testing for association in contingency tables with multiple column responses summary in many studies multiple categorical responses or measurements are made on members of different populations or treatment groups this arises often in surveys where individuals may mark all answers that apply when responding to a multiple choice question frequently it is of interest to determine whether the distributions of responses differ among groups in this situation the test statistic of the usual pearson chi square test no longer measures a scaled distance between observed and hypothesized cell counts in a contingency table and its distribution is no longer the familiar chisquare this paper presents a modification to the pearson statistic that measures the appropriate distance for multiple response tables the asymptotic distribution is shown to be that of a linear combination of chi square random variables with coefficients depending on the true probabilities a bootstrap resampling method is proposed instead to obtain a null hypothesis sampling distribution simulations show that this bootstrap method maintains its size under a variety of circumstances while a naively applied pearson chi square test is severely affected by multiple responses

block gauss elimination followed by a classical iterative method for the solution of linear systems in the last two decades many papers have appeared in which the application of an iterative method for the solution of a linear system is preceded by a step of the gauss elimination process in the hope that this will increase the rates of convergence of the iterative method this combination of methods has been proven successful especially when the matrix a of the system is an m matrix the purpose of this paper is to extend the idea of one to more gauss elimination steps consider other classes of matrices a e g p cyclic consistently ordered and generalize and improve the asymptotic convergence rates of some of the methods known so far

extrapolated gauss seidel i and sor methods for least squares problems recently special attention has been given in the literature to the problems of accurately computing the least squares solution of very largescale over determined systems of linear equations which occur in geodetic applications in particular it has been suggested that one can solve such problems iteratively by applying the block sor successive overrelaxation and egs1 extrapolated gauss seidel 1 plus semi iterative methods to a linear system with coefficient matrix 2 cyclic or 3 cyclic the comparison of 2 block sor and 3 block sor was made in 1 and showed that the 2 block sor is better in 6 the authors also proved that 3 block egs1 si is better than 3 block sor here we first show that the 2 block dj double jacobi si gs si and egs1 si methods are equivalent and all of them are equivalent to the 3 block egs1 si method then we prove that the composite methods and 2 block sor have the same asymptotic rate of convergence but the former has a better average rate of convergence finally numerical experiments are reported and confirm that the 3 block egs1 si is better than the 2 block sor

a young eidson s type algorithm for complex p cyclic sor spectra abstract in a recent work of ours we have solved the problem of the minimization of the spectral radius of the iteration matrix of a p cyclic successive overrelaxation sor method for the solution of the linear system ax b when the matrix a is block p cyclic consistently ordered for what is known as the one point problem for any p 3 particular cases of the one point problem were solved by young varga kjellberg kredell russell and others in the present work we develop a theory using the results of our previous one and solve first the two point problem special cases of which were solved by wrigley eiermann niethammer ruttan noutsos and others secondly we generalize and extend our theory to cover the many point problem and develop a young eidson s type algorithm for its solution as possible application areas we mention among others the best block p cyclic repartitioning for the sor method are the solution of large scale systems arising in queueing network problems in markov analysis

the numerical solution of equality constrained quadratic programming problems this paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem the constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints among this class of convergent iterative schemes are generalizations of the relaxed jacobi gauss seidel and symmetric gauss seidel schemes

a survey of the iterative methods for the solution of linear systems by extrapolation relaxation and other techniques abstract for the iterative solution of linear systems of algebraic equations ax b 1 with a c n n x b c n and det a 0 numerous methods exist although a classification of them seems not to be possible one may note that the first step for the construction of an iterative method usually begins with a splitting of a in 1 thus a is written as a m n where det m 0 and m is easily inverted so that 1 is equivalent to x tx c 2 t m 1n c m 1 the discussion will be restricted to the so called linear stationary iterative methods although some nonstationary ones will be mentioned and emphasis will be given to those which fall into the categories of extrapolation relaxation and similar ones for this it will be assumed that the spectrum t of t in 2 is contained in a well defined compact region r whose complement with respect to the complex plane is simply connected and 1 r under these assumptions and for each specific class of methods described each time an attempt will be made to present the optimum one the optimum is the one out of the class of methods for which the sequence of vectors yielded converges asymptotically to the unique solution of 1 as fast as possible

