valid methods for comparing predictive power of 2 biomarkers or risk score systems harrell c statistic has been routinely used as a global adequacy assessment of a risk score system and the difference of 2 harrell c statistics as a test statistic has been suggested in recent literature for comparison of predictive power of 2 biomarkers for censored outcome in this study we showed that such a test can have severely inflated type i errors as the difference between the 2 harrell c statistics does not converge to zero under the null hypothesis of equal predictive power measured by concordance probabilities as illustrated by 2 counterexamples and corresponding numerical simulations we further investigate a necessary and sufficient condition under which the difference of 2 harrell c statistics converges to zero under the null hypothesis

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

predictive capability of proportional hazards regression a measure of the predictive capability of a proportional hazards regression is derived the measure is based on the residuals appropriate to proportional hazards regression a population version is presented and can be seen not to depend on the censoring mechanism under the provision that any such censoring be independent or conditionally independent of the failure mechanism given the covariate for the special case of a weibull regression model for which the covariate distribution follows binary uniform normal or exponential laws we derive analytic results these alone give credence to the measure which can be seen to reflect strength of regression effect as quantified by the parameter estimate although on a scale between 0 and 1 independently of the intercept or shape parameter of the particular weibull law and only weakly dependent on the covariate distribution extensions to partial and multiple measures of predictive ability are straightforward an example 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

size of ordered binary decision diagrams representing threshold functions an ordered binary decision diagram obdd is a graph representation of a boolean function in this paper the size of ordered binary decision diagrams representing threshold functions is discussed we consider two cases the case when a variable ordering is given and the case when it is adaptively chosen we show 1 o 2n2 upper bound for both cases 2 2n2 lower bound for the former case and 3 n2 n2 lower bound for the latter case we also show some relations between the variable ordering and the size of obdds representing threshold functions

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

data analytic methods for matched case control studies the recent introduction of complex multivariate statistical models in matched case control studies is a mixed blessing their use can lead to a better understanding of the way in which many variables contribute to the risk of disease on the other hand these powerful methods can obscure salient features in the data that might have been detected by other less sophisticated methods this shortcoming is due to a lack of support methodology for the routine use of these models satisfactory computation of estimated relative risks and their standard errors is not sufficient justification for the fitted model goodness of fit must be examined if inferences are to be trusted this paper is concerned with the analysis of matched case control studies with logistic models analogies of these models to linear regression models are emphasized in particular basic concepts such as analysis of variance multiple correlation coefficient one degree of freedom tests and residual analysis are discussed the fairly new field of regression diagnostics is also introduced all procedures are illustrated on a study of bladder cancer in males

on the use of binary decision diagrams for solving problems on simple games abstract simple games are yes no cooperative games which arise in many practical applications recently we have used reduced ordered binary decision diagrams and quasi reduced ordered binary decision diagrams abbreviated as robdd s and qobdd s respectively for the representation of simple games and for the computation of some power indices in the present paper we continue this work we show how further important computational problems on simple games can be solved using qobdd s viz the identification of some key players the computation of the desirability relation on individuals the test whether a simple game is proper and strong respectively and the computation of qobdd representations for the sets of all minimal winning coalitions all shift minimal winning coalitions and all blocking coalitions respectively applications of these solutions include the computation of recent power indices based on shift minimal winning coalitions and the test for linear separability of a directed simple game

control of pattern formation by time delay feedback with global and local contributions we consider the suppression of spatiotemporal chaos in the complex ginzburglandau equation by a combined global and local time delay feedback feedback terms are implemented as a control scheme i e they are proportional to the difference between the time delayed state of the system and its current state we perform a linear stability analysis of uniform oscillations with respect to space dependent perturbations and compare with numerical simulations similarly for the fixed point solution that corresponds to amplitude death in the spatially extended system a linear stability analysis with respect to space dependent perturbations is performed and complemented by numerical simulations

estimating regression parameters using linear rank tests for censored data a class of estimates for regression parameters in a linear model with right censored data is proposed these estimates are derived by using linear rank tests for right censored data as estimating equations they are shown to be consistent and asymptotically normal with covariance matrix for which estimates are proposed efficient estimates within this class are derived together with conditions when they are fully efficient 0 introduction the problem of estimating regression coefficients using ranks has received much attention in the literature for uncensored data and the results are summarized very nicely in hettmansperger 1984 there has also been some work with censored data for the two sample problem specifically louis 1981 considered estimation of a scale change between two distributions with the use of a log rank test whereas wei and gail 1983 generalize to other linear rank tests as well we shall consider the general problem of multiple linear regression and show how linear rank tests can be used as estimating equations the resulting linear rank estimates will be shown to be asymptotically normal with variances that can be estimated consistently we shall also look at the efficiency of these estimates under various situations

