on a common circle natural scenes and gestalt rules to understand how the human visual system analyzes images it is essential to know the structure of the visual environment in particular natural images display consistent statistical properties that distinguish them from random luminance distributions we have studied the geometric regularities of oriented elements edges or line segments present in an ensemble of visual scenes asking how much information the presence of a segment in a particular location of the visual scene carries about the presence of a second segment at different relative positions and orientations we observed strong long range correlations in the distribution of oriented segments that extend over the whole visual field we further show that a very simple geometric rule cocircularity predicts the arrangement of segments in natural scenes and that different geometrical arrangements show relevant differences in their scaling properties our results show similarities to geometric features of previous physiological and psychophysical studies we discuss the implications of these findings for theories of early vision

uniform approximations to finite hilbert transform and its derivative interpolatory integration rules of numerical stability are presented for approximating cauchy principal value p v integrals 11 f t t c dt and hadamard finite part f p integrals 11f t t c 2 dt 1 c 1 respectively for a given smooth function f t present quadrature rules consist of interpolating f t at abscissae in the interval of integration 1 1 except for the pole c where neither the function value f c nor its derivative f c is required followed by subtracting out the singularities we demonstrate that the use of both endpoints 1 as abscissae in interpolating f t is essential for uniformly approximating the integrals namely for bounding the approximation errors independently of the values of c in fact for the f p integrals the use of double abscissae at both endpoints 1 as well as simple abscissae in 1 1 enables the uniform approximations while the use of simple abscissae at both endpoints 1 and those in 1 1 is sufficient for the p v integrals these facts suggest that finite hilbert transforms p v integrals and their derivatives f p integrals with varied values of c could be approximated efficiently with the same number of abscissae respectively some numerical examples are given

identifying the odds ratio estimated by a two stage instrumental variable analysis with a logistic regression model an adjustment for an uncorrelated covariate in a logistic regression changes the true value of an odds ratio for a unit increase in a risk factor even when there is no variation due to covariates the odds ratio for a unit increase in a risk factor also depends on the distribution of the risk factor we can use an instrumental variable to consistently estimate a causal effect in the presence of arbitrary confounding with a logistic outcome model we show that the simple ratio or two stage instrumental variable estimate is consistent for the odds ratio of an increase in the population distribution of the risk factor equal to the change due to a unit increase in the instrument divided by the average change in the risk factor due to the increase in the instrument this odds ratio is conditional within the strata of the instrumental variable but marginal across all other covariates and is averaged across the population distribution of the risk factor where the proportion of variance in the risk factor explained by the instrument is small this is similar to the odds ratio from a rct without adjustment for any covariates where the intervention corresponds to the effect of a change in the population distribution of the risk factor this implies that the ratio or two stage instrumental variable method is not biased as has been suggested but estimates a different quantity to the conditional odds ratio from an adjusted multiple regression a quantity that has arguably more relevance to an epidemiologist or a policy maker especially in the context of mendelian randomization

a new criterion for confounder selection we propose a new criterion for confounder selection when the underlying causal structure is unknown and only limited knowledge is available we assume all covariates being considered are pretreatment variables and that for each covariate it is known i whether the covariate is a cause of treatment and ii whether the covariate is a cause of the outcome the causal relationships the covariates have with one another is assumed unknown we propose that control be made for any covariate that is either a cause of treatment or of the outcome or both we show that irrespective of the actual underlying causal structure if any subset of the observed covariates suffices to control for confounding then the set of covariates chosen by our criterion will also suffice we show that other commonly used criteria for confounding control do not have this property we use formal theory concerning causal diagrams to prove our result but the application of the result does not rely on familiarity with causal diagrams an investigator simply need ask is the covariate a cause of the treatment and is the covariate a cause of the outcome if the answer to either question is yes then the covariate is included for confounder control we discuss some additional covariate selection results that preserve unconfoundedness and that may be of interest when used with our criterion

uniform approximation to cauchy principal value integrals with logarithmic singularity abstract an approximation of clenshaw curtis type is given for cauchy principal value integrals of logarithmically singular functions i f c 1 1 f x log x c x c d x c 1 1 with a given function f using a polynomial p n of degree n interpolating f at the chebyshev nodes we obtain an approximation i p n c i f c we expand p n in terms of chebyshev polynomials with o n log n computations by using the fast fourier transform our method is efficient for smooth functions f for which p n converges to f fast as n grows and so simple to implement this is achieved by exploiting three term inhomogeneous recurrence relations in three stages to evaluate i p n c for f z analytic on the interval 1 1 in the complex plane z the error of the approximation i p n c is shown to be bounded uniformly using numerical examples we demonstrate the performance of the present method

causal diagrams for empirical research summary the primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject matter information in particular the paper develops a principled nonparametric framework for causal inference in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data if so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions otherwise the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained

the numerical evaluation of hadamard finite part integrals a quadrature rule is described for the numerical evaluation of hadamard finite part integrals with a double pole singularity within the range of integration the rule is based upon the observation that such an integral is the derivative of a cauchy principal value integral

