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

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

a weighted concordance correlation coefficient for repeated measurement designs the need to quantify agreement between two raters or two methods of measuring a response often arises in research kappa statistics unweighted and weighted are appropriate when the data are nominal or ordinal whereas the concordance correlation coefficient is more appropriate when the data are measured on a continuous scale we develop weighted product moment and concordance correlation coefficients which are applicable for repeated measurements study designs we consider two distinct situations in which the repeated measurements are paired or unpaired over time we illustrate the methodology with examples comparing 1 two assays for measuring serum cholesterol 2 two estimates of dietary intake from a food frequency questionnaire and dietary recalls and 3 two measurements of percentage body fat from skinfold calipers and dual energy x ray absorptiometry

goodman and kruskal s λ a new look at an old measure of association we examine goodman and kruskal s lambda using efron s approach to regression and analysis of variance anova for zero one outcome data for a binary response cross classified by a single nominal predictor we present a computationally simple anova table in which lambda is analogous to pearson s r square we characterize the relationship between lambda and the commonly used apparent error rate in logistic regression and show that lambda is based implicitly on a prediction rule for a saturated model with classification level 0 5 this relationship suggests that we can correct the apparent error rate for chance by defining a natural generalization of lambda that we call pre the proportional reduction in error we illustrate the use of lambda and pre in an analysis of prognostic factors for one year survival in children with the acquired immunodeficiency syndrome aids

detecting outlying cells in two way contingency table via backwards stepping when fitting a model to a contingency table a significant lack of fit can sometimes be caused by a few outlier cells with the model fitting the remaining cells well these cells can be identified by using deleted residuals the residual from the expected count with the cell deleted and tested using the drop in likelihood ratio goodness of fit statistic from the model with the cell included to the model with the cell deleted with the cells being tested from least extreme to most extreme backwards stepping this article shows that using a bonferroni bound for the outlier test at each step results in a conservative test with good power to detect multiple outliers backwards stepping and the use of deleted residuals results in the limiting of both masking and swamping effects the procedure generalizes easily to complicated probability models

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

analysis of dispersion of multinomial responses abstract in analogy to analysis of variance for linear models analyses of dispersion for loglinear models for multinomial responses are constructed the analyses which are based on the entropy and concentration measures are used to construct tests of independence and measures of association

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 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

small time gaussian behavior of symmetric diffusion semi groups this work is involved with the short time asymptotics of diffusion semigroups in a general setting a generalization of fang s version of varadhan s formula is proven for general dirichlet spaces that are local and conservative the intrinsic metric appearing in the formula is characterized by pointwise distance for canonical dirichlet spaces on loop groups

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

generating mapping class groups with elements of fixed finite order abstract we show that for k 6 and g sufficiently large the mapping class group of a surface of genus g can be generated by three elements of order k we also show that this can be done with four elements of order 5 when g is at least 8

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

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

a diagnostic for cox regression and general conditional likelihoods abstract diagnostics for changes in the maximum likelihood estimate of due to deletion of single observations in linear and logistic regression models are equivalent to diagnostics obtained from fitting augmented regression models for the cox proportional hazards model and in conditional models for arbitrarily matched or stratified data diagnostics derived from augmented regression models are again useful and easy to compute these diagnostics are applicable to different formulations for the multiplicative covariable effects over time or over strata an exact or approximate treatment of tied failure times or multiple case matching and time dependent covariates

modelling ordered categorical data recent advances and future challenges this article summarizes recent advances in the modelling of ordered categorical ordinal response variables we begin by reviewing some models for ordinal data introduced in the literature in the past 25 years we then survey recent extensions of these models and related methodology for special types of applications such as for repeated measurement and other forms of clustering we also survey other aspects of ordinal modelling such as small sample analyses power and sample size considerations and availability of software throughout we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data

further results on the explained variation in proportional hazards regression summary two competing measures of the proportion of variation of possibly censored survival times explained by a given proportional hazards model are compared in a monte carlo study it is shown that the validity of a very quick likelihood based measure magee 1990 depends on a proportionality assumption the computationally more involved measure v2 of schemper 1990 is robust in this respect both measures produce very close results if the cox model assumptions are perfectly met

using second order generalized estimating equations to model heterogeneous intraclass correlation in cluster randomized trials in cluster randomized trials it is commonly assumed that the magnitude of the correlation among subjects within a cluster is constant across clusters however the correlation may in fact be heterogeneous and depend on cluster characteristics accurate modeling of the correlation has the potential to improve inference we use second order generalized estimating equations to model heterogeneous correlation in cluster randomized trials using simulation studies we show that accurate modeling of heterogeneous correlation can improve inference when the correlation is high or varies by cluster size we apply the methods to a cluster randomized trial of an intervention to promote breast cancer screening

every mapping class group is generated by 6 involutions abstract let mod g b denote the mapping class group of a surface of genus g with b punctures luo asked in torsion elements in the mapping class group of a surface math gt 0004048 v1 8apr2000 if there is a universal upper bound independent of genus for the number of torsion elements needed to generate mod g b we answer luo s question by proving that 3 torsion elements suffice to generate mod g 0 we also prove the more delicate result that there is an upper bound independent of genus not only for the number of torsion elements needed to generate mod g b but also for the order of those elements in particular our main result is that 6 involutions i e orientation preserving diffeomorphisms of order two suffice to generate mod g b for every genus g 3 b 0 and g 4 b 1

