global stability ofxn 1 a xpn 1 x1 pn 1 we obtain a general global attractivity result for a difference equation of the formxn 1 f xn xn 1 n 0 1 wheref x y is strictly decreasing in both arguments the result applies to the difference equationxn 1 a xpn 1 x1 pn 1 n 0 1 wherex 1 x0 a 0 andp 1

finding factors influencing risk comparing bayesian stochastic search and standard variable selection methods applied to logistic regression models of cases and controls when modeling the risk of a disease the very act of selecting the factors to be included can heavily impact the results this study compares the performance of several variable selection techniques applied to logistic regression we performed realistic simulation studies to compare five methods of variable selection 1 a confidence interval ci approach for significant coefficients 2 backward selection 3 forward selection 4 stepwise selection and 5 bayesian stochastic search variable selection ssvs using both informed and uniformed priors we defined our simulated diseases mimicking odds ratios for cancer risk found in the literature for environmental factors such as smoking dietary risk factors such as fiber genetic risk factors such as xpd and interactions we modeled the distribution of our covariates including correlation after the reported empirical distributions of these risk factors we also used a null data set to calibrate the priors of the bayesian method and evaluate its sensitivity of the standard methods 95 per cent ci backward forward and stepwise selection the ci approach resulted in the highest average per cent of correct associations and the lowest average per cent of incorrect associations ssvs with an informed prior had a higher average per cent of correct associations and a lower average per cent of incorrect associations than the ci approach this study shows that the bayesian methods offer a way to use prior information to both increase power and decrease false positive results when selecting factors to model complex disease risk

prediction error and its estimation for subset selected models strategies are compared for development of a linear regression model and the subsequent assessment of its predictive ability simulations were performed as a designed experiment over a range of data structures approaches using a forward selection of variables resulted in slightly smaller prediction errors and less biased estimators of predictive accuracy than all possible subsets selection but often did not improve on the full model random and balanced data splitting resulted in increased prediction errors and estimators with large mean squared error to maximize predictive accuracy while retaining a reliable estimate of that accuracy it is recommended that the entire sample usually he used for model development and assessment

large sample bayesian inference on the parameters of the proportional hazard models this paper considers large sample bayesian analysis of the proportional hazards model when interest is in inference on the parameters and estimation of the log relative risk for specified covariate vectors rather than on prediction of the survival function we use a normal prior distribution for the parameters and make inferences based on the derived posterior distribution the suggested approach is much simpler than alternative bayesian analyses previously suggested for the proportional hazards models using simulated data we compare estimates obtained from the bayesian analysis with those obtained from the full proportional hazards model and the reduced model after backwards elimination we show that under a wider range of assumptions the bayesian analysis provides reduced estimation errors and improved rejection of noise variables finally we illustrate the methodology using data from a large study of prognostic markers in breast cancer

when should epidemiologic regressions use random coefficients summary regression models with random coefficients arise naturally in both frequentist and bayesian approaches to estimation problems they are becoming widely available in standard computer packages under the headings of generalized linear mixed models hierarchical models and multilevel models i here argue that such models offer a more scientifically defensible framework for epidemiologic analysis than the fixed effects models now prevalent in epidemiology the argument invokes an antiparsimony principle attributed to l j savage which is that models should be rich enough to reflect the complexity of the relations under study it also invokes the countervailing principle that you cannot estimate anything if you try to estimate everything often used to justify parsimony regression with random coefficients offers a rational compromise between these principles as well as an alternative to analyses based on standard variable selection algorithms and their attendant distortion of uncertainty assessments these points are illustrated with an analysis of data on diet nutrition and breast cancer

approximate bayes factors and accounting for model uncertainty in generalised linear models summary ways of obtaining approximate bayes factors for generalised linear models are described based on the laplace method for integrals we propose a new approximation which uses only the output of standard computer programs for estimating generalised linear models this appears to be quite accurate a reference set of proper priors is suggested both to represent the situation where there is not much prior information and to assess the sensitivity of the results to the prior distribution the methods can be used when the dispersion parameter is unknown when there is overdispersion to compare link functions and to compare error distributions and variance functions the methods can be used to implement the bayesian approach to accounting for model uncertainty we describe an application to inference about relative risks in the presence of control factors where model uncertainty is large and important software to implement the methods is available at no cost from statlib

