weighted composition operators on hardy spaces let be analytic functions defined on d such that d d the operator given by f f is called a weighted composition operator in this paper we deal with the boundedness compactness weak compactness and complete continuity of weighted composition operators on hardy spaces hp 1 p in particular we prove that such an operator is compact on h1 if and only if it is weakly compact on this space this result depends on a technique which passes the weak compactness from an operator t to operators dominated in norm by t

testing against ordered alternatives for censored survival data in animal carcinogenesis experiments or comparative clinical trials where the r groups correspond to increasing dosages of an agent or an increasing number of additional modalities and the primary outcome is time to a certain event e g tumor occurrence death t ordered alternatives of the form h1 s1 s2 sr where si t pr ti t is the survival function for group are of special interest using the counting process formulation of the problem and generalizing earlier results we show that the class of all two sample weighted logrank statistics can be extended to test specifically for ordered alternatives ordered versions of the gehan wilcoxon unweighted logrank and peto prentice wilcoxon tests are investigated in detail asymptotic relative efficiencies under lehmann and scale alternatives are evaluated analytically and numerically among these three tests and against tarone s trend test 1975 for weibull gamma and lognormal survival distributions effects of censoring and unequa

maximum of the weighted kaplan meier tests with application to cancer prevention and screening trials summary a class of maximum weighted kaplan meier test statistics is described where the weight functions are chosen from a family of smooth functions the investigated test statistic is robust and sensitive to a variety of alternatives that are often observed in cancer prevention and screening trials a simulation study is performed to compare the size and power properties between the proposed test statistics and some existing ones we illustrate the procedure using data from a clinical trial of a breast cancer screening program

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

power of the mantel haenszel and other tests for discrete or grouped time to event data under a chained binomial model power for time to event analyses is usually assessed under continuous time models often however times are discrete or grouped as when the event is only observed when a procedure is performed wallenstein and wittes biometrics 1993 describe the power of the mantel haenszel test for discrete life tables under their chained binomial model for specified vectors of event probabilities over intervals of time herein the expressions for these probabilities are derived under a piecewise exponential model allowing for staggered entry and losses to follow up radhakrishna biometrics 1965 showed that the mantel haenszel test is maximally efficient under the alternative of a constant odds ratio and derived the optimal weighted test under other alternatives lachin biostatistical methods the assessment of relative risks 2011 describes the power function of this family of weighted mantel haenszel tests prentice and gloeckler biometrics 1978 describe a generalization of the proportional hazards model for grouped time data and the corresponding maximally efficient score test their test is also shown to be a weighted mantel haenszel test and its power function is likewise obtained there is trivial loss in power under the discrete chained binomial model relative to the continuous time case provided that there is a modest number of periodic evaluations relative to the case of homogeneity of odds ratios there can be substantial loss in power when there is substantial heterogeneity of odds ratios especially when heterogeneity occurs early in a study when most subjects are at risk but little loss in power when there is heterogeneity late in a study

rank based inference for the accelerated failure time model summary a broad class of rank based monotone estimating functions is developed for the semiparametric accelerated failure time model with censored observations the corresponding estimators can be obtained via linear programming and are shown to be consistent and asymptotically normal the limiting covariance matrices can be estimated by a resampling technique which does not involve nonparametric density estimation or numerical derivatives the new estimators represent consistent roots of the non monotone estimating equations based on the familiar weighted log rank statistics simulation studies demonstrate that the proposed methods perform well in practical settings two real examples are provided

complete decomposition of sub optimal regulators for singularly perturbed systems the linear regulator problem is studied for systems with slow and fast modes using the descriptor variable method the design of sub optimal regulators for such systems is completely decomposed it is shown that the sub optimal regulator is composed of fast and slow regulators and that both of these mutually decoupled can be designed independently a relevant computational procedure is also presented the procedure is applicable to designs of sub optimal regulators with either finite or infinite terminal time and for both singular and normal singularly perturbed systems

