on the sample size for studies based upon mcnemar s test when computing the sample size for studies using mcnemar s test one needs to know the probability of discordance and the odds ratio to be detected in many studies the investigator is unable to specify the probability of discordance but can state at least approximately the marginal probabilities of each variable this information leads to restrictions on the possible values of the cell probabilities and provides a range of admissible values for the off diagonal cells we compute the sample size needed in these circumstances and compare them to the results cited by schlesselman and connett et al these sample sizes for the method are quite close to those found in the monte carlo study of connett et al

on the laplacian eigenvalues of a graph abstract in the note we present an upper bound for the spectral radius of laplacian matrix of a graph in terms of a 2 degree of a vertex

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

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

the use of cusums and other techniques in modelling continuous covariates in logistic regression the assessment of continuous covariates singly as possible predictors in a multivariable logistic regression model is an important first step in the analysis an approach to plotting which uses a cusum cumulative sum of the binary response variable is described extreme deviation statistics associated with the cusum may be used to detect monotonic and non monotonic trends probability plots of the covariate in the two groups defined by the response variable may help to determine the appropriate scale transformation of the covariate and to anticipate possible problems with the logistic fit the ratio of the variances in the response non response groups is informative about the need for a quadratic term in the logistic model smoothed scatterplots of the response are valuable in displaying the observed and fitted values the techniques are illustrated with two data sets

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

sample size requirements for stratified cluster randomization designs sample size requirements are provided for designs of studies in which clusters are randomized within each of several strata where cluster size itself may be a stratifying factor the approach generalizes a formula derived by woolson et al which provides sample size requirements for the cochran mantel haenszel statistic issues of data analysis are also discussed

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

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

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

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

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

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

fixed width sequential stopping rules for a class of stochastic programs monte carlo sampling based methods are frequently used in stochastic programming when exact solution is not possible a critical component of monte carlo sampling based methods is determining when to stop sampling to ensure the desired quality of the solutions in this paper we develop stopping rules for sequential sampling procedures that depend on the width of an optimality gap confidence interval estimator the procedures solve a sequence of sampling approximations with increasing sample size to generate solutions and stop when the width of the confidence interval on the current solution s optimality gap plus an inflation factor fall below a prespecified value epsilon we first present a method that takes the schedule of sample sizes as an input and provide guidelines on the growth of sample sizes then we present a method that increases the sample sizes according to the current estimates of the optimality gap the larger the estimates the larger the increases in sample sizes we provide conditi

optimum sample size determination in stratified case control studies with cost considerations we investigate sample size determination for cochran s test for stratified case control studies when samples of cases and controls are allocated to maximize the asymptotic efficiency of cochran s test subject to fixed total cost with cost per control varying by strata we consider two situations typical of strata matched case control studies when one samples both cases and controls and when cases are given and one samples controls in each situation we develop and study an asymptotic method for finding the sample size required for a specific power under the optimum allocation proposed by nam and fears also for the second situation we investigate an asymptotic method for determining the common ratio k in one to k strata matched case control studies without cost consideration for a given power when cases are given neither the optimum nor the standard control sample sizes appear in a closed form we present numerical methods for calculating these sample sizes and illustrate them with examples we find the reduction in total cost obtained under the optimum allocation compared to standard allocation more pronounced as the differences in stratum specific costs of sampling controls increase

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

on some forests determined by their laplacian or signless laplacian spectrum we consider the class of graphs whose each component is either a proper subgraph of some smith graphs or belongs to a precized subset of smith graphs we classify the graphs from the considered class into those which are determined or not determined by laplacian or signless laplacian spectrum

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

rejoinder to modification of the computational procedure in parker and bregman s method of calculating sample size from matched case control studies with a dichotomous exposure

reducing mean squared error in the analysis of stratified epidemiologic studies kalish 1990 biometrics 46 493 499 proposed an approximately optimal estimator of a common odds ratio for pair matched case control studies his approach is easily extended to general stratified studies for unmatched stratified studies the form of the approximately optimal estimator is very simple and may often correspond to no more than a small correction to the ordinary stratified estimator