the dynamics of some discrete population models in this paper we are interested in providing sufficient conditions guaranteeing the fact that all positive solutions of the discrete delay difference equation a ii 1 aa f a e converge as n 00 in e a is a constant real number with 0 0 sufficiently small thus formally equation e comes from x t h x t h yx t d x t r for small h if we set z f x ht then the preceding equation for t n and r h m becomes yh 1 1 h y n hd y n m which is of the form e where a 1 yh and f u hd u u 2 0 clearly h 0 is small if and only if a e 0 1 1069 h

on a system of two nonlinear difference equations abstract in this paper we study the oscillatory behavior the boundedness of the solutions and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equationsxn 1 a yn xn p yn 1 a xn yn q n 0 1 p qare positive integers

the performance of random coefficient regression in accounting for residual confounding greenland 2000 biometrics 56 915 921 describes the use of random coefficient regression to adjust for residual confounding in a particular setting we examine this setting further giving theoretical and empirical results concerning the frequentist and bayesian performance of random coefficient regression particularly we compare estimators based on this adjustment for residual confounding to estimators based on the assumption of no residual confounding this devolves to comparing an estimator from a nonidentified but more realistic model to an estimator from a less realistic but identified model the approach described by gustafson 2005 statistical science 20 111 140 is used to quantify the performance of a bayesian estimator arising from a nonidentified model from both theoretical calculations and simulations we find support for the idea that superior performance can be obtained by replacing unrealistic identifying constraints with priors that allow modest departures from those constraints in terms of point estimator bias this superiority arises when the extent of residual confounding is substantial but the advantage is much broader in terms of interval estimation the benefit from modeling residual confounding is maintained when the prior distributions employed only roughly correspond to reality for the standard identifying constraints are equivalent to priors that typically correspond much worse

on the recursive sequence xn 1 a xnp b xn 1q c xn 2s we investigate the boundedness persistance and asymptotic behaviour of the positive solutions of the equation in the title where a b p q email protected 0 email protected 0

droplet structure in ising and potts models the structure of a recently introduced droplet in the q state potts model is analysed the authors derive exact relations from which it follows that the incipient infinite droplet at the critical temperature is a self similar fractal made of links and blobs just as recently found in random percolation the number of links nlinks between two points separated by a distance of the order b is given by nlinks varies as b1 nu b q where nu b q is the connectedness length exponent when t tc q and pb is used as an independent variable this result and the available calculations of nu b q indicate that the number of links decreases as q increases an analysis of the structure of the usual clusters made of nearest neighbour sites in a given potts site configuration is also made in particular it shows that for d 2 the incipient infinite cluster is made only of blobs and no links

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

the dynamics of χn 1 α βχna bχn cxn 1 facts and conjectures abstract we investigate the global character of solutions of the equation in the title with non negative parameters and nonnegative initial conditions

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

bayesian subset analysis as a means of assessing the importance of variation in treatment effect among patient subsets we derived posterior distributions for subset specific treatment effects the effects are represented by combinations of terms for treatment and treatment by covariate interaction effects in familiar regression models exchange ability among the interactions is a key assumption thus the results are of interest primarily in the context of examining a collection of subsets with no definite a priori distinction relative to treatment effect exchangeability leads to a shrinking of the posterior distributions of the interaction terms toward the natural origin of 0 offsetting the tendency of the estimated effects to disperse the method is applied to parameter estimates from a proportional hazards regression analysis of survival data from a clinical trial invoking the approximate multivariate normal distribution of the estimates no subjective prior distributions are required vague priors are used for all of the regression coefficients except the treatment by covariate interactions which are assumed to follow a normal distribution