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

a general class of nonparametric tests for survival analysis tarone and ware 1977 biometrika 64 156 160 developed a general class of s sample test statistics for right censored survival data that includes the log rank and modified wilcoxon procedures subsequently many authors have considered two and s sample classes in detail in this paper a family of nonparametric statistics is shown to unify existing and generate new test statistics for the s greater than or equal to 2 sample s sample trend and single continuous covariate problems

signed rank tests for censored matched pairs abstract i consider the problem of testing bivariate symmetry in matched pair experiments where the observations are subject to univariate censoring thus the observable random variables are given by y 1 y 2 and 1 2 where y j min x j c and j i x j c j 1 2 here x 1 x 2 is a random pair of partially observable lifetimes and c is a fixed or random censoring variable the hypothesis to be tested is that x 1 x 2 and x 2 x 1 have the same distribution following woolson and lachenbruch 1980 i consider censored data generalizations of signed rank tests such as the sign signed wilcoxon and signed normal scores tests i derive the asymptotic distribution of these test statistics under fixed and contiguous alternatives the efficiencies of the signed rank tests are considered in a log linear model and compared with efficiencies of the paired prentice wilcoxon and log rank tests

accelerated hazards regression model and its adequacy for censored survival data summary the accelerated hazards regression model is introduced to study the relationship between survival times and covariates through a scale change between hazard functions the model is also compared with several other popular classes of regression models for censored survival data in statistical literature test statistics are proposed and studied to assess the model s adequacy actual data from a randomized clinical trial of biodegradable carmustine polymer for treatment of brain cancer are analyzed to demonstrate the potential application of the regression model and the proposed test statistics

rank tests and a one sample logrank test for comparing observed survival data to a standard population for cox s proportional hazards model test procedures are described for the comparison of randomly right censored observed and expected standard population survival data it is observed that the class of linear rank tests proposed by prentice 1978 biometrika 65 167 179 may be employed to test regression hypotheses in this one sample setting in order to compare globally the observed and expected survival data the one sample limit of mantel s logrank test is formally derived the statistic derived is identical to one proposed from a diflerent viewpoint by breslow 1975 intemational statistical review 43 45 58 throughout it is assumed that the standard population survival function is a continuous function of time and that the survival or censoring time for each study sample individual is exact and is not grouped into categories

asymptotic properties of censored linear rank tests a conjecture of prentice is established which states that for censored linear rank test exact scores based on conditional expectations can be replaced by approximate scores obtained by evaluating the score function at an estimate of the survival function we show that under minimal conditions asymptotically equivalent tests are obtained when either the kaplan meier altshuler or moment estimator of the survival function is used asymptotic normality is also established for a general random censorship model under the null hypothesis and for contiguous alternatives this is used to calculate efficacies and when the censoring times are i i d an expression for the asymptotic relative efficiency is given which is a natural generalization of the one for classical uncensored linear rank tests 1 introduction prentice 1978 has constructed a general class of linear rank tests for a regression model with censored data his approach was to specify a score function for a classical linear rank test in the absence of censoring chernoff and savage 1958 hajek and sidak 1967 to construct all possible rankings of the unobserved uncensored values which were consistent with observed censored sample and then to assign to each observation the average of all possible scores it could have received in the absence of censoring giving equal weighting to each possible uncensored ranking except for a few special scoring functions notably logarithmic or power law scores these averages are unwieldy when there is appreciable censoring and prentice suggested that asymptotically equivalent tests might arise if the score function was evaluated at some suitable estimate of the survivor function this is well known to be true for uncensored data but requires further justification when censoring is present because the estimate of the survivor function is more complicated in this work we show that prentice s conjecture is true under minimal conditions and also we show that it remains true if prentice s moment estimator of the survivor function is replaced by either the kaplan meier estimator or altshuler s estimator asymptotic normality of these tests is established under a random censorship model for the null hypothesis and contiguous alternatives by martingale methods this is used to compute efficacies for a general random censorship model and asymptotic relative efficiencies for i i d potential censoring times where fully efficient linear rank tests exist asymptotic normality for general alternatives has been established in cuzick 1982 by more complicated methods