a note on bias of measures of explained variation for survival data papers evaluating measures of explained variation or similar indices almost invariably use independence from censoring as the most important criterion and they always end up suggesting that some measures meet this criterion and some do not most of the time leading to a conclusion that the first is better than the second as a consequence users are offered measures that cannot be used with time dependent covariates and effects not to mention extensions to repeated events or multi state models we explain in this paper that the aforementioned criterion is of no use in studying such measures because it simply favors those that make an implicit assumption of a model being valid everywhere measures not making such an assumption are disqualified even though they are better in every other respect we show that if these allegedly inferior measures are allowed to make the same assumption they are easily corrected to satisfy the independent from censoring criterion even better it is enough to make such an assumption only for the times greater than the last observed failure time which in contrast with the preferred measures makes it possible to use all the modeling flexibility up to and assume whatever one wants after as a consequence we claim that some of the measures being preferred as better in the existing reviews are in fact inferior

overall c as a measure of discrimination in survival analysis model specific population value and confidence interval estimation the assessment of the discrimination ability of a survival analysis model is a problem of considerable theoretical interest and important practical applications this issue is however more complex than evaluating the performance of a linear or logistic regression several different measures have been proposed in the biostatistical literature in this paper we investigate the properties of the overall c index introduced by harrell as a natural extension of the roc curve area to survival analysis we develop the overall c index as a parameter describing the performance of a given model applied to the population under consideration and discuss the statistic used as its sample estimate we discover a relationship between the overall c and the modified kendall s tau and construct a confidence interval for our measure based on the asymptotic normality of its estimate then we investigate via simulations the length and coverage probability of this interval finally we present a real life example evaluating the performance of a framingham heart study model

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

comparison of time delayed feedback schemes for spatiotemporal control of chaos in a reaction diffusion system with global coupling time delayed feedback control for stabilizing time periodic spatial patterns is investigated in a generic reaction diffusion system with global coupling we focus on the case of low dimensional chaos where unstable patterns admit only a single unstable mode spatial degrees of freedom are taken into account to define different control schemes the efficiency of these schemes is discussed where control forces are motivated by physical requirements as well as by the possibility of obtaining analytically exact results we find that control schemes that contain the full feedback of the inhibitor variable may finally destroy the control performance thus schemes that omit the inhibitor might be more efficient our numerical findings are explained in terms of floquet spectra and compared with analytical solutions of particular coupling schemes

empirical bifurcation analysis of optical pattern formation the experimental characterization of pattern forming bifurcations is difficult since the unstable solutions are generally not accessible in experiments in nonlinear optics a novel control scheme allows one to select and stabilize generic patterns and thus to track these solutions in parameter space this fourier space scheme is applied to a single feedback system and the amplitudes of roll square and hexagon patterns are determined experimentally even though the bifurcation is imperfect the coefficients of a prototype amplitude equation are recovered the coefficients show satisfactory agreement with theory and with numerical simulations which are performed for comparison the simulations also clarify the origin of the imperfect bifurcation in the experiment boundaries and speckles appear to have an unexpected strong influence

investigating the effects of ties on measures of concordance the concordance between predicted and observed outcomes referred to as the c index or c statistic is frequently used to quantify the discriminatory ability of a prognostic model it also commonly serves as a basis for distinguishing predictive strength between different models two alternative formulations of the c index are widely used one completely excluding ties from the computation c tied out and the other including ties c tied in however there has been little research concerning the effects of ties on these two measures in this paper we characterize changes in the performance of c tied in and c tied out for progressively less coarse or more coarse partitions of the data our theoretical and simulation results show that both measures can be heavily dependent on the number of tied pairs and their results can be substantially divergent we examine potential ambiguities that can occur when the two measures diverge in the presence of a substantial proportion of tied pairs we recommend that both c indices be computed as lower and upper bounds and that when feasible the c indices should be computed with respect to partitions corresponding to the same percentiles for the models being compared when it is desired to estimate the overall concordance the average value of the two c indices might also be considered we use the data from the hemodialysis clinical trial to illustrate our evaluation and methods

on the obdd representation of general boolean functions the behavior of ordered binary decision diagrams obdd for general boolean functions is studied a tight upper bound of 2 sup n n 2 epsilon for the worst case obdd size is derived although the size of an obdd is dependent on the ordering of decision variables it is shown that almost all functions are not sensitive to variable ordering

