rationality problem of three dimensional monomial group actions abstract let k be a field of characteristic not two and k x y z the rational function field over k with three variables x y z let g be a finite group acting on k x y z by monomial k automorphisms we consider the rationality problem of the fixed field k x y z g under the action of g namely whether k x y z g is rational that is purely transcendental over k or not we may assume that g is a subgroup of gl 3 z and the problem is determined up to conjugacy in gl 3 z there are 73 conjugacy classes of g in gl 3 z by results of endo miyata voskresenski lenstra saltman hajja kang and yamasaki 8 conjugacy classes of 2 groups in gl 3 z have negative answers to the problem under certain monomial actions over some base field k and the necessary and sufficient condition for the rationality of k x y z g over k is given in this paper we show that the fixed field k x y z g under monomial action of g is rational over k except for possibly negative 8 cases of 2 groups and unknown one case of the alternating group of degree four moreover we give explicit transcendental bases of the fixed fields over k for the unknown case we obtain an affirmative solution to the problem under some conditions in particular we show that if k is quadratically closed field then k x y z g is rational over k we also give an application of the result to 4 dimensional linear noether s problem

conservative hypothesis tests and confidence intervals using importance sampling importance sampling is a common technique for monte carlo approximation including that of p values here it is shown that a simple correction of the usual importance sampling p values provides valid p values meaning that a hypothesis test created by rejecting the null hypothesis when the p value is at most will also have a type i error rate of at most this correction uses the importance weight of the original observation which gives valuable diagnostic information under the null hypothesis using the corrected p values can be crucial for multiple testing and also in problems where evaluating the accuracy of importance sampling approximations is difficult inverting the corrected p values provides a useful way to create monte carlo confidence intervals that maintain the nominal significance level and use only a single monte carlo sample

multivariate non parametric methods for mann whitney statistics to analyse cross over studies with two treatment sequences a non parametric strategy for the analysis of ordinal data from cross over studies with two treatment sequences and d or 2 periods is examined through mann whitney rank measures of association for each period these statistics estimate the probability of larger response for a randomly selected patient in one group relative to a randomly selected patient in the other group such estimates are as well formed for comparisons between groups for u pairs of periods with the same treatment methods for u statistics are used to produce a consistent estimate of the covariance matrix for the d u mann whitney estimates the effects of periods and treatments on the respective mann whitney estimates are evaluated through linear or log linear models for estimation of the parameters in these models a modified weighted least squares method is applied through a 2d 1 or d u dimensional basis which effectively addresses potentially near singularities in the estimated covariance matrix of the mann whitney estimates the proposed methods are applicable to response variables with an interval or an ordered categorical scale their scope additionally has capabilities for controlling strata in the design of a cross over study or concomitant variables for which covariance adjustment is of interest for reduction of variance applications of the methods are illustrated through three cross over studies with different specifications for the two sequences of two treatments during two to four periods

parsimonious analysis of time dependent effects in the cox model cox s proportional hazards model can be extended to accommodate time dependent effects of prognostic factors we briefly review these extensions along with their varying degrees of freedom spending more degrees of freedom with conventional procedures a priori defined interactions with simple functions of time restricted natural splines piecewise estimation for partitions of the time axis allows the fitting of almost any shape of time dependence but at an increased risk of over fit this results in increased width of confidence intervals of time dependent hazard ratios and in reduced power to confirm any time dependent effect or even any effect of a prognostic factor by means of comparative empirical studies the consequences of over fitting time dependent effects have been explored we conclude that fractional polynomials and similarly penalized likelihood approaches today are the methods of choice avoiding over fit by parsimonious use of degrees of freedom but also permitting flexible modelling if time dependence of a usually a priori unknown shape is present in a data set the paradigm of a parsimonious analysis of time dependent effects is exemplified by means of a gastric cancer study

assessing time by covariate interactions in proportional hazards regression models using cubic spline functions proportional hazards or cox regression is a popular method for modelling the effects of prognostic factors on survival use of cubic spline functions to model time by covariate interactions in cox regression allows investigation of the shape of a possible covariate time dependence without having to specify a specific functional form cubic spline functions allow one to graph such time by covariate interactions to test formally for the proportional hazards assumption and also to test for non linearity of the time by covariate interaction the functions can be fitted with existing software using relatively few parameters the regression coefficients are estimated using standard maximum likelihood methods

an overall strategy based on regression models to estimate relative survival and model the effects of prognostic factors in cancer survival studies relative survival provides a measure of the proportion of patients dying from the disease under study without requiring the knowledge of the cause of death we propose an overall strategy based on regression models to estimate the relative survival and model the effects of potential prognostic factors the baseline hazard was modelled until 10 years follow up using parametric continuous functions six models including cubic regression splines were considered and the akaike information criterion was used to select the final model this approach yielded smooth and reliable estimates of mortality hazard and allowed us to deal with sparse data taking into account all the available information splines were also used to model simultaneously non linear effects of continuous covariates and time dependent hazard ratios this led to a graphical representation of the hazard ratio that can be useful for clinical interpretation estimates of these models were obtained by likelihood maximization we showed that these estimates could be also obtained using standard algorithms for poisson regression

