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

parsimonious analysis of time dependent effects in the cox model cox s proportional hazards model can be extended to accommodate time dependent effects of prognostic factors we briefly review these extensions along with their varying degrees of freedom spending more degrees of freedom with conventional procedures a priori defined interactions with simple functions of time restricted natural splines piecewise estimation for partitions of the time axis allows the fitting of almost any shape of time dependence but at an increased risk of over fit this results in increased width of confidence intervals of time dependent hazard ratios and in reduced power to confirm any time dependent effect or even any effect of a prognostic factor by means of comparative empirical studies the consequences of over fitting time dependent effects have been explored we conclude that fractional polynomials and similarly penalized likelihood approaches today are the methods of choice avoiding over fit by parsimonious use of degrees of freedom but also permitting flexible modelling if time dependence of a usually a priori unknown shape is present in a data set the paradigm of a parsimonious analysis of time dependent effects is exemplified by means of a gastric cancer study

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

studies in numerical nonlinear instability ii a new look at u t uu x 0 abstract it is shown that in general leap frog schemes any particular unstable solution behaves as an attractor of other solutions for a leap frog discretization of u t uu x 0 a particular kind of unstable solution is constructed which generically attracts any other solution estimates of the overflow time are presented and related to the notions of stability threshold and restricted stability

assessing time by covariate interactions in proportional hazards regression models using cubic spline functions proportional hazards or cox regression is a popular method for modelling the effects of prognostic factors on survival use of cubic spline functions to model time by covariate interactions in cox regression allows investigation of the shape of a possible covariate time dependence without having to specify a specific functional form cubic spline functions allow one to graph such time by covariate interactions to test formally for the proportional hazards assumption and also to test for non linearity of the time by covariate interaction the functions can be fitted with existing software using relatively few parameters the regression coefficients are estimated using standard maximum likelihood methods

a flexible approach to the crossing hazards problem we propose a simple and flexible framework for the crossing hazards problem the method is not confined to two sample problems but may also work with continuous exposure variables whose effect changes its sign at some time point of the observed follow up time penalized partial likelihood estimation relies upon the assumption of a smooth hazard ratio via low rank basis splines with a conventional difference penalty to ensure smoothness and additional ad hoc penalties to obtain restricted estimates useful in the context of crossing hazards the framework naturally also leads to a statistical test that has good power for revealing a global effect under several alternatives including crossing hazards we provide the results from a real data analysis and from some simulations to illustrate empirically the performance of the proposed approach as compared with the possible alternatives

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

phase transitions in experimental systems abstract scaling functions of the support and of the measure have been used to characterize the scaling behavior of a dynamical system while scaling functions for the scaling of the measure have been calculated for a number of experimental systems examples of scaling functions for the scaling of the support are difficult to obtain in this contribution we report on a phase transition like effect of an experimental p doped germanium semiconductor sample it is found that the results obtained from the dynamical scaling function agree with those obtained by horita et al from model maps indicating that scaling functions for the scaling of the support are a powerful method of characterizing experimental dynamical systems

a numerical energy conserving method for the dnls equation an implicit numerical energy conserving method is developed for the derivative nonlinear schrodinger dnls equation for periodic boundary conditions we find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the dnls equation if the modulation is small and k0 a22 2 t k0 is the wavenumber and a the amplitude the numerical scheme is used to follow the nonlinear behavior of the dnls modulational instability the numerical code is also tested by the evolution for one soliton initial data these tests show that if the modulation is not small compared to the background wave amplitude new nonlinear numerical instabilities are introduced

a strategy for modelling the effect of a continuous covariate in medicine and epidemiology low dimensional parametric models are well understood straightforward to communicate to other workers have very smooth curves and may easily be checked for consistency with background scientific knowledge or understanding they should therefore be ideal tools with which to represent smooth relationships between a continuous predictor and an outcome variable in medicine and epidemiology unfortunately a seriously restricted set of such models is used routinely in practical data analysis typically linear quadratic or occasionally cubic polynomials or sometimes a power or logarithmic transformation of a covariate since their flexibility is limited it is not surprising that the fit of such models is often poor royston and altman s recent work on fractional polynomials has extended the range of available functions it is clearly crucial that the chosen final model fits the data well achieving a good fit with minimal restriction on the functional form has been the motivation behind the major recent research effort on non parametric curve fitting techniques here i propose that one such model a possibly over fitted cubic smoothing spline may be used to define a suitable reference curve against which the fit of a parametric model may be checked i suggest a significance test for the purpose and examine its type i error and power in a small simulation study several families of parametric models including some with sigmoid curves are considered their suitability in fitting regression relationships found in several real data sets is investigated with all the example data sets a simple parametric model can be found which fits the data approximately as well as a cubic smoothing spline but without the latter s tendency towards artefacts in the fitted curve