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

empirical extensions of the lasso penalty to reduce the false discovery rate in high dimensional cox regression models correct selection of prognostic biomarkers among multiple candidates is becoming increasingly challenging as the dimensionality of biological data becomes higher therefore minimizing the false discovery rate fdr is of primary importance while a low false negative rate fnr is a complementary measure the lasso is a popular selection method in cox regression but its results depend heavily on the penalty parameter usually is chosen using maximum cross validated log likelihood max cvl however this method has often a very high fdr we review methods for a more conservative choice of we propose an empirical extension of the cvl by adding a penalization term which trades off between the goodness of fit and the parsimony of the model leading to the selection of fewer biomarkers and as we show to the reduction of the fdr without large increase in fnr we conducted a simulation study considering null and moderately sparse alternative scenarios and compared our approach with the standard lasso and 10 other competitors akaike information criterion aic corrected aic bayesian information criterion bic extended bic hannan and quinn information criterion hqic risk information criterion ric one standard error rule adaptive lasso stability selection and percentile lasso our extension achieved the best compromise across all the scenarios between a reduction of the fdr and a limited raise of the fnr followed by the aic the ric and the adaptive lasso which performed well in some settings we illustrate the methods using gene expression data of 523 breast cancer patients in conclusion we propose to apply our extension to the lasso whenever a stringent fdr with a limited fnr is targeted copyright 2016 john wiley sons ltd

logarithmic sobolev inequalities and spectral gaps perturbation theory abstract we prove the logarithmic sobolev inequality for a diffusion operator on the wiener space the operator we consider is a perturbation of the ornstein uhlenbeck operator we use the logarithmic sobolev inequality for the ornstein uhlenbeck operator our method is based on dirichlet forms and it can be applied to more general diffusions

generating the surface mapping class group by two elements wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements we prove that one of these generators can be taken as a dehn twist we also prove that the extended mapping class group is generated by two elements again one of which is a dehn twist another result we prove is that the mapping class groups are also generated by two elements of finite order

a concordance correlation coefficient to evaluate reproducibility a new reproducibility index is developed and studied this index is the correlation between the two readings that fall on the 45 degree line through the origin it is simple to use and possesses desirable properties the statistical properties of this estimate can be satisfactorily evaluated using an inverse hyperbolic tangent transformation a monte carlo experiment with 5 000 runs was performed to confirm the estimate s validity an application using actual data is given

applying r 2 type measures to ordered categorical data the concentration and entropy measures for categorical data tend to be highly dependent on the choice of response categories if the categorization of the response variable is arbitiary but it is reasonable to assume an underlying continuous distribution then an adaptation of the regression r 2 measure can be useful for describing multiple association

summarizing the goodness of fit of generalized linear models for longitudinal data this paper extends four goodness of fit measures of a generalized linear model glm to random effects and marginal models for longitudinal data the four measures are the proportional reduction in entropy measure the proportional reduction in deviance measure the concordance correlation coefficient and the concordance index the extended measures satisfy the basic requirements for measures of association two examples illustrate their use in model selection

large deviations for stochastic nonlinear beam equations abstract we establish a large deviation principle for the solutions of stochastic partial differential equations for nonlinear vibration of elastic panels also called stochastic nonlinear beam equations

predictive value of statistical models a review is given of different ways of estimating the error rate of a prediction rule based on a statistical model a distinction is drawn between apparent optimum and actual error rates moreover it is shown how cross validation can be used to obtain an adjusted predictor with smaller error rate a detailed discussion is given for ordinary least squares logistic regression and cox regression in survival analysis finally the splitsample approach is discussed and demonstrated on two data sets

logistic regression diagnostics a maximum likelihood fit of a logistic regression model and other similar models is extremely sensitive to outlying responses and extreme points in the design space we develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit the elements of the fitting process which constitute the usual output parameter estimates standard errors residuals etc will be used for this purpose with a properly designed computing package for fitting the usual maximum likelihood model the diagnostics are essentially free for the asking in particular good data analysis for logistic regression models need not be expensive or time consuming

