variance calculations and confidence intervals for estimates of the attributable risk based on logistic models the attributable risk ar defined as ar pr disease pr disease no exposure pr disease measures the proportion of disease risk that is attributable to an exposure recently bruzzi et al 1985 american journal of epidemiology 122 904 914 presented point estimates of ar based on logistic models for case control data to allow for confounding factors and secondary exposures to produce confidence intervals we derived variance estimates for ar under the logistic model and for various designs for sampling controls calculations for discrete exposure and confounding factors require covariances between estimates of the risk parameters of the logistic model and the proportions of cases with given levels of exposure and confounding factors these covariances are estimated from taylor series expansions applied to implicit functions similar calculations for continuous exposures are derived using influence functions simulations indicate that those asymptotic procedures yield reliable variance estimates and confidence intervals with near nominal coverage an example illustrates the usefulness of variance calculations in selecting a logistic model that is neither so simplified as to exhibit systematic lack of fit nor so complicated as to inflate the variance of the estimate of ar

nonreduced extended affine weyl groups abstract in this paper we study the structure of the weyl groups of nonreduced extended affine root systems we show that similar to the case of reduced types an extended affine weyl group w of type bcl is semidirect product of a finite weyl group w of type bl and a heisenberg like normal subgroup h which is also a characteristic subgroup of w moreover h is of the form h h h 0 where both h and h 0 are normal subgroups of h with h h 0 1 h is naturally isomorphic to the root lattice of a finite root system of type bcl furthermore the semidirect product of w and h is isomorphic to the weyl group of a kac moody affine subroot system of r of type bcl

integrable representations of toroidal lie algebras co ordinatized by rational quantum tori for any positive integers d n with d 1 and n 1 we fix an n by n complex matrix q qij satisfying qii 1 and qij qji 1 with all qij roots of unity let d q be the universal central extension of the lie subalgebra sld cq of gld cq with trace in cq cq where cq is the rational quantum torus associated to q and let d q be the lie algebra by adding the n degree derivations to d q with respect to the n non commuting variables in cq the lie algebra d q called the toroidal lie algebra co ordinatized by the rational quantum torus cq has an n dimensional center c in this paper we obtain a classification of irreducible integrable modules with finite dimensional weight spaces and with non zero center action over the toroidal lie algebra d q

comparison of adjusted attributable risk estimators the estimation of attributable risk in the presence of confounding and effect modification is studied in this paper different adjustment methods for the attributable risk are reviewed the results of a stimulation study comparing these methods under the unrestricted multinomial sampling model are reported from the study it is concluded that the maximum likelihood estimator resulting from the case load weighting of the stratum specific attributable risk estimates will be the best overall choice in all practical situations with relatively large sample sizes other adjusted attributable risk estimators depend heavily on the underlying structure of the multinomial model

coordinate algebras of extended affine lie algebras of type a1 abstract the cores of extended affine lie algebras of reduced types were classified except for type a1 in this paper we determine the coordinate algebra of extended affine lie algebras of type a1 it turns out that such an algebra is a unital z n graded jordan algebra of a certain type called a jordan torus we classify jordan tori and get five types of jordan tori

score confidence intervals and sample sizes for stratified comparisons of binomial proportions in a series of articles gart and nam construct the efficient score tests and confidence intervals with or without skewness correction for stratified comparisons of binomial proportions on the risk difference relative risk and odds ratio effect metrics however the stratified score methods and their properties are not well understood we rederive the efficient score tests which reveals their theoretical relationship with the contrast based score tests and provides a basis for adapting the method by using other weighting schemes the inverse variance weight is optimal for a common treatment effect in large samples we explore the behavior of the score approach in the presence of extreme outcomes when either no or all subjects in some strata are responders and provide guidance on the choice of weights in the analysis of rare events the score method is recommended for studies with a small number of moderate or large sized strata a general framework is proposed to calculate the asymptotic power and sample size for the score test in superiority noninferiority and equivalence clinical trials or case control studies we also describe a nearly exact procedure that underestimates the exact power but the degree of underestimation can be controlled to a negligible level the proposed methods are illustrated by numerical examples

approximately exact inference for the common odds ratio in several 2 2 tables abstract the conditional maximum likelihood estimator mle of the common odds ratio in a sequence of independent 2 2 tables is known to be superior to the mantel haenszel estimator in terms of asymptotic efficiency and has the further advantage that its exact distribution is known however a long standing barrier to the widespread use of this estimator has been computational intractability in particular the calculation of significance levels confidence sets and power based on the exact distribution requires fast and efficient algorithms an important class of such algorithms form the basis of statxact a software package able to solve various aspects of the exact inference problem for a sequence of several 2 2 tables in real time we provide an alternative methodology by developing several useful saddlepoint approximations to the exact distribution of the conditional mle the approximations are derived from an interesting representation for hypergeometric random variables as a sum of independent