confidence intervals for multinomial logistic regression in sparse data logistic regression is one of the most widely used regression models in practice but alternatives to conventional maximum likelihood estimation methods may be more appropriate for small or sparse samples modification of the logistic regression score function to remove first order bias is equivalent to penalizing the likelihood by the jeffreys prior and yields penalized maximum likelihood estimates ples that always exist even in samples in which maximum likelihood estimates mles are infinite ples are an attractive alternative in small to moderate sized samples and are preferred to exact conditional mles when there are continuous covariates we present methods to construct confidence intervals ci in the penalized multinomial logistic regression model and compare ci coverage and length for the ple based methods to that of conventional mle based methods in trinomial logistic regressions with both binary and continuous covariates based on simulation studies in sparse data sets we recommend profile cis over asymptotic wald type intervals for the ples in all cases furthermore when finite sample bias and data separation are likely to occur we prefer ple profile cis over mle methods

on latent trait estimation in multidimensional compensatory item response models making inferences from irt based test scores requires accurate and reliable methods of person parameter estimation given an already calibrated set of item parameters the latent trait could be estimated either via maximum likelihood estimation mle or using bayesian methods such as maximum a posteriori map estimation or expected a posteriori eap estimation in addition warm s psychometrika 54 427 450 1989 weighted likelihood estimation method was proposed to reduce the bias of the latent trait estimate in unidimensional models in this paper we extend the weighted mle method to multidimensional models this new method denoted as multivariate weighted mle mwle is proposed to reduce the bias of the mle even for short tests mwle is compared to alternative estimators i e mle map and eap and shown both analytically and through simulations studies to be more accurate in terms of bias than mle while maintaining a similar variance in contrast bayesian estimators i e map and eap result in biased estimates with smaller variability

efficient monte carlo methods for conditional logistic regression abstract exact inference for the logistic regression model is based on generating the permutation distribution of the sufficient statistics for the regression parameters of interest conditional on the sufficient statistics for the remaining nuisance parameters despite the availability of fast numerical algorithms for the exact computations there are numerous instances where a data set is too large to be analyzed by the exact methods yet too sparse or unbalanced for the maximum likelihood approach to be reliable what is needed is a monte carlo alternative to the exact conditional approach which can bridge the gap between the exact and asymptotic methods of inference the problem is technically hard because conventional monte carlo methods lead to massive rejection of samples that do not satisfy the linear integer constraints of the conditional distribution we propose a network sampling approach to the monte carlo problem that eliminates rejection entirely its advantages over alternative saddlepoint

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

negative solutions to three dimensional monomial noether problem three dimensional monomial noether problem can have negative solutions for 8 groups by the suitable choice of the coefficients we find the necessary and sufficient condition for the coefficients to have a negative solution the results are obtained by two criteria of irrationality using galois cohomology

binary regression models for contaminated data on etudie la nature des observations aberrantes dans le contexte des donnees de regression binaires on propose un modele simple qui permet qu un petit nombre de reponses binaires soit mal enregistre

jackknife bias reduction for polychotomous logistic regression despite theoretical and empirical evidence that the usual mles can be misleading in finite samples and some evidence that bias reduced estimates are less biased and more efficient they have not seen a wide application in practice one can obtain bias reduced estimates by jackknife methods with or without full iteration or by use of higher order terms in a taylor series expansion of the log likelihood to approximate asymptotic bias we provide details of these methods for polychotomous logistic regression with a nominal categorical response we conducted a monte carlo comparison of the jackknife and taylor series estimates in moderate sample sizes in a general logistic regression setting to investigate dichotomous and trichotomous responses and a mixture of correlated and uncorrelated binary and normal covariates we found an approximate two step jackknife and the taylor series methods useful when the ratio of the number of observations to the number of parameters is greater than 15 but we cannot recommend the two step and the fully iterated jackknife estimates when this ratio is less than 20 especially when there are large effects binary covariates or multicollinearity in the covariates

computing distributions for exact logistic regression abstract logistic regression is a commonly used technique for the analysis of retrospective and prospective epidemiological and clinical studies with binary response variables usually this analysis is performed using large sample approximations when the sample size is small or the data structure sparse the accuracy of the asymptotic approximations is in question on other occasions singularity of the covariance matrix of parameter estimates precludes asymptotic analysis under these circumstances use of exact inferential procedures would seem to be a prudent alternative cox 1970 showed that exact inference on the parameters of a logistic model with binary response requires consideration of the distribution of sufficient statistics for these parameters to date however resorting to the exact method has not been computationally feasible except in a few special situations this article presents an efficient recursive algorithm that generates the joint and conditional distributions of the sufficient

exact logistic regression theory and examples we provide an alternative to the maximum likelihood method for making inferences about the parameters of the logistic regression model the method is based appropriate permutational distributions of sufficient statistics it is useful for analysing small or unbalanced binary data with covariates it also applies to small sample clustered binary data we illustrate the method by analysing several biomedical data sets

exact inference for categorical data recent advances and continuing controversies methods for exact small sample analyses with categorical data have been increasingly well developed in recent years a variety of exact methods exist primarily using the approach that eliminates unknown parameters by conditioning on their sufficient statistics in addition a variety of algorithms now exist for implementing the methods this paper briefly summarizes the exact approaches and describes recent developments controversy continues about the appropriateness of some exact methods primarily relating to their conservative nature because of discreteness this issue is examined for two simple problems in which discreteness can be severe interval estimation of a proportion and the odds ratio in general adjusted exact methods based on the mid p value seem a reasonable way of reducing the severity of this problem

on the scalar curvature of constant mean curvature hypersurfaces in space forms abstract in this paper we study the behavior of the scalar curvature s of a complete hypersurface immersed with constant mean curvature into a riemannian space form of constant curvature deriving a sharp estimate for the infimum of s our results will be an application of a weak omori yau maximum principle due to pigola rigoli setti 2005 17