crossed random effect models for multiple outcomes in a study of teratogenesis human teratogens often manifest themselves through a broad spectrum of adverse effects although often not serious when considered individually such outcomes taken together may represent a syndrome that can lead to serious developmental problems accordingly studies that investigate the effect of human teratogens on fetal development typically record the presence or absence of a multitude of abnormalities resulting in the data of multivariate binary form for each infant such studies typically have three objectives 1 estimate an overall effect of exposure across outcomes 2 identify subjects having the syndrome and 3 identify those outcomes that constitute the syndrome so that doctors know what to look for when diagnosing the syndrome in other exposed newborns this article proposes the use of a logistic regression model with crossed random effect structure to address all three questions simultaneously we use the proposed models to analyze data from a study investigating the effects of in uter

second stage least squares versus penalized quasi likelihood for fitting hierarchical models in epidemiologic analyses hierarchical regression analysis holds much promise for epidemiologic analysis but has as yet seen limited application because of lack of easily used software and the relatively lengthy run times of preferred fitting methods such as true maximum likelihood and bayesian approaches this paper compares three relatively simple choices for estimation of the regression coefficients maximum likelihood first stage combined with a weighted least squares second stage mlls joint iteratively reweighted least squares fitting of first and second stage jils and empirically penalized quasi likelihood epql these choices can be combined with various methods for estimating the second stage variance the two contrasted here are based on first and second stage residuals jils and epql yielded indistinguishable results and had small sample performance superior to mlls in larger samples there was little practical difference among the methods use of first stage residuals to estimate the prior variance required considerably more computation than use of second stage residuals but produced no discernible improvement in regression coefficient estimates all three methods performed well for estimation of first stage parameters but were less satisfactory for estimation of second stage parameters

application of bayesian statistics to decision making during a clinical trial we describe the application of bayesian methods to the monitoring and analysis of a trial of treatment for patients with advanced colorectal carcinoma we discuss the choice of prior distribution and justify the use of a truncated normal distribution with a probability mass at zero difference the stopping rule based on the trials of the posterior distribution and a chosen range of equivalence yields an upper boundary very close to the pocock group sequential boundary the bayes stopping rule is quite sensitive to the amount of probability mass at zero in the prior distribution

performance of analytical methods for overdispersed counts in cluster randomized trials sample size degree of clustering and imbalance many different methods have been proposed for the analysis of cluster randomized trials crts over the last 30 years however the evaluation of methods on overdispersed count data has been based mostly on the comparison of results using empiric data i e when the true model parameters are not known in this study we assess via simulation the performance of five methods for the analysis of counts in situations similar to real community intervention trials we used the negative binomial distribution to simulate overdispersed counts of crts with two study arms allowing the period of time under observation to vary among individuals we assessed different sample sizes degrees of clustering and degrees of cluster size imbalance the compared methods are i the two sample t test of cluster level rates ii generalized estimating equations gee with empirical covariance estimators iii gee with model based covariance estimators iv generalized linear mixed models glmm and v bayesian hierarchical models bayes hm variation in sample size and clustering led to differences between the methods in terms of coverage significance power and random effects estimation glmm and bayes hm performed better in general with bayes hm producing less dispersed results for random effects estimates although upward biased when clustering was low gee showed higher power but anticonservative coverage and elevated type i error rates imbalance affected the overall performance of the cluster level t test and the gee s coverage in small samples important effects arising from accounting for overdispersion are illustrated through the analysis of a community intervention trial on solar water disinfection in rural bolivia

on the recursive sequence yn 1 p yn 1 qyn yn 1 abstract we investigate the periodic character and the global stability of solutions of the equation y n 1 p y n 1 qy n y n 1 with positive parameters and positive initial conditions

lattice shapes and scaling functions for bond random percolation on honeycomb lattices we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the bond random percolation model on honeycomb lattices with aspect ratios of 0 5 and 2 0 we find that such different aspect ratios give quite different scaling functions near the critical region however they give a consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations some interesting theoretical problems related to this work are discussed