fast computation of exact confidence limits for the common odds ratio in a series of 2 2 tables abstract the odds ratio is widely used as a measure of association in epidemiologic studies and clinical trials we consider calculation of exact confidence limits for the common odds ratio in a series of independent 2 2 tables and propose three modifications of the network algorithm of mehta patel and gray 1 formulating and dealing with the problem in algebraic instead of graph theoretic terms 2 performing convolutions on the natural scale instead of the logarithmic scale and 3 using the secant method instead of binary search to compute roots of polynomial equations enhancement of computational efficiency exceeding an order of magnitude afforded by these modifications is empirically demonstrated we also compare the modified method with one based on the fast fourier transform fft further we show that the fft method can also result in considerable loss of numerical accuracy the modifications proposed in this article yield an algorithm that is not only fast and accurate but that combine

a new variance estimator for the mantel haenszel odds ratio many asymptotic formulas for estimating the variance of and confidence intervals for odds ratios in case control studies are available but most have limitations the method suggested by cornfield 1956 proceedings of the third berkeley symposium 4 135 148 for example is iterative and requires calculation of exact variances if the number of subjects within each stratum is small this paper presents a new asymptotic noniterative variance estimate for the mantel haenszel odds ratio which based on a series of monte carlo experiments compares favorably with other noniterative estimates when used for confidence interval estimation

lie algebras graded by finite root systems and intersection matrix algebras this paper classifies the lie algebras graded by doubly laced finite root systems and applies this classification to identify the intersection matrix algebras arising from multiply affinized cartan matrices of types b c f and g this completes the determination of the lie algebras graded by finite root systems initiated by berman and moody who studied the simply laced finite root systems of rank 2

estimation of a common effect parameter from sparse follow up data breslow 1981 biometrika 68 73 84 has shown that the mantel haenszel odds ratio is a consistent estimator of a common odds ratio in sparse stratifications for cohort studies however estimation of a common risk ratio or risk difference can be of greater interest under a binomial sparse data model the mantel haenszel risk ratio and risk difference estimators are consistent in sparse stratifications while the maximum likelihood and weighted least squares estimators are biased under poisson sparse data models the mantel haenszel and maximum likelihood rate ratio estimators have equal asymptotic variances under the null hypothesis and are consistent while the weighted least squares estimators are again biased similarly of the common rate difference estimators the weighted least squares estimators are biased while the estimator employing mantel haenszel weights is consistent in sparse data variance estimators that are consistent in both sparse data and large strata can be derived for all the mantel haenszel estimators

interval estimation of the common odds ratio from k 2 x 2 tables under cluster sampling we propose a simple correction factor for the variance of the logarithm of the common odds ratio estimated by the method of mantel and haenszel from a series of 2 2 tables when data are cluster correlated the adjustment is applied to the variance estimators proposed by hauck and by robins breslow and greenland for the log of the mantel haenszel common odds ratio and its performance is evaluated in a simulation study the key features of the proposed adjustment are i it has closed form ii it can accommodate covariates defined at the cluster specific level the site specific level or both and iii it does not require the user to specify a particular correlation structure for the response data the correction derives from liang and zeger s generalized estimating equations gee technique for logistic regression modelling via simulation we examine empirical versus nominal coverage probabilities for interval estimation of the common odds ratio using adjusted and unadjusted variance estimates and we present ratios of observed to estimated variances results are compared to those obtained from the fully iterated gee analysis the characteristics of the simulation study mimic scenarios common in the periodontal research setting with small numbers of subjects n 25 50 moderate numbers of sites per cluster m 4 16 32 and modest intracluster correlation levels 0 0 0 1 0 2 0 3 results show that adjusted confidence intervals applied to the hauck or the robins breslow greenland variance estimate provide coverage probabilities close to the nominal level for the mantel haenszel common odds ratio over a variety of cluster sizes and levels of correlation copyright 1999 john wiley sons ltd

derivations and invariant forms of jordan and alternative tori jordan and alternative tori are the coordinate algebras of extended affine lie algebras of types a 1 and a 2 in this paper we show that the derivation algebra of a jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations in the course of proving this result we investigate derivations of the more general class of division graded jordan and alternative algebras we also describe invariant forms of these algebras

mid p confidence intervals for the poisson expectation conventionally a confidence interval ci for the standardized mortality ratio is set using the conservative ci for a poisson expectation mu employing the mid p argument we present alternative cis that are shorter than the conventional ones the mid p intervals do not guarantee the nominal confidence level but the true coverage probability is only lower than the nominal level for a few short ranges of mu the implications for mid p confidence intervals of various proposed definitions of two sided tests for discrete data are discussed

generalization of the mantel haenszel estimator to nonconstant odds ratios nouvel estimateur issu de la generalisation de l estimateur de mantel haenszel etude de ses proprietes asymptotiques en fonction du nombre de tables fixe et du comptage des cellules

