detecting and describing heterogeneity in meta analysis the investigation of heterogeneity is a crucial part of any meta analysis while it has been stated that the test for heterogeneity has low power this has not been well quantified moreover the assumptions of normality implicit in the standard methods of meta analysis are often not scrutinized in practice here we simulate how the power of the test for heterogeneity depends on the number of studies included the total information that is total weight or inverse variance available and the distribution of weights among the different studies we show that the power increases with the total information available rather than simply the number of studies and that it is substantially lowered if as is quite common in practice one study comprises a large proportion of the total information we also describe normal plots that are useful in assessing whether the data conform to a fixed effect or random effects model together with appropriate tests and give an application to the analysis of a multi centre trial of blood pressure reduction we conclude that the test of heterogeneity should not be the sole determinant of model choice in meta analysis and inspection of relevant normal plots as well as clinical insight may be more relevant to both the investigation and modelling of heterogeneity

gauss seidel newton armijo approach for minimization problems on the non negative orthant application to spatial price equilibrium problems abstract to deal with the optimization problem min x 0 x we propose a gauss seidel type iterative approach where variables are modified sequentially one at a time gsna algorithm or by blocks bgsna algorithm relying on both the newton approach and an armijo rule an inaccurate line search is performed at each step to determine a step size insuring global convergence both robustness global convergence and efficiency rate of local convergence are analyzed under minimal hypotheses even weaker than those often required to prove the convergence of gauss seidel methods these procedures are applied to spatial price equilibrium problems and the numerical results indicate that they are competitive with minos

semiparametric bayesian analysis of matched case control studies with missing exposure this article considers bayesian analysis of matched case control problems when one of the covariates is partially missing within the likelihood context the standard approach to this problem is to posit a fully parametric model among the controls for the partially missing covariate as a function of the covariates in the model and the variables making up the strata sometimes the strata effects are ignored at this stage our approach differs not only in that it is bayesian but far more importantly in the manner in which it treats the strata effects we assume a dirichlet process prior with a normal base measure for the stratum effects and estimate all of the parameters in a bayesian framework three matched case control examples and a simulation study are considered to illustrate our methods and the computing scheme

conditional and unconditional categorical regression models with missing covariates summary we consider methods for analyzing categorical regression models when some covariates 2 are completely observed but other covariates x are missing for some subjects when data on x are missing at random i e when the probability that x is observed does not depend on the value of x itself we present a likelihood approach for the observed data that allows the same nuisance parameters to be eliminated in a conditional analysis as when data are complete an example of a matched casecontrol study is used to demonstrate our approach

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

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

a variable dimension solution approach for the general spatial price equilibrium problem algorithms are presented that are specifically designed for solving general nonlinear multicommodity spatial price equilibrium problems i e problems with nonlinear transportation cost functions nonlinear supply and demand functions inter commodity congestion effects intercommodity substitution and complementarity effects and interactions among transportation links and among spatially separated markets the algorithms are specializations of an iterative method for solving nonlinear complementarity problems that requires solving a system of nonlinear equations at each iteration the algorithms exploit the network structure of the problems to reduce the size of the system of equations to be solved at each iteration the decision rules for determining which equations are to be included in the system at each iteration are extremely simple and the remainder of the computational work is carried out by the nonlinear equation solver because of this the algorithms are very easy to implement with readily available software in addition since the decision rules only require sign information only the final system needs to be solved with precision

asymmetric variational inequality problems over product sets applications and iterative methods in this paper we i describe how several equilibrium problems can be uniformly modelled by a finite dimensional asymmetric variational inequality defined over a cartesian product of sets and ii investigate the local and global convergence of various iterative methods for solving such a variational inequality problem because of the special cartesian product structure these iterative methods decompose the original variational inequality problem into a sequence of simpler variational inequality subproblems in lower dimensions the resulting decomposition schemes often have a natural interpretation as some adjustment processes

computing nonlinear network equilibria this paper examines the problem of computing nonlinear network equilibria using a newton iteration by exploiting the network structure we are able to show that the iteration is globally convergent monotonic and locally quadratically convergent using only simple algebraic arguments

a general dynamic spatial price equilibrium model formulation solution and computational results abstract this paper presents a general dynamic finite horizon spatial price equilibrium model inventorying is allowed at both supply and demand markets and backordering is also permitted for as many time periods as dictated by the competitive equilibrium the case of perishable commodities where inventorying may be practicable for only one or several time periods is handled within the model by appropriately defined feasibility conditions alternative variational inequality formulations are given and then exploited akin to the work for static spatial price equilibrium problems to construct gauss seidel schemes computational experience for these schemes and for equilibration methods is given for a number of scenarios

