explicit advanced step point eas methods and the eas2 multistep scheme for the solution of non stiff initial value problems abstract in this work we comprehensively examine for the first time in a paper the eas2 methods which are part of the explicit advanced step point eas family of methods the eas formulae comprise three distinct schemes eas1 eas2 and eas3 in this paper we consider the eas2 methods which are meticulously studied and assessed and their superior regions of absolute stability are presented crucially the computational efficiency of eas2 is thoroughly examined and comparative numerical results are presented with the use of a variable step variable order eas2 code the efficiency of eas2 is measured against the established and powerful adams formulae as the latter were implemented in the shampine and gordon code the extensive numerical results provide good evidence that eas2 is competitive i e faster and more accurate on a majority of test problems with the well established adams methods for the numerical solution of non stiff initial value problems

a general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems this paper introduces a general technique for the construction of multistep methods capable of integrating without local truncation error homogeneous linear odes with constant coefficients including those in particular that result in oscillatory solutions moreover these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies even confluent ones that occur from nonhomogeneous terms in the differential equation our procedure allows the derivation of many of the existing codes with similar properties as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency the properties of the methods are studied within a general framework and numerical examples are presented these demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes particularly when integrating the class of equations with perturbed oscillatory solutions the methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known as occurs in satellite orbit propagation the underlying ideas have already been applied to the improvement of some chebyshev methods that are not multistep

a 5 3 pair of explicit arkn methods for the numerical integration of perturbed oscillators a new embedded pair of explicit rkn methods specially adapted to the numerical integration of perturbed oscillators is obtained this pair depends on a parameter v h 0 h is the integration step and is the dominant frequency and it has four stages and algebraic orders five and three the numerical experiments carried out show that the new pair is very competitive when it is compared with high quality codes proposed in the scientific literature

counter matching a stratified nested case control sampling method summary the counter matching design for sampling from risk sets is presented in which each sampled risk set includes the failure and a random sample of controls from sampling strata defined by exposure information available for all cohort subjects asymptotic relative efficiency comparisons indicate that this type of sampling has superior efficiency to simple nested case control sampling in situations of practical interest a simple extension of the method is given which allows for nonrepresentative sampling of failures analysis of data from counter matched studies may be performed using standard conditional logistic likelihood software which allows for an offset in the model

dissipative high phase lag order numerov type methods for the numerical solution of the schrodinger equation a generator of families of explicit hybrid methods with minimal phase lag is developed in this paper the methods of the generator have algebraic order six the main characteristic of the new methods is that they are dissipative i e they are not symmetric and they have not an interval of periodicity the generator is of dissipation order eight numerical results indicate that these new methods are more efficient than older ones i e the property of the phase lag is more crucial than the nonempty interval of periodicity for the construction of the numerical methods for the numerical solution of the schrodinger type equations

a fourth algebraic order trigonometrically fitted predictor corrector scheme for ivps with oscillating solutions a scheme of trigonometrically fitted predictor corrector p c adams bashforth moulton methods is constructed in this paper our new p c method is based on the third order adams bashforth scheme as predictor and on the fourth order adams moulton scheme as corrector we tested the efficiency of our newly developed scheme against well known methods with excellent results the numerical experimentation showed that our method is considerably more efficient compared to well known methods used for the numerical solution of initial value problems with oscillating solutions

a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

symplectic partitioned runge kutta methods with the phase lag property abstract in this work specially tuned symplectic partitioned runge kutta sprk methods with minimum phase lag and phase fitted have been considered the general framework for constructing sprk methods with minimum phase lag is given

properties and implementation of ρ adams methods based on mixed type interpolation we investigate the properties of the coefficients of modified r adams methods for the integration of odes the derivation of these methods is in contrast with the classical adams methods not based on a polynomial interpolation theory but rather starts from a mixed interpolation theory in which a parameter a is involved it will be shown that the coefficients of the modified methods possess properties which make these methods very attractive further we will discuss the role of so called over implicit modified r adams schemes in the construction of more general linear multistep methods our second goal is to show that the modified adams bashforth adams moulton methods axe very well suited to be implemented as a predictor corrector pair in particular we will discuss the choice of the interpolation parameter when such a method is applied to general systems of equations numerical tests are performed to support the theory

asymptotic distribution theory for cox type regression models with general relative risk form the theory and application of the cox 1972 failure time regression model has almost without exception assumed an exponential form for the dependence of the hazard function on regression variables other regression forms may be more natural or descriptive in some applications for example a linear relative risk regression model provides a convenient framework for studying epidemiologic risk factor interactions this note uses the counting process formulation of andersen and gill 1982 to develop asymptotic distribution theory for a class of intensity function regression models in which the usual exponential regression form is relaxed to an arbitrary non negative twice differentiable form some stability and regularity conditions beyond those of andersen and gill are required to show the consistency of the observed information matrix which in general need not be positive semidefinite

classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree abstract in this paper we classify the centers localized at the origin of coordinates and their isochronicity for the polynomial differential systems in r 2 of degree d that in complex notation z x i y can be written as z i z a z d n 1 2 z d n 1 2 b z d n 1 2 z d n 1 2 c z d 1 2 z d 1 2 d z d 2 j n 1 2 z d 2 j n 1 2 where j is either 0 or 1 if j 0 then d 5 is an odd integer and n is an even integer satisfying 2 n d 1 2 if j 1 then d 3 is an integer and n is an integer with converse parity with d and satisfying 0 n d 1 3 where denotes the integer part function furthermore r and a b c d c note that if d 3 and j 0 we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in k e malkin 1964 17 n i vulpe and k s sibirskii 1988 25 j llibre and c valls 2009 15 and if d 2 j 1 and c 0 we are also obtaining as a particular case the quadratic polynomial differential systems studied in n n bautin 1952 2 h zoladek 1994 26 so the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations

methods for the analysis of sampled cohort data in the cox proportional hazards model methods are provided for regression parameter and cumulative baseline hazard estimation in the cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law the key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes leading to estimation methods parallel to those for full cohort data these techniques are illustrated for a number of sampling designs including three novel designs counter matching with additional randomly sampled controls quota sampling of controls and nested case control sampling with number of controls dependent on the failure s exposure status general asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic distributions for estimators from a large class of designs

robust variance estimation for rate ratio parameter estimates from individually matched case control data the asymptotic variance and robust variance estimators of rate ratios estimated using conditional logistic regression from individually matched case control data are derived when the presumed proportional hazards model ismisspecified the robust variance estimators are easily computed using schoenfeld residuals generated from standard partial likelihood estimation software for failure time data simulation studies indicate that the robust variance estimators perform well for typical sizes and that the rare disease version should be adequate for all practical purposes it was also found that model misspecification must be quite extreme before the model based i e inverse information variance is significantly biased and that the robust variance estimators are somewhat more variable than the model based we conclude that the model based variance estimator can be used when model misspecification is not severe the robust estimator should be used when the presumed model clearly fits the data poorly

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

asymptotic theory for nested case control sampling in the cox regression model by providing a probabilistic model for nested case control sampling in epidemiologic cohort studies consistency and asymptotic normality of the maximum partial likelihood estimator of regression parameters in a cox proportional hazards model can be derived using process and martingale theory as in andersen and gill a general expression for the asymptotic variance is given and used to calculate asymptotic relative efficiencies relative to the full cohort variance in some important special cases

a generator of high order embedded p stable methods for the numerical solution of the schro dinger equation a generator of new embedded p stable methods of order 2n 2 where n is the number of layers used by the embedded methods for the approximate numerical integration of the one dimensional schrodinger equation is developed in this paper these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for one dimensional differential equations of the schrodinger type show the validity of the developed theory

piecewise linearized methods for initial value problems with oscillating solutions two piecewise linearized methods for the numerical solution of nonlinear second order differential equations which contain the first order derivative are presented these methods provide piecewise analytical solutions in open intervals result in explicit finite difference equations for both the displacement and the velocity or only the displacement at nodal points and continuous solutions everywhere one of the piecewise linearized methods is a two level method that provides smooth solutions and is self starting the other is a non self starting three level technique the paper also presents several iterative schemes a time linearized three point technique and a time linearized numerov method an extensive assessment of the piecewise linearized methods is carried out in order to assess their accuracy in nine examples and it is shown that in some cases these techniques provide as accurate results as exponentially or trigonometrically fitting methods that do require an estimate of the dominant frequency of the solution the piecewise linearized methods presented here do not require such an estimate at all but may exhibit spurious fixed points attractors in nonlinear problems and their accuracy is a strong function of both the nonlinearities and the time step however they are applicable to nonlinear second order differential equations which contain the first order derivative

miscellanea comparison of case control to full cohort analyses under model misspecification we investigate the question of whether or not estimates of relative risk from matched case control studies well represent the population from which they were drawn we derive the formula for the asymptotic maximum partial likelihood estimators from nested case control studies when the cox proportional hazards model is misspecified and compare these to the corresponding full cohort values under various types of model misspecification such as covariate omission mismodelling available measurements and covariates measured with error we found that lack of covariate information as in covariate omission and measurement error cannot per se result in differences in case control and cohort estimates differences can only occur when the covariates used in the presumed model are mismodelled and we find that the magnitude of the misspecification must be unreasonably large in order to produce differences of practical importance

residuals for relative risk regression summary several possible definitions of residuals are given for relative risk regression with time varying covariates each such residual has a representation as an estimator of a stochastic integral with respect to the martingale arising from a subject s failure time counting process previously proposed residuals for individual study subjects and for specific time points are shown to be special cases of this definition as are previously derived regression diagnostics an illustration and various generalizations are also given able methods are required to detect various departures from modelling assumptions suitably defined residuals may play an important role in such identification however the nonparametric aspect of the model the possibility that modelled regression variables may be varying with follow up time and most importantly the usual presence of right censorship implies that specialized residual definitions are required the class of residuals considered here is most easily formulated using counting process notation for the failure time data let ni t yi t and zi t represent respectively for the ith subject the values of counting censoring and covariate processes at follow up time t i 1 n while ni u yi u zi u 0c u t specifies the corresponding histories for the ith subject prior to time t thus in a typical univariate failure time application ni with right continuous sample paths will take value zero prior to the time of failure on the ith subject and value one thereafter while yi with left continuous sample paths will take value one at times at which the ith subject is at risk for an observed failure and value zero otherwise the counting process ni can be uniquely decomposed so that for all t i

a parametric symmetric linear four step method for the efficient integration of the schrodinger equation and related oscillatory problems in this article we develop an explicit symmetric linear phase fitted four step method with a free coefficient as parameter the parameter is used for the optimization of the method in order to solve efficiently the schrodinger equation and related oscillatory problems we evaluate the local truncation error and the interval of periodicity as functions of the parameter we reveal a direct relationship between the periodicity interval and the local truncation error we also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature the analysis and the numerical results help us to determine the optimal values of the parameter which render the new method highly efficient

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

robust variance estimation for the case cohort design large cohort studies of rare outcomes require extensive data collection often for many relatively uninformative subjects sampling schemes have been proposed that oversample certain groups for example the case cohort design of prentice 1986 biometrika 73 1 11 provides an efficient method of analysis of failure time data however the variance estimate must explicitly correct for correlated score contributions a simple robust variance estimator is proposed that allows for more complicated sampling mechanisms the variance estimate uses a jackknife estimate of the variance of the individual influence function and is shown to be equivalent to a robust variance estimator proposed by lin and wei 1989 journal of the american statistical association 84 1074 1078 for the standard cox model simulation results indicate excellent agreement with corrected asymptotic estimates and appropriate test size the technique is illustrated with data evaluating the efficacy of mammography screening in reducing breast cancer mortality

trigonometrically fitted predictor corrector methods for ivps with oscillating solutions in this paper we develop a trigonometrically fitted predictor corrector p c scheme which is based on the well known two step second order adams bashforth method as predictor and on the third order adams moulton method as corrector numerical experiments show that the new trigonometrically fitted p c method is substantially more efficient than widely used methods for the numerical solution of initial value problems ivps with oscillating solutions

cox regression in nested case control studies with auxiliary covariates nested case control ncc design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors thomas maximum partial likelihood estimator is commonly used to estimate the regression parameters in cox s model for ncc data in this article we consider a situation in which failure censoring information and some crude covariates are available for the entire cohort in addition to ncc data and propose an improved estimator that is asymptotically more efficient than thomas estimator we adopt a projection approach that heretofore has only been employed in situations of random validation sampling and show that it can be well adapted to ncc designs where the sampling scheme is a dynamic process and is not independent for controls under certain conditions consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed furthermore a simplified approximate estimator is proposed when the disease is rare extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with thomas estimator and other competing estimators moreover sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated and we find that the biases are reasonably small in realistic situations we further demonstrate the proposed method with data from studies on wilms tumor

cox regression with incomplete covariate measurements abstract this article provides a general solution to the problem of missing covariate data under the cox regression model the estimating function for the vector of regression parameters is an approximation to the partial likelihood score function with full covariate measurements and reduces to the pseudolikelihood score function of self and prentice in the special setting of case cohort designs the resulting parameter estimator is consistent and asymptotically normal with a covariance matrix for which a simple and consistent estimator is provided extensive simulation studies show that the large sample approximations are adequate for practical use the proposed approach tends to be more efficient than the complete case analysis especially for large cohorts with infrequent failures for case cohort designs the new methodology offers a variance covariance estimator that is much easier to calculate than the existing ones and allows multiple subcohort augmentations to improve efficiency real data taken f

an eight step semi embedded predictor corrector method for orbital problems and related ivps with oscillatory solutions for which the frequency is unknown our new linear symmetric semi embedded predictor corrector method sepcm presented here is based on the multistep symmetric method of quinlan and tremaine 1990 with eight steps and eighth algebraic order and constructed to solve numerically the two dimensional kepler problem it can also be used to integrate related ivps with oscillatory solutions for which the frequency is unknown firstly we present a sepcm see panopoulos and simos 2013 36 37 in pair form this form has the advantage that reduces the computational expense from this form we construct a new symmetric eight step method the new scheme has constant coefficients and algebraic order ten we tested the efficiency of our newly developed scheme against some well known methods from the literature we measure the efficiency of the methods and conclude that the new scheme is the most efficient of all the compared methods and for all the problems solved