a class of integrable modules for the core of eala coordinatized by quantum tori extended affine lie algebras eala are higher dimensional generalizations of affine kac moody lie algebras introduced in 19 they have been further studied in 1 2 5 toroidal lie algebras which are universal central extensions of g c t 1 1 t 1 n g simple finite dimensional lie algebra and are prime examples of cores of ealas which are studied in 4 7 14 16 22 24 25 there are many ealas which allow not only laurent polynomial algebra c t 1 1 t 1 n as co ordinate algebra but also quantum tori jordan tori and the octonians tori as co ordinated algebras depending on type of the lie algebra see 1 3 5 6 26 the structure theory of the eala of type ad 1 is tied up with lie algebra gld c cq where cq is the quantum torus see section 1 quantum tori defined in 21 are non commutative analogue of laurent polynomial algebras in this paper we study integrable modules for the core of the eala corresponding to gld c cq which we call we will also add derivations d1 d2 to and call it here cq is two variable quantum torus defined as the ring of non commutating laurent polynomials c t 1 1 t 1 2 with relation to t1t2 qt2t1 note that if q is generic then is an eala we classify theorem 5 2 irreducible integrable modules see definition 1 9 for where the zero degree center acts non trivially we prove that any such module is isomorphic to a highest weight module or a lowest module up to an automorphism see section 5 let and aff be finite and affine root system of sld c and its affinization consider aff z 2 and consider the subalgebra b corresponding to see section 1 by highest weight modules we mean a module with a weight vector killed by b and the highest weight h c t t 1 given as in theorem 4 1 for any such we define a highest weight module

generalized mantel haenszel estimators for k 2 x j tables mickey and elashoff 1985 biometrics 41 623 635 gave an extension of mantel haenszel estimation to log linear models for 2 x j x k tables their extension yields two generalizations of the mantel haenszel odds ratio estimator to k 2 x j tables this paper provides variance and covariance estimators for these generalized mantel haenszel estimators that are dually consistent i e consistent in both large strata and sparse data and presents comparisons of the efficiency of the generalized mantel haenszel estimators

summary attributable risk estimation from unmatched case control data abstract we propose an alternative method to obtain summary estimators variances and confidence intervals for attributable risk measures this method combines weighted exposure prevalences for cases and controls across strata formed by the cross classification of relevant covariates to form estimates of attributable risk among the exposed and attributable risk in the target population the major benefit of this approach over those previously proposed in the literature is that it operates on data summed across strata rather than on estimation of statistics within each stratum this alternative method for attributable risk measures utilizes the mantel haenszel estimate of an average odds ratio and can be implemented using the matrix procedure in sas this method is appropriate even when the within stratum sample sizes are too small for other methods to be valid simulation results indicate that this method is superior to others with respect to bias and coverage probability for confidence intervals

estimators and confidence intervals for the marginal odds ratio using logistic regression and propensity score stratification propensity score methods are widely used to estimate treatment or exposure effects in observational studies in studies with binary response the effect can be described as an odds ratio and the mantel haenszel estimator is traditionally used for stratified data although propensity score methods are designed for marginal treatment effects it has been shown that the mantel haenszel estimator stratified for propensity score is a questionable estimator for the marginal odds ratio which describes the change in odds of response if everybody versus nobody were treated we studied recently proposed alternative estimators for the marginal odds ratio one stratified for propensity score the other derived from logistic regression additionally we adapted the methodology of the logistic regression based estimator for the derivation of a marginal odds ratio estimator to covariate adjustment by the propensity score we also derived corresponding variance estimators using the delta method the estimators were illustrated and compared to the inverse probability weighted estimator and the stratified mantel haenszel estimator in a study dealing with respiratory tract infections in children in germany furthermore simulation studies that were carried out to investigate relative bias variance and coverage probability showed reasonable performance of marginal odds ratio estimators if response rates or regression based approaches were used their variances were accurately estimated in contrast the stratified mantel haenszel estimator was substantially biased in some situations due to problems of non collapsibility and thus it is generally inappropriate for a reliable estimation of the marginal odds ratio

estimators of the variance of the mantel haenszel log odds ratio estimate summary a new approach to large sample studies of the variance of the log odds ratio estimate of mantel and haenszel is developed and used to derive formulas that lead to useful standard error formulas for the mantel haenszel estimate the variance estimator of robins breslow and greenland 1986 biometrics 42 311 324 is derived in a new way and a new variance estimator valid only for the case of 1 m matching is also derived using the new approach

a characterization of affine kac moody lie algebras we give a new characterization of the affine kac moody algebras in terms of extended affine lie algebras we also present new realizations of the twisted affine kac moody algebras