confidence distributions and a unifying framework for meta analysis this article develops a unifying framework as well as robust meta analysis approaches for combining studies from independent sources the device used in this combination is a confidence distribution cd which uses a distribution function instead of a point point estimator or an interval confidence interval to estimate a parameter of interest a cd function contains a wealth of information for inferences and it is a useful device for combining studies from different sources the proposed combining framework not only unifies most existing meta analysis approaches but also leads to development of new approaches we illustrate in this article that this combining framework can include both the classical methods of combining p values and modern model based meta analysis approaches we also develop under the unifying framework two new robust meta analysis approaches with supporting asymptotic theory in one approach each study size goes to infinity and in the other approach the number of studies

a parametric linear complementarity technique for the computation of equilibrium prices in a single commodity spatial model this paper presents a parametric linear complementarity technique for the computation of equilibrium prices in a single commodity spatial model we first reformulate the model as a linear complementarity problem and then apply the parametric principal pivoting algorithm for its solution this reformulation leads to the study of an arc arc weighted adjacency matrix associated with a simple digraph having weights on the nodes several basic properties of such a matrix are derived using these properties we show how the parametric principal pivoting algorithm can be greatly simplified in this application finally we report some computational experience with the proposed technique for solving some large problems

odds ratio inference with dependent data summary when data can be presented as a series of k 2 x 2 tables with cell counts xi ni xi yi mi yi it is often assumed that xi and yi are binomially distributed this paper deals with inference for the common odds ratio i l when the binomial assumption is invalid when k increases the consistency and asymptotic normality of the mantel haenszel 1959 estimator is derived the conditional maximum likelihood estimator is shown to be inconsistent and the asymptotic bias is computed when either the first order markov chain or beta binomial model is assumed the mantel haenszel test for testing f 1 is also shown to be inappropriate through some simulation studies two consistent test statistics are proposed and shown to be comparable to each other in terms of size and efficiency some possible further work is described

bayesian semiparametric modeling for matched case control studies with multiple disease states summary we present a bayesian approach to analyze matched case control data with multiple disease states the probability of disease development is described by a multinomial logistic regression model the exposure distribution depends on the disease state and could vary across strata in such a model the number of stratum effect parameters grows in direct proportion to the sample size leading to inconsistent mles for the parameters of interest even when one uses a retrospective conditional likelihood we adopt a semiparametric bayesian framework instead assuming a dirichlet process prior with a mixing normal distribution on the distribution of the stratum effects we also account for possible missingness in the exposure variable in our model the actual estimation is carried out through a markov chain monte carlo numerical integration scheme the proposed methodology is illustrated through simulation and an example of a matched study on low birth weight of newborns hosmer d a and lemeshow s 2000 applied logistic regression with two possible disease groups matched with a control group

a two sample censored data rank test for acceleration a score test for the null hypothesis of proportional hazards against rank regression alternatives is proposed as a complement to the logrank test for comparing censored survival curves the test statistic has an asymptotic normal distribution that is independent of the logrank distribution under the null hypothesis and its power is good against acceleration alternatives i e with crossing hazards where the logrank test fails monte carlo studies indicate that its small sample properties are comparable to those of the logrank and ranksum procedures

inferences about exposure disease associations using probability of exposure information unconditional and conditional likelihood methods are proposed for modeling odds and odds ratios relating a binary disease variable and an exposure variable that is only known probabilistically e g via measurement of a surrogate exposure variable while adjusting appropriately for a vector of covariables assuming nondifferential errors a probabilistic structure is developed that permits analysis of the marginal relation between disease and the surrogate while maintaining important properties of the marginal relation between disease and true exposure in particular analyses conditioned on marginal totals in surrogate disease tables remove the same nuisance parameters as are removed in a conditional analysis of true exposure disease tables in addition a relation between the probability of exposure poe in the disease and disease free groups is derived permitting the use of information about exposure from populations with structure different from the study population examples are presented illustr

conditional association and unidimensionality in monotone latent variable models latent variable models represent the joint distribution of observable variables in terms of a simple structure involving unobserved or latent variables usually assuming the conditional independence of the observable variables given the latent variables these models play an important role in educational measurement and psychometrics in sociology and in population genetics and are implicit in some work on systems reliability we study a broad class of latent variable models namely the monotone unidimensional models in which the latent variable is a scalar the observable variables are conditionally independent given the latent variable and the conditional distribution of the observables given the latent variable is stochastically increasing in the latent variable all models in this class imply a new strong form of positive dependence among the observable variables namely conditional positive association this positive dependence condition may be used to test whether any model in this class can provide an adequate fit to observed data various applications generalizations and a numerical example are discussed