on the convergence of product formulas for the numerical evaluation of derivatives of cauchy principal value integrals product rules of interpolatory type on the zeros of generalized smooth jacobi polynomials for the numerical approximation of cauchy principal value integrals and their derivatives are considered convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the density function

strong collapsibility of association measures in linear models a measure of association in linear models is strongly collapsible over a discrete background variable if it remains unchanged no matter how the background variable is partially pooled and if it also coincides with the corresponding marginal measure of association strong collapsibility implies that the measure of association can be studied no matter how the background variable is categorized and no matter whether or not it is recorded in this paper necessary and sufficient conditions are given for strong collapsibility of the measure of association in linear models

a new algorithm for cauchy principal value and hadamard finite part integrals an algorithm for the approximate evaluation of integrals defined by cauchy principal value or by hadamard finite part has been proposed the convergence of the procedure is proved the stability of the algorithm is considered and some numerical examples are given

calculating singular integrals as an ill posed problem applying numerical quadrature to singular integrals in a straightforward way leads to uncontrolled instability with respect to data errors in this paper we describe how to control this instability

a novel fast modification of the adomian decomposition method to solve integral equations of the first kind with hypersingular kernels abstract in this paper using the modified adomian decomposition method a novel fast scheme has been presented being interactive to be used for solving a class of hypersingular integral equations of the first kind compared to the standard adomian decomposition method this method expedites the quick accumulation of a series of solutions different problems for exemplification have been solved in order to display the accuracy and validity of the aforementioned method according to the results it gets known that the novel fast modified adomian decomposition method is so efficient and simple which gives the exact solution and also shuns the complicated function of long theoretical calculations of collocation methods based on polynomial producing the exact solution

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

distortion of effects caused by indirect confounding summary undetected confounding may severely distort the effect of an explanatory variable on a response variable as defined by a stepwise data generating process the best known type of distortion which we call direct confounding arises from an unobserved explanatory variable common to a response and its main explanatory variable of interest it is relevant mainly for observational studies since it is avoided by successful randomization by contrast indirect confounding which we identify in this paper is an issue also for intervention studies for general stepwise generating processes we provide matrix and graphical criteria to decide which types of distortion may be present when they are absent and how they are avoided we then turn to linear systems without other types of distortion but with indirect confounding for such systems the magnitude of distortion in a least squares regression coefficient is derived and shown to be estimable so that it becomes possible to recover the effect of the generating process from the distorted coefficient

conditionally specified continuous distributions a distribution is conditionally specified when its model constraints are expressed conditionally for example besag s 1974 spatial model was specified conditioned on the neighbouring states and pseudolikelihood is intended to approximate the likelihood using conditional likelihoods there are three issues of interest existence uniqueness and computation of a joint distribution in the literature most results and proofs are for discrete probabilities here we exclusively study distributions with continuous state space we examine all three issues using the dependence functions derived from decomposition of the conditional densities we show that certain dependence functions of the joint density are shared with its conditional densities therefore two conditional densities involving the same set of variables are compatible if their overlapping dependence functions are identical we prove that the joint density is unique when the set of dependence functions is both compatible and complete in addition a joint density apart from a constant can be computed from the dependence functions in closed form since all of the results are expressed in terms of dependence functions we consider our approach to be dependence based whereas methods in the literature are generally density based applications of the dependence based formulation are discussed

an incomplete data approach to the analysis of covariance structures in this paper linear structural equation models with latent variables are considered it is shown how many common models arise from incomplete observation of a relatively simple system subclasses of models with conditional independence interpretations are also discussed using an incomplete data point of view the relationships between the incomplete and complete data likelihoods assuming normality are highlighted for computing maximum likelihood estimates the em algorithm and alternatives are surveyed for the alternative algorithms simplified expressions for computing function values and derivatives are given likelihood ratio tests based on complete and incomplete data are related and an example on using their relationship to improve the fit of a model is given

propensity score methods for bias reduction in the comparison of a treatment to a non randomized control group in observational studies investigators have no control over the treatment assignment the treated and non treated that is control groups may have large differences on their observed covariates and these differences can lead to biased estimates of treatment effects even traditional covariance analysis adjustments may be inadequate to eliminate this bias the propensity score defined as the conditional probability of being treated given the covariates can be used to balance the covariates in the two groups and therefore reduce this bias in order to estimate the propensity score one must model the distribution of the treatment indicator variable given the observed covariates once estimated the propensity score can be used to reduce bias through matching stratification subclassification regression adjustment or some combination of all three in this tutorial we discuss the uses of propensity score methods for bias reduction give references to the literature and illustrate the uses through applied examples