conditional rank tests for the two sample problem under random censorship for the two sample problem with randomly censored data there exists a general asymptotic theory of rank statistics which are functionals of stochastic integrals with respect to certain empirical martingales in the present paper a conditional counterpart of this theory is developed the conditional martingales are versions of the original ones reduced to the unit interval having their jumps at fixed lattice points the resulting conditional tests are strictly distribution free under the null hypothesis of randomness if the censoring distributions in both samples are equal and are asymptotically equivalent to their unconditional counterparts even if the censoring distributions are different simulations for linear rank statistics and kolmogorov smirnov type statistics show superiority of the conditional versions over their unconditional counterparts with respect to size and robustness under unequal censoring in both samples at the same time the power of the conditional and unconditional tests is very similar in most cases

the algebraic structure of non commutative analytic toeplitz algebras the non commutative analytic toeplitz algebra is the wot closed algebra generated by the left regular representation of the free semigroup on n generators we develop a detailed picture of the algebraic structure of this algebra in particular we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n ball the k dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of h over the unit disk in 6 17 18 20 a good case is made that the appropriate analogue for the analytic toeplitz algebra in n non commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators the papers cited obtain a compelling analogue of beurling s theorem and inner outer factorization in this paper we add further evidence the main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of cn the kernel is the subgroup of quasi inner automorphisms which are trivial modulo the wot closed commutator ideal additional evidence of analytic properties comes from the structure of k dimensional completely contractive representations which have a structure very similar to the fibration of the maximal ideal space of h over the unit disk an important tool in our analysis is a detailed structure theory for wot closed right ideals curiously left ideals remain more obscure the non commutative analytic toeplitz algebra ln is determined by the left regular representation of the free semigroup fn on n generators z1 zn which acts on 2 fn by w v wv for v w in fn in particular the algebra ln is the unital wot closed algebra generated by the isometries li zi for 1 i n this algebra and its norm closed version the noncommutative disk algebra were introduced by popescu 19 in an abstract sense in connection with a non commutative von neumann inequality and 1991 mathematics subject classification 47d25 march 9 1997 october 9 1997 final draft first author partially supported by an nserc grant and a killam research fellowship second author partially supported by an nsf grant

a chi squared goodness of fit test for randomly censored data abstract in this article procedures analogous to karl pearson s well known chi squared goodness of fit test for a simple null hypothesis are developed under the random censorship model it is shown that one straightforward analog of pearson s statistic is diminished in applicability due to the form of its limiting distribution this leads to the development of an asymptotically exact test based on a wald type statistic with a chi squared limiting null distribution this test is compared and contrasted theoretically and via a simulation with akritas test with respect to significance levels asymptotic local powers and finite sample powers the general conclusions from the simulation study are that the proposed test usually achieves the desired significance levels when the probability of observing a censored or an uncensored value in the last interval is not small whereas akritas test tends to be a bit anticonservative on the other hand akritas test is more powerful than the proposed test in a model

the robustness properties of the linear quadratic regulators for singular systems the lqr linear quadratic regulator robustness properties of singular systems are investigated it is shown that the optimal regulators for singular systems with an appropriate selection of the feedback gain have the same robustness properties as those in regular systems namely at least 60 degrees guaranteed phase margin infinite gain margin and 50 gain reduction margin

