sample size determination for an exponential survival model with an unrestricted covariate we derive formulae for estimating sample size and power for detecting the effect of an unrestricted covariate on survival time these are useful in designing survival studies with different patterns of recruitment and follow up when survival time is exponentially distributed we use the asymptotic covariance matrix conditional expectation and taylor s expansion techniques to develop these formulae computer simulations indicate that the asymptotic approximations used in developing the formulae are good over a range of parameter values and different patterns of recruitment and follow up that are relevant to survival studies

combining galerkin approximation techniques with the principle of hashin and shtrikman to derive a new fft based numerical method for the homogenization of composites we report on the mathematical analysis of two different fft based numerical schemes for the homogenization of composite media within the framework of linear elasticity the basic scheme of moulinec and suquet 1994 1998 9 and 10 and the energy based scheme of brisard and dormieux 2010 13 casting these two schemes as galerkin approximations of the same variational problem allows us to assert their well posedness and convergence more importantly we extend in this work their domains of application by relieving some stringent conditions on the reference material which were previously thought necessary the origins of the flaws of each scheme are identified and a third scheme is proposed which seems to combine the strengths of the basic and energy based schemes while leaving out their weaknesses finally a rule is proposed for handling heterogeneous pixels voxels a situation frequently met when images of real materials are used as input to these schemes

effective sample sizes for confidence intervals for survival probabilities we examine various methods to estimate the effective sample size for construction of confidence intervals for survival probabilities we compare the effective sample sizes of cutler and ederer1 and peto et al 2 as well as a modified cutler ederer effective sample size we investigate the use of these effective sample sizes in the common situation of many censored observations that intervene between the time point of interest and the last death before this time we note that there is no a priori reason to treat upper and lower confidence intervals in a symmetric fashion since censored survival data are by nature asymmetric we recommend the use of the cutler ederer effective sample size in construction of upper confidence intervals and the peto effective sample size in construction of lower confidence intervals two examples with real data demonstrate the differences between confidence intervals formed with different effective sample sizes this study also illustrates the need for caution in the application of simulation studies to real problems

strong approximations of some biometric estimates under random censorship the empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample it is not however appropriate when the observations are incomplete this is the case for example when the observations have been randomly censored on the right in the latter case the so called product limit estimate of kaplan and meier 16 has been generally accepted as a substitute for the empirical distribution function and it possesses many of the same properties cf meier 19 developing the corresponding statement of efron 8 weak convergence over a finite interval of the kaplan meier product limit estimate to a gaussian process when there is censorship on the right has been obtained by breslow and crowley 3 cf also meier 19 breslow and crowley obtained their result via the limiting distribution of the closely related cumulative hazard process under random censorship a generalisation of the efron breslow crowley theorem was formulated by yang 24 for the case when censorship on the right is performed by more competing risks these results have numerous statistical applications in areas such as medical follow up studies life testing actuarial sciences and demography koziol and green 18 have computed tables for the limiting distribution of the cram6r von mises statistic based on the kaplan meier product limit estimate under a specific random censorship model a point will be stressed in this paper however that by a certain transformation which seems to have escaped many researchers attention and is due to efron 8 it is always possible to obtain the standard wiener process in the limit the distributions of many functionals of which are well known this transformation was only recently used in a stimulating note by gillespie and fisher 14 to construct asymptotic confidence bands for the unknown distribution function in the kaplan meier model a general scheme of random censorship can be defined in the following way let x be a real random variable with distribution function f t