on the weights of certain quadratures for the numerical evaluation of cauchy principal value integrals and their derivatives we consider interpolatory quadrature rules of the type begin gathered frac d k d lambda k int limits 1 1 1 x alpha 1 x beta frac f x x lambda dx sum limits i 1 n w ni k lambda f x ni r n k f hfill k 0 1 alpha beta 0 hfill end gathered where the nodes x ni are the zeros of the jacobi polynomialp n x in particular we prove that sum limits i 1 n left w ni k lambda right o n k log n uniformly in 1 e 1 e furthermore we show that whenk 1 f x c p 1 1 p 1 andf p x h 1 1 0 1 we have r n 1 f o n p mu 1 delta 0 small as we like uniformly in 1 e 1 e

superconvergence of the composite simpson s rule for a certain finite part integral and its applications the superconvergence phenomenon of the composite simpson s rule for the finite part integral with a third order singularity is studied the superconvergence points are located and the superconvergence estimate is obtained some applications of the superconvergence result including the evaluation of the finite part integrals and the solution of a certain finite part integral equation are also discussed and two algorithms are suggested numerical experiments are presented to confirm the superconvergence analysis and to show the efficiency of the algorithms

signed directed acyclic graphs for causal inference formal rules governing signed edges on causal directed acyclic graphs are described in this paper and it is shown how these rules can be useful in reasoning about causality specifically the notions of a monotonic effect a weak monotonic effect and a signed edge are introduced results are developed relating these monotonic effects and signed edges to the sign of the causal effect of an intervention in the presence of intermediate variables the incorporation of signed edges into the directed acyclic graph causal framework furthermore allows for the development of rules governing the relationship between monotonic effects and the sign of the covariance between two variables it is shown that when certain assumptions about monotonic effects can be made then these results can be used to draw conclusions about the presence of causal effects even when data is missing on confounding variables

criteria for surrogate end points based on causal distributions summary when a treatment has a positive average causal effect ace on an intermediate variable or surrogate end point which in turn has a positive ace on a true end point the treatment may have a negative ace on the true end point due to the presence of unobserved confounders which is called the surrogate paradox a criterion for surrogate end points based on aces has recently been proposed to avoid the surrogate paradox for a continuous or ordinal discrete end point the distributional causal effect dce may be a more appropriate measure for a causal effect than the ace we discuss criteria for surrogate end points based on dces we show that commonly used models such as generalized linear models and cox s proportional hazard models can make the sign of the dce of the treatment on the true end point determinable by the sign of the dce of the treatment on the surrogate even if the models include unobserved confounders furthermore for a general distribution without any assumption of parametric models we give a sufficient condition for a distributionally consistent surrogate and prove that it is almost necessary

on the identification of path analysis models with one hidden variable summary we study criteria for identifiability of path analysis models with one hidden variable we first derive sufficient criteria for identification of models in which marginalisation is carried out over the hidden variable the sufficient criteria are based on the structure of the directed acyclic graph associated with the path analysis model and can be derived from the graph we treat further the identification of models when the hidden variable is conditioned on and establish connections with the extended skew normal distribution finally it is shown that the derived conditions extend the existing graphical criteria for identification

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

numerical evaluation of hypersingular integrals abstract boundary integral equations which employ integrals which exist only if defined in the cauchy principal value sense or as the hadamard finite part are currently used with success to solve many two and three dimensional problems of applied mechanics we will recall definitions and main properties of these integrals examine some numerical approaches for their evaluation and present several new results

quadrature rules with multiple nodes for evaluating integrals with strong singularities we present a method based on the chakalov popoviciu quadrature formula of lobatto type a rather general case of quadrature with multiple nodes for approximating integrals defined by cauchy principal values or by hadamard finite parts as a starting point we use the results obtained by l gori and e santi cf on the evaluation of hilbert transforms by means of a particular class of turan quadrature rules numer algorithms 10 1995 27 39 quadrature rules based on s orthogonal polynomials for evaluating integrals with strong singularities oberwolfach proceedings applications and computation of orthogonal polynomials isnm 131 birkhauser basel 1999 pp 109 119 we generalize their results by using some of our numerical procedures for stable calculation of the quadrature formula with multiple nodes of gaussian type and proposed methods for estimating the remainder term in such type of quadrature formulae numerical examples illustrations and comparisons are also shown

a sinc quadrature rule for hadamard finite part integrals a sinc quadrature rule is presented for the evaluation of hadamard finite part integrals of analytic functions integration over a general are in the complex plane is considered special treatment is given to integrals over the interval 1 1 theoretical error estimates are derived and numerical examples are included

collapsibility of logistic regression coefficients summary in this paper we discuss collapsibility of logistic regression coefficients over a background variable and present necessary and sufficient conditions for collapsibility