accuracy of preconditioned cg type methods for least squares problems abstract the conjugate gradient method with imgs an incomplete modified version of gram schmidt orthogonalization to obtain an incomplete orthogonal factorization preconditioner applied to the normal equations pcgls is often used as the basic iterative method to solve the linear least squares problems in this paper a detailed analysis is given for understanding the effect of rounding errors on imgs and determining the accuracy of computed solutions of pcgls with imgs for linear least squares problems in finite precision it is shown that for a consistent system the difference between the true residuals and the updated approximate residual vectors generated depends on the machine precision e on the maximum growth in norm of the iterates over their initial values the norm of the true solution and the condition number of r which is affected by the drop set in incomplete gram schmidt factorization similar results are obtained for the difference between the true and computed solution for inconsistent systems numerical tests are carried out to confirm the theoretical conclusions

jackknifing and bootstrapping goodness of fit statistics in sparse multinomials abstract many goodness of fit problems can be put in the following form n n 1 nk is a multinomial n vector and it is desired to test the composite null hypothesis h 0 p against all possible alternatives the usual tests used are pearson s 1900 statistic or the likelihood ratio statistic neyman and pearson 1928 cressie and read 1984 pointed out that both of these statistics are in the power family of statistics with 1 and 0 respectively they suggested an alternative statistic with although all of these statistics are asymptotically 2 in the usual situation of k fixed and n this is not the case if the multinomial is sparse specifically morris 1975 showed that under certain regularity conditions with k and n both x 2 and g 2 are asymptotically normal with different mean and variance under a simple null hypothesis cressie and read 1984 extended these results to the general 2ni family although these results have not been proven for composite nul

minimization of l 2 norms of the sor and msor operators the average behavior of the sor method and the msor method is studied for matrices whose jacobian matrix is skew symmetric matrix the objective is to find the values of optimal parameters to minimize the l 2 norm of the sor and msor operators based on the eigenvalue functional given by yin and yuan appl math comput 127 2002 327 333

order reducing conjugate gradients versus block aor for constrained least squares problems abstract we compare the convergence properties of two iterative algorithms for solving equality constrained least squares problems the first algorithm due to barlow nichols and plemmons applies a variation of the conjugate gradient algorithm to a symmetric positive definite system which is smaller than the original problem the second block accelerated overrelaxation is a two parameter generalization of block sor barlow nichols and plemmons have proven that their order reducing conjugate gradient algorithm converges faster than block sor we extend their result to show that the algorithm is also superior to block aor numerical experiments on some structural engineering problems support the analysis

on computational procedures for the force method it is known that the matrix force method has certain advantages over the displacement method for a class of structural problems it is also known that the force method when carried out by the conventional gauss jordan procedure tends to fill in the problem data making the method unattractive for large size sparse problems this poor fill in property however is not necessarily inherent to the method and the sparsity may be maintained if one uses what we call the turn back lu procedure the purpose of this paper is two fold first it is shown that there exist some close relationships between the force method and the least squares problem and that many existing algebraic procedures to perform the force method can be regarded as applications extensions of certain well known matrix factorization schemes for the least squares problem secondly it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint included in this unification is the turn back lu procedure which was originally proposed by topcu in his thesis 8 it is explained why this procedure tends to produce sparse and banded self stress and flexibility matrices with small band width some computational results are presented to demonstrate the superiority of the turn back lu procedure over the other schemes considered in this paper

optimal p cyclic sor for complex spectra abstract we consider the successive overrelaxation sor method for the solution of a linear system ax b when the matrix a has a block p p partitioned p cyclic form and its associated block jacobi matrix jp is weakly cyclic of index p following the pioneering work by young and varga in the 1950s many researchers have considered various cases for the spectrum jp and have determined optimal values for the relaxation factor 0 2 so that the sor method converges as fast as possible after the most recent work on the best block p cyclic repartitioning and that on the solution of large scale systems arising in queueing network problems in markov analysis the optimization of the convergence of the p cyclic sor for more complex spectra jp has become more demanding here we state the one point problem for the general p cyclic complex sor case the existence and the uniqueness of its solution are established by analyzing and developing further the theory of the associated hypocycloidal curves for the determination of the optimal parameter s an algorithm is presented and a number of illustrative numerical examples are given

