the asymptotic properties of nonparametric tests for comparing survival distributions the asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary and possibly unequal right censoring the test statistics include equivalents to the log rank statistic the modified wilcoxon statistic and the class of rank invariant test procedures introduced by peto peto when there are equal censoring distributions and the hazard functions are proportional the sample size formula for the f test used to compare exponential samples is shown to be valid for the log rank test in certain situations the power of the log rank test falls as the amount of censoring decreases

inference for a nonlinear counting process regression model martingale and counting process techniques are applied to the problem of inference for general conditional hazard functions this problem was first studied by beran who introduced a class of estimators for the conditional cumulative hazard and survival functions in the special case of time independent covariates here the covariate can be time dependent the classical i i d assumptions are relaxed by replacing them with certain asymptotic stability assumptions and models involving recurrent failures are included this is done within the framework of a general nonparametric counting process regression model important examples of the model include right censored survival data semi markov processes an illness death process with duration dependence and age dependent birth and death processes

nonparametric sequential tests against ordered alternatives in multiple armed clinical trials summary in a multiple armed clinical trial where one agent is administered at several dose levels or where each treatment regimen is formed by adding more agents to the preceding combination it is desirable to use test statistics that are sensitive to alternatives of stochastically ordered distributions liu et al 1992 recently developed a class of such test statistics by taking linear combinations of the familiar two sample weighted log rank statistics in the present paper we derive the asymptotic joint distributions of their test statistics calculated at different points in time applications to sequential survival studies with staggered patient entry are emphasized a colon cancer example is provided

parameter estimation following group sequential hypothesis testing summary parameter estimation techniques which fail to adjust for the interim analyses of group sequential test designs will introduce bias in much the same way that the repeated use of single sample hypothesis testing causes inflation of the type one statistical error rate methods based on the duality of hypothesis testing and interval estimation require definition of an ordering for the outcome space for the test statistic in this paper estimation following a group sequential hypothesis test for the mean of a normal distribution with known variance is investigated a proposed ordering of the sample space based on the maximum likelihood estimate of the mean is found to result in estimates which compare favourably with estimates computed from orderings investigated by tsiatis rosner mehta 1984 and chang o brien 1986 for a variety of group sequential designs the proposed ordering is then adapted for use when the sizes of groups accrued between analyses is random

noisy collective behaviour in deterministic cellular automata we investigate cellular automata in four and five dimensions for which chate and manneville recently have found nontrivial collective behaviour more precisely though being fully deterministic the average magnetization seems to be periodic respectively quasiperiodic with superimposed noise whose amplitude decreases with system size we confirm this behaviour on very large systems and over very large times we analyse in detail the statistical properties of the noise systems on small lattices and or subject to additional external noise are metastable arguments by grinstein et al suggest that in the periodic case the infinite deterministic systems should be metastable too these arguments are generalized to quasiperiodic systems we find evidence that they do indeed apply but we find no direct evidence for metastability of large systems

analysis of factorial survival experiments several new methodological issues that arise within two way factorial designs for survival experiments are discussed within the framework of asymptotic theory for the proportional hazards model with two binary treatment covariates these issues include the proper formulation of null hypotheses and alternatives the choice among log rank and adjusted or stratified log rank statistics the asymptotic correlation between test statistics for the separate main effects the asymptotic power under the various possible methods of analysis of tests to detect main effects and interactions the comparison of power to detect main effects within a 2 x 2 factorial design with power in a three group trial where no patients are randomized simultaneously to both treatments and the problems of analysis arising when accrual or exposure to one of the treatments is terminated early for ethical reasons

use of logrank tests and group sequential methods at fixed calendar times this paper presents simulations to determine the operating characteristics of several group sequential boundaries when applied to the repeated analysis of survival data at equal intervals of calendar time because the group sequential boundaries of pocock 1977 biometrika 64 191 199 and o brien and fleming 1979 biometrics 35 549 556 were constructed on the assumption that each interim analysis provides an equal increment of statistical information these boundaries are not theoretically appropriate for interim analysis at prescheduled calendar times nonetheless our simulations show that these boundaries yield size and power near nominal levels for repeated logrank analyses at equal intervals of calendar time