a non parametric procedure for evaluating treatment effect in the meta analysis of survival data this paper addresses the problem of combining information from independent clinical trials which compare survival distributions of two treatment groups current meta analytic methods which take censoring into account are often not feasible for meta analyses which synthesize summarized results in published or unpublished references as these methods require information usually not reported the paper presents methodology which uses the log log survival function difference i e log logs2 t log logs1 t as the contrast index to represent the multiplicative treatment effect on survival in independent trials this article shows by the second mean value theorem for integrals that this contrast index denoted as theta is interpretable as a weighted average on a natural logarithmic scale of hazard ratios within the interval 0 t in a trial when the within trial proportional hazards assumption is true theta is the logarithm of the proportionality constant for the common hazard ratio for the interval considered within the trial in this situation an important advantage of using theta as a contrast index in the proposed methodology is that the estimation of theta is not affected by length of follow up time other commonly used indices such as the odds ratio risk ratio and risk differences do not have this invariance property under the proportional hazard model since their estimation may be affected by length of follow up time as a technical artefact thus the proposed methodology obviates problems which often occur in survival meta analysis because trials do not report survival at the same length of follow up time even when the within trial proportional hazards assumption is not realistic the proposed methodology has the capability of testing a global null hypothesis of no multiplicative treatment effect on the survival distributions of two groups for all studies a discussion of weighting schemes for meta analysis is provided in particular a weighting scheme based on effective sample sizes is suggested for the meta analysis of time to event data which involves censoring a medical example illustrating the methodology is given a simulation investigation suggested that the methodology performs well in the presence of moderate censoring

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

a conservative and convergent scheme for undercompressive shock waves undercompressive shock waves arise in numerous physical applications we propose a class of conservative finite volume type schemes to approximate weak solutions of conservation laws that contain undercompressive shock waves we prove the convergence of a subsequence of approximate solutions towards a generalized entropy solution if the mesh width tends to zero the proof relies on a refined bv compactness analysis which accounts for the effect of the kinetic relation that drives the undercompressive wave at the same time we establish a new proof for the existence of solutions to the underlying model numerical experiments supplement the analytical results

variance of the kaplan meier estimator and its quantiles under certain fixed censoring models for fixed censoring models that contain at most one intermediate censoring point we obtain exact algebraic expressions for the asymptotic variances of i the quantiles of the kaplan meier km 1958 survival estimator and ii the km estimator itself at fixed time points the relationship between i and ii is found to be the same as the one derived by sander 1975 and reid 1981b for the random censorship model confidence intervals for the quantiles based on i are briefly discussed and compared to previously known procedures although greenwood s formula is recommended over ii in practice because of its desirable conditioning on the observed censoring pattern ii is of theoretical interest as an asymptotic limit for greenwood s formula in closed form

estimating the median survival time summary two methods are proposed for setting confidence intervals for the median survival time it is shown how these are related to bootstrap methods for censored data and some numerical comparisons are presented in many medical follow up studies the survival experience of a group of patients is summarized by the product limit or kaplan meier estimate of the survival distribution this may follow an analysis of the relation between survival time and some explanatory variables by for example the proportional hazards model the median of the estimated survival distribution is sometimes a convenient summary statistic and may be preferred to the mean because it is less sensitive to long tails in the estimated survival distribution this paper is concerned with estimating the distribution of the estimated median and providing confidence intervals for the true median two methods are considered in 2 they are both based in different ways on generalizations of the solution for the uncensored case these methods turn out to be connected to two ways of extending bootstrap methods for use with censored data this connexion is discussed in 3 thinking about extensions of the bootstrap to censored data highlights the need for careful consideration of the role of the censoring mechanism and how it should be incorporated into the data analysis section 4 provides some examples and monte carlo simulations the work is closely related to that of efron 1981 and emerson 1981

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

on the problem of linear elasticity for an infinite region containing a finite number of non intersecting spherical inhomogeneities abstract the problem of linear elasticity for an infinite number of non intersecting spherical and more generally ellipsoidal inhomogeneities is attacked the approach taken does not misrepresent geometry of inhomogeneities although the continuity conditions at the intefaces are only approximately satisfied the principal idea of the approach is to extend the method of kachanov 1985 int j fracture28 r11 r19 1987 int j solids structures23 23 43 for interacting cracks to the realm of the eshelby equivalent inclusion method the application to a test problem for two spherical cavities suggests that the approach can be useful for predictions of the overall response of composite materials and interfacial stress concentrations

simultaneous non parametric confidence intervals for survival probabilities from censored data in medical studies with censored data kaplan and meier s product limit estimator has frequent use as the estimate of the survival function simultaneous confidence intervals for the survival function at various time points constitute a useful addition to the analysis this study compares several such methods we consider in a simulation investigation two whole curve confidence bands and four methods based on the bonferroni inequality the results show that three bonferroni type methods are essentially equivalent all being better than the other methods when the number of time points is small 3 or 5

the moving grid method for nonlinear hyperbolic conservation laws numerically moving grids are often found to improve the resolution of approximate solutions to nonlinear hyperbolic conservation laws in this paper we show how to modify many standard numerical flux functions to incorporate this idea a generalized godunov type scheme is shown in the scalar case to converge to the entropy satisfying solution as the mesh size tends to zero a courant number of one half is found necessary in general and proved to be sufficient to obtain convergence convergence for methods based upon any two point conservative and monotone numerical flux function is also shown in the scalar case

symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data we describe existing tests and introduce two new tests concerning the value of a survival function these tests may be used to construct a confidence interval for the survival probability at a given time or for a quantile of the survival distribution simulation studies show that error rates can differ substantially from their nominal values particularly at survival probabilities close to zero or one we recommend our new constrained bootstrap test for its good overall performance

a note on confidence limits for quartiles with right censored data two new methods are proposed for constructing the confidence limits for quartiles the small sample behaviour of these two methods is compared with the jennison and turnbull modified brookmeyer crowley method at three quartiles the simulation study indicates that one of the new methods the smoothed modified reflected method is preferred over the other methods when the censoring rate is less than 40 per cent while the jennison and turnbull method is preferred for higher censoring the results for the upper quartile are similar to the median for the lower quartile with small sample and high censoring semi infinite intervals may often occur the correct practice is to permit the semi infinite intervals without modifying them since it is more informative and gives closer to nominal level coverage

nonparametwic confidence intervals for the median in the presence of right censoriilg a method is proposed for constructing nonpararnetric confidence intervals for the median of a survival distribution the method applies when survival data are subject to random right censoring the confidence interval is constructed by inverting a hypothesis test the test used being one that generalizes the binomial test to accommodate censoring a nonparametric estimate of the eventual failure time for each censored observation modifies the usual binomial test statistic the estimates are computed using the kaplan meier product limit estimate of the survival function a large simulation study indicates that the nonparametric method performs well for various survival distributions of different shapes two existing techniques that assume exponentiality perform well for exponential data they give good coverage probabilities with intervals that are shorter than those provided by the nonparametric approach the nonparametric intervals exhibit superior coverage probabilities when data are generated from nonexponential distributions this is the case for weibull data with both increasing and decreasing hazard functions the nonparametric construction is appropriate for samples of size 10 or greater with up to 50 censoring a monte carlo study supports the use of a binomial distribution to approximate the exact discrete distribution of the test statistic median survival time is often used for evaluation of the efficacy of a treatment for a chronic disease one would like to provide a confidence interval for the median as well as a point estimate when little is known about the shape of the survival distribution an interval is needed for which no particular parametric form for the distribution is assumed a nonparametric confidence interval for the median is easily formed when all observations are complete the problem increases in subtlety and complexity when incomplete or censored observations exist in this paper a method is proposed for constructing nonparametric confidence intervals for the median when observations may be right censored the distribution of survival time may be either discrete or continuous and it need not have finite moments the method does assume however that the censoring mechanism is noninformative in that censoring acts independently of survival time see lagakos 1979 in the absence of censoring the usual construction of a nonparametric confidence interval for the median inverts the sign test this test is based upon the number of failures that occur beyond the hypothesized value of the median the present paper generalizes the procedure to allow for right censoring by using the kaplan meier estimate of survival to approximate the number of failures beyond the hypothesized median confidence interval estimation when data are censored has received recent attention in

applications of crude incidence curves crude incidence curves display the cumulative number of failures of interest as a function of time with competing causes of failure they are distinct from cause specific incidence curves that treat secondary types of failures as censored observations after briefly reviewing their definition and estimation we present five applications of crude incidence curves to show their utility in a broad range of studies in some of these applications it is helpful to model survival time distributions with use of two different time metameters for example time from diagnosis and age of the patient we describe how one can incorporate published vital statistics into the models when secondary types of failure correspond to common causes of death

a confidence interval for the median survival time a nonparametric asymptotic confidence interval for the median survival time is developed for the case where data are subject to arbitrary right censoring this is accomplished by inverting a generalization of the sign test for censored data a simulation study shows that this nonparametric confidence interval performs well for a variety of underlying survival functions the procedure is applied to data from a clinical trial that compared f our dosage regimens of 5 uorouracil

approximate confidence intervals for probabilities of survival and quantiles in life table analysis for survival probabilities with censored data rothman 1978 journal of chronic diseases 31 557 560 has recommended the use of quadratic confidence limits based on the assumption that the product of the effective sample size at time t and the life table estimate of the survival probability past time t follows a binomial distribution this paper shows that the proposed confidence limits are asymptotically correct for continuous survival data these intervals as well as those based on the arcsine transformation the logit transformation and the log log transformation are compared by simulation to those based on greenwood s formula the usual method of interval estimation in life table analysis with large amounts of data the alternatives to the greenwood method all produce acceptable intervals on the basis of overall performance the intervals suggested by rothman are preferred for smaller samples any of these methods may be used to generate confidence sets for the median survival time or for any other quantile

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

the problem of elastic inclusions at finite concentration abstract this paper is aimed at determining the overall properties and the local stresses of a composite material the material is constituted of elastic ellipsoidal inclusions suspended in a homogeneous elastic matrix an analytical approximate method is proposed to account for the interaction between the inclusions at finite concentration this approach is based on a formulation of the problem of heterogeneous elasticity proposed by zeller r and dederichs p h 1973 elastic constant of polycrystals phys status solidi b 55 831 842 the zero order approximation of our scheme coincides with the mori tanaka method when the inclusions are spherical but improves significantly this method when higher order approximations are considered the morphological and spatial distribution of the inclusions is accounted for in our scheme through different examples it is shown how our results compare to existing analytical or numerical solutions

a comparison of reflected versus test based confidence intervals for the median survival time based on censored data the small sample performance of some recently proposed nonparametric methods of constructing confidence intervals for the median survival time based on randomly right censored data is compared with that of two new methods most of these methods are equivalent for large samples all proposed intervals are either test based or reflected intervals in the sense defined in the paper coverage probabilities for the interval estimates were obtained by exact calculation for uncensored data and by stimulation for three life distributions and four censoring patterns in the range of situations studied test based methods often have less than nominal coverage while the coverage of the new reflected confidence intervals is closer to nominal although somewhat conservative and these intervals are easy to compute

numerical approximations to nonlinear conservation laws with locally varying time and space grids an explicit time differencing technique is introduced to approximate nonlinear conservation laws this differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments previous stability requirements i e the cfl condition placed a global bound on the size of the time increments for scalar monotone approximations in one space dimension using this variable step time differencing convergence to the correct physical solution is proven given only a local cfl condition

bootstrap type confidence intervals for quantiles of the survival distribution in this paper we outline and illustrate an easy to program method for analytically calculating both parametric and non parametric bootstrap type confidence intervals for quantiles of the survival distribution based on right censored data this new approach allows for the incorporation of covariates within the framework of parametric models the procedure is based upon the notion of fractional order statistics and is carried forth using a simple beta transformation of the estimated survival function parametric or non parametric it is the only direct method currently available in the sense that all other methods are based on inverting test statistics or employing confidence intervals for other survival quantities we illustrate that the new method has favourable coverage probabilities for median confidence intervals as compared to six other competing methods

a variational form of the equivalent inclusion method for numerical homogenization abstract due to its relatively low computational cost the equivalent inclusion method is an attractive alternative to traditional full field computations of heterogeneous materials formed of simple inhomogeneities spherical ellipsoidal embedded in a homogeneous matrix the method can be seen as the discretization of the lippmann schwinger equation with piecewise polynomials contrary to the original approach of moschovidis and mura who discretized the strong form of the lippmann schwinger equation through taylor expansions we propose in the present paper a galerkin discretization of the weak form of this equation combined with the new mixed boundary conditions recently introduced by the authors the resulting method is particularly well suited to homogenization it is shown that this new variational approach has a number of benefits i the resulting linear system is well posed ii the numerical solution converges to the exact solution as the maximum degree of the polynomials tends to infinity and iii the method can provide rigorous bounds on the apparent properties of the statistical volume element provided that the matrix is stiffer or softer than all inhomogeneities this paper presents the formulation and implementation of the new variational form of the equivalent inclusion method its efficiency is investigated through numerical applications in 2d and 3d elasticity