evaluation of cox s model and logistic regression for matched case control data with time dependent covariates a simulation study case control studies are typically analysed using the conventional logistic model which does not directly account for changes in the covariate values over time yet many exposures may vary over time the most natural alternative to handle such exposures would be to use the cox model with time dependent covariates however its application to case control data opens the question of how to manipulate the risk sets through a simulation study we investigate how the accuracy of the estimates of cox s model depends on the operational definition of risk sets and or on some aspects of the time varying exposure we also assess the estimates obtained from conventional logistic regression the lifetime experience of a hypothetical population is first generated and a matched case control study is then simulated from this population we control the frequency the age at initiation and the total duration of exposure as well as the strengths of their effects all models considered include a fixed in time covariate and one or two time dependent covariate s the indicator of current exposure and or the exposure duration simulation results show that none of the models always performs well the discrepancies between the odds ratios yielded by logistic regression and the true hazard ratio depend on both the type of the covariate and the strength of its effect in addition it seems that logistic regression has difficulty separating the effects of inter correlated time dependent covariates by contrast each of the two versions of cox s model systematically induces either a serious under estimation or a moderate over estimation bias the magnitude of the latter bias is proportional to the true effect suggesting that an improved manipulation of the risk sets may eliminate or at least reduce the bias

robustness of multipredictor adaptive regulators musmar abstract effects of multipredictor information in adaptive control are studied particularly from the standpoint of plant structural uncertainties and unmodelled dynamics by an ode analysis it is shown that for an armax plant the equilibria of a multipredictor based lq self tuning regulator namely the musmar algorithm are the extrema of a receding horizon variant of the cost further if the control horizon is large enough the musmar possible convergence points tightly approximate the local minima of the unconditional quadratic cost constrained to the chosen regulator regressor

alternative variance and efficiency calculations for the case cohort design summary we define a bootstrap sampling plan for the case cohort design and present bootstrap variance estimates and confidence intervals for the log relative hazard the coverages of these confidence intervals are found to be near nominal levels in simulations a simple null variance calculation based on a superpopulation model is shown to provide good estimates of the power and efficiency of the case cohort design

inverse probability weighting in nested case control studies with additional matching a simulation study nested case control designs are inevitably less efficient than full cohort designs and it is important to use available information as efficiently as possible reuse of controls by inverse probability weighting may be one way to obtain efficiency improvements and it can be particularly advantageous when two or more endpoints are analyzed in the same cohort the controls in a nested case control design are often matched on additional factors than at risk status and this should be taken into account when reusing controls although some studies have suggested methods for handling additional matching a thorough investigation of how this affects parameter estimates and weights is lacking our aim is to provide such a discussion to help developing guidelines for practitioners we demonstrate that it is important to adjust for the matching variables in regression analyses when the matching is broken we present three types of estimators for the inverse sampling probabilities accounting for additional matching one of these estimators was somewhat biased when the cases and controls were matched very closely we investigated how additional matching affected estimates of interest with varying degree of association between the matching variables and exposure outcome strong associations introduced only a small bias when the matching variables were properly adjusted for sometimes exposure variables for example blood samples are analyzed in batches rather strong batch effects had to be present before this introduced much bias when the matching was broken all simulations are based on a study of prostate cancer and vitamin d copyright 2013 john wiley sons ltd

bootstrap for the case cohort design the case cohort design facilitates economical investigation of risk factors in a large survival study with covariate data collected only from the cases and a simple random subset of the full cohort methods that accommodate the design have been developed for various semiparametric models but most inference procedures are based on asymptotic distribution theory such inference can be cumbersome to derive and implement and does not permit confidence band construction while bootstrap is an obvious alternative how to resample is unclear because of complications from the two stage sampling design we establish an equivalent sampling scheme and propose a novel and versatile nonparametric bootstrap for robust inference with an appealingly simple single stage resampling theoretical justification and numerical assessment are provided for a number of procedures under the proportional hazards model

more efficient estimators for case cohort studies the case cohort study design used to reduce costs in large cohort studies is a random sample of the entire cohort named the subcohort augmented with subjects having the disease of interest but not in the subcohort sample when several diseases are of interest several case cohort studies may be conducted using the same subcohort with each disease analyzed separately ignoring the additional exposure measurements collected on subjects with the other diseases this is not an efficient use of the data and in this paper we propose more efficient estimators we consider both joint and separate analyses for the multiple diseases we propose an estimating equation approach with a new weight function and we establish the consistency and asymptotic normality of the resulting estimator simulation studies show that the proposed methods using all available information gain efficiency we apply our proposed method to the data from the busselton health study