on conjugacy of cartan subalgebras in extended affine lie algebras abstract that finite dimensional simple lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as coxeter dynkin diagrams and indecomposable irreducible root systems is arguably one of the most elegant results in mathematics the definition of the root system is done by fixing a cartan subalgebra of the given lie algebra the remarkable fact is that up to isomorphism this construction is independent of the choice of the cartan subalgebra the modern way of establishing this fact is by showing that all cartan subalgebras are conjugate for symmetrizable kac moody lie algebras with the appropriate definition of cartan subalgebra conjugacy has been established by peterson and kac an immediate consequence of this result is that the root systems and generalized cartan matrices are invariants of the kac moody lie algebras the purpose of this paper is to establish conjugacy of cartan subalgebras for extended affine lie algebras a natural class of lie algebras that generalizes the finite dimensional simple lie algebra and affine kac moody lie algebras

notes on interval estimation of the attributable risk in cross sectional sampling the attributable risk ar is probably the most useful and commonly used epidemiologic index to measure the importance of a risk factor in public health issues this paper focuses the discussion on interval estimation of the ar in cross sectional studies and compares the finite sample performance of five asymptotic interval estimators of the ar by calculating the coverage probability and the average length in a variety of situations this paper notes that the coverage probability of the two interval estimators proposed by leung and kupper including the one that combines the interval estimator on the basis of wald s test statistic can be substantially less than the desired confidence level when the underlying risk ratio equals 1 as long as the sample size is reasonably large 100 and the probability of exposure is moderate 0 20 the interval estimator suggested by fleiss can consistently perform well with respect to the coverage probability in a variety of situations considered here however using this interval estimator tends to generally lose efficiency this paper also finds that with respect to the coverage probability the interval estimator using fieller s theorem is generally preferable to the interval estimator on the basis of wald s test statistic when the prevalence rate ratio rr between the exposure and the non exposure groups is 2 finally this paper notes that if we know that the underlying parameter rr is large 4 and the probability of exposure is not small 0 05 the interval estimator suggested by leung and kupper will probably be preferable to all the other estimators considered here copyright 2001 john wiley sons ltd

the design of case control studies the effect of confounding on sample size requirements this paper considers the extent to which confounding effects of covariates which are not controlled for by matching in the design may influence the sample size necessary for case control studies the quantitative calculations are performed for an age matched case control study on lung cancer and air pollution and are based on different evaluation methods for illustrative purposes attention is confined to a dichotomous risk factor and a single dichotomous covariate by using the numerical values of a pilot study investigating lung cancer and air pollution it turns out that the sample size required for detecting a relative risk as close as 1 15 to 1 is only slightly influenced by the strength of the association between confounder and risk factor for reasonable variations around our empirical values on the other hand sample size considerably increases with increasing relative risk of a confounder even when the association remains small the sample size required for an individually matched analysis practically equals that for an age stratified analysis when the relative risk of the covariate is one with a relative risk greater than one however the size for a matched analysis exceeds that for a stratified analysis and the ratio between them increases with increasing relative risk

bias in methods for deriving standardized morbidity ratio and attributable fraction estimates this paper examines several methods for deriving standardized morbidity ratios smr and attributable fraction attributable risk percentage estimates we show that some of the proposed methods will in general produce biased estimators although the low variance of certain estimators sometimes compensates for their bias a few methods are based on statistical fallacies and should be avoided especially the method of deriving the expected number of cases from a regression equation that does not include the exposure under study

statistical methods for estimating attributable risk from retrospective data this paper extends levin s measure of attributable risk to adjust for confounding by aetiologic factors other than the exposure of interest one can estimate this extended measure from case control data provided either i from the control data one can estimate exposure prevalence within each stratum of the confounding factor or ii one has additional information available concerning the confounder distribution and the stratum specific disease rates in both cases we give maximum likelihood estimates and their estimated asymptotic variances and show them to be independent of the sampling design matched vs random computer simulations investigate the behaviour of these estimates and of three types of confidence intervals when sample size is small relative to the number of confounder strata the simulations indicate that attributable risk estimates tend to be too low the bias is not serious except when exposure prevalence is high among controls in this case the estimates and their standard error estimates are also highly unstable in general the asymptotic standard error estimates performed quite well even in small samples and even when the true asymptotic standard error was too small by contrast the bootstrap estimate tended to be too large none of the three confidence intervals proved superior in accuracy to the other two thus there appears no advantage in using the log based interval suggested by walter which is always longer than the simpler symmetric interval

on the relation of extended affine weyl groups and indefinite weyl groups abstract extended affine weyl groups are the weyl groups of root systems of a new class of lie algebras called extended affine lie algebras in this paper we show that a reduced extended affine weyl group is the homomorphic image of some indefinite kac moody weyl group where the homomorphism and its kernel are given explicitly