sequential linear rank tests for two sample censored survival data under extremely general patterns of patient arrival allocation to treatment and loss to follow up in randomized clinical trial settings the sequentially computed logrank statistic mantel 1966 is shown under the null hypothesis of identically distributed lifetimes to have exactly uncorrelated increments and is shown via rebolledo s 1980 martingale invariance principle to satisfy a functional central limit theorem justifying sequential logrank tests of jones and whitehead 1979 generalizations are made to other twosample rank tests for censored survival data and practical applicability to real randomized clinical trials is discussed

large sample behaviour of the product limit estimator on the whole line weak convergence results are proved for the product limit estimator on the whole line applications are given to confidence band construction estimation of mean lifetime and to the theory of q functions the results are obtained using stochastic calculus and in probability linear bounds for empirical processes

repeated significance testing in survival analysis using a litter matched design this paper shows that asymptotically a general class of linear rank procedures in a litter matched design form a gauss markov process over time this result is used to develop a procedure for repeated significance testing for time to response differences in litter matched studies in addition an important class of two sample linear rank procedures is shown to have uncorrelated increments over time an example is given using a litter matched cancer study empirical indications of the accuracy of a method for approximating multivariate normal critical values are also given

sequential log rank tests adjusting for covariates with the accelerated life model summary this paper presents a simple sequential method for comparing two treatments in randomized clinical trials which allows for the adjustment of other covariates with the accelerated life model the new procedure is the same as the ordinary sequential log rank test except that the test statistic is calculated on an adjusted time scale this covariate adjustment can lead to more efficient designs of group sequential trials especially when the effects of covariates being adjusted for are strong an aids example is provided

period doubling route to chaos for a global variable of a probabilistic automata network as a function of a parameter characterizing the degree of mixing of site values the density of nonzero sites of some one dimensional cellular automata is shown to exhibit a sequence of period doubling bifurcations and to behave chaotically when the degree of mixing is sufficiently large the automata network rules which are considered appear to be useful to model complex systems as in epidemiology in which the motion of the individuals is believed to play an important role

sequential analysis for censored regression data abstract motivated by buckley and james modification of the least squares procedure for censored regression data we derive a score process that incorporates the time evolution and handles staggered entry data in a natural way by censored regression data we mean that the responses are subject to a possible right censorship such cases often arise from clinical trials and industrial life tests the score process can be interpreted as a weighted comparison of transformed survival times this is especially suitable for the accelerated failure time regression model by expressing it in an appropriate form so that the counting process and its associate martingale theory can be applied we show that the score process is approximated by a mean 0 multidimensional gaussian process a consistent estimator of its covariance matrix function is provided based on the covariance matrix estimator and the sequential test of slud and wei a repeated significance test is then proposed usefulness of the procedure is il

monitoring a general class of two sample survival statistics with applications summary this paper considers a general class of statistics for testing the equality of two survival distributions in clinical trials with sequential monitoring the tests can be expressed as lebesgue stieltjes integrals of a weight function with respect to the difference between two survival distributions prominent members of this class include the two sample difference in kaplan meier estimates the test of medians brookmeyer crowley 1982 a truncated version of efron s 1967 test and the pepe fleming statistic pepe fleming 1989 1991 statistics in this class are shown to converge to a gaussian process indexed by information time under both null and local alternatives even if different statistics are used at different information times properly standardised statistics in a subclass converge to gaussian processes with independent increments so that the usual group sequential techniques for monitoring a clinical trial can be applied the design of a trial comparing two treatments with respect to mother to newborn transmission of hiv is used to illustrate practical aspects of monitoring