analysis of overfitting in the regularized cox model the cox proportional hazards model is ubiquitous in the analysis of time to event data however when the data dimension p is comparable to the sample size n maximum likelihood estimates for its regression parameters are known to be biased or break down entirely due to overfitting this prompted the introduction of the so called regularized cox model in this paper we use the replica method from statistical physics to investigate the relationship between the true and inferred regression parameters in regularized multivariate cox regression with lalt subagt 2alt subagt regularization in the regime where both p and n are large but with p n 1 we thereby generalize a recent study from maximum likelihood to maximum a posteriori inference we also establish a relationship between the optimal regularization parameter and allowing for straightforward overfitting corrections in time to event analysis

information criteria for firth s penalized partial likelihood approach in cox regression models in the estimation of cox regression models maximum partial likelihood estimates might be infinite in a monotone likelihood setting where partial likelihood converges to a finite value and parameter estimates converge to infinite values to address monotone likelihood previous studies have applied firth s bias correction method to cox regression models however while the model selection criteria for firth s penalized partial likelihood approach have not yet been studied a heuristic aic type information criterion can be used in a statistical package application of the heuristic information criterion to data obtained from a prospective observational study of patients with multiple brain metastases indicated that the heuristic information criterion selects models with many parameters and ignores the adequacy of the model moreover we showed that the heuristic information criterion tends to select models with many regression parameters as the sample size increases thereby in the present study we propose an alternative aic type information criterion based on the risk function a bayesian information criterion type was also evaluated further the presented simulation results confirm that the proposed criteria performed well in a monotone likelihood setting the proposed aic type criterion was applied to prospective observational study data copyright 2017 john wiley sons ltd

bias reduced and separation proof conditional logistic regression with small or sparse data sets conditional logistic regression is used for the analysis of binary outcomes when subjects are stratified into several subsets e g matched pairs or blocks log odds ratio estimates are usually found by maximizing the conditional likelihood this approach eliminates all strata specific parameters by conditioning on the number of events within each stratum however in the analyses of both an animal experiment and a lung cancer case control study conditional maximum likelihood cml resulted in infinite odds ratio estimates and monotone likelihood estimation can be improved by using cytel inc s well known logxact software which provides a median unbiased estimate and exact or mid p confidence intervals here we suggest and outline point and interval estimation based on maximization of a penalized conditional likelihood in the spirit of firth s biometrika 1993 80 27 38 bias correction method cfl we present comparative analyses of both studies demonstrating some advantages of cfl over competitors we report on a small sample simulation study where cfl log odds ratio estimates were almost unbiased whereas logxact estimates showed some bias and cml estimates exhibited serious bias confidence intervals and tests based on the penalized conditional likelihood had close to nominal coverage rates and yielded highest power among all methods compared respectively therefore we propose cfl as an attractive solution to the stratified analysis of binary data irrespective of the occurrence of monotone likelihood a sas program implementing cfl is available at http www muw ac at msi biometrie programs

penalization bias reduction and default priors in logistic and related categorical and survival regressions penalization is a very general method of stabilizing or regularizing estimates which has both frequentist and bayesian rationales we consider some questions that arise when considering alternative penalties for logistic regression and related models the most widely programmed penalty appears to be the firth small sample bias reduction method albeit with small differences among implementations and the results they provide which corresponds to using the log density of the jeffreys invariant prior distribution as a penalty function the latter representation raises some serious contextual objections to the firth reduction which also apply to alternative penalties based on t distributions including cauchy priors taking simplicity of implementation and interpretation as our chief criteria we propose that the log f 1 1 prior provides a better default penalty than other proposals penalization based on more general log f priors is trivial to implement and facilitates mean squared error reduction and sensitivity analyses of penalty strength by varying the number of prior degrees of freedom we caution however against penalization of intercepts which are unduly sensitive to covariate coding and design idiosyncrasies copyright 2015 john wiley sons ltd

on obtaining permutation distributions in polynomial time abstract polynomial time algorithms are presented for finding the permutation distribution of any statistic that is a linear combination of some function of either the original observations or the ranks this class of statistics includes the original fisher two sample location statistic and such common nonparametric statistics as the wilcoxon ansari bradley savage and many others the algorithms are presented for the two sample problem and it is shown how to extend them to the multisample problem for example to find the distribution of the kruskal wallis and other extensions of the wilcoxon and to the single sample situation stratification ties and censored observations are also easily handled by the algorithms the algorithms require polynomial time as opposed to complete enumeration algorithms which require exponential time this savings is effected by first calculating and then inverting the characteristic function of the statistic

bias reduction of maximum likelihood estimates summary it is shown how in regular parametric problems the first order term is removed from the asymptotic bias of maximum likelihood estimates by a suitable modification of the score function in exponential families with canonical parameterization the effect is to penalize the likelihood by the jeffreys invariant prior in binomial logistic models poisson log linear models and certain other generalized linear models the jeffreys prior penalty function can be imposed in standard regression software using a scheme of iterative adjustments to the data

computing exact distributions for polytomous response data abstract this article presents an efficient method for computing exact conditional distributions of the sufficient statistics for the parameters of four polytomous response models for nominal response two baseline category logit models and for ordinal response two adjacent categories logit models are considered the method consists of recursive generation of the joint distribution of the sufficient statistics augmented by a technique to slice off portions of the evolving sample space so as to eventually yield the required conditional distribution two actual data sets are analyzed to illustrate the method