beta binomial model for meta analysis of odds ratios in meta analysis of odds ratios ors heterogeneity between the studies is usually modelled via the additive random effects model rem an alternative multiplicative rem for ors uses overdispersion the multiplicative factor in this overdispersion model odm can be interpreted as an intra class correlation icc parameter this model naturally arises when the probabilities of an event in one or both arms of a comparative study are themselves beta distributed resulting in beta binomial distributions we propose two new estimators of the icc for meta analysis in this setting one is based on the inverted breslow day test and the other on the improved gamma approximation by kulinskaya and dollinger 2015 p 26 to the distribution of cochran s q the performance of these and several other estimators of icc on bias and coverage is studied by simulation additionally the mantel haenszel approach to estimation of ors is extended to the beta binomial model and we study performance of various icc estimators when used in the mantel haenszel or the inverse variance method to combine ors in meta analysis the results of the simulations show that the improved gamma based estimator of icc is superior for small sample sizes and the breslow day based estimator is the best for n 100 the mantel haenszel based estimator of or is very biased and is not recommended the inverse variance approach is also somewhat biased for ors 1 but this bias is not very large in practical settings developed methods and r programs provided in the web appendix make the beta binomial model a feasible alternative to the standard rem for meta analysis of ors 2017 the authors statistics in medicine published by john wiley sons ltd

classification and construction of quasisimple lie algebras abstract we study a class of possibly infinite dimensional lie algebras called the quasisimple lie algebras qsla s and generalizing semisimple and affine kac moody lie algebras they are characterized by the existence of a finite dimensional cartan subalgebra a non degenerate symmetric ad invariant killing form and nilpotent rootspaces attached to non isotropic roots we are then able to derive a classification theorem for the possible irreducible elliptic quasisimple root systems moreover we construct explicit realizations of some of them as untwisted and twisted current algebras generalizing the affine loop algebras

a note on statistical hypothesis testing based on log transformation of the mantel haenszel common odds ratio for differential item functioning classification when differential item functioning dif is investigated dif classification is made using statistical test results and estimated dif sizes in practice one of the well known dif classifications is that of the educational testing service ets a negligible dif b medium dif and c large dif rules this article provides a clarifying note on a a sketch of the proof of the asymptotic normality of what is known as the mantel haenszel mh delta which provides the basis of a point and an interval null hypothesis test based on the mh delta and b how to conduct an interval null hypothesis test using the mh delta which is necessary for the c dif classification

resolvability and the upper dimension of graphs abstract for an ordered set w w 1 w 2 w k of vertices and a vertex v in a connected graph g the metric representation of v with respect to w is the k vector r v w d v w 1 d v w 2 d v w k where d x y represents the distance between the vertices x and y the set w is a resolving set for g if distinct vertices of g have distinct representations a new sharp lower bound for the dimension of a graph g in terms of its maximum degree is presented a resolving set of minimum cardinality is a basis for g and the number of vertices in a basis is its metric dimension dim g a resolving set s of g is a minimal resolving set if no proper subset of s is a resolving set the maximum cardinality of a minimal resolving set is the upper dimension dim g the resolving number res g of a connected graph g is the minimum k such that every k set w of vertices of g is also a resolving set of g then 1 dim g dim g res g n 1 for every nontrivial connected graph g of order n it is shown that dim g res g n 1 if and only if g k n while dim g res g 2 if and only if g is a path of order at least 4 or an odd cycle the resolving numbers and upper dimensions of some well known graphs are determined it is shown that for every pair a b of integers with 2 a b there exists a connected graph g with dim g dim g a and res g b also for every positive integer n there exists a connected graph g with res g dim g n and dim g dim g n

attributable risk estimation from matched case control data a methodology is proposed for obtaining summary estimators variances and confidence intervals for attributable risk measures from data obtained through a case control study design where one or more controls have been matched to each case the sampling design for obtaining these data is conceptualized as a simple random sample of cases being equivalent to a simple random sample of matched sets by combining information across the strata determined by the matched sets this approach provides all of the benefits associated with the mantel haenszel procedure for the estimators of attributable risk among the exposed and population attributable risk asymptotic variances are derived under the assumption that the frequencies of the unique response patterns follow the multinomial distribution simulation results indicate that these methods fare very well with respect to bias and coverage probability

the alternative torus and the structure of elliptic quasi simple lie algebras of type we present the complete classification of the tame irreducible elliptic quasi simple lie algebras of type a2 and in particular specialize on the case where the coordinates are not associative here the coordinates are cayley dickson algebras over laurent polynomial rings in v 3 variables which we call alternative tori in giving our classification we need to present much information on these alternative tori and the lie algebras coordinatized by them