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

uniqueness of local minima for linear quadratic control design the problem of minimizing a weighted sum of the input and output variances from a linear scalar system is considered this is viewed as an optimization problem with the parameters of the regulator as unknowns and it is proved that if the regulator is flexible enough then every local minimum to this problem is a global optimum this result is useful if a gradient method is used to find the optimal regulator

weighted estimating equations for semiparametric transformation models with censored data from a case cohort design summary in a case cohort design introduced by prentice 1986 covariates are assembled only for a subcohort randomly selected from the entire cohort and any additional cases outside the subcohort semiparametric transformation models are considered here for failure time data from the case cohort design weighted estimating equations are proposed for esti mation of the regression parameters the estimation procedure of survival probability at given covariate levels is also provided asymptotic properties are derived for the estimators using finite population sampling theory u statistics theory and martingale convergence results the finite sample properties of the proposed estimators as well as the efficiency relative to the full cohort estimators are assessed via simulation studies a case cohort dataset from the atherosclerosis risk in communities study is used to illustrate the estimating procedure

influence functions for proportional hazards regression summary influence functions for the regression parameters in the proportional hazards model are presented it is suggested that empirical influence functions computed for each observation and each covariate can be useful in an informal way to identify influential observations this is illustrated on the stanford heart transplant data and two other examples

variable selection for case cohort studies with failure time outcome case cohort designs are widely used in large cohort studies to reduce the cost associated with covariate measurement in many such studies the number of covariates is very large so an efficient variable selection method is necessary in this paper we study the properties of a variable selection procedure using the smoothly clipped absolute deviation penalty in a case cohort design with a diverging number of parameters we establish the consistency and asymptotic normality of the maximum penalized pseudo partial likelihood estimator and show that the proposed variable selection method is consistent and has an asymptotic oracle property simulation studies compare the finite sample performance of the procedure with tuning parameter selection methods based on the akaike information criterion and the bayesian information criterion we make recommendations for use of the proposed procedures in case cohort studies and apply them to the busselton health study

efficient estimation of semiparametric transformation models for two phase cohort studies under two phase cohort designs such as case cohort and nested case control sampling information on observed event times event indicators and inexpensive covariates is collected in the first phase and the first phase information is used to select subjects for measurements of expensive covariates in the second phase inexpensive covariates are also used in the data analysis to control for confounding and to evaluate interactions this article provides efficient estimation of semiparametric transformation models for such designs accommodating both discrete and continuous covariates and allowing inexpensive and expensive covariates to be correlated the estimation is based on the maximization of a modified nonparametric likelihood function through a generalization of the expectation maximization algorithm the resulting estimators are shown to be consistent asymptotically normal and asymptotically efficient with easily estimated variances simulation studies demonstrate that the asymptotic approximati

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

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

the existence theorem for the steady navier stokes problem in exterior axially symmetric 3d domains we study the nonhomogeneous boundary value problem for the navier stokes equations of steady motion of a viscous incompressible fluid in a three dimensional exterior domain with multiply connected boundary we prove that this problem has a solution for axially symmetric domains and data without any smallness restrictions on the fluxes our main tool is a recent version of the morse sard theorem for sobolev functions obtained by bourgain et al rev mat iberoam 29 1 1 23 2013

smooth approximations without critical points of continuous mappings between banach spaces and diffeomorphic extractions of sets abstract let e f be separable hilbert spaces and assume that e is infinite dimensional we show that for every continuous mapping f e f and every continuous function e e 0 there exists a c mapping g e f such that f x g x e x and d g x e f is a surjective linear operator for every x e we also provide a version of this result where e can be replaced with a banach space from a large class including all the classical spaces with smooth norms such as c 0 l p or l p 1 p and f can be taken to be any banach space such that there exists a bounded linear operator from e onto f in particular for such e f every continuous mapping f e f can be uniformly approximated by smooth open mappings part of the proof provides results of independent interest that improve some known theorems about diffeomorphic extractions of closed sets from banach spaces or hilbert manifolds

regression calibration in failure time regression in this paper we study a regression calibration method for failure time regression analysis when data on some covariates are missing or mismeasured the method estimates the missing data based on the data structure estimated from a validation data set a random subsample of the study cohort in which covariates are always observed ordinary cox 1972 journal of the royal statistical society series b 34 187 220 regression is then applied to estimate the regression coefficients using the observed covariates in the validation data set and the estimated covariates in the nonvalidation data set the method can be easily implemented we present the asymptotic theory of the proposed estimator finite sample performance is examined and compared with an estimated partial likelihood estimator and other related methods via simulation studies where the proposed method performs well even though it is technically inconsistent finally we illustrate the method with a mouse leukemia data set

a bridge between dubovitskiĭ federer theorems and the coarea formula abstract the morse sard theorem requires that a mapping v r n r m is of class c k k max n m 0 in 1957 dubovitski generalized this result by proving that almost all level sets for a c k mapping have h s negligible intersection with its critical set where s max n m k 1 0 here the critical set or m critical set is defined as z v m x r n rank v x m another generalization was obtained independently by dubovitski and federer in 1966 namely for c k mappings v r n r d and integers m d they proved that the set of m critical values v z v m is h q negligible for q m 1 n m 1 k they also established the sharpness of these results within the c k category here we prove that dubovitski s theorem can be generalized to the case of continuous mappings of the sobolev lorentz class w p 1 k r n r d p n k this is the minimal integrability assumption that guarantees the continuity of mappings in this situation the mappings need not to be everywhere differentiable and in order to handle the set of nondifferentiability points we establish for such mappings an analog of the luzin n property with respect to lower dimensional hausdorff content finally we formulate and prove a bridge theorem that includes all the above results as particular cases as a limiting case in this bridge theorem we also establish a new coarea type formula if e x r n rank v x m then e j m v x d x r d h n m e v 1 y d h m y the mapping v is r d valued with arbitrary d and the formula is obtained without any restrictions on the image v r n such as m rectifiability or finiteness with respect to the m hausdorff measure these last results are new also for smooth mappings but are presented here in the general sobolev context the proofs of the results are based on our previous joint papers with j bourgain 2013 2015

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

marginal hazards model for case cohort studies with multiple disease outcomes case cohort study designs are widely used to reduce the cost of large cohort studies while achieving the same goals especially when the disease rate is low a key advantage of the case cohort study design is its capacity to use the same subcohort for several diseases or for several subtypes of disease in order to compare the effect of a risk factor on different types of diseases times to different events need to be modelled simultaneously valid statistical methods that take the correlations among the outcomes from the same subject into account need to be developed to this end we consider marginal proportional hazards regression models for case cohort studies with multiple disease outcomes we also consider generalized case cohort designs that do not require sampling all the cases which is more realistic for multiple disease outcomes we propose an estimating equation approach for parameter estimation with two different types of weights consistency and asymptotic normality of the proposed estimators are established large sample approximation works well in small samples in simulation studies the proposed methods are applied to the busselton health study

global measures of local influence for proportional hazards regression models cox proportional hazards regression models have been extended to multiple failure times and other situations by using a robust covariance matrix instead of the usual inverse of the information matrix we modify an existing measure of individual influence by substituting the robust covariance matrix for the inverse of the information matrix this provides a scalar measure of influence with known mean and bounded range on the interval 0 1 the measure is applicable to marginal multiple failure time models two examples are presented

wiener filter design using polynomial equations a simplified way of deriving realizable and explicit wiener filters is presented discrete time problems are discussed in a polynomial equation framework optimal filters predictors and smoothers are calculated by means of spectral factorizations and linear polynomial equations a tool for obtaining these equations for a given problem structure is described it is based on the evaluation of orthogonality in the frequency domain by means of canceling stable poles with zeros comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical wiener solution the simplicity of the proposed derivation method is particularly evident in multistage filtering problems to illustrate two examples are discussed a filtering and a generalized deconvolution problem a new solvability condition for linear polynomial equation appearing in scalar problems is also presented

proportional hazards regression for cancer studies there has been some recent work in the statistical literature for modeling the relationship between the size of cancers and probability of detecting metastasis i e aggressive disease methods for assessing covariate effects in these studies are limited in this article we formulate the problem as assessing covariate effects on a right censored variable subject to two types of sampling bias the first is the length biased sampling that is inherent in screening studies the second is the two phase design in which a fraction of tumors are measured we construct estimation procedures for the proportional hazards model that account for these two sampling issues in addition a nelson aalen type estimator is proposed as a summary statistic asymptotic results for the regression methodology are provided the methods are illustrated by application to data from an observational cancer study as well as to simulated data

on the steady navier stokes equations in 2d exterior domains abstract we study the boundary value problem for the stationary navier stokes system in two dimensional exterior domain we prove that any solution of this problem with finite dirichlet integral is uniformly bounded also we prove the existence theorem under zero total flux assumption

generalized case cohort sampling a class of cohort sampling designs including nested case control case cohort and classical case control designs involving survival data is studied through a unified approach using cox s proportional hazards model by finding an optimal sample reuse method via local averaging a closed form estimating function is obtained leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case cohort and nested case control studies a semiparametric efficient estimator can also be found with some further computation in addition the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available

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

weighted semiparametric likelihood method for fitting a proportional odds regression model to data from the case cohort design the problem of fitting a proportional odds regression model to data from the case cohort design proposed by prentice is considered a weighted semiparametric likelihood method is proposed under the proportional odds model the maximum weighted semiparametric likelihood estimators of both the regression parameter and the transformation function are shown to be consistent and normally distributed the applicability of the weighted semiparametric likelihood method to the semiparametric transformation regression models is also discussed in particular when the proportional hazards regression model is fitted estimators proposed by chen and lo can be generated by the weighted semiparametric likelihood method under different weighting schemes a simulation study suggests that the case cohort design is also useful under the proportional odds regression model and the proposed method performs well with practical finite sample sizes

sample size power calculation for stratified case cohort design the case cohort cc study design usually has been used for risk factor assessment in epidemiologic studies or disease prevention trials for rare diseases the sample size power calculation for a stratified cc scc design has not been addressed before this article derives such result based on a stratified test statistic simulation studies show that the proposed test for the scc design utilizing small sub cohort sampling fractions is valid and efficient for situations where the disease rate is low furthermore optimization of sampling in the scc design is discussed and compared with proportional and balanced sampling techniques an epidemiological study is provided to illustrate the sample size calculation under the scc design

comparison of case cohort estimators based on data on premature death of adult adoptees a case cohort sample of adoptees was collected to investigate genetic and environmental influences on premature death which motivated us to supplement existing simulation results to explore the performance of various estimators proposed for case cohort samples of survival data we studied six regression coefficients estimators which differ with regard to the weighting scheme used in a pseudo likelihood function and two different estimators of their variances compared to earlier simulation studies we changed the following conditions type of explanatory variable the distribution of lifetimes and the percentage of deaths in the full cohort the latter condition affected the performance of the estimated variances of the regression coefficients where we found a systematic bias of the estimator proposed by self and prentice dependent on the percentages of deaths this dependence of percentages of death was different for different sizes of case cohort studies a robust variance estimator showed a better overall performance the estimators of regression coefficients compared did not differ much the estimators proposed by kalbfleisch and lawless and by prentice performing very well results of the case cohort data of adoptees were not in conflict with earlier findings of a moderate genetic influence on premature death in adulthood copyright 2003 john wiley sons ltd

nonsmooth morse sard theorems abstract we prove that every function f r n r satisfies that the image of the set of critical points at which the function f has taylor expansions of order n 1 and non empty subdifferentials of order n is a lebesgue null set as a by product of our proof for the proximal subdifferential p we see that for every lower semicontinuous function f r 2 r the set f x r 2 0 p f x is l 1 null

competing risks analysis of the case cohort design the case cohort design has been put forward as an alternative design in epidemiological follow up studies which is less expensive than a full scale cohort study in this design all cases are assembled together with a randomly selected sub cohort which is then used as a comparison group one advantage being that the same sub cohort may serve as a comparison group for a number of different case series however the fact that all case groups are compared with the same sub cohort creates a correlation between estimated effects of exposures on the different outcomes this asymptotic correlation is derived using martingale central limit theory by studying competing risks data sampled using a case cohort design in a monte carlo simulation study we study effects on the correlation of factors such as number of cases and size of sub cohort

fitting semiparametric transformation regression models to data from a modified case cohort design summary we consider the problem of fitting semiparametric transformation regression models to data from a modified case cohort study in which the censoring times of all the censored subjects in the cohort are observed we propose to maximise a conditional profile likelihood to obtain the estimator of the regression parameter under the assumption that the censoring is independent of the covariates the estimator is shown to be consistent and asymptotically normally distributed numerical studies suggest that the relative efficiency of the estimator is very high and that the estimator is often less biased than the estimator from the complete case analysis and more accurate than the pseudolikelihood estimator

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

retrospective likelihood based methods for analyzing case cohort genetic association studies summary the case cohort cch design is a cost effective design for assessing genetic susceptibility with time to event data especially when the event rate is low in this work we propose a powerful pseudo score test for assessing the association between a single nucleotide polymorphism snp and the event time under the cch design the pseudo score is derived from a pseudo likelihood which is an estimated retrospective likelihood that treats the snp genotype as the dependent variable and time to event outcome and other covariates as independent variables it exploits the fact that the genetic variable is often distributed independent of covariates or only related to a low dimensional subset estimates of hazard ratio parameters for association can be obtained by maximizing the pseudo likelihood a unique advantage of our method is that it allows the censoring distribution to depend on covariates that are only measured for the cch sample while not requiring the knowledge of follow up or covariate information on subjects not selected into the cch sample in addition to these flexibilities the proposed method has high relative efficiency compared with commonly used alternative approaches we study large sample properties of this method and assess its finite sample performance using both simulated and real data examples