avoiding infinite estimates of time dependent effects in small sample survival studies we address the phenomenon of monotone likelihood in cox regression with time dependent effects monotone likelihood occurs in the fitting process of a cox model if at least one parameter estimate diverges to infinity we show that the probability of monotone likelihood is increased by the inclusion of time dependent effects particularly in small samples with several unbalanced and highly predictive covariates and with a high percentage of censoring firth s bias reduction procedure was shown to provide an ideal solution to monotone likelihood here we extend his idea to cox regression with time dependent effects by penalized maximum likelihood estimation finite hazard ratio estimates of constant and time dependent effects can be obtained penalized likelihood ratio tests and profile penalized likelihood confidence intervals are proposed as tools for inference a monte carlo study of cox regression with time dependent effects confirms advantages of firth corrected fc over standard cox analysis in terms of average bias and median absolute deviation we also compare the fc and standard cox approaches by means of analyses of two studies with time dependent effects an sas macro and an r package for fc cox regression with time varying covariates and time dependent effects are available at http www muw ac at msi biometrie programs copyright 2008 john wiley sons ltd

blocked designs for experiments with correlated non normal response many experiments measure a response that cannot be adequately described by a linear model with normally distributed errors and are often run in blocks of homogeneous experimental units we develop the first general methods of obtaining efficient blocked designs for experiments with an exponential family response described by a marginal model fitted via generalized estimating equations this methodology is appropriate when the blocking factor is a nuisance variable as often occurs in industrial experiments a d optimality criterion is developed for finding designs robust to the values of the marginal model parameters and applied using two strategies unrestricted algorithmic search and blocking of an optimal design for the corresponding generalized linear model glm designs from each strategy are critically compared and shown to be more efficient than use of a random allocation to blocks of the points from an optimal glm design the designs are compared for a range of values of the intra block working c

comment on bias reduction in conditional logistic regression by j x sun s sinha s wang and t maiti statistics in medicine 2010 doi 10 1002 sim 4105

an algorithm for exact logistic regression abstract an efficient algorithm is given for the exact logistic analysis of a single parameter hypothesis tests and confidence intervals for a logistic regression parameter are obtained the method provides for the control of the effect of a stratification factor when making inferences concerning the regression parameter

on the existence of maximum likelihood estimates in logistic regression models summary the problems of existence uniqueness and location of maximum likelihood estimates in log linear models have received special attention in the literature haberman 1974 chapter 2 wedderburn 1976 silvapulle 1981 for multinomial logistic regression models we prove existence theorems by considering the possible patterns of data points which fall into three mutually exclusive and exhaustive categories complete separation quasicomplete separation and overlap our results suggest general rules for identifying infinite parameter estimates in log linear models for frequency tables

multiple imputation of industry and occupation codes in census public use samples using bayesian logistic regression abstract we describe methods used to create a new census data base that can be used to study comparability of industry and occupation classification systems this project represents the most extensive application of multiple imputation to date and the modeling effort was considerable as well hundreds of logistic regressions were estimated one goal of this article is to summarize the strategies used in the project so that researchers can better understand how the new data bases were created another goal is to show how modifications of maximum likelihood methods were made for the modeling and imputation phases of the project to multiply impute 1980 census comparable codes for industries and occupations in two 1970 census public use samples logistic regression models were estimated with flattening constants for many of the regression models considered the data were too sparse to support conventional maximum likelihood analysis so some alternative had to be employed these methods solve existence and

quasi monomial actions and some 4 dimensional rationality problems abstract let g be a finite group acting on k x 1 x n the rational function field of n variables over a field k the action is called a purely monomial action if x j 1 i n x i a i j for all g for 1 j n where a i j 1 i j n gl n z the main question is that under what situations the fixed field k x 1 x n g is rational purely transcendental over k this rationality problem has been studied by hajja kang hoshi rikuna when n 3 in this paper we will prove that k x 1 x 2 x 3 x 4 g is rational over k provided that the purely monomial action is decomposable to prove this result we introduce a new notion the quasi monomial action which is a generalization of previous notions of multiplicative group actions moreover we determine the rationality problem of purely quasi monomial actions of k x y g over k where k k g

approximate monte carlo conditional inference in exponential families summary this article presents an algorithm for approximate frequentist conditional inference on two or more parameters for any regression model in the generalized linear model glim family we thereby extend highly accurate inference beyond the cases of logistic regression and contingency tables implimented in commercially available software the method makes use of the double saddlepoint approximations of skovgaard 1987 journal of applied probability24 875 887 and jensen 1992 biometrika79 693 703 to the conditional cumulative distribution function of a sufficient statistic given the remaining sufficient statistics this approximation is then used in conjunction with noniterative monte carlo methods to generate a sample from a distribution that approximates the joint distribution of the sufficient statistics associated with the parameters of interest conditional on the observed values of the sufficient statistics associated with the nuisance parameters this algorithm is an alternate approach to that presented by kolassa and tanner 1994 journal of the american statistical association89 697 702 in which a markov chain is generated whose equilibrium distribution under certain regularity conditions approximates the joint distribution of interest in kolassa and tanner 1994 the gibbs sampler was used in conjunction with these univariate conditional distribution function approximations the method of this paper does not require the construction and simulation of a markov chain thus avoiding the need to develop regularity conditions under which the algorithm converges and the need for the data analyst to check convergence of the particular chain examples involving logistic and truncated poisson regression are presented