penalized likelihood in cox regression in a cox regression model instability of the estimated regression coefficients can be reduced by maximizing a penalized partial log likelihood where a penalty function of the regression coefficients is substracted from the partial log likelihood in this paper we choose the optimal weight of the penalty function by maximizing the predictive value of the model as measured by the crossvalidated partial log likelihood our methods are illustrated by a study of ovarian cancer survival and by a study of centre effects in kidney graft survival

on the small time asymptotics of diffusion processes on hilbert spaces in this paper we establish a small time large deviation principle and obtain the following small time asymptotics lim t 0 2t log p x 0 b x t c d 2 b c for diffusion processes on hilbert spaces where d b c is the intrinsic metric between two subsets b and c associated with the diffusions the case of perturbed ornstein uhlenbeck processes is treated separately at the end of the paper

the explained variation in proportional hazards regression summary two new measures of the proportion of the variation of possibly censored survival times explained by a given proportional hazards model are presented it is shown that the previously suggested proportion of explained log likelihood is not a useful measure of predictive precision the new measures are intuitively appealing and sufficiently simple for routine application

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

on the small time behavior of ornstein uhlenbeck processes with unbounded linear drifts in this paper we study the small time behavior of ornstein ulenbeck processes with unbounded linear drifts on hilbert spaces the large deviations estimates are obtained the general theory of dirichlet forms lyons zheng s decompositions and the convolution representation of the processes play an important role

a study of a class of stochastic differential equations with non lipschitzian coefficients we study a class of stochastic differential equations with non lipschitz coefficients a unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved the dependence with respect to the initial values is investigated to obtain a continuous version of solutions the modulus of continuity of coefficients is assumed to be less than x y log open image in new window finally a large deviation principle of freidlin wentzell type is also established in the paper

an r2 statistic for fixed effects in the linear mixed model statisticians most often use the linear mixed model to analyze gaussian longitudinal data the value and familiarity of the r2 statistic in the linear univariate model naturally creates great interest in extending it to the linear mixed model we define and describe how to compute a model r2 statistic for the linear mixed model by using only a single model the proposed r2 statistic measures multivariate association between the repeated outcomes and the fixed effects in the linear mixed model the r2 statistic arises as a 1 1 function of an appropriate f statistic for testing all fixed effects except typically the intercept in a full model the statistic compares the full model with a null model with all fixed effects deleted except typically the intercept while retaining exactly the same covariance structure furthermore the r2 statistic leads immediately to a natural definition of a partial r2 statistic a mixed model in which ethnicity gives a very small p value as a longitudinal predictor of blood pressure bp compellingly illustrates the value of the statistic in sharp contrast to the extreme p value a very small r2 a measure of statistical and scientific importance indicates that ethnicity has an almost negligible association with the repeated bp outcomes for the study copyright 2008 john wiley sons ltd

stochastic 3d tamed navier stokes equations existence uniqueness and small time large deviation principles abstract in this paper we first give a direct approach to the existence and uniqueness of strong solutions to stochastic 3d tamed navier stokes equations in case of dirichlet boundary conditions for the stokes laplacian then we prove a small time large deviation principle for the solutions

the dynamics of the stochastic shadow gierer meinhardt system we consider the dynamics of the stochastic shadow gierer meinhardt system with one dimensional standard brownian motion we establish the global existence and uniqueness of solutions we also prove a large deviation result

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates

penalized partial likelihood regression for right censored data with bootstrap selection of the penalty parameter le modele de cox a risque proportionnel est souvent utilise pour estimer l association entre des covariables et un delai de survenue d un evenement qui peut etre censure les estimateurs bases sur la vraisemblance partielle sont utilises pour l estimation et la prediction du risque relatif de l evenement cependant ces estimateurs sont instables et ont une grande variance en cas de colinearite entre les covariables ou quand le nombre d evenements n est pas nettement plus grand que le nombre de covariables une vraisemblance partielle penalisee est proposee pour donner des estimateurs du risque relatif plus precis nous montrons qu asymptotiquement il existe toujours un parametre de penalite pour la vraisemblance partielle penalisee qui reduit le carre moyen de l erreur d estimation du logarithme du risque relatif nous proposons une methode de reechantillonnage pour choisir le parametre de penalite des simulations et un exemple montrent que des estimateurs bases sur la vraisemblance partielle penalisee et la methode du bootstrap peuvent dans certaines circonstances avoir un biais plus petit que celui des estimateurs bases sur la vraisemblance partielle ainsi qu un carre moyeu plus petit de l erreur d estimation et de prediction du logarithme du risque relatif nous illustrons ces methodes avec des donnees d une etude sur le myelome multiple du eastern cooperative oncology group ecog