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

the analysis of longitudinal polytomous data generalized estimating equations and connections with weighted least squares in recent years methods have been developed for modelling repeated observations of a categorical response obtained over time on the same individual although situations in which the repeated response is binary or poisson have been studied extensively relatively little attention has been given to polytomous categorical response variable in this paper we extend the estimating equations initially developed for clustered discrete data by liang and zeger 1986 biometrika 73 13 22 and subsequently extended by prentice 1988 biometrics 44 1033 1048 to polytomous response variables under certain assumptions we illustrate that these estimating equations simplify to the weighted least squares wls equations formalized by koch et al 1977 biometrics 33 133 158 this connection provides a formal framework for obtaining iterated weighted least squares model parameter estimates cumulative logit models are developed and applied to a representative longitudinal data set simulation results comparing wls an iterative form of wls and independence estimating equations using a robust estimate of the variance are presented

testing for conditional multiple marginal independence summary survey respondents are often prompted to pick any number of responses from a set of possible responses categorical variables that summarize this kind of data are called pick any c variables counts from surveys that contain a pick any c variable along with a group variable r levels and stratification variable q levels can be marginally summarized into an r c q contingency table a question that may naturally arise from this setup is to determine if the group and pick any c variable are marginally independent given the stratification variable a test for conditional multiple marginal independence cmmi can be used to answer this question since subjects may pick any number out of c possible responses the cochran 1954 biometrics10 417 451 and mantel and haenszel 1959 journal of the national cancer institute22 719 748 tests cannot be used directly because they assume that units in the contingency table are independent of each other therefore new testing methods are developed cochran s test statistic is extended to r 2 q tables and a modified version of this statistic is proposed to test cmmi its sampling distribution can be approximated through bootstrapping other cmmi testing methods discussed are bootstrap p value combination methods and bonferroni adjustments simulation findings suggest that the proposed bootstrap procedures and the bonferroni adjustments consistently hold the correct size and provide power against various alternatives

the analysis ofk step iterative methods for linear systems from summability theory using the theory of euler methods from summability theory we investigate general iterative methods for solving linear systems of equations in particular for a given euler method a regions of the complex plane is determined such that ak step iterative method converges if the eigenvalues of an iteration operatort are contained ins for a givens optimal methods are described and upper and lower bounds are derived for the associated asymptotic rate of convergence special attention is given to two step methods with complex parameters

modeling a categorical variable allowing arbitrarily many category choices this article discusses the modeling of a categorical variable for which subjects can select any number of categories for c categories the response variable consists of a cross classification of c binary components one pertaining to each category using data from a survey loughin t m and scherer p n 1998 biometrics 54 630 637 in which kansas farmers indicated their primary sources of veterinary information we discuss simultaneous logit modeling of the binary components of the multivariate response the use of maximum likelihood or quasi likelihood fitting provides chi squared tests with degrees of freedom df c r 1 for testing the independence between each of the c response components and an explanatory variable with r categories these tests are alternatives to the weighted chi squared test and the bootstrap test proposed by loughin and scherer for this hypothesis

a likelihood based method for analysing longitudinal binary responses summary in this paper we discuss a likelihood based method for analysing correlated binary responses based on a multivariate model it is related to the pseudo maximum likelihood approach suggested recently by zhao prentice 1990 their parameterization results in a simple pairwise model in which the association between responses is modelled in terms of correlations while the present paper uses conditional log odds ratios with this approach higher order associations can be incorporated in a natural way one important advantage of this parameterization is that the maximum likelihood estimates of the marginal mean parameters are robust to misspecification of the time dependence we describe an iterative two stage procedure for obtaining the maximum likelihood estimates two examples are presented to illustrate this methodology

convergence of a direct iterative method for large scale least squares problems abstract in 1975 chen and gentleman suggested a 3 block sor method for solving least squares problems based on a partitioning scheme for the observation matrix a into a a 1 a 2 where a1 is square and nonsingular in many cases a1 is obvious from the nature of the problem this combined direct iterative method was discussed further and applied to angle adjustment problems in geodesy where a1 is easily formed and is large and sparse by plemmons in 1979 recently niethammer de pillis and varga have rekindled interest in this method by correcting and extending the sor convergence interval the purpose of our paper is to discuss an alternative formulation of the problem leading to a 2 block sor method for this formulation it is shown that the resulting direct iterative method always converges for sufficiently small sor parameter in contrast to the 3 block formulation formulas for the optimum sor parameter and the resulting asymptotic convergence factor based upon a2a 11 2 are given furthermore it is shown that this 2 cyclic block sor method always gives better convergence results than the 3 cyclic one for the same amount of work per iteration the direct part of the algorithm requires only a sparse matrix factorization of a1 our purpose here is to establish theoretical convergence results in line with the purpose of the recent paper by niethammer de pillis and varga practical considerations of choosing a1 in certain applications and of estimating the resulting a2a 11 2 will be addressed elsewhere