characterizations of linear weingarten submanifolds abstract in this paper we deal with complete linear weingarten submanifolds m n immersed with parallel normalized mean curvature vector field in a riemannian space form q c n p of constant sectional curvature c and with n 4 we recall that a submanifold is called linear weingarten when its mean and scalar curvatures are linearly related in this setting we establish a suitable extension of the generalized maximum principle of omori yau in order to show that such a submanifold m n must be either totally umbilical or isometric to a clifford torus s 1 1 r 2 s n 1 r when c 1 a circular cylinder r s n 1 r when c 0 or a hyperbolic cylinder h 1 1 r 2 s n 1 r when c 1 we also study the parabolicity of these submanifolds with respect to a cheng yau modified operator

a comparison of exact mid p and score tests for matched case control studies a recently developed algorithm for generating the distribution of sufficient statistics for conditional logistic models can be put to a twofold use first it provides an avenue for performing inference for matched case control studies that does not rely on the assumption of a large sample size second joint distributions generated by this algorithm can be used to make comparisons of various inferential procedures that are free from monte carlo sampling errors in this paper these two features of the algorithm are utilized to compare small sample properties of the exact mid p value and score tests for a conditional logistic model with two unmatched binary covariates both uniparametric and multiparametric tests performed at a nominal significance level of 05 were studied it was found that the actual significance levels of the mid p test tend to be closer to the nominal level when compared with those of the other two tests

improving on exact tests by approximate conditioning summary the often conservative nature for discrete data of so called exact tests seems usually the result of unnecessarily precise conditioning we consider avoiding this by conditioning only approximately on the sufficient statistics for nuisance parameters modest relaxation of conditioning results in small loss in terms of the rationale for conditional inference but can greatly reduce the difficulties caused by discreteness exact calculation of p values based on approximate conditioning is possible but unattractive both in terms of the amount of calculation involved and in requiring explicit specification of the extent to which conditioning is to be relaxed it is shown that there is a highly accurate easily computed and very natural asymptotic approximation that avoids these difficulties

bias reduction and a solution for separation of logistic regression with missing covariates logistic regression is an important statistical procedure used in many disciplines the standard software packages for data analysis are generally equipped with this procedure where the maximum likelihood estimates of the regression coefficients are obtained iteratively it is well known that the estimates from the analyses of small or medium sized samples are biased also in finding such estimates often a separation is encountered in which the likelihood converges but at least one of the parameter estimates diverges to infinity standard approaches of finding such estimates do not take care of these problems moreover the missingness in the covariates adds an extra layer of complexity to the whole process in this article we address these three practical issues bias separation and missing covariates by means of simple adjustments we have applied the proposed technique using real and simulated data the proposed method always finds a solution and the estimates are less biased a sas macro that implements the proposed method can be obtained from the authors

bias reduced and separation proof gee with small or sparse longitudinal binary data generalized estimating equation gee is a popular approach for analyzing correlated binary data however the problems of separation in gee are still unknown the separation created by a covariate often occurs in small correlated binary data and even in large data with rare outcome and or high intra cluster correlation and a number of influential covariates this paper investigated the consequences of separation in gee and addressed them by introducing a penalized gee termed as pgee the pgee is obtained by adding firth type penalty term which was originally proposed for generalized linear model score equation to standard gee and shown to achieve convergence and provide finite estimate of the regression coefficient in the presence of separation which are not often possible in gee further a small sample bias correction to the sandwich covariance estimator of the pgee estimator is suggested simulations also showed that the gee failed to achieve convergence and or provided infinitely large estimate of the regression coefficient in the presence of complete or quasi complete separation whereas the pgee showed significant improvement by achieving convergence and providing finite estimate even in the presence of near to separation the pgee also showed superior properties over the gee furthermore the bias corrected sandwich estimator for the pgee estimator showed substantial improvement over the standard sandwich estimator by reducing bias in estimating type i error rate an illustration using real data also supported the findings of simulation the pgee with bias corrected sandwich covariance estimator is recommended to use for small to moderate size sample n 50 and even can be used for large sample if there is any evidence of separation or near to separation

semiparametric probit models with univariate and bivariate current status data summary multivariate current status data are frequently encountered in biomedical and public health studies semiparametric regression models have been extensively studied for univariate current status data but most existing estimation procedures are computationally intensive involving either penalization or smoothing techniques it becomes more challenging for the analysis of multivariate current status data in this article we study the maximum likelihood estimations for univariate and bivariate current status data under the semiparametric probit regression models we present a simple computational procedure combining the expectation maximization algorithm with the pool adjacent violators algorithm for solving the monotone constraint on the baseline function asymptotic properties of the maximum likelihood estimators are investigated including the calculation of the explicit information bound for univariate current status data as well as the asymptotic consistency and convergence rate for bivariate current status data extensive simulation studies showed that the proposed computational procedures performed well under small or moderate sample sizes we demonstrate the estimation procedure with two real data examples in the areas of diabetic and hiv research

likelihood calculations for matched case control studies and survival studies with tied death times summary a recursion is presented which permits rapid computation of a conditional likelihood function which arises in matched case control studies and survival studies with tied death times an example with seven parameters and ten large strata the largest of which contains 29 cases and 52 controls illustrates the feasibility of the conditional maximum likelihood solution these methods may be specialized to permit rapid calculation of the conditional maximum likelihood estimate of the common odds ratio for several 2 x 2 tables