instrumental variables as bias amplifiers with general outcome and confounding drawing causal inference with observational studies is the central pillar of many disciplines one sufficient condition for identifying the causal effect is that the treatment outcome relationship is unconfounded conditional on the observed covariates it is often believed that the more covariates we condition on the more plausible this unconfoundedness assumption is this belief has had a huge impact on practical causal inference suggesting that we should adjust for all pretreatment covariates however when there is unmeasured confounding between the treatment and outcome estimators adjusting for some pretreatment covariate might have greater bias than estimators without adjusting for this covariate this kind of covariate is called a bias amplifier and includes instrumental variables that are independent of the confounder and affect the outcome only through the treatment previously theoretical results for this phenomenon have been established only for linear models we fill in this gap in the literature by providing a general theory showing that this phenomenon happens under a wide class of models satisfying certain monotonicity assumptions we further show that when the treatment follows an additive or multiplicative model conditional on the instrumental variable and the confounder these monotonicity assumptions can be interpreted as the signs of the arrows of the causal diagrams

patterns and symmetries in the visual cortex and in natural images as borders between different entities lines are an important element of natural images indeed the neurons of the mammalian visual cortex are tuned to respond best to lines of a given orientation this preferred orientation varies continuously across most of the cortex but also has vortex like singularities known as pinwheels in attempting to describe such patterns of orientation preference we are led to consider underlying rotation symmetries oriented segments in natural images tend to be collinear neurons are more likely to be connected if their preferred orientations are aligned to their topographic separation these are indications of a reduced symmetry requiring joint rotations of both orientation preference and the underlying topography this is verified by direct statistical tests in both natural images and in cortical maps using the statistics of natural scenes we construct filters that are best suited to extracting information from such images and find qualitative similarities to mammalian vision

the cauchy principal value and the hadamard finite part integral as values of absolutely convergent integrals the divergent integral abf x x x0 n 1dx for a x0 b and n 0 1 2 is assigned under certain conditions the value equal to the simple average of the contour integrals c f z z x0 n 1dz where c c is a path that starts from a and ends at b and which passes above below the pole at x0 it is shown that this value which we refer to as the analytic principal value is equal to the cauchy principal value for n 0 and to the hadamard finite part of the divergent integral for positive integer n this implies that where the conditions apply the cauchy principal value and the hadamard finite part integral are in fact values of absolutely convergent integrals moreover it leads to the replacement of the boundary values in the sokhotski plemelj fox theorem with integrals along some arbitrary paths the utility of the analytic principal value in the numerical analytical and asymptotic evaluations of the principal value and the finite part integral is discussed and demonstrated

algorithms for approximating finite hilbert transform with end point singularities and its derivatives abstract algorithms are proposed for the numerical evaluation of cauchy principal value integrals 1 1 w t f t t x d t 1 x 1 with weight functions of jacobi type singularities w t 1 t 1 t where 1 2 and 1 2 for a given function f t and hadamard finite part integrals 1 1 w t f t t x 2 d t the function f is interpolated by using a finite sum of chebyshev polynomials the present algorithms require o n log n arithmetic operations where n is the order of the interpolation polynomial it is shown that the present scheme gives uniform approximations namely the errors are bounded independently of x and is very efficient for smooth f further we discuss approximations of hyper singular integrals 1 1 w t f t t x n d t n 3 and show their uniform convergences numerical examples are given to demonstrate the performance of the present schemes

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

collapsibility of relative risk in contingency tables with a response variable we consider the collapsibility of relative risks in contingency tables with a polytomous response variable simple collapsibility of relative risks means that they are consistent over all levels of a collapsed variable and also coincide with the corresponding marginal relative risk strong collapsibility of relative risks means that they retain simple collapsibility no matter how the variable is partially collapsed we present necessary and sufficient conditions for simple and strong collapsibilities of relative risks

a note on a paper by g mastroianni and g monegato recently mastroianni and monegato derived error estimates for a numerical approach to evaluate the integral ab 11 f x y x y dxdy where a b 1 1 or a b a 1 or a b 1 b and f x y is a smooth function see g mastroianni and g monegato error estimates in the numerical evaluation of some bem singular integrals math comp 70 2001 251 267 the error bounds for the quadrature rule approximating the inner integral given in theorems 3 4 of that paper are not correct according to the proof however following a different approach we are able to improve the pointwise error estimates given in that paper

on a generalization of a result of w g cochran summary a relationship due to w g cochran showing the effect on least squares regression coefficients of marginalizing over or conditioning on an explanatory variable is generalized to quantile regression coefficients the condition under which conditioning does not induce interaction or effect reversal is shown examples are given the discussion is simplest when all variables are continuous the extension to discrete variables is outlined

on marginal and conditional parameters in logistic regression models a fundamental research question is how much a variation in a covariate influences a binary response variable in a logistic regression model both directly or through mediators we derive the exact formula linking the parameters of marginal and conditional regression models with binary mediators when no conditional independence assumptions can be made the formula has the appealing property of being the sum of terms that vanish whenever parameters of the conditional models vanish thereby recovering well known results as particular cases it also permits to quantify the distortion induced by omission of some relevant covariates opening the way to sensitivity analysis also in this case as the parameters of the conditional models are multiplied by terms that are always positive or bounded the formula may be used to construct reasonable bounds on the parameters of interest we assume that conditionally on a set of covariates the data generating process can be represented by a directed acyclic graph we also show how the results here presented lead to the extension of path analysis to a system of binary random variables