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

rank based estimating equations with general weight for accelerated failure time models an induced smoothing approach the induced smoothing technique overcomes the difficulties caused by the non smoothness in rank based estimating functions for accelerated failure time models but it is only natural when the estimating function has gehan s weight for a general weight the induced smoothing method does not provide smooth estimating functions that can be easily evaluated we propose an iterative induced smoothing procedure for general weights with the estimator from gehan s weight initial value the resulting estimators have the same asymptotic properties as those from the non smooth estimating equations with the same weight their variances are estimated with an efficient resampling approach that avoids solving estimating equations repeatedly the methodology is generalized to incorporate an additional weight to accommodate missing data and various sampling schemes in a numerical study the proposed estimators were obtained much faster without losing accuracy in comparison to those from non smooth estimating equations and the variance estimators provided good approximation of the variation in estimation the methodology was applied to two real datasets the first one from an adolescent depression study and the second one from a cancer study with missing covariates by design the implementation is available in an r package aftgee

a relationship between two forms of linear rank procedures for censored data summary this paper gives a useful relationship between two popular classes of linear rank statistics for arbitrarily right censored data the relationship allows a linear rank procedure with certain optimal properties to be transformed into the form of a procedure having established asymptotic properties under proportional hazards alternatives

weighted log rank statistics for comparing two distributions summary the derivation of locally most powerful rank statistics for comparing two censored samples against parametric alternatives is reformulated for more general classes of alternatives a particular class is considered which includes the case of two weibull distributions differing in shape parameter

maximum weighted partial likelihood estimators for the cox model new estimators for cox regression that maximize a weighted version of the partial likelihood are considered their asymptotic properties on the model are sketched estimators with bounded influence curves are identified and shown to have good asymptotic relative efficiency the results of a small simulation study are included the estimators are applied to a well known data set the possible influence of an outlying observation on the cox estimator can be of any magnitude individuals who survive longest have a high potential of being influential in analogy with weighted linear rank tests we introduce estimators of the regression coefficients that maximize a weighted partial likelihood for instance by using weights proportional to the number at risk one obtains a wilcoxon type estimator this estimator has bounded influence provided the covariate space is compact and in a wide variety of situations its asymptotic relative efficiency is at least as good as that of the wilcoxon test asymptotic prop

estimation and testing in a two sample generalized odds rate model abstract the generalized odds ratio for a survival variable t is defined as t t c c 1 1 sc t sc t for c 0 and log s t for c 0 where s t pr t t this ratio coincides with the integrated hazard for c 0 and the odds ratio for c 1 when the distribution of t depends on a covariate vector we assume that conditionally on the covariates log t t c is linear in the covariates this model is a generalization of the proportional hazard model phm which has an interpretation both as a phm with random nuisance effects clayton and cuzick 1986 and as a proportional odds rate model with the odds rate defined from the response times of series systems harrington and fleming 1982 and bickel 1986a considered rank tests for this semiparametric model clayton and cuzick 1986 considered estimation we use the odds rate representation to define a class of estimates of the proportionality parameter in the two sample case we show that the estimates are consistent and asymptotica

descriptor variable systems and optimal state regulation linear systems of the form e dot x ax bu with e singular are treated it is desired to find a control which drives the system asymptotically to the origin minimizing a quadratic cost functional no restrictions are placed on initial conditions the cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order riccati equation

sample size calculation for cluster randomization trials with a time to event endpoint cluster randomization trials randomize groups called clusters of subjects called subunits between intervention arms and observations are collected from each subject in this case subunits within each cluster share common frailties so that the observations from subunits of each cluster tend to be correlated oftentimes the outcome of a cluster randomization trial is a time to event endpoint with censoring in this article we propose a closed form sample size formula for weighted rank tests to compare the marginal survival distributions between intervention arms under cluster randomization with possibly variable cluster sizes extensive simulation studies are conducted to evaluate the performance of our sample size formula under various design settings real study examples are taken to demonstrate our method

a comparison of estimated proportional hazards models and regression trees we present examples of the usage of regression trees for censored response via two real world datasets one a rheumatoid arthritis survival study and the other a hip replacement study and draw comparisons with the results of cox proportional hazards modelling the two methods pursue different goals motivation of the tree techniques is the desire to extract meaningful prognostic groups while the proportional hazards model enables assessment of the impact of risk factors the methods are thus complementary for the arthritis study the two techniques corroborate one another although the flavour of the conclusions derived differ for the hip replacement study however the regression tree approach reveals structure that would not emerge from a routine proportional hazards analysis we also discuss the treatment of data analytic issues such as the handling of missing values and influence in the presence of non uniform censoring