rationality of three dimensional quotients by monomial actions let g be a finite 2 group and k be a field satisfying that i chark 2 and ii a k for any a k if g acts on the rational function field k x y z by monomial k automorphisms then the fixed field k x y z g is rational purely transcendental over k applications of this theorem will be given

dynamic cox modelling based on fractional polynomials time variations in gastric cancer prognosis the most popular model used for survival analysis is the proportional hazards regression model proposed by cox this is mainly due to its exceptional simplicity nevertheless the fundamental assumption of the cox model is the proportionality of the hazards for many applications however this assumption is doubtful proposals to extend the cox model for non proportional hazards to allow for dynamic effect structures usually either depend on prespecifications or require non standard estimation techniques and are thus not favoured in application moreover tests to verify the dynamic effect structures are not straightforward or lack omnibus power in this paper we propose a flexible method for modelling dynamic effects in survival data within the cox framework the method is based on fractional polynomials as introduced by royston and altman this allows for a transformation of the dynamic predictor which leads back to the conventional cox model and hence fitting is straightforward using standard estimation techniques in addition it offers the possibility of easily verifying the existence of time variation we describe a model selection algorithm which selects time varying effects only when evidence is given in the data we illustrate the properties of the approach in a simulation study and compare it with other methods in a survival study of gastric cancer patients we apply the approach to analyse dynamic structures in the effects of prognostic factors

submanifolds with constant scalar curvature in a space form abstract we deal with complete submanifolds m n having constant positive scalar curvature and immersed with parallel normalized mean curvature vector field in a riemannian space form q c n p of constant sectional curvature c 1 0 1 in this setting we show that such a submanifold m n must be either totally umbilical or isometric to a clifford torus s 1 1 r 2 s n 1 r when c 1 a circular cylinder r s n 1 r when c 0 or a hyperbolic cylinder h 1 1 r 2 s n 1 r when c 1 this characterization theorem corresponds to a natural improvement of previous ones due to alias garcia martinez and rigoli 2 cheng 4 and guo and li 6

a solution to the problem of monotone likelihood in cox regression the phenomenon of monotone likelihood is observed in the fitting process of a cox model if the likelihood converges to a finite value while at least one parameter estimate diverges to infinity monotone likelihood primarily occurs in small samples with substantial censoring of survival times and several highly predictive covariates previous options to deal with monotone likelihood have been unsatisfactory the solution we suggest is an adaptation of a procedure by firth 1993 biometrika 80 27 38 originally developed to reduce the bias of maximum likelihood estimates this procedure produces finite parameter estimates by means of penalized maximum likelihood estimation corresponding wald type tests and confidence intervals are available but it is shown that penalized likelihood ratio tests and profile penalized likelihood confidence intervals are often preferable an empirical study of the suggested procedures confirms satisfactory performance of both estimation and inference the advantage of the procedure over previous options of analysis is finally exemplified in the analysis of a breast cancer study

small sample confidence regions in exponential families this article presents an algorithm for small sample conditional confidence regions for two or more parameters for any discrete regression model in the generalized linear interactive model family regions are constructed by careful inversion of conditional hypothesis tests this method presupposes the use of approximate or exact techniques for enumerating the sample space for some components of the vector of sufficient statistics conditional on other components such enumeration may be performed exactly or by exact or approximate monte carlo including the algorithms of kolassa and tanner 1994 journal of the american statistical association 89 697 702 1999 biometrics 55 246 251 this method also assumes that one can compute certain conditional probabilities for a fixed value of the parameter vector because of a property of exponential families one can use this set of conditional probabilities to directly compute the conditional probabilities associated with any other value of the vector of the parameters of interest this observation dramatically reduces the computational effort required to invert the hypothesis test to obtain the confidence region to construct a region with confidence level 1 alpha the algorithm begins with a grid of values for the parameters of interest for each parameter vector on the grid corresponding to the current null hypothesis one transforms the initial set of conditional probabilities using exponential tilting and then calculates the p value for this current null hypothesis the confidence region is the set of parameter values for which the p value is at least alpha

joint estimation of time dependent and non linear effects of continuous covariates on survival in order to yield more flexible models the cox regression model lambda t x lambda 0 t exp betax has been generalized using different non parametric model estimation techniques one generalization is the relaxation of log linearity in x lambda t x lambda 0 t exp r x another is the relaxation of the proportional hazards assumption lambda t x lambda 0 t exp beta t x these generalizations are typically considered independently of each other we propose the product model lambda t x lambda 0 t exp beta t r x which allows for joint estimation of both effects and investigate its properties the functions describing the time dependent beta t and non linear r x effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood likelihood ratio tests are proposed to compare alternative models simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model by contrast type i error rates may be highly inflated and the estimates considerably biased if the model is misspecified applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

tests of significance using regression models for ordered categorical data regression models of the type proposed by mccullagh 1980 journal of the royal statistical society series b 42 109 142 are a general and powerful method of analyzing ordered categorical responses assuming categorization of an unknown continuous response of a specified distribution type tests of significance with these models are generally based on likelihood ratio statistics that have asymptotic chi 2 distributions therefore investigators with small data sets may be concerned with the small sample behavior of these tests in a monte carlo sampling study significance tests based on the ordinal model are found to be powerful but a modified test procedure using an f distribution with a finite number of degrees of freedom for the denominator is suggested such that the empirical significance level agrees more closely with the nominal significance level in small sample situations we also discuss the parallels between an ordinal regression model assuming underlying normality and conventional multiple regression we illustrate the model with two data sets one from a study investigating the relationship between phosphorus in soil and plant available phosphorus in corn grown in that soil and the other from a clinical trial comparing analgesic drugs