on quadrature for cauchy principal value integrals of oscillatory functions the authors develop an algorithm for the numerical evaluation of cauchy principal value integrals of oscillatory functions the method is based on an interpolatory procedure at the zeros of the orthogonal polynomials with respect to a jacobi weight a numerically stable procedure is obtained and the corresponding algorithm can be implemented in a fast way yielding satisfactory numerical results bounds of the error and of the amplification factor are also provided

comments on choice of time scale and its effect on significance of predictors in longitudinal studies by michael j pencina martin g larson and ralph b d agostino statistics in medicine 2007 26 1343 1359 the paper by pencina larson and d agostino 1 abbreviated pld raises a number of important practical issues regarding the choice of time scale and adjustment for age at entry into longitudinal cohort studies pld compare several approaches to the preferred entry age adjusted age scale model with hazard h a a0 z h0 a a0 exp z 1 where a is attained age a0 is the age at entry into the cohort and z is a vector of covariates in simulations pld used the baseline weibull hazard h0 a a0 a 1 which is independent of a0 but sampled from the left truncated weibull distribution with truncation at a0 and survival s a a0 z exp exp z a a0 for a a0 they found serious bias only for what they called the unadjusted age scale model which does not include age at entry as a covariate and which also ignores left truncation in analyzing left truncated data on the attained age scale one must decide whether or not to adjust for age at entry by including a0 as a covariate in the model or by stratifying the baseline hazard by categories of a0 analysts agree that one must account for left truncation in data based on attained age either by using a left truncated likelihood for parametric models or by including in the risk sets only persons at risk after their age at entry in a cox model analysis indeed this latter option is included in widely distributed survival software such as the entrytime statement in sas proc phreg which can be found at http v8doc sas com sashtml stat chap49 sect9 htm having accounted for left truncation the analyst must decide whether to allow the baseline hazard to depend on a0 by stratification on a0 as in korn graubard and midthune 2 or by including a0 as a covariate that affects the nuisance hazard we call these procedures covariate adjustment on a0 to distinguish them from left truncation adjustment on a0 since the unadjusted age scale model in pld fails to adjust for left truncation it results in a large bias table i in pld even for a model that requires no covariate adjustment on a0 because by construction in the simulation h0 a a0 a 1 does not depend on a0 table i estimated hazard ratios without and with adjustment for age at entry into cohort a0 according to categories of body mass index bmi for three ranges of attained age data are from the combined nhanes i ii iii data set used in reference 4 we would insist on the need to adjust for left truncation but whether or not it is necessary to adjust for age at entry a0 in addition depends on circumstance thiebaut and benichou 3 found for models in which h0 a a0 was independent of a0 only adjustment for left truncation was required to obtain unbiased estimates of even if z and a0 were correlated adjustment on a0 is needed if it is required for the validity of the proportional hazards assumption on z in equation 1 for example if the study tends to accrue healthy subjects the hazard may be reduced initially thus a 60 year old man who was accrued at age 40 may have a higher hazard than a 60 year old man with the same z who was accrued at age 58 because the latter man is more subject to favorable selection bias from recent accrual in this context equation 1 may hold only if the baseline hazard is allowed to depend on a0 these issues are of practical importance for example our group reported 4 on associations between body mass index bmi and mortality in the u s our analyses were based on attained age and adjusted for left truncation but no covariate adjustment was used for age at entry we have now repeated the analysis with adjustment for age at entry a0 by including it as an additional covariate in the cox model this adjustment had negligible effect compared to the analysis without covariate adjustment on a0 either on estimated relative risks table i or on the estimated numbers of deaths in the u s associated with various levels of bmi table ii table ii estimated excess deaths in the u s associated with categories of body mass index bmi without and with adjustment for age at entry into cohort a0 estimates are for all ages combined and from the combined nhanes i ii iii data set used in reference

joint response graphs and separation induced by triangular systems summary we consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions the independences are captured by missing edges in a directed graph a matrix form of such a graph called the generating edge matrix is triangular so the distributions that are generated over such graphs are called triangular systems we study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models i e for alternative recursive factorizations of the given density using joint conditional distributions for this we introduce families of linear triangular equations which do not require assumptions of distributional form the strength of the associations that are implied by such linear families for chain graph models is derived the edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix it is shown how induced independences and dependences can be studied by graphs by edge matrix calculations and via the properties of densities some ways of using the results are illustrated

natural scene statistics and the structure of orientation maps in the visual cortex visual activity after eye opening influences feature map structure in primary visual cortex v1 for instance rearing cats in an environment of stripes of one orientation yields an over representation of that orientation in v1 however whether such changes also affect the higher order statistics of orientation maps is unknown a statistical bias of orientation maps in normally raised animals is that the probability of the angular difference in orientation preference between each pair of points in the cortex depends on the angle of the line joining those points relative to a fixed but arbitrary set of axes natural images show an analogous statistical bias however whether this drives the development of comparable structure in v1 is unknown we examined these statistics for normal stripe reared and dark reared cats and found that the biases present were not consistently related to those present in the input or to genetic relationships we compared these results with two computational models of orientation map development an analytical model and a hebbian model the analytical model failed to reproduce the experimentally observed statistics in the hebbian model while orientation difference statistics could be strongly driven by the input statistics similar to those seen in experimental maps arose only when symmetry breaking was allowed to occur spontaneously these results suggest that these statistical biases of orientation maps arise primarily spontaneously rather than being governed by either input statistics or genetic mechanisms