a descriptor system approach to singular perturbation of linear regulators the relationship between the regulator problem for a singularly perturbed system and the analogous one for a descriptor system is investigated meanwhile a decomposition approach is provided for solution of the linear quadratic regulator problem for singularly perturbed systems this approach is a significant advance over the two stage method

two sample asymptotically distribution free tests for incomplete multivariate observations abstract asymptotically distribution free tests for equality of two multivariate distributions based on censored observations are proposed by choosing appropriate test scores we obtain the multivariate versions of the gehan and log rank tests an important application of the proposed tests is in the area in which multiple observations are collected on study subjects some of whom may have missing observations the patterns of missing observations are allowed to differ for the two groups to be compared these tests are particularly useful for longitudinal studies for example the analysis of repeated measurements and growth curves real examples are also provided for illustration

an algorithm for solving the singularly perturbed h algebraic riccati equation abstract the algebraic riccati equation of singularly perturbed h linear quadratic optimal control problems is decoupled into two completely independent reduced order pure slow and pure fast h algebraic riccati equations even though the algebraic riccati equations obtained are nonsymmetric they are efficiently solved in terms of lyapunov iterations by using the newton method the presented algorithm can be used for h parallel filtering and regulation of systems displaying two time scale dynamics

a generalization of the weibull distribution with application to the analysis of survival data abstract the weibull distribution which is frequently used for modeling survival data is embedded in a larger family obtained by introducing an additional shape parameter this generalized family not only contains distributions with unimodal and bathtub hazard shapes but also allows for a broader class of monotone hazard rates furthermore the distributions in this family are analytically tractable and computationally manageable the modeling and analysis of survival data using this family is discussed and illustrated in terms of a lifetime dataset and the results of a two arm clinical trial

a note on the relation between two forms of linear rank statistics for right censored and grouped data we show that two forms of linear rank statistics for right censored and grouped data are equivalent in the case of no tied value among uncensored observations we provide some intuition about the equivalence

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

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

a method for determining groups in multiple survival curves survival analysis includes a wide variety of methods for analyzing time to event data one basic but important goal in survival analysis is the comparison of survival curves between groups several nonparametric methods have been proposed in the literature to test for the equality of survival curves for censored data when the null hypothesis of equality of curves is rejected leading to the clear conclusion that at least one curve is different it can be interesting to ascertain whether curves can be grouped or if all these curves are different from each other a method is proposed that allows determining groups with an automatic selection of their number the validity and behavior of the proposed method was evaluated through simulation studies the applicability of the proposed method is illustrated using real data software in the form of an r package has been developed implementing the proposed method

a monte carlo study of some two sample rank tests with censored data abstract the powers of several nonparametric tests for the two sample problem with censored data are compared by simulation the tests studied include gehan s efron s and peto s and prentice s generalized wilcoxon tests along with the logrank test the test with the greatest power changes with the sample sizes censoring mechanism and distribution of the random variables the test that performed the best overall was the peto prentice generalized wilcoxon statistic with an asymptotic variance estimate

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

weighted log rank type statistics for comparing survival curves when there is a time lag in the effectiveness of treatment summary in certain long term treatment trials one expects some lag period before the treatment is fully effective this paper presents two weighted log rank type statistics designed to have good efficiency over a wide range of lags a maximin efficiency robust statistic and a simplified version of this statistic both statistics can be computed as easily as the log rank statistic asymptotic efficiency calculations supported by small sample simulations show that the two proposed statistics are substantially more efficient than the conventional log rank statistic in certain lag situations with comparatively little efficiency loss relative to the log rank statistic when no lag exists we recommend the maximin statistic for situations where a lag is expected but cannot be specified precisely in advance

linear rank tests for the nonresponders problem with censored data summary in a two sample setting a two parameter mixture model is postulated for the treated patients where a proportion of the patients improve and the remaining proportion of nonresponders behave like the control patients the response is subject to random censoring censored data versions of linear rank statistics are developed in order to detect improvement due to treatment the tests are compared to standard tests such as the logrank test in terms of asymptotic relative efficiency empirical power and performance on real data