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

information bias and adjusted profile likelihoods summary the bias and information bias of the ordinary profile score statistic are both typically of order 0 1 several additive adjustments to the profile score statistic that reduce its bias to order o n have been proposed in certain situations the information bias of these adjusted profile score statistics is also reduced to order o n q specifically the modified profile likelihood of barndorff nielsen yields an adjusted profile score statistic having both reduced bias and reduced information bias in general however a bias reducing adjustment to the profile score statistic will not automatically reduce the order of the information bias as well in this paper an analytical formula is obtained for the information bias of a general bias adjusted profile score statistic this formula is used to compare bias reducing adjustments to the profile score statistic as well as to construct further adjustments that reduce the information bias to o n 1 several examples are presented to illustrate use of the formula for information bias in particular the information bias formula may be utilized in a criterion for choosing between orthogonal parameterizations for the conditional profile likelihood of cox and reid

weighted profile likelihood based confidence interval for the difference between two proportions with paired binomial data inference on the difference between two binomial proportions in the paired binomial setting is often an important problem in many biomedical investigations tang et al 2010 statistics in medicine discussed six methods to construct confidence intervals henceforth we abbreviate it as ci for the difference between two proportions in paired binomial setting using method of variance estimates recovery in this article we propose weighted profile likelihood based cis for the difference between proportions of a paired binomial distribution however instead of the usual likelihood we use weighted likelihood that is essentially making adjustments to the cell frequencies of a 2 2 table in the spirit of agresti and min 2005 statistics in medicine we then conduct numerical studies to compare the performances of the proposed cis with that of tang et al and agresti and min in terms of coverage probabilities and expected lengths our numerical study clearly indicates that the weighted profile likelihood based intervals and jeffreys interval cf tang et al are superior in terms of achieving the nominal level and in terms of expected lengths they are competitive finally we illustrate the use of the proposed cis with real life examples copyright 2014 john wiley sons ltd

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

the incidence of monotone likelihood in the cox model in estimating covariate coefficients for the cox proportional hazards model there is a nonzero probability for any finite sample that the maximum likelihood estimate will be infinite this implies that in monte carlo studies the variance is not a suitable estimator of precision this also may necessitate a stratified analysis when multiple covariates are involved a simple method is given for detecting and handling these infinite estimates

bias correction in maximum likelihood logistic regression this paper provides an expression for bias of the maximum likelihood logistic regression estimates for use with small sample sizes this bias correction is based on an expansion of the maximum likelihood equation simulation results show these corrections to be highly effective in small samples

a permutation test for inference in logistic regression with small and moderate sized data sets inference based on large sample results can be highly inaccurate if applied to logistic regression with small data sets furthermore maximum likelihood estimates for the regression parameters will on occasion not exist and large sample results will be invalid exact conditional logistic regression is an alternative that can be used whether or not maximum likelihood estimates exist but can be overly conservative this approach also requires grouping the values of continuous variables corresponding to nuisance parameters and inference can depend on how this is done a simple permutation test of the hypothesis that a regression parameter is zero can overcome these limitations the variable of interest is replaced by the residuals from a linear regression of it on all other independent variables logistic regressions are then done for permutations of these residuals and a p value is computed by comparing the resulting likelihood ratio statistics to the original observed value simulations of binary outcome data with two independent variables that have binary or lognormal distributions yield the following results a in small data sets consisting of 20 observations type i error is well controlled by the permutation test but poorly controlled by the asymptotic likelihood ratio test b in large data sets consisting of 1000 observations performance of the permutation test appears equivalent to that of the asymptotic test and c in small data sets the p value for the permutation test is usually similar to the mid p value for exact conditional logistic regression

confidence intervals after multiple imputation combining profile likelihood information from logistic regressions in the logistic regression analysis of a small sized case control study on alzheimer s disease some of the risk factors exhibited missing values motivating the use of multiple imputation usually rubin s rules rr for combining point estimates and variances would then be used to estimate symmetric confidence intervals cis on the assumption that the regression coefficients were distributed normally yet rarely is this assumption tested with or without transformation in analyses of small sparse or nearly separated data sets such symmetric ci may not be reliable thus rr alternatives have been considered for example bayesian sampling methods but not yet those that combine profile likelihoods particularly penalized profile likelihoods which can remove first order biases and guarantee convergence of parameter estimation to fill the gap we consider the combination of penalized likelihood profiles clip by expressing them as posterior cumulative distribution functions cdfs obtained via a chi squared approximation to the penalized likelihood ratio statistic cdfs from multiple imputations can then easily be averaged into a combined cdf c allowing confidence limits for a parameter at level 1 to be identified as those and that satisfy cdf c 2 and cdf c 1 2 we demonstrate that the clip method outperforms rr in analyzing both simulated data and data from our motivating example clip can also be useful as a confirmatory tool should it show that the simpler rr are adequate for extended analysis we also compare the performance of clip to bayesian sampling methods using markov chain monte carlo clip is available in the r package logistf

an empirical comparison of statistical tests for assessing the proportional hazards assumption of cox s model in the analysis of survival data using the cox proportional hazard ph model it is important to verify that the explanatory variables analysed satisfy the proportional hazard assumption of the model this paper presents results of a simulation study that compares five test statistics to check the proportional hazard assumption of cox s model the test statistics were evaluated under proportional hazards and the following types of departures from the proportional hazard assumption increasing relative hazards decreasing relative hazards crossing hazards diverging hazards and non monotonic hazards the test statistics compared include those based on partitioning of failure time and those that do not require partitioning of failure time the simulation results demonstrate that the time dependent covariate test the weighted residuals score test and the linear correlation test have equally good power for detection of non proportionality in the varieties of non proportional hazards studied using illustrative data from the literature these test statistics performed similarly