a general condition for avoiding effect reversal after marginalization summary the paper examines the effect of marginalizing over a possibly unobserved background variable on the conditional relation between a response and an explanatory variable in particular it is shown that some conclusions derived from least squares regression theory apply in general to testing independence for arbitrary distributions it is also shown that the general condition of independence of the explanatory variable and the background ensures that monotonicity of dependence is preserved after marginalization relations with effect reversal and with collapsibility are sketched

statistics of natural images scaling in the woods we study the statistics of an ensemble of images taken in the woods distributions of local quantities such as contrast are scale invariant and have nearly exponential tails power spectra exhibit scaling with a nontrivial exponent these data limit the information content of natural images and point to the importance of gain control strategies in visual processing

towards a theory of early visual processing we propose a theory of the early processing in the mammalian visual pathway the theory is formulated in the language of information theory and hypothesizes that the goal of this processing is to recode in order to reduce a generalized redundancy subject to a constraint that specifies the amount of average information preserved in the limit of no noise this theory becomes equivalent to barlow s redundancy reduction hypothesis but it leads to very different computational strategies when noise is present a tractable approach for finding the optimal encoding is to solve the problem in successive stages where at each stage the optimization is performed within a restricted class of transfer functions we explicitly find the solution for the class of encodings to which the parvocellular retinal processing belongs namely linear and nondivergent transformations the solution shows agreement with the experimentally observed transfer functions at all levels of signal to noise

gauss quadrature rules for finite part integrals we construct a set of polynomials n x which are orthogonal with respect to w x x 2 where wx is a weight function these polynomials can be used for the definition of a gauss quadrature formula for the finite part integral the process is exactly the same as the one used for the extraction of the classical gauss formula for the riemann integrals three different methods are derived the first and most accurate quadrature formula is successfully tested in some numerical examples the proposed quadrature formulas have many applications in problems of mathematical physics mechanics etc

uniform convergence of derivatives of lagrange interpolation abstract uniform convergence of derivatives of lagrange interpolation at the union of zeros of jacobi polynomials and some additional points is investigated

the effect of pooling across strata in perfectly balanced studies pair matching frequency matching and stratification are used to obtain perfect exposure balance within strata defined by levels of confounders we consider the consequences of pooling such balanced data across strata in the analysis we examine model based inference for a variety of response variables and generalized linear regressions as well as for the cox 1972 journal of the royal statistical society series b 34 187 220 model if the data are pooled so that stratum effects are omitted from the regression certain models retain nominal size including all poisson models and all normal models with known variance the model based tests have supranominal size for all exponential models and subnominal size for all bernoulli models and for the cox model these results contrast with a previous analysis of simple randomization which only ensures exposure balance on average with simple randomization omission of stratum effects from these models leads to model based significance tests with supranominal size except for the cox and bernoulli models which retain nominal size just as for simple randomization however it is wise to include stratum effects in the analysis of perfectly balanced data to avoid biased estimation of exposure effects and to increase the efficiency of the comparison though motivated by cohort data these results also apply to balanced case control data under the logistic model

moderating effects of subgroups in linear models summary possibilities for moderating effects of a subgrouping variable on strength or direction of an association have been much discussed by social scientists but have not been given satisfactory statistical formulations the results concern directed measures of associations in linear models containing just three variables some key words analysis of covariance analysis of variance cg distribution conditional independence graphical chain model parallel regressions yule simpson paradox 1 introduction linear models are commonly used as a framework to estimate and test how a continuous response variable depends on potential influencing variables this paper is concerned with the situation in which two influences are random variables as well which may be discrete or continuous the continuous random variables are to stand for quantitative properties of observational units while the discrete variables either capture qualitative properties or they represent subgroups of the population to which the observational units belong to think of all variables as random variables is usually appropriate for data obtained from observational studies where one cannot control which levels or values of the influencing variables are to be observed the linear model permits an analysis conditional on fixed levels and values of the influences but frequently the association structure among the influences is in social science applications of interest itself and as it will turn out it is important for a correct interpretation of how the response depends on the influences awareness of such problems in interpretation is widespread among social scientists as is documented by extensive discussions of moderating effects see for example saunders 1956 zedeck 1971 and baron kenny 1986 there a variable is called a moderator if its presence changes the strength or direction of an association however methods recommended to social scientists for identifying moderating effects given in the above literature or by cohen cohen 1983 pp 310 4 have been shown to be seriously deficient wermuth 1988 thus there is need for clarification we treat the case of a potential discrete moderator variable and call its categories or levels the subgroups in the linear model a moderating effect is closely tied to the notion of consistent results on an association wermuth 1987 in the present context results are said to be weakly consistent if the associations within subgroups coincide in direction and strongly consistent if the associ ations within subgroups coincide in direction and strength this notion depends typically on the chosen measure of association in linear models we look at directed measures of

collapsibility and response variables in contingency tables summary various definitions of the collapsibility of a hierarchical log linear model for a multidimensional contingency table are considered and shown to be equivalent necessary and sufficient conditions for collapsibility are found in terms of the generating class it is shown that log linear models are appropriate for tables with response and explanatory variables if and only if they are collapsible onto the explanatory variables some key word8 collapsibility contingency table graphical model interaction graph log linear model response variable s sufficiency shown to be closely related some models have the property that relations between a set of the classifying factors may be studied by examination of the table of marginal totals formed by summing over the remaining factors such models are said to be collapsible onto the given set of factors collapsibility has important consequences for hypothesis testing and model selection and can be useful in data reduction we consider various definitions of collapsibility and show their equivalence furthermore necessary and sufficient conditions for collapsi bility are found in terms of the generating class many tables analysed in practice involve response variables simple examples one of which is given in 3 suffice to show the importance of distinguishing between response and explanatory variables first that inappropriate models may be avoided and second that natural and relevant models that are not log linear may be considered this paper characterizes appropriate and inappropriate log linear models for tables with response variables and some alternative approaches for the analysis of such tables are briefly considered we consider a multidimensional contingency table n based on a set of classifying factors f for a given subset a of f we are interested in the table of marginal totals na that is to say the table of cell counts summed over the remaining factors ac that is the complement of a in f we identify a hierarchical log linear model l that is the set of probabilities p e l with its generating class whose elements generators are given in square brackets thus for example the model ab bcd for a 4 way table corresponds

hilbert and hadamard transforms by generalized chebyshev expansion abstract an automatic quadrature is presented for approximating hadamard finite part fp integrals of a smooth function with a double pole singularity within the range of integration the quadrature rule is derived from the differentiation of an approximation to a cauchy principal value integral or the hilbert transform the approximation to the fp integral is represented as a function of the value of pole by using chebyshev polynomials of the second kind since the error can be estimated independently of the value of pole a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance with the same number of function evaluations numerical examples are also included to illustrate the performance of the methods

efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels abstract in this paper we present a special clenshaw curtis filon ccf type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels the non oscillatory and nonsingular part of the integrand is replaced by a special hermite interpolation polynomial error bounds with respect to the frequency k and the number of the clenshaw curtis points n are considered the overall computational complexity for the scheme is o nlog n numerical experiments support the theoretical analysis

approximate tests and interval estimation of the common relative risk in the combination of 2 2 tables consider i pairs of mutually independent binomial variates xoi and x1i with corresponding parameters poi and p1i and sample sizes noi and n1 where i 1 i we assume that the relative risk is constant that is b pt izpoi i 1 i further let qji 1 pji and we assume throughout that 0 pji qji 1 for i 1 i and j 0 1 we consider asymptotic results as noi s and n1j s become large but i is fixed extending the results of cochran 1954 for the common odds ratio radhakrishna 1965 derived an asymptotically efficient test of the hypothesis b 1 recently for i 1 koopman 1984 heuristically derived a useful approximate method for finding confidence limits for 0 in the present paper we use the methods of score statistics particularly as given by bartlett 1953a 1953b 1955 to derive for the general case the asymptotically efficient tests of b 1 and of its homogeneity over the i tables as well as approximate confidence limits for the common 0 the test of 0 1 is identical with that of radhakrishna and koopman s limits are a special case of these more general results thus this rather simple theory gives a coherent structured methodology for analysing ratios of binomial probabilities parallel to that usually employed for common odds ratios see for example gart 1970 1971

the amalgamation and geometry of two by two contingency tables if a pair of two by two contingency tables are amalgamated by addition it can happen that a measure of association for the amalgamated table lies outside the interval between the association measures of the individual tables we call this the amalgamation paradox and we show how it can be avoided by suitable designs of the sampling experiments we also study the conditions for the homogeneity of two subpopulations with respect to various measures of association some of the proofs have interesting geometrical interpretations

two trigonometric quadrature formulae for evaluating hypersingular integrals two trigonometric quadrature formulae one of non interpolatory type and one of interpolatory type for computing the hypersingular integral are developed on the basis of trigonometric quadrature formulae for cauchy principal value integrals the formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae fast three term recurrence relations for evaluating the quadrature weights are derived numerical tests are carried out using the current formula as applications two simple crack problems are considered one is a semi infinite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions it is found that the present method generally gives superior results copyright 2002 john wiley sons ltd

on the issue of imposing boundary conditions on quantum fields an interesting example of the deep interrelation between physics and mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields this procedure has recently been re examined with care comments on that and previous analysis are provided here together with considerations on the results of the purely mathematical zeta function method in an attempt at clarifying the issue hadamard regularization is invoked in order to fill the gap between the infinities appearing in the qft renormalized results and the finite values obtained in the literature with other procedures

the superconvergence of the composite trapezoidal rule for hadamard finite part integrals the composite trapezoidal rule has been well studied and widely applied for numerical integrations and numerical solution of integral equations with smooth or weakly singular kernels however this quadrature rule has been less employed for hadamard finite part integrals due to the fact that its global convergence rate for hadamard finite part integrals with p 1 order singularity is p order lower than that for the riemann integrals in general in this paper we study the superconvergence of the composite trapezoidal rule for hadamard finite part integrals with the second order and the third order singularity respectively we obtain superconvergence estimates at some special points and prove the uniqueness of the superconvergence points numerical experiments confirm our theoretical analysis and show that the composite trapezoidal rule is efficient for hadamard finite part integrals by noting the superconvergence phenomenon

on substantive research hypotheses conditional independence graphs and graphical chain models summary graphs consisting of points and lines or arrows as connections between selected pairs of points are used to formulate hypotheses about relations between variables points stand for variables connections represent associations when a missing connection is interpreted as a conditional independence the graph characterizes a conditional independence structure as well statistical models called graphical chain models correspond to special types of graphs which are interpreted in this fashion examples are used to illustrate how conditional independences are reflected in summary statistics derived from the models and how the graphs help to identify analogies and equivalences between different models graphical chain models are shown to provide a unifying concept for many statistical techniques that in the past have proven to be useful in analyses of data they also provide tools for new types of analysis

collapsibility of distribution dependence summary cox and wermuth proposed that the partial derivative of the conditional distribution function of a random variable y given another x is used for measuring association between two variables with arbitrary distributions the paper presents a necessary and sufficient condition for uniform collapsibility of this association measure over a third variable w

testing collapsibility in contingency tables on etudie en particulier le cas important des tables 2 2 k d ou il decoule les extensions aux tables plus grandes

criteria for surrogate end points summary a surrogate end point is often used to evaluate the effects of treatments or exposures on the true end point in medical researches various criteria for the statistical surrogate principal surrogate and strong surrogate have been proposed we first illustrate that with a surrogate end point that is defined by these criteria it is possible that a treatment has a positive effect on the surrogate which in turn has a positive effect on the true end point but the treatment has a negative effect on the true end point we define such a phenomenon as a surrogate paradox the surrogate paradox also means that the sign of the treatment effect on the true end point is unpredictable by the effect signs of both the treatment on the surrogate and the surrogate on the true end point then we propose two notions for a consistent surrogate and a strictly consistent surrogate to avoid the surrogate paradox with the causal network that was presented by lauritzen we discuss the conditions for a strong surrogate to be a consistent surrogate and a strictly consistent surrogate

graphical and recursive models for contingency tables summary we discuss two classes of models for contingency tables graphical and recursive models both of which arise from restrictions that are expressible as conditional independencies of variable pairs the first of these is a subclass of hierarchical log linear models each of its models can be represented by an undirected graph in the second class each model corresponds to a particular kind of a directed graph instead and can be characterized by a nontrivial factorization of the joint distribution in terms of response variables we derive decomposable or multiplicative models as the intersecting class this result has useful consequences for exploratory types of analysis as well as for the model interpretation we can give an aid for detecting well fitting decomposable models in a transformation of the observed contingency table and each decomposable model may be interpreted with the help of an undirected or directed graph

criteria for confounders in epidemiological studies the paper addresses a formal definition of a confounder based on the qualitative definition that is commonly used in standard epidemiology text books to derive the criterion for a factor to be a confounder given by miettinen and cook and to clarify inconsistency between various criteria for a confounder we introduce the concepts of an irrelevant factor an occasional confounder and a uniformly irrelevant factor we discuss criteria for checking these and show that miettinen and cook s criterion can also be applied to occasional confounders moreover we consider situations with multiple potential confounders and we obtain two necessary conditions that are satisfied by each confounder set none of the definitions and results presented in this paper require the ignorability and sufficient control confounding assumptions which are commonly employed in observational and epidemiological studies

on uniform approximations to hypersingular finite part integrals abstract in this paper new uniform approximation schemes for computation of hypersingular finite part integrals are studied the methods are verified to be supremely qualified for oscillatory integrands in addition based on the results on chebyshev approximations the uniform convergence and computational error bounds are considered preliminary numerical results show the stability and efficiency of the schemes

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals in a previous paper the authors proposed a modified gaussian rule m wf t to compute the integral wf t in the cauchy principal value sense associated with the weightw and they proved the convergence in closed sets contained in the integration interval the main purpose of the present work is to prove uniform convergence of the sequence m wf t on the whole integration interval and to give estimates for the remainder term the same results are shown for particular subsequences of the gaussian rules m wf t for the evaluation of cauchy principal value integrals a result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made