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

comparing two samples extensions of the t rank sum and log rank tests abstract i consider the problem of testing that two populations are identical with respect to the distribution of a continuous variable against the alternative that values tend to be larger in one population i observe that the t rank sum and log rank tests are insensitive for a large class of alternatives that may be expected to occur commonly in practice propose a criterion for identifying this problem in a particular data set and propose corresponding extensions of the conventional methods the proposed methods should be useful for both identifying and interpreting group differences

rank tests for survival analysis a comparison by analogy with games several commonly used two sample linear rank tests for censored data have been shown to be closely linked in mathematical structure this pedagogical paper elucidates the relationships among the tests by introducing a videogame played between a boys team and a girls team with different rules of payment by the consecutive losers considered depending on the rule of payment the total amount of money the girls win as a team becomes the wilcoxon statistic the savage statistic or their generalizations under censoring

rank tests for clustered survival data when dependent subunits are randomized in clustered survival data subunits within each cluster share similar characteristics so that observations made from them tend to be positively correlated in clinical trials the correlated subunits from the same cluster are often randomized to different treatment groups in this case the variance formulas of the standard rank tests such as the logrank gehan wilcoxon or prentice wilcoxon proposed for independent samples need to be adjusted for intracluster correlations both within and between treatment groups for testing equality of marginal survival distributions in this paper we derive a general form of simple variance formulas of the rank tests when subunits from the same cluster are randomized into different treatment groups extensive simulation studies are conducted to investigate small sample performance of the variance formulas we compare our non parametric rank tests based on the adjusted variances with one from a shared frailty model which is an optimal semi parametric testing procedure when the intracluster correlations within and between groups are the same

estimating regression parameters using linear rank tests for censored data a class of estimates for regression parameters in a linear model with right censored data is proposed these estimates are derived by using linear rank tests for right censored data as estimating equations they are shown to be consistent and asymptotically normal with covariance matrix for which estimates are proposed efficient estimates within this class are derived together with conditions when they are fully efficient 0 introduction the problem of estimating regression coefficients using ranks has received much attention in the literature for uncensored data and the results are summarized very nicely in hettmansperger 1984 there has also been some work with censored data for the two sample problem specifically louis 1981 considered estimation of a scale change between two distributions with the use of a log rank test whereas wei and gail 1983 generalize to other linear rank tests as well we shall consider the general problem of multiple linear regression and show how linear rank tests can be used as estimating equations the resulting linear rank estimates will be shown to be asymptotically normal with variances that can be estimated consistently we shall also look at the efficiency of these estimates under various situations

the problem of a covariate time qualitative interaction in a survival study we introduce a test for the equality of two survival distributions against the specific alternative of crossing hazards although this kind of alternative is somewhat rare designing a test specifically aimed at detecting such departures from the null hypothesis in this direction leads to powerful procedures upon which we can call in those few cases where such departures are suspected furthermore the proposed test and an approximate version of the test are seen to suffer only moderate losses in power when compared with their optimal counterparts should the alternative be one of proportional hazards our interest in the problem is motivated by clinical studies on the role of acute graft versus host disease as a risk factor in leukemic children and we discuss the analysis of this study in detail the model we use in this work is a special case of the one introduced by anderson and senthilselvan 1982 applied statistics 31 44 51 we propose overcoming an inferential problem stemming from their model by using the methods of davies 1977 biometrika 64 247 254 1987 biometrika 74 33 43 backed up by resampling techniques we also look at an approach relying directly on resampling techniques the distributional aspects of this approach under the null hypothesis are interesting but practically its behaviour is such that its use cannot be generally recommended outlines of the necessary asymptotic theory are presented and for this we use the tools of martingale theory