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

conditional regression analysis of the exposure disease odds ratio using known probability of exposure values conditional inference methods are proposed for the odds ratio between binary exposure and disease variables when only the probability of exposure is known for each study subject we develop a conditional likelihood approach that removes nuisance parameters and permits inferences to be made about important parameters in log odds ratio regression models we also discuss a heuristic procedure based on estimating the unknown number of truly exposed individuals this procedure provides a simple framework for interpreting our likelihood based statistics and leads to a mantel haenszel type estimator and a goodness of fit test as an example of the use of this methodology we present an analysis of some genetic data of swift et al 1976 cancer research 36 209 215

testing the conditional independence and monotonicity assumptions of item response theory when item characteristic curves are nondecreasing functions of a latent variable the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses this general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves the proposed tests are simple have known asymptotic null distributions and possess certain optimal properties in an example the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves

a comparison of tests of homogeneity of odds ratios in k 2 2 tables we conducted a simulation study to determine the performance of nine procedures for testing the homogeneity of odds ratios in k 2 2 contingency tables we recommend tarone s approximate score test based on the mantel haenszel estimator of the common odds ratio for use in practice we also recommend a non iterative statistic developed by gart and based on the modified woolf estimator of the common odds ratio for very large samples in balanced or mildly unbalanced designs we base our recommendation of a statistic on its performance in terms of size and power in comparison with the other statistics considered

meta analysis of quantile intervals from different studies with an application to a pulmonary tuberculosis data after the completion of many studies experimental results are reported in terms of distribution free confidence intervals that may involve pairs of order statistics this article considers a meta analysis procedure to combine these confidence intervals from independent studies to estimate or construct a confidence interval for the true quantile of the population distribution data synthesis is made under both fixed effect and random effect meta analysis models we show that mean square error mse of the combined quantile estimator is considerably smaller than that of the best individual quantile estimator we also show that the coverage probability of the meta analysis confidence interval is quite close to the nominal confidence level the random effect meta analysis model yields a better coverage probability and smaller mse than the fixed effect meta analysis model the meta analysis method is then used to synthesize medians of patient delays in pulmonary tuberculosis diagnosis in china to provide an illustration of the proposed methodology

the generalized linear complementarity problem revisited given a vertical block matrixa we consider in this paper the generalized linear complementarity problem vlcp q a introduced by cottle and dantzig we formulate this problem as a linear complementarity problem with a square matrixm a formulation which is different from a similar formulation given earlier by lemke our formulation helps in extending many well known results in linear complementarity to the generalized linear complementarity problem we also show that the class of vertical block matrices which cottle and dantzig s algorithm can process is the same as the class of equivalent square matrices which lemke s algorithm can process we also present some degree theoretic results on a vertical block matrix

a semiparametric odds ratio model for measuring association we propose a semiparametric odds ratio model to measure the association between two variables taking discrete values continuous values or a mixture of both methods for estimation and inference with varying degrees of robustness to model assumptions are studied semiparametric efficient estimation and inference procedures are also considered the estimation methods are compared in a simulation study and applied to the study of associations among genital tract bacterial counts in hiv infected women

generalization of the mantel haenszel estimating function for sparse clustered binary data we extend the mantel haenszel estimating function to estimate both the intra cluster pairwise correlation and the main effects for sparse clustered binary data we propose both a composite likelihood approach and an estimating function approach for the analysis of such data the proposed estimators are consistent and asymptotically normally distributed simulation results demonstrate that the two approaches are comparable in terms of bias and efficiency however the estimating equation approach is computationally simpler analysis of the georgia high blood pressure survey is used for illustration

an iterative sor algorithm for multi period spatial equilibria abstract this paper presents an iterative algorithm for the solution of multi period spatial equilibrium problems formulated in a net import format we first present a net import formulation for the problem and then apply the sor based iterative algorithm for its solution we later show that the algorithm can be extended to problems with sparse network and with congestion on the flows finally we report computational experience with the proposed algorithm for the solution of some large problems

the large sample relative efficiency of the mantel haenszel estimator in the fixed strata case the large sample relative efficiency of the mantel haenszel estimator psi mh of an odds ratio is investigated for the case in which the number of strata is fixed and the sample sizes within each stratum increase indefinitely the results show that psi mh is very efficient over a wide range of designs likely to occur in practice however conditions are identified under which the relative efficiency of this widely used estimator can be unusually low