a note on two block sor methods for sparse least squares problems abstract we compare two recently proposed block sor methods for the solution of large least squares problems with the conjugate gradient algorithm for solving the specially preconditioned normal equations by proving that all three methods are based on the same krylov sequence we show that the conjugate gradient algorithm is generally preferable to the two sor approaches

successive overrelaxation methods for solving the rank deficient linear squares problem abstract we develop successive overrelaxation sor methods for finding the least squares solution of minimal norm to the system ax b where a is an m n matrix of rank r the methods are obtained by first augmenting the system to a block 4 4 consistent system of linear equations the augmented coefficient matrix is then split by a subproper sor splitting an interval for the relaxation parameter in which the subproper sor iteration matrix is semiconvergent is determined along with the optimal relaxation parameter which minimizes the modulus of the controlling eigenvalue of the sor matrix since the scheme computes at first only a solution to the augmented system it is subsequently shown how to transform such a solution to the unique solution of minimal 2 norm analysis of the practical implementation of the algorithms developed here will be given elsewhere

a parallelizable sor like method systems with plus shaped and linear spectra abstract the classical sor method replaces an invertible cyclic linear system in b x f with a family of systems i n b x f where for some optimal scalar b the spectrum of b b is circular the eigenvalues of b are linearly placed falling on the real line moreover the spectral radius of b b is less than that of b similarly this paper explores a tensor mechanism which converts more general systems in b x f to equivalent systems i 2n b x f where the spectrum of b may be plus shaped the spectrum of b is circular and x col 0 x as in the sor theory the spectral radius of b is less than that of b among other things we recapture the results of the sor theory improve on some known results and extend our theory directly to matrices in b which need not be symmetric yet have real plus and minus valued eigenvalues

a simple test of association for contingency tables with multiple column responses loughin and scherer 1998 biometrics54 630 637 investigated tests of association in two way tables when one of the categorical variables allows for multiple category responses from individual respondents standard chi squared tests are invalid in this case and they developed a bootstrap test procedure that provides good control of test levels under the null hypothesis this procedure and some others that have been proposed are computationally involved and are based on techniques that are relatively unfamiliar to many practitioners in this paper the methods introduced by rao and scott 1981 journal of the american statistical association76 221 230 for analyzing complex survey data are used to develop a simple test based on a corrected chi squared statistic

generalized central limit theorem and renormalization group we introduce a simple instance of the renormalization group transformation in the banach space of probability densities by changing the scaling of the renormalized variables we obtain as fixed points of the transformation the levy strictly stable laws we also investigate the behavior of the transformation around these fixed points and the domain of attraction for different values of the scaling parameter the physical interest of a renormalization group approach to the generalized central limit theorem is discussed

superior convergence domains for the p cyclic ssor majorizer abstract thus far for an n n complex nonsingular matrix a the symmetric successive overrelaxation ssor majorizing operator has been used to establish convergence properties of the ssor method mostly in the case where a is an h matrix in this paper we use actually a similarity transformation of the ssor majorizer to investigate convergence properties of the block ssor method when a is a block p cyclic matrix let ja denote the block jacobi method and let j a we establish regions in the plane where s a q a l a here s a is the block ssor iteration operator associated with a l a is the block successive overrelaxation sor iteration operator associated with a and q a is a convenient similarity transformation of the majorizing operator for s a of special interest to us are the values of for which the above inequality holds for the corresponding values of the relaxation parameter a 2 1 the latter being an important quantity in the sor ssor theory for h matrices

simultaneously modeling joint and marginal distributions of multivariate categorical responses abstract we discuss model fitting methods for analyzing simultaneously the joint and marginal distributions of multivariate categorical responses the models are members of a broad class of generalized logit and loglinear models we fit them by improving a maximum likelihood algorithm that uses lagrange s method of undetermined multipliers and a newton raphson iterative scheme we also discuss goodness of fit tests and adjusted residuals and give asymptotic distributions of model parameter estimators for this class of models inferences are equivalent for poisson and multinomial sampling assumptions simultaneous models for joint and marginal distributions may be useful in a variety of applications including studies dealing with longitudinal data multiple indicators in opinion research cross over designs social mobility and inter rater agreement the models are illustrated for one such application using data from a recent general social survey regarding opinions about various types of government s

convergence of block iterative methods applied to sparse least squares problems abstract recently special attention has been given in the mathematical literature to the problems of accurately computing the least squares solutions of very large scale overdetermined systems of linear equations such as those arising in geodetical network problems in particular it has been suggested that one solve such problems iteratively by applying the block sor successive overrelaxation iterative method to a consistently ordered block jacobi matrix that is weakly cyclic of index 3 here we obtain new results theorem 1 giving the exact convergence and divergence domains for such iterative applications it is then shown how these results extend and correct the literature on such applications in addition analogous results theorem 2 are given for the case when the eigenvalues of the associated block jacobi matrix are nonnegative

convergence domains of the ssor method for a class of generalized consistently ordered matrices abstract the functional equation relating the eigenvalues of the block symmetric successive overrelaxation ssor iteration matrix with those of the block jacobi iteration matrix found by chong and cai 1985 is used in order to obtain precise domains of convergence of the block ssor iteration method associated with a class of generalized consistently ordered gco k p k matrices a p 2 k 1 p 1 we show that the domain of convergence depends only on the relaxation parameter the spectral radius of the block jacobi iteration matrix and the value of l k p unlike the case k p k 1 p 1 p 3 which was studied in an earlier paper by the authors 1989 beyond certain critical values of l the domain of convergence does not grow monotonically with l and manifests various sorts of behavior however as in the case k p k 1 p 1 it is shown that the intersection of the convergence domains taken over all pairs p k coincides with the exact domain of convergence of the point ssor iteration method associated with nonsingular h matrices a that is matrices which can be diagonally scaled to being strictly diagonally dominant

bootstrap recycling a monte carlo alternative to the nested bootstrap abstract a monte carlo algorithm is described that can be used in place of the nested bootstrap it is particularly advantageous when there is a premium on the number of bootstrap samples either because samples are hard to generate or because expensive computations are applied to each sample this recycling algorithm is useful because it enables inference procedures like prepivoting and bootstrap iteration in models where nested bootstrapping is computationally impractical implementation of the recycling algorithm is quite straightforward as a replacement of the double bootstrap for example bootstrap recycling involves two stages of sampling as does the double bootstrap the first stage of both algorithms is the same simulate from the fitted model in the second stage of recycling one batch of samples is simulated from one measure a measure dominating all the first stage fits these samples are recycled with each first stage sample to yield estimated adjustments to the original inference procedur

on the first order rao scott correction of the umesh loughin scherer statistic summary decady and thomas 2000 biometrics56 893 896 propose a first order corrected umesh loughin scherer statistic to test for association in an r c contingency table with multiple column responses agresti and liu 1999 biometrics55 936 943 point out that such statistics are not invariant to the arbitrary designation of a zero or one to a positive response this paper shows that in addition the proposed testing procedure does not hold the correct size when there are strong pairwise associations between responses

testing for marginal independence between two categorical variables with multiple responses questions that ask respondents to choose all that apply from a set of items occur frequently in surveys categorical variables that summarize this type of survey data are called both pick any c variables and multiple response categorical variables it is often of interest to test for independence between two categorical variables when both categorical variables can have multiple responses traditional pearson chi square tests for independence should not be used because of the within subject dependence among responses an intuitively constructed version of the pearson statistic is proposed to perform the test using bootstrap procedures to approximate its sampling distribution first and second order adjustments to the proposed statistic are given in order to use a chi square distribution approximation a bonferroni adjustment is proposed to perform the test when the joint set of responses for individual subjects is unavailable simulations show that the bootstrap procedures hold the correct size more consistently than the other procedures

optimality relationships forp cyclic sor the optimality question for blockp cyclic sor iterations discussed in young and varga is answered under natural conditions on the spectrum of the block jacobi matrix in particular it is shown that repartitioning a blockp cyclic matrix into a blockq cyclic form q p results in asymptotically faster sor convergence for the same amount of work per iteration as a consequence block 2 cyclic sor is optimal under these conditions