multiloop realization of extended affine lie algebras and lie tori an important theorem in the theory of infinite dimensional lie algebras states that any affine kac moody algebra can be realized that is to say constructed explicitly using loop algebras in this paper we consider the corresponding problem for a class of lie algebras called extended affine lie algebras ealas that generalize affine algebras ealas occur in families that are constructed from centreless lie tori so the realization problem for ealas reduces to the realization problem for centreless lie tori we show that all but one family of centreless lie tori can be realized using multiloop algebras in place of loop algebras we also obtain necessary and sufficient conditions for two centreless lie tori realized in this way to be isotopic a relation that corresponds to isomorphism of the corresponding families of ealas

k metric antidimension the study and analysis of social graphs impacts on a wide range of applications such as community decision making support and recommender systems with the boom of online social networks such analyses are benefiting from a massive collection and publication of social graphs at large scale unfortunately individuals privacy right might be inadvertently violated when publishing this type of data in this article we introduce k anonymity a novel privacy measure aimed at evaluating the resistance of social graphs to active attacks k anonymity is based on a new problem in graph theory the k metric antidimension defined as follows let g v e be a simple connected graph and s w 1 w t v an ordered subset of vertices the metric representation of a vertex u v with respect to s is the t vector r u s d g u w 1 d g u w t where dg u v represents the length of a shortest u v path in g we call s a k antiresolving set if k is the largest positive integer such that for every vertex v v s there exist other k 1 different vertices v 1 v k 1 v s with r v s r v 1 s r v k 1 s the k metric antidimension of g is the minimum cardinality among all the k antiresolving sets for g we address the k metric antidimension problem by proposing a true biased algorithm with success rate above 80 when considering random graphs of size at most 100 the proposed algorithm is used to determine the privacy guarantees offered by two real life social graphs with respect to k anonymity we also investigate theoretical properties of the k metric antidimension of graphs in particular we focus on paths cycles complete bipartite graphs and trees

fixed point subalgebras of extended affine lie algebras abstract it is a well known result that the fixed point subalgebra of a finite dimensional complex simple lie algebra under a finite order automorphism is a reductive lie algebra so it is a direct sum of finite dimensional simple lie subalgebras and an abelian subalgebra we consider this for the class of extended affine lie algebras and are able to show that the fixed point subalgebra of an extended affine lie algebra under a finite order automorphism which satisfies certain natural properties is a sum of extended affine lie algebras up to existence of some isolated root spaces a subspace of the center and a subspace which is contained in the centralizer of the core moreover we show that the core of the fixed point subalgebra modulo its center is isomorphic to the direct sum of the cores modulo centers of the involved summands

inference for odds ratio regression models with sparse dependent data suppose the number of 2 x 2 tables is large relative to the average table size and the observations within a given table are dependent as occurs in longitudinal or family based case control studies we consider fitting regression models to the odds ratios using table level covariates the focus is on methods to obtain valid inferences for the regression parameters beta when the dependence structure is unknown in this setting liang 1985 biometrika 72 678 682 has shown that inference based on the noncentral hypergeometric likelihood is sensitive to misspecification of the dependence structure in contrast estimating functions based on the mantel haenszel method yield consistent estimators of beta we show here that under the estimating function approach wald s confidence interval for beta performs well in multiplicative regression models but unfortunately has poor coverage probabilities when an additive regression model is adopted as an alternative to wald inference we present a mantel haenszel quasi likelihood function based on integrating the mantel haenszel estimating function a simulation study demonstrates that in medium sized samples the mantel haenszel quasi likelihood approach yields better inferences than other methods under an additive regression model and inferences comparable to wald s method under a multiplicative model we illustrate the use of this quasi likelihood method in a study of the familial risk of schizophrenia

quantum tori and the structure of elliptic quasi simple lie algebras abstract we study and classify those tame irreducible elliptic quasi simple lie algebras which are simply laced and of rankl 3 the first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates when the type isdorethe coordinates are laurent polynomials in variables while for typeathe coordinates can be any quantum torus in variables the next step is to study the universal central extension as well as the derivation algebra of the core these are related to the first connes cyclic homology group of the coordinates the final step is to use this information to give constructions of lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras

an estimating equation approach for the analysis of case control studies with exposure measured at several levels liang gave an extension of the mantel haenszel estimating procedure for a common odds ratio to logistic regression models it is applicable to case control studies with multiple exposure levels which yield k 2 j tables this paper provides variance and covariance estimators which are consistent in both sparse data and large strata for liang s estimating functions in the k 2 j tables case and proposes an approximate confidence interval method for the common odds ratios

central extensions of lie algebras graded by finite root systems lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by berman and moody benkart and zelmanov and neher in this paper we determine the central extensions of these lie algebras and hence describe them completely up to isomorphism

extended affine weyl groups abstract in this paper we study the weyl groups of reduced extended affine root systems the root systems of extended affine lie algebras we start by describing the extended affine weyl group as a semidirect product of a finite weyl group and a heisenberg like normal subgroup this provides a unique expression for the weyl group elements in terms of some naturally arisen transformations which is crucial in the further study of extended affine weyl groups we use this to give a presentation called a presentation by conjugation for an important subclass of extended affine weyl groups using a new notion called the index which is an invariant of the extended affine root systems we show that one of the important features of finite and affine root systems related to weyl group holds for the class of extended affine root systems