nonparametric rank based methods for group sequential monitoring of paired censored survival data summary this research gives methods for nonparametric sequential monitoring of paired censored survival data in the two sample problem using paired weighted log rank statistics with adjustments for dependence in survival and censoring outcomes the joint asymptotic closed form distribution of these sequentially monitored statistics has a dependent increments structure simulations validating operating characteristics of the proposed methods highlight power and size consequences of ignoring even mildly correlated data a motivating example is presented via the early treatment diabetic retinopathy study

group sequential tests with censored survival data adjusting for covariates summary group sequential tests for the analysis of survival data that compare two treatments and allow for the adjustment of other concomitant variables are discussed these methods allow early termination of a study if large treatment differences occur based on simulation results we compare the operating characteristics of these adjusted tests with the unadjusted sequentially computed log rank test

repeated significance testing for a general class of statistics used in censored survival analysis abstract the asymptotic joint distribution of a class of sequentially computed statistics used in survival analysis is derived when the individuals under study enter serially and are subject to random loss to follow up these results are used for constructing repeated significance tests over real time

efficient energies and algorithms for parametric snakes parametric active contour models are one of the preferred approaches for image segmentation because of their computational efficiency and simplicity however they have a few drawbacks which limit their performance in this paper we identify some of these problems and propose efficient solutions to get around them the widely used gradient magnitude based energy is parameter dependent its use will negatively affect the parametrization of the curve and consequently its stiffness hence we introduce a new edge based energy that is independent of the parameterization it is also more robust since it takes into account the gradient direction as well we express this energy term as a surface integral thus unifying it naturally with the region based schemes the unified framework enables the user to tune the image energy to the application at hand we show that parametric snakes can guarantee low curvature curves but only if they are described in the curvilinear abscissa since normal curve evolution do not ensure constant arc length we propose a new internal energy term that will force this configuration the curve evolution can sometimes give rise to closed loops in the contour which will adversely interfere with the optimization algorithm we propose a curve evolution scheme that prevents this condition

towards a general asymptotic theory for cox model with staggered entry a general asymptotic theory is established for the two parameter cox score process with staggered entry data it extends in several directions the existing theory developed by sellke and siegmund slud and gu and lai an essential tool employed here is a modern empirical process theory as elucidated in a recent monograph by pollard

the asymptotic joint distribution of the efficient scores test for the proportional hazards model calculated over time the log rank test derived by mantel 1966 and peto 1972 is a special case of the efficient scores test for the proportional hazards model it is used frequently in the analysis of survival duration in clinical trials where patients entering the study are assigned at random to different treatments and are followed until they either fail or the study is terminated most survival studies are designed so that the log rank test is calculated at the termination of the study however it is not unusual for the investigator to monitor the experiment periodically it would be very useful to know the joint distribution of the test statistic at different points in time in order to adjust for the effect of repeated significance testing although the joint distribution has not been derived previously sequential procedures for the analysis of survival data in clinical trials have been given by several authors including armitage 1975 pp 140 6 jones whitehead 1979 and nagelkerke hart 1980 in this paper we establish the asymptotic joint distribution of the more general efficient scores test for the proportional hazards model calculated at different points in time and indicate how this can be used for repeated significance testing

sequential analysis of the proportional hazards model abstract for the proportional hazards model of survival analysis an appropriate large sample theory is developed for cases of staggered entry and sequential analysis the principal techniques involve an approximation of the score process by a suitable martingale and a random rescaling of time based on the observed fisher information as a result we show that the maximum partial likelihood estimator behaves asymptotically like brownian motion author

semiparametric efficiency and its implication on the design and analysis of group sequential studies abstract authors have shown that the time sequential joint distributions of many statistics used to analyze data arising from group sequential time to event and longitudinal studies are multivariate normal with an independent increments covariance structure in theorem 1 of this article we demonstrate that this limiting distribution arises naturally when one uses an efficient test statistic to test a single parameter in a semiparametric or parametric model because we are able to think of many of the statistics in the literature in this fashion the limiting distribution under investigation is just a special case of theorem 1 using this general structure we then develop an information based design and monitoring procedure that can be applied to any type of model for any type of group sequential study provided that there is a unique parameter of interest that can be efficiently tested