prediction accuracy measures for a nonlinear model and for right censored time to event data this article develops a pair of new prediction summary measures for a nonlinear prediction function with right censored time to event data the first measure defined as the proportion of explained

regression modelling strategies for improved prognostic prediction regression models such as the cox proportional hazards model have had increasing use in modelling and estimating the prognosis of patients with a variety of diseases many applications involve a large number of variables to be modelled using a relatively small patient sample problems of overfitting and of identifying important covariates are exacerbated in analysing prognosis because the accuracy of a model is more a function of the number of events than of the sample size we used a general index of predictive discrimination to measure the ability of a model developed on training samples of varying sizes to predict survival in an independent test sample of patients suspected of having coronary artery disease we compared three methods of model fitting 1 standard step up variable selection 2 incomplete principal components regression and 3 cox model regression after developing clinical indices from variable clusters we found regression using principal components to offer superior predictions in the test sample whereas regression using indices offers easily interpretable models nearly as good as the principal components models standard variable selection has a number of deficiencies

information gain and a general measure of correlation summary given a parametric model of dependence between two random quantities x and y the notion of information gain can be used to define a measure of correlation this definition of correlation generalizes both the usual product moment correlation coeffi cient for the bivariate normal model and the multiple correlation coefficient in the standard linear regression model the use of this information based correlation in a descriptive statistical analysis is examined and several examples are given if the dependence between two random quantities x and y is modelled parametri cally then the concept of information gain can be used to define a measure of correlation this correlation coefficient can appear in two possible contexts depending on whether one models the joint distribution of x and y or just the conditional distribution of y given x an important motivating feature for this information based correlation coefficient is the fact that it generalizes both the usual product moment correlation coefficient for the bivariate normal model and the usual multiple correlation coefficient for the standard multiple regression model with normal errors our intuition is well developed for these usual correlation coefficients and hopefully our intuition will still be applicable for the information based correlation in more general modelling situations of parametric dependence further since our correlation coefficient is based on information gain we might hope to extend our intuition to interpret the information gain in any statistical modelling situation where we want to assess how much better a more complicated model is than a simpler model the concept of information gain for general statistical models is described in 2 this concept is then used to define an information based measure of correlation the joint case is covered in 3 and the conditional case in 4 estimation of the correlation coefficient is carried out by estimating the corresponding information gain see 5 7 the use of information gain for the purpose of model choice in a descriptive statistical analysis is discussed in 8 and a comparison between our approach and akaike s information criterion is given in 9 some examples of the use of this information based correlation are given in 10 and 11

the bootstrap and identification of prognostic factors via cox s proportional hazards regression model this paper describes the use of the bootstrap 1 2 a new computer based statistical methodology to help validate a regression model resulting from the fitting of cox s proportional hazards model3 to a set of censored survival data as an example we define a prognostic model for outcome in childhood acute lymphocytic leukemia with the cox model and use of a training set of 224 patients to validate the accuracy of the model we use a bootstrap resampling technique to mimic the population under study in two stages first we select the important prognostic factors via a stepwise regression procedure with 100 bootstrap samples secondly we estimate the corresponding regression parameters for these important factors with 400 bootstrap samples the bootstrap result suggests that the model constructed from the training set is reasonable

on the interpretation and use of r 2 in regression analysis the coefficient of determination or squared multiple correlation coefficient r2 is a common output from computer regression packages we argue first that this statistic can be interpreted as an estimator of a population parameter only when the regressors are random in such a model the variation of r2 is discussed and a simple approximate confidence interval for the population coefficient of determination is proposed its use is illustrated on data from computerized tomography investigation of pigs

summarizing the predictive power of a generalized linear model this paper studies summary measures of the predictive power of a generalized linear model paying special attention to a generalization of the multiple correlation coefficient from ordinary linear regression the population value is the correlation between the response and its conditional expectation given the predictors and the sample value is the correlation between the observed response and the model predicted value we compare four estimators of the measure in terms of bias mean squared error and behaviour in the presence of overparameterization the sample estimator and a jack knife estimator usually behave adequately but a cross validation estimator has a large negative bias with large mean squared error one can use bootstrap methods to construct confidence intervals for the population value of the correlation measure and to estimate the degree to which a model selection procedure may provide an overly optimistic measure of the actual predictive power

measures of dependence for censored survival data in the linear regression model with normal errors the squared product moment correlation provides the standard measure of dependence between the explanatory variable and the response variable using the concept of information gain a measure of dependence can also be defined for more general regression models used in survival analysis such as the weibull regression model or cox s proportional hazards model further this measure of dependence can be conveniently estimated even when the response variable is subject to censoring and a simple approximation is available