median bias reduction of maximum likelihood estimates for regular parametric problems we show how median centring of the maximum likelihood estimate can be achieved by a simple modification of the score equation for a scalar parameter of interest the estimator is equivariant under interest respecting reparameterizations and is third order median unbiased with a vector parameter of interest componentwise equivariance and third order median centring are obtained like the implicit method of firth 1993 for bias reduction the new method does not require finiteness of the maximum likelihood estimate and is effective in preventing infinite estimates simulation results for continuous and discrete models including binary and beta regression confirm that the method succeeds in achieving componentwise median centring and in solving the boundary estimate problem while keeping comparable dispersion and the same approximate distribution as its main competitors

on the asymptotic inefficiency of certain noniterative estimators of a common relative risk or odds ratio nurminen 1981 gives conditions for the asymptotic efficiency of general weighted mean estimators his application is a noniterative estimator of a common relative risk for binomial variates this summary estimator and by implication the mantel haenszel summary odds ratio estimator are asserted to be asymptotically efficient we examine conditions for the asymptotic efficiency in the unconditional sample space of the mantelhaenszel estimator for a common odds ratio the estimator of a common relative risk considered by nurminen and tarone s 1981 estimator of a common relative risk for binomial variables contrary to nurminen we find that except for a few degenerate cases these estimators are inefficient whenever the common odds ratio or common relative risk is not equal to 1

projection method mantel haenszel estimator for k 2 j tables abstract a projection method is introduced to generalize the mantel haenszel estimator for estimating the common odds ratios in k 2 j j 2 tables the method provides estimators in an explicit form and needs no iterative computation it is shown that the generalized mantel haenszel estimator proposed by greenland and yanagawa and fujii is obtained by this method as a special case furthermore selecting appropriate weight it is shown that the method leads to an estimator that is asymptotically equivalent to the generalized mantel haenszel estimator proposed by liang a dually consistent estimator of the variance of the estimator is proposed the characteristics of these estimators have been evaluated by monte carlo means

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

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

effects of exposure misclassification on regression analyses of epidemiologic follow up study data in epidemiologic studies subjects are often misclassified as to their level of exposure ignoring this misclassification error in the analysis introduces bias in the estimates of certain parameters and invalidates many hypothesis tests for situations in which there is misclassification of exposure in a follow up study with categorical data we have developed a model that permits consideration of any number of exposure categories and any number of multiple category covariates when used with logistic and poisson regression procedures this model helps assess the potential for bias when misclassification is ignored when reliable ancillary information is available the model can be used to correct for misclassification bias in the estimates produced by these regression procedures

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

a class of polynomially solvable linear complementarity problems although the general linear complementarity problem lcp is np complete there are special classes that can be solved in polynomial time one example is the type where the defining matrix is nondegenerate and for which the n step property holds in this paper we consider an extension of the property to the degenerate case by introducing the concept of an extended n step vector and matrix it is shown that the lcp defined by such a matrix is polynomially solvable as well

pairwise reactive sor algorithms for quadratic programming of net import spatial equilibrium models an iterative method based on the successive overrelaxation sor is proposed to solve quadratic programming of net important spatial equilibrium models the algorithm solves the problem by updating the variables pairwise at each iteration the principal feature of the algorithm is that the lagrange multipliers corresponding to the constraints do not have to be calculated at each iteration as is the case in sor based algorithms yet the lagrange multipliers can easily be extracted from the solution values

bias corrected maximum likelihood estimator of a log common odds ratio summary a bias correction factor for the maximum likelihood estimator of the log common odds ratio for sparse data is derived which is applicable when the design is balanced and or when the number of strata is large the corrected maximum likelihood estimator is practically unbiased and its mean squared error is smaller than that of the uncorrected maximum likelihood estimator the correction factor for the maximum likelihood estimator approaches one as strata sizes increase examples demonstrate that the corrected maximum likelihood estimator and the exact conditional maximum likelihood estimator are essentially identical although the standard error of the former tends to be slightly smaller than that of the latter it is also demonstrated that a simple version of the correction factor is a satisfactory approximation the bias corrected maximum likelihood estimator proposed in this paper is the extension of breslow s 1981 and it can be useful particularly when the computational burden of the exact conditional method is excessive

interpretation and estimation of summary ratios under heterogeneity of the several types of summary odds ratios in use in epidemiology only those proposed by mantel and haenszel and miettinen do not assume homogeneity constancy of stratum specific parameters under heterogeneity miettinen s estimators consistently estimate meaningful epidemiologic parameters whereas the asymptotic expectation of the mantel haenszel odds ratio varies according to irrelevant features of the sampling design although one can modify the mantel haenszel summary to overcome the latter problem one can also extend the miettinen estimators to sparse data and thus limit the use of mantel haenszel summaries to situations in which the homogeneity hypothesis is reasonable this paper provides formulae for the extended miettinen estimators and large sample variances and confidence limits for summary ratios under heterogeneity