global attractivity of the difference equation xn 1 xn αxn 1 β xn our aim in this paper is to investigate the global attractivity of nonnegative solution of the difference equationx n 1 x n ax n 1 b x n n 0 1 where a 0 b 0 are real numbers the initial conditions x 1 x 0 are nonnegative real numbers and with b and x 1 not both zero we show that the positive equilibrium point of the equation is a global attractor

on the system of two difference equations xn 1 i 0kai yn ipi yn 1 i 0kbi xn iqi abstract in this paper we study the system of two difference equations of the form x n 1 i 0 k a i y n i p i y n 1 i 0 k b i x n i q i where a i b i i 0 1 k x i y i i k k 1 0 are positive numbers and p i q i i 0 k are positive constants more precisely we investigate the boundedness the persistence of the positive solutions the existence of a unique positive equilibrium and the global asymptotic stability of the positive equilibrium of the above system finally we find solutions of the system which do not oscillate about the positive equilibrium

global attractivity in a nonlinear difference equation abstract we consider the nonlinear difference equation x n a k 1 m b k x n k n 0 1 2 where a and bk k 1 2 m are nonnegative numbers with b mk 1bk 0 and we are interested in whether all positive solutions are attracted by the positive equilibrium l a 2 a 2 2 b

on the new solutions to the fuzzy difference equation xn 1 a bxn abstract in this paper using a generalization of division for fuzzy numbers we study the existence and global behavior of the fuzzy difference equation x n 1 a b x n n 0 1 where a and b are positive fuzzy numbers some examples are presented to illustrate the applicability of our results

on first order fuzzy ricatti difference equation this paper deals with the existence the boundedness and the asymptotic behavior of the positive solutions to a first order fuzzy ricatti difference equation xn 1 a xnb xn n 0 1 where xn is a sequence of positive fuzzy numbers a b and the initial value x0 are positive fuzzy numbers moreover an example is given to demonstrate the effectiveness of the results obtained

boundary conditions and scaling functions of percolation models we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the site percolation model on square lattices with free and periodic boundary conditions we find that different boundary conditions give quite different scaling functions near the critical region however they give the consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations similar results are found for other percolation models the implications of our calculated results for some theoretical problems of current interest are discussed

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

a new perspective on priors for generalized linear models abstract this article deals with specifications of informative prior distributions for generalized linear models our emphasis is on specifying distributions for selected points on the regression surface the prior distribution on regression coefficients is induced from this specification we believe that it is inherently easier to think about conditional means of observables given the regression variables than it is to think about model dependent regression coefficients previous use of conditional means priors seems to be restricted to logistic regression with one predictor variable and to normal theory regression we expand on the idea of conditional means priors and extend these to arbitrary generalized linear models we also consider data augmentation priors where the prior is of the same form as the likelihood we show that data augmentation priors are special cases of conditional means priors with current monte carlo methodology such as importance sampling and gibbs sampling our priors result in

percolation fractals and the finite size scaling of existence probability abstract based on the connection between phase transitions of thermodynamic systems and percolation transitions we obtain the geometrical meaning of critical exponents using finite size scaling of the percolation probability p and the existence probability e p for percolating clusters and percolating subgraphs respectively where e p is the ratio of the total probability weight for percolating subgraphs to the total probability weight for all subgraphs we use a histogram monte carlo simulation method to calculate e p for a q state bond correlated percolation model qbcpm which is corresponding to the q state potts model on the square lattice with various linear dimensions we find that for a given q e p has very good scaling behavior but e p for different q has different scaling functions