small sample bias of point estimators of the odds ratio from matched sets the bias of several point estimators of the odds ratio arising from matched pair data is investigated for small samples simple alternatives to the traditional maximum likelihood estimator are suggested on both the original scale and the logarithm scale in each case the suggested estimators possess a superior performance in terms of mean square error generalizations are given for 1 r matched data sets

bcn graded lie algebras arising from fermionic representations abstract we use fermionic representations to obtain a class of bc n graded lie algebras coordinatized by quantum tori with nontrivial central extensions

estimation of attributable risk for case control studies with multiple matching kuritz and landis considered case control studies with multiple matching and proposed an asymptotic interval estimator of the attributable risk based on wald s statistic using monte carlo simulation kuritz and landis demonstrated that their interval estimator could perform well when the number of matched sets was large or 100 however the number of matched sets may often be moderate or small in practice in this paper we evaluate the performance of kuritz and landis interval estimator in small or moderate number of matched sets and compare it with four other interval estimators we note that the coverage probability of kuritz and landis interval estimator tends to be less than the desired confidence level when the probability of exposure among cases is large in these cases the interval estimator using the logarithmic transformation and the two interval estimators derived from the quadratic equations developed here can generally improve the coverage probability of kuritz and landis interval estimator especially for the case of a small number of matched sets furthermore we find that an interval estimator derived from a quadratic equation is consistently more efficient than kuritz and landis interval estimator the interval estimator using the logit transformation although which performs poorly when the underlying odds ratio or is close to 1 can be useful when both the probability of exposure among cases and the underlying or are moderate or large

interval estimation of the attributable risk for multiple exposure levels in case control studies with confounders the attributable risk ar is one of the most important and commonly used epidemiological indices to assess the public health importance of an association between a risk factor and a disease when the underlying risk factor has multiple exposure levels in the presence of confounders we consider the case control studies using random sampling to collect the cases and controls here we develop four asymptotic interval estimators for ar including the interval estimator using wald s statistic the interval estimator using the logarithmic transformation the interval estimator using the logit transformation and the interval estimator derived from a quadratic equation we apply monte carlo simulation to evaluate the finite sample performance of these interval estimators in a variety of situations we demonstrate that given an adequately large sample size all the estimators developed here can actually perform reasonably well we note that the interval estimator using the logit transformation may be of limited use when the number of studied subjects is not large we also note that the interval estimator using the logarithmic transformation can lose efficiency compared to the interval estimator using wald s statistic or the interval estimator derived from a quadratic equation developed in this paper finally we use the data taken from a case ontrol tudy of the oral contraceptive use in myocardial nfarction patients with various smoking levels to illustrate he practical usefulness of these estimators copyright 2003 john wiley sons ltd

a note on the covariance of the mantel haenszel log odds ratio estimator and the sample marginal rates a simple technique developed in phillips unpublished ph d dissertation university of windsor windsor ontario 1987 is used to approximate cov theta mh pi i 1 2 where theta mh is the mantel haenszel log odds ratio estimator for a 2 x 2 x k table and the pi are the sample marginal proportions these results are then applied to obtain an approximate variance estimate of an adjusted risk difference based on the mantel haenszel odds ratio estimator

estimators of the mantel haenszel variance consistent in both sparse data and large strata limiting models summary this paper proposes a new estimator of the variance of the mantel haenszel estimator of common odds ratio that is easily computed and consistent in both sparse data and large strata limiting models monte carlo experiments compare its performance to that of previously proposed variance estimators

vertex operators arising from the homogeneous realization for we use the underlying fock space for the homogeneous vertex operator representation of the affine lie algebra to construct a family of vertex operators as an application an irreducible module for an extended affine lie algebra of type a n 1 coordinatized by a quantum torus q of 2 variables or 3 variables is obtained moreover this module turns out to be a highest weight module which is an analog of the basic module for affine lie algebras

mixed metric dimension of graphs abstract let g v e be a connected graph a vertex w v distinguishes two elements vertices or edges x y e v if dg w x dg w y a set s of vertices in a connected graph g is a mixed metric generator for g if every two distinct elements vertices or edges of g are distinguished by some vertex of s the smallest cardinality of a mixed metric generator for g is called the mixed metric dimension and is denoted by dim m g in this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds we also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph finally we prove that the problem of determining the mixed metric dimension of a graph is np hard in the general case