hilbert spaces of dirichlet series and their multipliers we consider various hilbert spaces of dirichlet series whose norms are given by weighted 2 norms of the dirichlet coefficients we describe the multiplier algebras of these spaces the functions in the multiplier algebra may or may not extend to be analytic on a larger half plane than the functions in the hilbert space

monotone estimating equations for censored data the monotone class rank test based estimating equations for regression models with right censored data is considered we introduce an estimator which is a solution of a monotone estimating equation that is an extension of the gehan test the estimator is easy to derive sqrt n consistent and asymptotically normal under minimal conditions all monotone estimating equations are characterized and a simulation study which shows that our suggested procedure performs well is included

a general view of multipliers and composition operators this paper defines a multiplier pair which is a very general setting in which the notions of multipliers and composition operators can be studied we prove some results in this general setting and we present many new interesting examples in which these notions can be interpreted

tests for location with k samples and censored data summary tests which are specific for location are developed in the k sample situation with right censored data the tests are based on general estimating functions and do not require full parametric assumptions for large samples but in general they are not based only on ranks one class of tests allows shapes to differ in the k populations and includes as a special case the median test of brookmeyer crowley 1982 if the distributions are assumed to differ at most in location modified tests are suggested in which survival function estimates are obtained by pooling the data from all samples prentice s 1978 median test with minor modifications is included in this class

a two sample test with interval censored data via multiple imputation interval censored data arise naturally in large scale panel studies where subjects can only be followed periodically and the event of interest can only be recorded as having occurred between two examination times in this paper we consider the problem of comparing two interval censored samples we propose to impute exact failure times from interval censored observations to obtain right censored data then apply existing techniques such as harrington and fleming s g rho tests to imputed right censored data to appropriately account for variability a multiple imputation algorithm based on the approximate bayesian bootstrap abb is discussed through simulation studies we find that it performs well the advantage of our proposal is its simplicity to implement and adaptability to incorporate many existing two sample comparison techniques for right censored data the method is illustrated by reanalysing the breast cosmesis study data set

linear rank statistics for truncated data summary motivated by prentice s 1978 generalization of linear rank statistics for censored data we introduce a similar kernel for deriving linear rank statistics for truncated data they take very similar forms to those of prentice s statistics moreover we also show that they can be expressed equivalently in weighted log rank type forms thus providing natural variance estimators explicit forms corresponding to wilcoxon and log rank tests are obtained extensions to data subject to both left truncation and right censoring are provided

three classes of censored data rank tests strengths and weaknesses under censoring summary nonparametric tests are useful when definite parametric alternatives are not available the behaviour under censoring of extensions of the tests is used to delineate three distinct classes of censored data rank tests asymptotically efficient approximately distribution free and approximately unbiased section 2 gives qualitative comparisons 3 gives some representative numerical comparisons the pitman efficiencies approximated by lee desu gehan 1975 from simulations with 50 observations in each group are shown to agree with asymptotic calculations 1 weighted log rank tests it is assumed that two samples of randomly censored observations are available that is that the true lifetimes yij have cumulative distribution functions f y i oj for j 1 n i i 1 2 that the censoring times cij have cumulative distribution functions h y i of and that the observable quantities are xij min yij c and dij xij yij where a denotes the indicator function of the set a it is assumed that 01 0 that f t 1 0 is continuous in 0 and that the null hypothesis is 61 02 0 the true value of 02 will be denoted by 0 and the functions f t i 0 and h t i 0 will be abbreviated to f and h survival distributions will be denoted by a bar so that f t 1 f t the hazard rates of f and f t 10 will be denoted by a and ao respectively the test statistics studied are functions of di t the number of deaths observed before time t in the ith sample ri t the number at risk at time t in the ith sample and aj an estimator of the cumulative hazard function for the ith sample these functions are defined formally by n i n i