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

bayesian multiple testing for two sample multivariate endpoints dans les essais cliniques ou l on s interesse a plusieurs variables il arrive souvent que des tests simultanes portent sur des resultats correles entre eux et concernent des hypotheses elles meme correlees entre elles cet article dans un cadre bayesien propose d utiliser comme loi de probabilite a priori des effets traitement une distribution multivariee ou la combinaison des parametres permet simultanement d affecter une probabilite strictement positive a l absence d effet traitement pour chacune des hypotheses et de modeliser differents types de correlations la correlation entre les effets traitements standardises c est a dire rapportes aux ecarts types des variables concernees la correlation entre les variables dichotomiques de presence ou d absence de l effet traitement la correlation enfin entre les statistiques de test conditionnellement aux effets reels nous developpons ensuite dans le cas de l etude multivariee de deux echantillons une methode de comparaisons multiples qui nous permet de calculer pour des variables prealablement specifiees les probabilites a posteriori d une absence d effet traitement sont aussi discutees dans le contexte d un exemple clinique reel les demarches relatives au choix de la probabilite a priori ainsi que les problemes de robustesse

global behavior of solutions of a nonlinear second order difference equation abstract our aim in this paper is to investigate the boundedness the persistence the global attractivity the existence of periodic solutions of period 2 and the oscillatory nature of solutions of the equation x n 1 f x n x n 1 n 0 1 where f is a continuous positive and decreasing function on 0 2 0 0 the results are applied to the difference equation y n 1 a y n p 1 y n 1 q n 0 1 where a p q 0

global attractivity for a class of higher order nonlinear difference equations our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equationx n 1 a bx email protected k i 0b ix n i n 0 1 where a b b email protected 0 a b 0 b k email protected 0 email protected 1 2 and the initial conditions x k x 0 are arbitrary real numbers we show that one positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

model selection an integral part of inference we argue that model selection uncertainty should be fully incorporated into statistical inference whenever estimation is sensitive to model choice and that choice is made with reference to the data we consider different philosophies for achieving this goal and suggest strategies for data analysis we illustrate our methods through three examples the first is a poisson regression of bird counts in which a choice is to be made between inclusion of one or both of two covariates the second is a line transect data set for which different models yield substantially different estimates of abundance the third is a simulated example in which truth is known

necessary and sufficient conditions for boundedness and stability of n order difference equation we investigate the boundedness persistence and asymptotic behaviour of the positive solutions of the equationx k 1 a 1x k p 1 a 2x k 1 p 2 a nx k n 1 p n where a 1 a 2 a 3 p i i 1 2 email protected 0 and a i i 4 5 email protected 0

putting background information about relative risks into conjugate prior distributions in bayesian and empirical bayes analyses of epidemiologic data the most easily implemented prior specifications use a multivariate normal distribution for the log relative risks or a conjugate distribution for the discrete response vector this article describes problems in translating background information about relative risks into conjugate priors and a solution traditionally conjugate priors have been specified through flattening constants an approach that leads to conflicts with the true prior covariance structure for the log relative risks one can however derive a conjugate prior consistent with that structure by using a data augmentation approximation to the true log relative risk prior although a rescaling step is needed to ensure the accuracy of the approximation these points are illustrated with a logistic regression analysis of neonatal death risk

inference for non random samples observational data are often analysed as if they had resulted from a controlled study and yet the tacit assumption of randomness can be crucial for the validity of inference we take some simple statistical models and supplement them by adding a parameter which reflects the degree of non randomness in the sample for a randomized study is known to be 0 we examine the profile log likelihood for and the sensitivity of inference to small non zero values of particular models cover the analysis of survey data with item non response the paired comparison t test and two group comparisons using observational data with covariates some practical examples are discussed allowing for sampling bias increases the uncertainty of estimation and weakens the significance of treatment effects sometimes substantially so

scaling functions of the q state potts model on planar lattices abstract based on the subgraph expansion of the q state potts model qpm in an external field it has been shown that the qpm is corresponding to a q state bond correlated percolation model qbcpm the histogram monte carlo simulation method proposed by hu is used to calculate the existence probability e p g p q and the percolation probability p g p q of the qbcpm on the honeycomb the kagome and the plane triangular lattices with various linear dimensions from e p g p q and p g p q we obtain scaling functions of the qpm and qbcpm we find that as q or the coordination number of the lattices increases the widths of the scaling functions also increase the implication of this study is discussed