likelihood ratio tests with three way tables likelihood ratio lr tests for association and for interaction are examined for three way contingency tables particularly the widely used 2 2 k table mutual information identities are used to characterize the information decomposition and the logical relationship between the omnibus lr test for conditional independence across k strata and its two independent components lr tests for interaction and for uniform association the latter two tests are logically connected to formulating a natural two step test for conditional independence the proposed two step test with reduced nominal levels is suggested instead of the breslow day test and the cochran mantel haenszel test this yields efficient interval estimation for both the interaction parameter and the common odds ratio compared with using the mantel haenszel estimate this allows the development of power analysis for testing general hypotheses of varied interactions using an invariant pythagorean law of relative entropy

alternative estimators of the common odds ratio for 2 x 2 tables hauck 1984 reported on the comparative efficiencies and biases as evident from simulation studies of three estimators of a common odds ratio or common log odds ratio these three estimators included both the conditional and the unconditional maximum likelihood ml estimators and the estimator based on the mantel haenszel mh 1959 procedure but particular care needs to be taken in the interpretation of seeming biases in the estimation of the common odds ratio reasonably we might look for unbiasedness in the log odds ratio rather than in the odds ratio itself odds ratio estimates are likely to be highly right skewed around the true parameter value while the log odds ratio estimate would tend to show more symmetric behavior if an estimator shows unbiasedness relative to the log odds ratio it will necessarily show positive bias relative to the odds ratio itself this is because in the presence of some variation the arithmetic mean will exceed the geometric mean if the geometric mean is the true odds ratio the arithmetic mean will exceed the true odds ratio another consequence is that if two estimators are unbiased relative to the log odds ratio then the one showing the greater variability in estimation of the log odds ratio will ordinarily show greater bias in the estimation of the odds ratio for log normally distributed estimates which is not exactly the present case the bias in the arithmetic mean due to variability in the logarithmic estimates is known to be the factor exp r 2 2 this kind of thing can be recognized from hauck s results where one can see that the bias in estimates of the odds ratio reflects both the bias and the variance in estimates of the log odds ratio

classification of division zn graded alternative algebras abstract the octonion torus or cayley torus appears as a coordinate algebra of extended affine lie algebras of type a 2 and f 4 a generalized version of the octonion torus called division z n graded alternative algebras is classified in this paper using the result we can complete the classification of division a 2 z n graded lie algebras up to central extensions which are a generalization of the cores of extended affine lie algebras of type a 2

extended mantel haenszel estimating procedure for multivariate logistic regression models a class of estimating functions is proposed for the estimation of multivariate relative risk in stratified case control studies it reduces to the well known mantel haenszel estimator when there is a single binary risk factor large sample properties of the solutions to the proposed estimating equations are established for two distinct situations efficiency calculations suggest that the proposed estimators are nearly fully efficient relative to the conditional maximum likelihood estimator for the parameters considered application of the proposed method to family data and longitudinal data where the conditional likelihood approach fails is discussed two examples from case control studies and one example from a study on familial aggregation are presented

saddlepoint approximations for small sample logistic regression problems double saddlepoint approximations provide quick and accurate approximations to exact conditional tail probabilities in a variety of situations this paper describes the use of these approximations in two logistic regression problems an investigation of regression analysis of the log odds ratio in a sequence or set of 2 2 tables via simulation studies shows that in practical settings the saddlepoint methods closely approximate exact conditional inference the double saddlepoint approximation in the test for trend in a sequence of binomial random variates is also shown via simulation studies to be an effective approximation to exact conditional inference copyright 2000 john wiley sons ltd

computing the k metric dimension of graphs abstract given a connected graph g v e a set s v is a k metric generator for g if for any two different vertices u v v there exist at least k vertices w 1 w k s such that dg u wi dg v wi for every i 1 k a metric generator of minimum cardinality is called a k metric basis and its cardinality the k metric dimension of g we make a study concerning the complexity of some k metric dimension problems for instance we show that the problem of computing the k metric dimension of graphs is np hard however the problem is solved in linear time for the particular case of trees

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices

attributable risk estimation from case control data via logistic regression by fitting an unconditional logistic regression model to unmatched case control data an estimate of the joint population attributable risk for the factor included is obtained this estimate and its asymptotic variance can easily be computed from the intercept parameter and its asymptotic variance a generalization to the analysis of stratified data with large strata enables the calculation of stratum specific attributable risks and their variances via stratum specific intercept parameters if sampling of cases is independent of strata an estimate of the summary attributable risk and its asymptotic variance may be obtained as a weighted sum of the stratum specific attributable risks

variance estimators for attributable fraction estimates consistent in both large strata and sparse data a number of variance formulae for the attributable fraction have been presented but none is consistent in sparse data such as found in individually matched case control studies this paper employs mantel haenszel estimation to derive variance estimators for attributable fractions that are dually consistent that is consistent in both sparse data and large strata the method may also be applied using conditional maximum likelihood extensions of these estimators to situations involving effect modification and preventive exposures are also derived examples of applications to individually matched case control studies are given