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

extension of the ablowitz ladik method to the derivative nonlinear schro dinger equation abstract the derivative nonlinear schrodinger equation is solved by application of the ablowitz ladik scheme to an equivalent equation the variations of the results due to modifications in the spatial grid size and time step are analyzed the scheme maintains the main properties of the original equation and allows the use of rather large time steps

nonparametric survival analysis with time dependent covariate effects a penalized partial likelihood approach techniques are developed for nonparametric analysis of data under a cox regression like model permitting time dependent covariate effects determined by a regression function beta 0 t estimators resulting from maximization of an appropriate penalized partial likelihood are shown to exist and a computational approach is outlined weak uniform consistency with a rate of convergence and pointwise asymptotic normality of the estimators are established under regularity conditions a consistent estimator of a common baseline hazard function is presented and used to construct a consistent estimator of the asymptotic variance of the estimator of the regression function extensions to multiple covariates general relative risk functions and time dependent covariates are discussed

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

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

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

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

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

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

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

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

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

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

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

finite difference schemes for nonlinear wave equation that inherit energy conservation property abstract we propose two general finite difference schemes that inherit energy conservation property from nonlinear wave equations such as the nonlinear klein gordon equation nlkge one of proposed schemes is implicit and another is explicit many studies exist on fdss that inherit energy conservation property from nlkge and we can derive all of their schemes from the proposed general schemes in this paper the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly because of this the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions we make new fdss for fermi pasta ulam equation string vibration equation shimoji kawai equation ske and ebihara equation and verify numerically the inheritance of the energy conservation property for nlkge and ske

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

focusing a mechanism for instability of nonlinear finite difference equations abstract a mechanism for the destabilization of numerical algorithms for partial differential equations is suggested the novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

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

grid resonances focusing and benjamin feir instabilities in leapfrog time discretizations abstract w l briggs a c newell and t sarie j comput phys 50 83 1983 studied the solutions of a class of discretizations of u t au x uu x 0 where a is a constant they discovered that the numerical solution of the discretization becomes unbounded despite the fact that theoretically the solution should remain bounded they attributed this anomalous behavior to a focusing mechanism in this paper we make use of a multiple scales analysis and show that the instabilities are caused by a resonance effect introduced by the discretizations the contributions from the space and time discretizations are analysed separately and in detail thus the structure of the focusing mechanism becomes transparent

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

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

on nonlinear instabilities in leap frog finite difference schemes abstract briggs newell and sarie j comput phys 51 83 1983 have discussed a mechanism for the destabilisation of finite difference approximations to nonlinear partial differential equations their ideas were developed using the leap frog approximation to the advection equation here the same situation is examined in a manner which compares the basic solution to a periodic wavetrain an investigation is made into the stability of the basic solution to small disturbances which take the form of side band fourier modes the relation between side band growth and envelope modulation is discussed

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

new code parameters from reed solomon subfield codes we determine the dimensions of subfield codes of reed solomon codes and construct certain extensions and lengthenings of these codes we start from the duals using the language of orthogonal arrays as a first result this allows us to obtain a fair number of improvements in the list of binary ternary and quaternary linear codes with largest known minimal distance

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

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

likelihood approaches for proportional likelihood ratio model with right censored data regression methods for survival data with right censoring have been extensively studied under semiparametric transformation models such as the cox regression model and the proportional odds model however their practical application could be limited because of possible violation of model assumption or lack of ready interpretation for the regression coefficients in some cases as an alternative in this paper the proportional likelihood ratio model introduced by luo and tsai is extended to flexibly model the relationship between survival outcome and covariates this model has a natural connection with many important semiparametric models such as generalized linear model and density ratio model and is closely related to biased sampling problems compared with the semiparametric transformation model the proportional likelihood ratio model is appealing and practical in many ways because of its model flexibility and quite direct clinical interpretation we present two likelihood approaches for the estimation and inference on the target regression parameters under independent and dependent censoring assumptions based on a conditional likelihood approach using uncensored failure times a numerically simple estimation procedure is developed by maximizing a pairwise pseudo likelihood we also develop a full likelihood approach and the most efficient maximum likelihood estimator is obtained by a profile likelihood simulation studies are conducted to assess the finite sample properties of the proposed estimators and compare the efficiency of the two likelihood approaches an application to survival data for bone marrow transplantation patients of acute leukemia is provided to illustrate the proposed method and other approaches for handling non proportionality the relative merits of these methods are discussed in concluding remarks

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

new bounds on binary linear codes of dimension eight corresp let n k d be the smallest integer n such that a binary linear code of length n dimension k and minimum distance at least d exists new results are given that improve the best previously known bounds on n 8 d

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

the weak nonlinear instability of euler explicit scheme for the convective equation abstract the weak nonlinear instability of the euler explicit scheme for the convective equation u uux vxx is studied analytically using perturbation expansion and discrete fourier transform techniques short time weak nonlinear growth of the fourier modes for the noise is found provided that the linear stability conditions are satisfied as long as the perturbation expansion is valid the maximum growth is approximately proportional to the inverse of the grid spacing

fourier spectral methods for pseudo parabolic equations fourier spectral methods of galerkin and collocation type for quasilinear pseudo parabolic equations are presented convergence with spectral accuracy is proved for both galerkin and collocation approximations unconditionally stable discretization in time by explicit one step methods is analyzed fast transform methods to compute the nonlinear terms are proposed

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

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

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

additive and multiplicative covariate regression models for relative survival incorporating fractional polynomials for time dependent effects relative survival is used to estimate patient survival excluding causes of death not related to the disease of interest rather than using cause of death information from death certificates which is often poorly recorded relative survival compares the observed survival to that expected in a matched group from the general population models for relative survival can be expressed on the hazard mortality rate scale as the sum of two components where the total mortality rate is the sum of the underlying baseline mortality rate and the excess mortality rate due to the disease of interest previous models for relative survival have assumed that covariate effects act multiplicatively and have thus provided relative effects of differences between groups using excess mortality rate ratios in this paper we consider i the use of an additive covariate model which provides estimates of the absolute difference in the excess mortality rate and ii the use of fractional polynomials in relative survival models for the baseline excess mortality rate and time dependent effects the approaches are illustrated using data on 115 331 female breast cancer patients diagnosed between 1 january 1986 and 31 december 1990 the use of additive covariate relative survival models can be useful in situations when the excess mortality rate is zero or slightly less than zero and can provide useful information from a public health perspective the use of fractional polynomials has advantages over the usual piecewise estimation by providing smooth estimates of the baseline excess mortality rate and time dependent effects for both the multiplicative and additive covariate models all models presented in this paper can be estimated within a generalized linear models framework and thus can be implemented using standard software copyright 2005 john wiley sons ltd

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

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

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

time dependent effects of fixed covariates in cox regression a nonparametric modification is proposed for cox s proportional hazards model cox 1972 journal of the royal statistical society series b 34 187 220 where the covariates are fixed but their effects are allowed to vary in time parameters are introduced for the covariate effects at the uncensored survival times estimates are obtained by maximizing the penalized partial log likelihood that arises if a penalty function of the first order differences of consecutive parameters is subtracted from the partial log likelihood the choice of the smoothing parameter which determines the weight of the penalty is based on akaike s information criterion the methods are illustrated in a set of ovarian cancer data and in a set of kidney transplantation data

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

on d schoenfeld s approach for testing the proportional hazards assumption summary a goodness of fit score statistic for the proportional hazards assumption is derived from a partial likelihood and is shown to be a simplification of schoenfeld s 1980 statistic in the case of the several group problem the test statistic may be formulated as a simple sum of quadratic expressions a conservative form is also given in this case

the smallest length of eight dimensional binary linear codes with prescribed minimum distance let n 8 d be the smallest integer n for which a binary linear code of length n dimension 8 and minimum distance d exists we prove that n 8 18 42 n 8 26 58 n 8 28 61 n 8 30 65 n 8 34 74 n 8 36 77 n 8 38 81 n 8 42 89 and n 8 60 124 after these results all values of n 8 d are known

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

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

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

spline based tests in survival analysis this paper examines a method for testing hypotheses on covariate effects in a proportional hazards model and also on how effects change over time in regression analysis of survival data the technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models the basic idea is to formulate a flexible parametric alternative using fixed knot splines together with a penalty function that penalizes noisy alternatives more than smooth ones to focus the power of the tests toward smooth alternatives the test statistics are the analogs of ordinary likelihood based statistics only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log likelihood large sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases numerical results suggest these approximations may be adequate with moderate sized samples

network meta analysis of longitudinal data using fractional polynomials abstract network meta analysis of randomized controlled trials rcts are often based on one treatment effect measure per study however many studies report data at multiple time points furthermore not all studies measure the outcomes at the same time points as an alternative to a network meta analysis based on a synthesis of the results at one time point a network meta analysis method is presented that allows for the simultaneous analysis of outcomes at multiple time points the development of outcomes over time of interventions compared in an rct is modeled with fractional polynomials and the differences between the parameters of these polynomials within a trial are synthesized across studies with a bayesian network meta analysis the proposed models are illustrated with an analysis of rcts evaluating interventions for osteoarthritis of the knee fixed and random effects second order fractional polynomials were applied to the case study network meta analysis with models that represent the treatment effects in terms of several parameters using fractional polynomials can be considered a useful addition to models for network meta analysis of repeated measures previously proposed when rcts report treatment effects at multiple follow up times these models can be used to synthesize the results even if reporting times differ across the studies

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