generalized conjugate priors for bayesian analysis of risk and survival regressions le choix pour les analyses bayesiennes portant sur les risques relatifs de distributions a priori conjuguees peut etre tres limitant du fait que la forme de cette distribution depend du parametre de valeur centrale cependant en introduisant separement ce parametre on generalise ces distributions a priori ce qui permet de modeliser une tres large palette d opinions a priori tout en conservant les proprietes de simplicite des calculs propres aux analyses conjuguees cet article illustre les resultats fournis par ces analyses avec des conjuguees generalisees a partir d exemples sur des enquetes cas x temoins dont le but etait d etudier le lien des amenagements en c bles electriques wire codes en zone residentielle et des champs magnetiques avec les cas de leucemie chez l enfant

general position space renormalisation group for correlated percolation a position space renormalisation group psrg approach has been developed for general site percolation problems in which the site occupancies are correlated rather than independent random variables psrg parameters are systematically introduced to describe multiple site correlations at the two and four site levels of approximation in the square lattice the authors find only two physical fixed points one characterises random or pure percolation and has a universality class which is shown to encompass a variety of locally correlated site problems including unfrustrated plaquettes and four coordinated sites in a random bond model the second describes ising correlated percolation and at the two site level yields an excellent estimate to within 2 for the nearest neighbour spin correlation function of the ising model

a nondegenerate penalized likelihood estimator for variance parameters in multilevel models group level variance estimates of zero often arise when fitting multilevel or hierarchical linear models especially when the number of groups is small for situations where zero variances are implausible a priori we propose a maximum penalized likelihood approach to avoid such boundary estimates this approach is equivalent to estimating variance parameters by their posterior mode given a weakly informative prior distribution by choosing the penalty from the log gamma family with shape parameter greater than 1 we ensure that the estimated variance will be positive we suggest a default log gamma 2 penalty with 0 which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero thus remaining consistent with the data while being nondegenerate we also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior

on the recursive sequence xn 1 α βxn γxn 1bxn cxn 1 we investigate the global character of solutions of the equation in the title with nonnegative parameters and positive initial conditions we give a detailed description of the semicycles of solutions and prove that the equilibrium of the equation is globally asymptotically stable

approximate bayesian evaluation of multiple treatment effects summary we propose an approximate bayesian method for comparing an experimental treatment to a control based on a randomized clinical trial with multivariate patient outcomes overall treatment effect is characterized by a vector of parameters corresponding to effects on the individual patient outcomes we partition the parameter space into four sets where respectively the experimental treatment is superior to the control the control is superior to the experimental the two treatments are equivalent and the treatment effects are discordant we compute posterior probabilities of the parameter sets by treating an estimator of the parameter vector like a random variable in the bayesian paradigm the approximation may be used in any setting where a consistent asymptotically normal estimator of the parameter vector is available the method is illustrated by application to a breast cancer data set consisting of multiple time to event outcomes with covariates and to count data arising from a cross classification of response infection and treatment in an acute leukemia trial

stopping rules for clinical trials incorporating clinical opinion a method is described of eliciting a range of equivalence i e a range of differences between two treatments over which a group of clinical trial participants would have no clear preference for either treatment this range of equivalence is then incorporated into a formal stopping rule for the trial using an extension of the group sequential design tables for the implementation of the design are presented the method is discussed in the context of other sequential trial designs

data augmentation priors for bayesian and semi bayes analyses of conditional logistic and proportional hazards regression data augmentation priors have a long history in bayesian data analysis formulae for such priors have been derived for generalized linear models but their accuracy depends on two approximation steps this note presents a method for using offsets as well as scaling factors to improve the accuracy of the approximations in logistic regression this method produces an exceptionally simple form of data augmentation that allows it to be used with any standard package for conditional logistic or proportional hazards regression to perform bayesian and semi bayes analyses of matched and survival data the method is illustrated with an analysis of a matched case control study of diet and breast cancer

approximate bayesian inference in conditionally independent hierarchical models parametric empirical bayes models abstract we consider two stage models of the kind used in parametric empirical bayes peb methodology calling them conditionally independent hierarchical models we suppose that there are k units which may be experimental subjects cities study centers etcetera at the first stage the observation vectors yi for units i 1 k are independently distributed with densities p yi i or more generally p yi i at the second stage the unit specific parameter vectors i are iid with densities p i the peb approach proceeds by regarding the second stage distribution as a prior and noting that if were known inference about could be based on its posterior since is not known the simplest peb methods estimate the parameter by maximum likelihood or some variant and then treat as if it were known to be equal to this estimate although this procedure is sometimes satisfactory a well known defect is that it neglects the uncertainty due to the estimation of in this article w