repeated confidence intervals for the median survival time summary we describe methods for the construction of a confidence interval for median survival time based on right censored data these methods are extended to the construction of repeated confidence intervals for the median based on accumulating data here the overall probability that all intervals contain the true median is guaranteed at a fixed level the use of repeated confidence intervals for median survival time in post marketing surveillance is discussed a confidence interval for median survival time provides a useful summary of the survival experience of a group of patients if confidence intervals are calculated repeatedly as data accumulates the probability that at least one interval fails to contain the median may be much higher than the error rate for a single interval and if these confidence intervals are used in a decision making process the probability of an incorrect decision increases accordingly jennison turnbull 1984 have proposed methods for calculating repeated confidence intervals appropriate to such situations similar ideas have also been discussed by lai 1984 in 2 we propose a new form of single sample nonparametric confidence interval this interval has asymptotically correct coverage probability and monte carlo simulations suggest it is superior to its competitors for small sample sizes repeated confidence intervals for the median are presented in 3 and their stnall sample size performance is assessed by monte carlo simulation an example of their use is given in 4 all the methods considered can easily be modified to give confidence intervals for other quantiles or for the survival probability at a fixed time

two sample repeated significance tests based on the modified wilcoxon statistic abstract the asymptotic distribution theory of sequentially computed modified wilcoxon scores is developed for two sample survival data with random staggered entry and random loss to follow up the asymptotic covariance indicates generally dependent modified wilcoxon increments contradicting the authors reading of jones and whitehead 1979 a repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory the early stopping properties of this procedure are illustrated by a prostate cancer example

simultaneous group sequential analysis of rank based and weighted kaplan meier tests for paired censored survival data this research sequentially monitors paired survival differences using a new class of nonparametric tests based on functionals of standardized paired weighted log rank pwlr and standardized paired weighted kaplan meier pwkm tests during a trial these tests may alternately assume the role of the more extreme statistic by monitoring pemax the maximum between the absolute values of the standardized pwlr and pwkm one combines advantages of rank based rb and non rb paired testing paradigms simulations show that monitoring treatment differences using pemax maintains type i error and is nearly as powerful as using the more advantageous of the two tests in proportional hazards ph as well as non ph situations hence pemax preserves power more robustly than individually monitored pwlr and pwkm while maintaining a reasonably simple approach to design and analysis of results an example from the early treatment diabetic retinopathy study etdrs is given

group sequential designs for monitoring survival probabilities in the present paper we derive the asymptotic joint distribution of the nelson aalen estimators for the cumulative hazard function or the kaplan meier estimators for the survival function calculated at different possibly random calendar time points of a survival study this result is used to develop group sequential designs for clinical trials in which patients are enrolled over an extended period and the survival probability at a given failure time point is the parameter of primary interest the proposed designs allow monitoring the survival probability or probabilities at interim looks so that the study can be terminated when there is sufficient evidence for a positive or negative result a detailed illustration is provided with a problem from the national wilms tumor study group which motivated this research

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

sequential monitoring of clinical trials the role of information and brownian motion sequential monitoring has been a topic of major interest in clinical trials methodology over the past two decades this paper presents a unified conceptual framework for sequential monitoring that covers a wide variety of monitoring procedures in a wide variety of clinical trial settings the central elements of this framework consist of a suitable concept of statistical information and a scheme for using this concept as a basis for summarizing the accumulating results of a trial in a standardized form through a stochastic process that can be shown to approximate classical brownian motion the ideas are developed in a simple step by step fashion and illustrated by several practical examples

on the bias of maximum likelihood estimation following a sequential test summary tlhe bias of maximum likelihood estimates calculated at the end of a sequential procedure is investigated for the two sequential designs considered in detail the sequential probability ratio test and the triangular test this bias is appreciable a method of calculating an adjusted estimate with reduced bias is described and an approximation to the standard error of the new estimate is provided examples of the implementation of the method are given and its advantages and disadvantages relative to alternative approaches are discussed