adjusted score functions for monotone likelihood in the cox regression model standard inference procedures for the cox model involve maximizing the partial likelihood function monotone partial likelihood is an issue that frequently happens in the analysis of health science studies monotone likelihood mainly occurs in samples with substantial censoring of survival times and is associated with categorical covariates in particular and more frequently it usually happens when one level of a categorical covariate has just experienced censoring times in order to overcome this problem heinze and schemper proposed an adjusted partial likelihood score function obtained by suitably adapting the general approach of firth for mean bias reduction the procedure is effective in preventing infinite estimates as an alternative solution we propose an approach based on the adjusted score function recently suggested by kenne pagui et al for median bias reduction this procedure also solves the infinite estimate problem and has an additional advantage of being invariant under componentwise reparameterizations this latter fact is fundamental under cox model since hazards ratio interpretation is obtained by exponentiating parameter estimates numerical studies of the proposed method suggest better inference properties than those of the mean bias reduction a real data application related to a melanoma skin dataset is used as illustration for a comparison basis of the methods

conditional logistic regression with sandwich estimators application to a meta analysis motivated by a meta analysis of animal experiments on the effect of dietary fat and total caloric intake on mammary tumorigenesis we explore the use of sandwich estimators of variance with conditional logistic regression classical conditional logistic regression assumes that the parameters are fixed effects across all clusters while the sandwich estimator gives appropriate inferences for either fixed effects or random effects however inference using the standard wald test with the sandwich estimator requires that each parameter is estimated using information from a large number of clusters since our example violates this condition we introduce two modifications to the standard wald test first we reduce the bias of the empirical variance estimator the middle of the sandwich by using standardized residuals second we approximately account for the variance of these estimators by using the t distribution instead of the normal distribution where the degrees of freedom are estimated using satterthwaite s approximation through simulations we show that these sandwich estimators perform almost as well as classical estimators when the true effects are fixed and much better than the classical estimators when the true effects are random we achieve simulated nominal coverage for these sandwich estimators even when some parameters are estimated from a small number of clusters

a comparative investigation of methods for logistic regression with separated or nearly separated data in logistic regression analysis of small or sparse data sets results obtained by classical maximum likelihood methods cannot be generally trusted in such analyses it may even happen that the likelihood meets the convergence criteria while at least one parameter estimate diverges to infinity this situation has been termed separation and it typically occurs whenever no events are observed in one of the two groups defined by a dichotomous covariate more generally separation is caused by a linear combination of continuous or dichotomous covariates that perfectly separates events from non events separation implies infinite or zero maximum likelihood estimates of odds ratios which are usually considered unrealistic i provide some examples of separation and near separation in clinical data sets and discuss some options to analyse such data including exact logistic regression analysis and a penalized likelihood approach both methods supply finite point estimates in case of separation profile penalized likelihood confidence intervals for parameters show excellent behaviour in terms of coverage probability and provide higher power than exact confidence intervals general advantages of the penalized likelihood approach are discussed

hypersurfaces with constant scalar curvature in space forms let rn l c be an n 1 dimensional riemannian manifold with constant sectional curvature c we also call it space form when c 1 r l c s l i e n 1 dimensional unit sphere space when c o rn t c e n l i e n 1 dimensional euclidean space let m be an n dimensional compact hypersurface in rn l c and el e a local orthonormal frame field on m col con its dual coframe field then the second fundamental form of m is

mle s bias pathology model updated mle and wallace s minimum message length method the inherent bias pathology of the maximum likelihood estimation method is confirmed for models with unknown parameters and when maximum likelihood estimate mle is function of mle to reduce s bias the likelihood equation to be solved for is updated using the model for the data y in it for various models with a shape parameter model updated mu mle mu reduces s bias for the pareto model mu reduces in addition s variance the results explain the difference that puzzled fisher between biased and the unbiased estimate he obtained for two models with the abandoned two stage procedure used in mumle s implementation mu is also obtained with the minimum message length method thus motivating the use of priors in frequentist inference

an optimal estimating equation based on the first three cumulants for improving on quasilikelihood estimation two types of quadratic estimating equations have been proposed one based on the edgeworth expansion the other on the generalisation of the quasi score the first requires that the skewness of observations has a small departure from the exponential family the second requires the knowledge of both skewness and kurtosis we introduce an optimal quadratic estimating equation applicable when the skewness is not small and the kurtosis is unknown apart from optimality the manner in which skewness is incorporated ensures that its misspecification does not affect n consistency and that the estimator enjoys an invariance property akin to that of the bias corrected maximum likelihood estimate simulations indicate a solid improvement in accuracy

bias correction in generalized linear models summary in this paper we derive general formulae for first order biases of maximum likelihood estimates of the linear parameters linear predictors the dispersion parameter and fitted values in generalized linear models these formulae may be implemented in the glim program to compute bias corrected maximum likelihood estimates to order n 1 where n is the sample size with minimal effort by means of a supplementary weighted regression for linear logistic models it is shown that the asymptotic bias vector of j8 is almost collinear with f3 the approximate formula flp m for the bias of s in logistic models where p dim and m e mi is the sum of the binomial indices is derived and checked numerically