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

universal scaling functions and quantities in percolation models abstract we briefly review recent work on universal finite size scaling functions ufssfs and quantities in percolation models the topics under discussion include a ufssfs for the existence probability also called crossing probability e p the percolation probability p and the probability w n of the appearance of n percolating clusters b universal slope for average number of percolating clusters c ufssfs for a q state bond correlated percolation model corresponding to the q state potts model we also briefly mention some very recent related developments and discuss implications of our results

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

geometrical factor and thermal properties of the potts model based on the connection between the q state potts model qpm and a q state bond correlated percolation model qbcpm we define a non percolating geometrical factor g and percolating geometrical factor gp which depend only on q and geometrical properties of the system the thermal properties of the qpm could be derived simply from g and g the idea is confirmed by an exact calculation for one dimensional qpm and may be extended to other ising like spin models our formulation also provides a geometrical meaning for the finite size scaling and broadening at first order phase transitions of the spin model

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

the global attractivity of difference equations of nonincreasing nonlinearities with applications abstract the global attractivity of the unique positive equilibrium of the equation x n 1 g x n k 1 x n k 2 x n k m n 0 1 2 is investigated where g is a nonincreasing continuous function in each of its arguments and the k are positive integers we are able to use our results to extract some important global attractivity criteria of the positive equilibria of the discrete clark model x n 1 x n f x n k and the rational recursive sequence x n 1 a bx n a x n 1 as well

bayesian subset analysis in a colorectal cancer clinical trial subset analysis is the examination of treatment comparisons within groups of patients with restricted levels of patient characteristics such analyses are vulnerable to multiplicity effects we examine the problem in the context of a proportional hazards model with terms for treatment each of several dichotomous covariates representing the patient characteristics of interest and treatment by covariate interaction effects parametrically a subset specific treatment effect is equal to the treatment effect term plus a linear combination of the interaction terms we present bayesian point and interval estimates under the assumption that the interaction terms are exchangeable and the prior distributions for the other regression parameters are locally uniform this produces a shrinking of the estimated interaction effects towards zero thereby discounting them and dealing in a natural way with multiplicity we illustrate the method using results of a recent north central cancer treatment group mayo clinic study in advanced colorectal cancer

a semi bayes approach to the analysis of correlated multiple associations with an application to an occupational cancer mortality study thomas et al presented the application of empirical bayes methods to the problem of multiple inference in epidemiologic studies one limitation of their approach which they noted was the need to assume exchangeable log relative risk parameters and independent relative risk estimates numerical integration was also required here i generalize their approach to allow for non exchangeable parameters and non independent estimates the resulting method is bayesian in so far as some feature of the prior distribution are specified from prior information but is empirical bayes in so far as some explicit parameters in the prior distribution are estimated from the data estimation is based on approximations to the posterior distribution this allows one to implement the approach with standard software packages for matrix algebra the method is illustrated in an occupational mortality study of 84 exposure cancer associations

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

clusters and droplets in the q state potts model a potts correlated polychromatic percolation is studied the clusters are made of sites corresponding to a given value of the q state potts variables connected by bonds being active with probability pb to treat this problem an s state potts hamiltonian diluted with q state potts variables instead of lattice gas variables is introduced to which the the migdal kadanoff renormalisation group is applied it is found for a particular choice of pb 1 e k where k is the potts coupling constant divided by the boltzmann factor that these clusters called droplets diverge at the potts critical point with potts exponents

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea