choice of column scores for testing independence in ordered 2 x k contingency tables the numerous statistical methods for testing no association between a binary response rows and k ordered categories columns group naturally into two classes those that require preassigned numerical column scores and those that do not an example of the former would be a logistic regression analysis and of the latter would be a wilcoxon rank sum test in this paper we demonstrate that the perceived advantage of not preassigning scores is illusory we do this by presenting an example from our consulting experience in which the midrank scores used by the rank tests that do not require preassigned scores are clearly inappropriate our recommendations are to assign reasonable column scores whenever possible and to consider equally spaced scores when the choice is not apparent midranks as scores should always be examined for their appropriateness before a rank test is applied

importance sampling for estimating exact probabilities in permutational inference abstract this article discusses importance sampling as an alternative to conventional monte carlo sampling for estimating exact significance levels in a broad class of two sample tests including all of the linear rank tests with or without censoring homogeneity tests based on the chi squared hypergeometric and likelihood ratio statistics the mantel haenszel trend test and zelen s test for a common odds ratio in several 2 2 contingency tables inference is based on randomly selecting 2 k contingency tables from a reference set of all such tables with fixed marginals through a network algorithm the tables are selected in proportion to their importance for reducing the variance of the estimated monte carlo p value spectacular gains up to four orders of magnitude are achieved relative to conventional monte carlo sampling the technique is illustrated on four real data sets

a network algorithm for performing fisher s exact test in r c contingency tables abstract an exact test of significance of the hypothesis that the row and column effects are independent in an r c contingency table can be executed in principle by generalizing fisher s exact treatment of the 2 2 contingency table each table in a conditional reference set of r c tables with fixed marginal sums is assigned a generalized hypergeometric probability the significance level is then computed by summing the probabilities of all tables that are no larger on the probability scale than the observed table however the computational effort required to generate all r c contingency tables with fixed marginal sums severely limits the use of fisher s exact test a novel technique that considerably extends the bounds of computational feasibility of the exact test is proposed here the problem is transformed into one of identifying all paths through a directed acyclic network that equal or exceed a fixed length some interesting new optimization theorems are developed in the process the numer

exact approaches for testing hypotheses based on the intra class kappa coefficient testing involving the intra class kappa coefficient is commonly performed in order to assess agreement involving categorical ratings a number of procedures have been proposed which make use of the limiting null distribution as the sample size goes to infinity in order to compute the observed significance as with many tests based on asymptotic null distributions these tests are associated with problematic type i error control for selected sample sizes and points in the parameter space we propose and study a collection of exact testing approaches for both the one sample and k sample scenarios for the one sample case p values are obtained using the exact distribution of the test statistic conditional on a sufficient statistic in addition unconditional approaches are considered on the basis of maximization across the nuisance parameter space numerical evaluation reveals advantages with the exact unconditional procedures copyright 2014 john wiley sons ltd

weak compactness and uniform convergence of operators in spaces of bochner integrable functions the classical characterization of weak compactness in the space l p over a measure space x c p states that a bounded set kc l p is relatively weakly compact iff it is uniformly o additive or equivalently uniformly inegrable or uniformly p absolutely continuous in case p l x 00 1 in this paper we give a new characterization of weak compactness in l p which is completely different from the classical one it is stated in terms of uniform weak convergence of certain admissible sequences of operators theorem 6 it is interesting to make a comparative analysis of these two characterizations of weak compactness

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

explicit solutions to dynamic diffusion type equations and their time integrals this paper deals with solutions of diffusion type partial dynamic equations on discrete space domains we provide two methods for finding explicit solutions examine their asymptotic behavior and time integrability these properties depend significantly not only on the underlying time structure but also on the dimension and symmetry of the problem throughout the paper the results are interpreted in the context of random walks and related stochastic processes

an ordering of dependence for contingency tables abstract a majorization ordering is defined on matrices with the same row and column sums this ordering is used as an ordering of dependence for contingency tables results are derived for maximal and minimal matrices with respect to the majorization ordering this theory can be used to maximize and minimize schur concave functions defined over matrices when there are row and column sum constraints in this paper it is applied to the algorithm of mehta and patel 1983 for finding the p value of fisher s exact test

loglinear multidimensional irt models for polytomously scored items a loglinear irt model is proposed that relates polytomously scored item responses to a multidimensional latent space the analyst may specify a response function for each response indicating which latent abilities are necessary to arrive at that response each item may have a different number of response categories so that free response items are more easily analyzed conditional maximum likelihood estimates are derived and the models may be tested generally or against alternative loglinear irt models

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

exact person fit indexes for the rasch model for arbitrary alternatives a new algorithm for obtaining exact person fit indexes for the rasch model is introduced which realizes most powerful tests for a very general family of alternative hypotheses including tests concerning dif as well as model deviating item correlations the method is also used as a goodness of fit test for whole data sets where the item parameters are assumed to be known for tests with 30 items at most exact values are obtained for longer tests a monte carlo algorithm is proposed simulated examples and an empirical investigation demonstrate test power and applicability to item elimination

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

methods for exact goodness of fit tests abstract numerous goodness of fit tests with asymptotic chi squared distributions have been proposed for discrete multivariate data and there has been much discussion about using asymptotic results for computing critical values when there are small expected cell values although exact methods would be preferred in these situations it generally is believed that such methods are computationally intractable we propose methods for calculating exact distributions and significance levels for goodness of fit statistics that are computationally feasible over a wide range of models in particular the distribution for a simple multinomial model can be evaluated in polynomial time for composite null hypotheses we calculate the distribution conditional on the sufficient statistics for the nuisance parameters we calculate the characteristic function of a distribution and invert the characteristic function using the fast fourier transform fft our approach emphasizes the relationship between exact methods and

constructing the exact significance level for a person fit statistic this paper uses an extension of the network algorithm originally introduced by mehta and patel to construct exact tail probabilities for testing the general hypothesis that item responses are distributed according to the rasch model by assuming that item difficulties are known the algorithm is applicable to the statistical tests either given the maximum likelihood ability estimate or conditioned on the total score a simulation study indicates that the network algorithm is an efficient tool for computing the significance level of a person fit statistic based on test lengths of 30 items or less

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

a quasi exact test for comparing two binomial proportions the use of the fisher exact test for comparing two independent binomial proportions has spawned an extensive controversy in the statistical literature many critics have faulted this test for being highly conservative partly in response to such criticism some statisticians have suggested the use of a modified non randomized version of this test namely the mid p value test this paper examines the actual type i error rates of this test for both one sided and two sided tests and for a wide range of sample sizes we show that the actual levels of significance of the mid p test tend to be closer to the nominal level as compared with various classical tests the computational effort required for the mid p test is no more than that needed for the fisher exact test further the basis for its modification is a natural adjustment for discreteness thus the test easily generalizes to r c contingency tables and other discrete data problems

dichotomies structure and concentration in normed spaces abstract we use probabilistic topological and combinatorial methods to establish the following deviation inequality for any normed space x r n there exists an invertible linear map t r n r n with p t g e t g e e t g c exp c max e 2 e log n e 0 where g is the standard n dimensional gaussian vector and c c 0 are universal constants it follows that for every e 0 1 and for every normed space x r n there exists a k dimensional subspace of x which is 1 e euclidean and k c e log n log 1 e this improves by a logarithmic on e term the best previously known result due to g schechtman

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

permutation distributions via generating functions with applications to sensitivity analysis of discrete data abstract generating functions provide a simple and elegant way to describe probability or frequency distributions of discrete statistics and in particular permutation distributions they are also a computational tool many efficient algorithms including those described as fast fourier transform methods network methods and multiple shift methods are different implementations of the recursions needed to evaluate generating functions efficiently our goals here are twofold first we make the relationship between these efficient methods and generating functions explicit this establishes a language for looking at other questions in randomization exact inference and may help in finding more efficient implementations second we propose methods to examine the sensitivity of results of exact analysis of discrete data to small perturbations in the data specifically we consider two settings how the analysis would change if one outcome changed and how the analysis would change if one observation was adde

random unconditional convergence of vector valued dirichlet series abstract we study random unconditionality of dirichlet series in vector valued hardy spaces h p x it is shown that a banach space x has type 2 respectively cotype 2 if and only if for every choice x n n x it follows that x n n s n is random unconditionally convergent respectively divergent in h 2 x the analogous question on h p x spaces for p 2 is also explored we also provide explicit examples exhibiting the differences between the unconditionality of x n n s n in h p x and that of x n z n n in h p x

an algorithm for finding the exact significance levels of r c contingency tables abstract an algorithm is presented for calculating the exact permutation significance value for r c contingency tables in general the algorithm does not require the total enumeration of all tables consistent with the given marginals as a result it is faster than existing algorithms

algebraic algorithms for sampling from conditional distributions we construct markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic examples include contingency tables logistic regression and spectral analysis of permutation data the algorithms involve computations in polynomial rings using grobner bases

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

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

estimating exact p values by the method of control variates or monte carlo rescue abstract despite algorithmic advances in exact nonparametric inference problems often occur that are too large for exact p value computations but too sparse for reliable asymptotic results in these situations monte carlo methods are a good compromise they bound the true p value within a confidence interval but a factor discouraging the use of monte carlo p values is their sensitivity to the random number sequence one can overcome this drawback by computing a 99 c i confidence interval of width less than 0001 then the estimated p values become insensitive to the random number sequence up to three decimals for all practical purposes these estimates are invariant to random number sequences the usual monte carlo method requires millions of samples to yield such an invariant estimate the monte carlo scheme presented here decreases the sample size by two to three orders of magnitude we illustrate this method with tests for r c tables two sample survival data and stratified 2 2 tables

all linear and integer programs are slim 3 way transportation programs we show that any rational convex polytope is polynomial time representable as a 3 way line sum transportation polytope of slim r c 3 format this universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure we provide a polynomial time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs our construction resolves several standing problems on 3 way transportation polytopes for example it demonstrates that unlike the case of 2 way contingency tables the range of values an entry can attain in any slim 3 way contingency table with specified 2 margins can contain arbitrary gaps our smallest such example has format 6 4 3 our construction provides a powerful automatic tool for studying concrete questions about transportation polytopes and contingency tables for example it automatically provides new proofs for some classical results including a well known real feasible but integer infeasible 6 4 3 transportation polytope of m vlach and bitransportation programs where any feasible bitransportation must have an arbitrarily large prescribed denominator

dynamic diffusion type equations on discrete space domains abstract we consider a class of partial dynamic equations on discrete space domains which includes as a special case the discrete space versions of the diffusion heat equation we focus on initial value problems and study the existence and uniqueness of forward and backward solutions moreover we discuss other topics such as sum and sign preservation maximum and minimum principles or symmetry of solutions

computing an exact confidence interval for the common odds ratio in several 2 2 contingency tables abstract a quadratic time network algorithm is provided for computing an exact confidence interval for the common odds ratio in several 2 2 independent contingency tables the algorithm is shown to be a considerable improvement on an existing algorithm developed by thomas 1975 which relies on exhaustive enumeration problems that would formerly have consumed several cpu hours can now be solved in a few cpu seconds the algorithm can easily handle sparse data sets where asymptotic results are suspect the network approach on which the algorithm is based is also a powerful tool for exact statistical inference in other settings

multiresolution image classification by hierarchical modeling with two dimensional hidden markov models this paper treats a multiresolution hidden markov model for classifying images each image is represented by feature vectors at several resolutions which are statistically dependent as modeled by the underlying state process a multiscale markov mesh unknowns in the model are estimated by maximum likelihood in particular by employing the expectation maximization algorithm an image is classified by finding the optimal set of states with maximum a posteriori probability states are then mapped into classes the multiresolution model enables multiscale information about context to be incorporated into classification suboptimal algorithms based on the model provide progressive classification that is much faster than the algorithm based on single resolution hidden markov models

partial dynamic equations on time scales in this work we generalize existing ideas of the univariate case of the time scales calculus to the bivariate case formal definitions of partial derivatives and iterated integrals are offered and bivariate partial differential operators are examined in particular solutions of the homogeneous and nonhomogeneous heat and wave operators are found when initial distributions given are in terms of elementary functions by means of the generalized laplace transform for the time scale setting finally the so termed mixed time scale setting is discussed examples are given and solutions are provided in tabular form

nonlinear model validation using correlation tests new higher order correlation tests which use model residuals combined with system inputs and outputs are presented to check the validity of a general class of nonlinear models the new method is illustrated by testing both simple and complex nonlinear system models

goodness of fit tests for large sparse multinomial distributions abstract a goodness of fit statistic d 2 is introduced for use in multinomial distributions pearson s x2 and d 2 are both approximately normally distributed when the sample size n is not large relative to the number of multinomial categories k under sequences of local alternative hypotheses the test based on d 2 exhibits moderate power when the x2 test is biased application is made to the analysis of large sparse contingency tables a theorem is presented that describes the likelihood ratio for testing a simple multinomial distribution against a mixture of multinomial distributions a wide variety of mixing distributions is considered and the d 2 statistic is a special case of log when testing for dirichlet mixtures of multinomial distributions in the case where n is large relative to k x2 and d 2 k behave approximately as chi squared random variables and differ by a very small amount under the null hypothesis in this situation x2 and d 2 k will generally yield the same inference with a l

the unconditional basic sequence problem we construct a banach space that does not contain any infinite un conditional basic sequence and investigate further properties of this space for example it has no subspace that can be written as a topological direct sum of two infinite dimensional spaces this property implies that every operator on the space is a strictly singular perturbation of a multiple of the identity in par ticular it is either strictly singular or fredholm with index zero this implies that the space is not isomorphic to any proper subspace trinity college cambridge cb2 1tq england current address department of mathematics university college london gower street lon don wc1e 6bt england e mail address wtg l o phx cam ac uk universit paris vii u f r de mathtmatiques 2 place jussieu 75251 paris cedex 05 france current address equipe d analyse et mathematiques appliquees universite de mamne la vallee 93160 noisy le grand france e mail address maurey logiquejussieu fr this content downloaded from 157 55 39 105 on fri 07 oct 2016 04 30 48 utc all use subject to http about jstor org terms

approximate conditional inference in generalized linear models on developpe des approximations precises et facilement calculables pour les densites conditionnelles et les distributions des statistiques exhaustives dans les modeles lineaires generalises avec des fonctions de liaison canoniques

exact significance testing to establish treatment equivalence with ordered categorical data this communication concerns the problem of establishing the therapeutic equivalence of two treatments that are being compared on the basis of ordered categorical data the problem is formulated as a significance test in which the null hypothesis specifies a treatment difference an efficient numerical algorithm for computing the exact significance level is provided along with a simple method for obtaining the asymptotic significance level both methods are applied to a clinical trial of a new agent versus an active control guidelines for when to use the exact procedure and when to rely on asymptotic theory are provided

testing hardy weinberg equilibrium an objective bayesian analysis we analyze the general multiallelic hardy weinberg equilibrium problem from an objective bayesian testing standpoint we argue that for small or moderate sample sizes the answer is rather sensitive to the prior chosen and this suggests to carry out a sensitivity analysis with respect to the prior this goal is achieved through the identification of a class of priors specifically designed for this testing problem in this paper we consider the class of intrinsic priors under the full model indexed by a tuning quantity the training sample size these priors are objective satisfy savage s continuity condition and have proved to behave extremely well for many statistical testing problems we compute the posterior probability of the hardy weinberg equilibrium model for the class of intrinsic priors assess robustness over the range of plausible answers as well as stability of the decision in favor of either hypothesis copyright 2010 john wiley sons ltd

saddlepoint approximations of the distribution of the person parameter in the two parameter logistic model large sample theory states the asymptotic normality of the maximum likelihood estimator of the person parameter in the two parameter logistic 2pl model in short tests however the assumption of normality can be grossly wrong as a consequence intended coverage rates may be exceeded and confidence intervals are revealed to be overly conservative methods belonging to the higher order theory more specifically saddlepoint approximations are a convenient way to deal with small sample problems confidence bounds obtained by these means hold the approximate confidence level for a broad range of the person parameter moreover an approximation to the exact distribution permits to compute median unbiased estimates mue that are as likely to overestimate as to underestimate the true person parameter additionally in small samples these mue are less mean biased than the often used maximum likelihood estimator

the complexity of three way statistical tables multiway tables with specified marginals arise in a variety of applications in statistics and operations research we provide a comprehensive complexity classification of three fundamental computational problems on tables existence counting and entry security one outcome of our work is that each of the following problems is intractable already for slim 3 tables with constant number 3 of rows 1 deciding existence of 3 tables with specified 2 marginals 2 counting all 3 tables with specified 2 marginals 3 deciding whether a specified value is attained in a specified entry by at least one of the 3 tables having the same 2 marginals as a given table this implies that a characterization of feasible marginals for such slim tables sought by much recent research is unlikely to exist another consequence of our study is a systematic efficient way of embedding the set of 3 tables satisfying any given 1 marginals and entry upper bounds in a set of slim 3 tables satisfying suitable 2 marginals with no entry bounds this provides a valuable tool for studying multi index transportation problems and multi index transportation polytopes remarkably it enables us to automatically recover a famous example due to vlach of a real feasible integer infeasible collection of 2 marginals for 3 tables of smallest possible size 3 4 6

existence and uniqueness of solutions to discrete diffusion equations abstract motivated by the classical one dimensional diffusion equation u t ku xx f x t x r t 0 u x 0 x x r we find the unique solution of the partial difference equation t u x t k 2 x x u x 1 t f x t x z t 0 u x 0 x x z where k 0 1 our approach is to construct the green s function for the appropriate partial difference operator and almost immediately we establish an existence and uniqueness result for the discrete nonhomogenous problem

methods for the analysis of contingency tables with large and small cell counts abstract the traditional practice has been to analyze contingency tables using asymptotic 2 approximations for the tail probability of certain test statistics even when the approximation is known to be poor for two dimensional tables pearson s x 2 statistic is most commonly used whereas for multidimensional tables the likelihood ratio test is preferred because it makes it easy to test series of nested hypotheses when asymptotic approximations are not adequate exact tests are preferred recent advances in algorithms have broadened the range of two dimensional tables for which satisfactory answers can be obtained baglivo olivier and pagano 1985 mehta and patel 1983 1986 pagano and taylor halvorsen 1981 in this article we propose algorithms that use a mixture of exact and asymptotic techniques to calculate tail probabilities our methods can be used with several different test statistics for two dimensional and multidimensional contingency table models and hence further broaden the set of tables

performing the exact test of hardy weinberg proportion for multiple alleles summary the hardy weinberg law plays an important role in the field of population genetics and often serves as a basis for genetic inference because of its importance much attention has been devoted to tests of hardy weinberg proportions hwp over the decades it has long been recognized that largesample goodness of fit tests can sometimes lead to spurious results when the sample size and or some genotypic frequencies are small although a complete enumeration algorithm for the exact test has been proposed it is not of practical use for loci with more than a few alleles due to the amount of computation required we propose two algorithms to estimate the significance level for a test of hwp the algorithms are easily applicable to loci with multiple alleles both are remarkably simple and computationally fast relative efficiency and merits of the two algorithms are compared guidelines regarding their usage are given numerical examples are given to illustrate the practicality of the algorithms

a bayesian independence test for small datasets we propose a bayesian test for independence among signals where only a small dataset is available traditional frequentist approaches often fail in this case due to inaccurate estimation of either the source statistical models or the threshold used by the test statistics in addition these frequentist methods cannot incorporate prior information into the computation of the test statistics our procedure renders parametric the nonparametric problem of testing for independence by quantizing the observed data samples into a table of cell counts the test statistic is based on the likelihood of the observed cell counts under the independence hypothesis where the marginal cell probabilities are modeled by independent symmetric dirichlet priors we apply our bayesian test to validate the solutions to the problem of blind source separation with small datasets using both synthetic and real life benchmark data the experimental results indicate that our approach can overcome the scarcity of data samples and significantly outperform the standard frequentist parametric methods with a proper selection of the prior parameters

higher order approximations to conditional distribution functions this paper derives higher order terms in the double saddlepoint expansion of skovgaard for a unidimensional conditional cumulative distribution function expansions for continuous and lattice random variables are derived results are applied to the sufficient statistic in logistic regression

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

using path induction to evaluate sequential allocation procedures path induction is a technique used to speed the process of making multiple exact evaluations of a sequential allocation procedure where the options are discrete and their outcomes follow a discrete distribution multiple evaluations are needed for determining criteria such as maxima or minima over parameter regions where the location of the extremal value is unknown in advance for visualizing characteristics such as robustness or for obtaining the distribution of a statistic rather than just its mean by using an initial phase to determine the number of paths reaching each terminal state the subsequent evaluations are far faster than repeated use of standard evaluation techniques algorithms are given for fully sequential and staged sequential procedures and the procedures can be either deterministic or random the procedures can be generated by any technique including dynamic programming or ad hoc approaches and the evaluations performed can be quite flexible and need not be related to the method of obtaining the procedure while the emphasis is on path induction the techniques used to speed up the analyses of staged allocation procedures can also be used to improve backward induction for such procedures if multiple evaluations need to be carried out however path induction will still be far superior for each parameter configuration to be evaluated one reduces the time by a factor of n where n is the size of the experiment by using path induction rather than the standard technique of backward induction in some settings the savings is significantly greater than n

exact inference for proportions from a stratified finite population abstract auditors and others often encounter finite populations with a dichotomous characteristic from which they draw stratified samples in auditing the dichotomy arises when a population item is classified as either in error or in compliance with some rule or regulation usually the proportion of errors is small the auditing objective may require calculation of a p value for the sample outcome relative to a hypothesis or a confidence bound for the proportion or total number of errors in the population in sampling from l strata with hypotheses concerning the total number of errors in the population the calculation of p values is not straightforward the complication arises because the parameter of the null hypothesis does not completely specify the distribution of the test statistic this distribution depends on an l 1 dimensional nuisance parameter consisting of the number of errors in each stratum because confidence bounds can be obtained by inverting the hypothesis test the same difficulty

approximating the conditional distribution of person fit indexes for checking the rasch model we present a simple approximation to the conditional distribution of goodness of fit statistics for the rasch model assuming that the item difficulties are known the approximation is easily programmed and gives relatively accurate assessments of conditionalp values for tests of length 10 or more a few generalizations are discussed

a disequilibrium coefficient approach to hardy weinberg testing a comparison was made of various tests for hardy weinberg equilibrium with emphasis on methods for multiple alleles for an overall test of deviations from equilibrium the classical chi square goodness of fit test generally performs well with continuity corrections needed for extreme gene frequencies or extreme departures from equilibrium for small samples probability tests are preferable and for multiple alleles these probability tests may be performed on a sample of all possible sets of genotypic frequencies having a fixed set of sample gene frequencies numerical work showed that the continuity corrected chi square was the most conservative test procedure and the uncorrected chi square the least conservative with multiple alleles a better appreciation of the nature of departures from equilibrium is given by the use of disequilibrium coefficients defined for each heterozygote as the difference between observed and expected frequencies likelihood ratio tests can be used to test each of these coefficients individually but a satisfactory procedure is to divide the squared estimate of each coefficient by its estimated variance and regard the ratio as a single degree of freedom chi square numerical studies confirmed the validity of this approach which has the great advantage of not requiring solutions of nonlinear equations

sequential implementation of monte carlo tests with uniformly bounded resampling risk this paper introduces an open ended sequential algorithm for computing the p value of a test using monte carlo simulation it guarantees that the resampling risk the probability of a different decision than the one based on the theoretical p value is uniformly bounded by an arbitrarily small constant previously suggested sequential or nonsequential algorithms using a bounded sample size do not have this property although the algorithm is open ended the expected number of steps is finite except when the p value is on the threshold between rejecting and not rejecting the algorithm is suitable as standard for implementing tests that require re sampling it can also be used in other situations to check whether a test is conservative iteratively to implement double bootstrap tests and to determine the sample size required for a certain power an r package implementing the sequential algorithm is available online

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

random generation of 2 2 2 j contingency tables we propose two markov chains for sampling m 1 dimensional contingency tables indexed by 1 2 m 1 2 n stationary distributions of our chains are the uniform distribution and a conditional multinomial distribution which is equivalent to the hypergeometric distribution if m 1 mixing times of our chains are bounded by n n 1 ln n 2me n n 1 ln dn e where d is the average of the values in the cells and e is a given error bound we use the path coupling method for estimating the mixing times of our chains and showed that our chains mix rapidly

the conditional distribution of goodness of fit statistics for discrete data abstract i consider the distribution of pearson s statistic and of the likelihood ratio goodness of fit statistic for discrete data in the important case where the data are extensive but sparse it is argued that the appropriate reference distribution is conditional on the sufficient statistic for the unknown regression parameters the first three conditional asymptotic cumulants are derived by edgeworth expansion and these are used for the computation of tail probabilities the principal advantage of the limit considered here as opposed to the more usual x 2 limit is that the cell counts need not be large

an exact test for hardy weinberg and multiple alleles algorithms for generating the exact distribution of a finite sample drawn from a population in hardy weinberg equilibrium are given for multiple alleles the finite sampling distribution is derived analogously to fisher s 2 x 2 exact distribution and is equivalent to levene s conditional finite sampling distribution for hardy weinberg populations the algorithms presented are fast computationally and allow for quick alternatives to standard methods requiring corrections and approximations computation time is on the order of a few seconds for three allele examples and up to 2 minutes for four allele examples on an ibm 3081 machine

exact inference for contingency tables with ordered categories abstract this article proposes an efficient numerical algorithm for small sample exact inferences in contingency tables having ordinal classifications the inferences which apply conditional on the observed marginal totals also provide an exact analysis for the log linear model of linear by linear association for cell probabilities an exact test of independence has a one sided p value equal to the null probability that model based maximum likelihood estimates of odds ratios are at least as large as the observed estimates the conditional nonnull distribution yields confidence intervals for odds ratios having a linear by linear structure the computations utilize an extension of the network algorithm proposed by mehta and patel 1983

computational tools for exact conditional logistic regression logistic regression analyses are often challenged by the inability of unconditional likelihood based approximations to yield consistent valid estimates and p values for model parameters this can be due to sparseness or separability in the data conditional logistic regression though useful in such situations can also be computationally unfeasible when the sample size or number of explanatory covariates is large we review recent developments that allow efficient approximate conditional inference including monte carlo sampling and saddlepoint approximations we demonstrate through real examples that these methods enable the analysis of significantly larger and more complex data sets we find in this investigation that for these moderately large data sets monte carlo seems a better alternative as it provides unbiased estimates of the exact results and can be executed in less cpu time than can the single saddlepoint approximation moreover the double saddlepoint approximation while computationally the easiest to obtain offers little practical advantage it produces unreliable results and cannot be computed when a maximum likelihood solution does not exist

an empirical assessment of saddlepoint approximations for testing a logistic regression parameter saddlepoint methods provide quick and easy approximations to significance levels for conditional tests of logistic regression parameters we evaluate the accuracies of saddlepoint approximations for three well known conditional tests bartlett s test for no three factor interaction in a 2 x 2 x 2 table the test for trend in a series of probabilities and the exact test of no association in stratified 2 x 2 tables with a common odds ratio general recommendations are suggested regarding the use of saddlepoint approximations for exact conditional significance levels

bayesian methods for examining hardy weinberg equilibrium testing for hardy weinberg equilibrium is ubiquitous and has traditionally been carried out via frequentist approaches however the discreteness of the sample space means that uniformity of p values under the null cannot be assumed with enumeration of all possible counts conditional on the minor allele count offering a computationally expensive way of p value calibration in addition the interpretation of the subsequent p values and choice of significance threshold depends critically on sample size because equilibrium will always be rejected at conventional levels with large sample sizes we argue for a bayesian approach using both bayes factors and the examination of posterior distributions we describe simple conjugate approaches and methods based on importance sampling monte carlo the former are convenient because they yield closed form expressions for bayes factors which allow their application to a large number of single nucleotide polymorphisms snps in particular in genome wide contexts we also describe straightforward direct sampling methods for examining posterior distributions of parameters of interest for large numbers of alleles at a locus we resort to markov chain monte carlo we discuss a number of possibilities for prior specification and apply the suggested methods to a number of real datasets

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

a hereditarily indecomposable mathcal l _ infty space that solves the scalar plus compact problem we construct a hereditarily indecomposable banach space with dual space isomorphic to l1 every bounded linear operator on this space is expressible as i k with a scalar and k compact

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

tukey classification of some ideals on ω and the lattices of weakly compact sets in banach spaces abstract we study the lattice structure of the family of weakly compact subsets of the unit ball b x of a separable banach space x equipped with the inclusion relation this structure is denoted by k b x and also with the parametrized family of almost inclusion relations k l e b x where e 0 this structure is denoted by ak b x tukey equivalence between partially ordered sets and a suitable extension to deal with ak b x are used assuming the axiom of analytic determinacy we prove that separable banach spaces fall into four categories namely k b x is equivalent either to a singleton or to or to the family k q of compact subsets of the rational numbers or to the family c of all finite subsets of the continuum also under the axiom of analytic determinacy a similar classification of ak b x is obtained for separable banach spaces not containing l 1 we prove in zfc that k b x ak b x are equivalent to either 0 k q or c the lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied

the variance for the disequilibrium coefficient in the individual hardy weinberg test summary it is of interest to detect the deviation from hardy weinberg proportion hwp for a particular heterozygote hernandez and weir 1989 biometrics45 53 70 suggested a disequilibrium coefficient approach and proposed a 1 d f 2 test this note derives the appropriate variance under the null hypothesis for performing this individual hwp test examples of applying the test to human genetic disease data are presented

a warning on the use of chi squared statistics with frequency tables with small expected cell counts abstract when applied to frequency tables with small expected cell counts pearson chi squared test statistics may be asymptotically inconsistent even in cases in which a satisfactory chi squared approximation exists for the distribution under the null hypothesis this problem is particularly important in cases in which the number of cells is large and the expected cell counts are quite variable to illustrate this bias of the chi squared test this article considers the pearson chi squared test of the hypothesis that the cell probabilities for a multinomial frequency table have specified values in this case the expected value and variance of the pearson chi square may be evaluated under both the null and alternative hypotheses when the number of cells is large normal approximations and discrete edgeworth expansions may also be used to assess the size and power of the pearson chi squared test these analyses show that unless all cell probabilities are equal it is possible to select a significance lev

markov chains for monte carlo tests of genetic equilibrium in multidimensional contingency tables hardy weinberg equilibrium and linkage equilibrium are fundamental concepts in population genetics in practice testing linkage equilibrium in haplotype data is equivalent to testing independence in a large sparse multidimensional contingency table testing hardy weinberg and linkage equilibrium simultaneously on multilocus genotype data introduces the additional complications of missing information and symmetry constraints on marginal probabilities to avoid unreliable large sample approximations for sparse contingency tables one can use exact tests like fisher s classical test that condition on observed marginal totals unfortunately computing p values for exact tests is often infeasible because of the large number of tables consistent with the marginal totals of an observed table we develop here markov chains for sampling from the appropriate conditional distributions for testing genetic equilibrium these chains compare favorably with a parallel independent sampling method that we present for n haplotype observations on j loci the markov chains converge to their stationary distributions in j 1 n ln n 2 0 n steps and can be an efficient tool for estimating p values our theoretical treatment of these results involves strong stationary stopping times order statistics large deviations and the embedding of poisson processes we include some general results on the application of strong stationary times to bounding the precision and bias of sample average estimators

on uniform generation of two way tables with fixed margins and the conditional volume test of diaconis and efron two efficient monte carlo algorithms are proposed for uniformly generating two way contingency tables with fixed margins these permit some improvements on recent work of diaconis efron and gangolli especially concerning estimates of the total number of such tables

high order accurate methods for retrospective sampling problems in this paper we discuss the relationship between prospective and retrospective sampling problems estimates of the parameter of interest may be obtained by solving suitable estimating equations under both sampling schemes most common examples of such estimates include the maximum likelihood estimates some classical results and more recent development of the first order asymptotic relationship between the estimators are reviewed high order expansions are given for the distributions of the retrospective estimators expansions for the marginal distributions of interest are obtained for both prospective and retrospective data furthermore it is shown that the two expansions are asymptotically equal at least up to order o n 1 this implies that readily available prospective saddlepoint methods may be applied to the analysis of retrospective data without loss of high order accuracy the results are also briefly illustrated numerically