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

power and sample size evaluation for the mcnemar test with application to matched case control studies various expressions have appeared for sample size calculation based on the power function of mcnemar s test for paired or matched proportions especially with reference to a matched case control study these differ principally with respect to the expression for the variance of the statistic under the alternative hypothesis in addition to the conditional power function i identify and compare four distinct unconditional expressions i show that the unconditional calculation of schlesselman for the matched case control study can be expressed as a first order unconditional calculation as described by miettinen corrections to schlesselman s unconditional expression presented by fleiss and levin and by dupont which use different models to describe exposure association among matched cases and controls are also equivalent to a first order unconditional calculation i present a simplification of these corrections that directly provides the underlying table of cell probabilities from which one can perform any of the alternative sample size calculations also i compare the four unconditional sample size expressions relative to the exact power function the conclusion is that miettinen s first order expression tends to underestimate sample size while his second order expression is usually fairly accurate though possibly slightly anti conservative a multinomial based expression presented by connor among others is also fairly accurate and is usually slightly conservative finally a local unconditional expression of mitra among others tends to be excessively conservative

sample size for individually matched case control studies the standard formulas used to calculate sample size for an individually matched case control study assume a constant probability of exposure throughout the pool of possible controls we propose new formulas that allow for heterogeneity in the probability of exposure among controls in different matched sets since matching factors are suspected of being confounders they are expected to divide the total population into subgroups with different proportions exposed thus the assumption of homogeneity of exposure among controls made by the currently used formulas is inconsistent with the assumptions used to design a matched study the proposed formulas avoid this inconsistency we present an example to illustrate how heterogeneity can affect the required sample size

a note on the spectral characterization of θ graphs we consider graphs that is graphs obtained by subdividing the edges of the multigraph consisting of 3 parallel edges it is shown that any graph g is determined by the spectrum the multiset of eigenvalues except possibly when it contains a unique 4 cycle

power calculations for matched case control studies power calculations are derived for matched case control studies in terms of the probability po of exposure among the control patients the correlation coefficient phi for exposure between matched case and control patients and the odds ratio psi for exposure in case and control patients for given type i and type ii error probabilities alpha and beta the odds ratio that can be detected with a given sample size is derived as well as the sample size needed to detect a specified value of the odds ratio graphs are presented for paired designs that show the relationship between sample size and power for alpha 05 beta 2 and different values of po phi and psi the sample size needed for designs involving m matched control patients can be derived from these graphs by means of a simple equation these results quantify the loss of power associated with increasing correlation between the exposure status of matched case and control patients sample size requirements are also greatly increased for values of po near 0 or 1 the relationship between sample size psi phi and po is discussed and illustrated by examples

on the sample size for one sided equivalence of sensitivities based upon mcnemar s test in this paper we formalize the problem of testing the one sided equivalence in the sensitivities of two medical diagnostic tests under a matched pair study design we derive conditional and unconditional sample size formulae which are decreasing functions of the probability of being diagnosed by both tests we calculate upper boundary and midpoint sample sizes results of a monte carlo simulation study that compares the proposed sample size formulae with that of lachenbruch suggest that our midpoint conditional sample size is the best choice to obtain the desired power for the type of equivalence studies discussed in the paper

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

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

modification of the computational procedure in parker and bregman s method of calculating sample size from matched case control studies with a dichotomous exposure summary several authors have addressed the problem of calculating sample size for a matched case control study with a dichotomous exposure the approach of parker and bregman 1986 biometrics42 919 926 is in our view one of the most satisfactory since it requires specification of quantities that are often easily available to the investigator however its recommended implementation involves a computational approximation we show here that the approximation performs poorly in extreme situations and can be easily replaced with a more exact calculation

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

control sample size when cases are given in constant ratio stratum matched case control studies strata matched case control studies based on a given number of cases and k times as many controls are common we obtain a necessary and sufficient condition for there to exist a numerical solution for k with a desired level of a power we derive the maximum power of the cochran test which may be less than one when the number of cases is fixed we provide an approximate formula for the minimum number of cases required for a specified power there exists a numerical solution that is convergence of an iterative method for k if the number of cases given is greater than this minimum number and no such a solution otherwise we also show that the incremental gain in relative efficiency of the test with respect to k is diminished as k gets large

optimal allocation of simulation experiments in discrete stochastic optimization and approximative algorithms approximate solutions for discrete stochastic optimization problems are often obtained via simulation it is reasonable to complement these solutions by confidence regions for the argmin set we address the question how a certain total number of random draws should be distributed among the set of alternatives two goals are considered the minimization of the costs caused by using a statistical estimate of the true argmin and the minimization of the expected size of the confidence sets we show that an asymptotically optimal sampling strategy in the case of normal errors can be obtained by solving a convex optimization problem to reduce the computational effort we propose a regularization that leads to a simple one step allocation rule

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

power and sample size for a collection of 2 x 2 tables a simple approximation is presented for the asymptotic power of the mantel haenszel procedure for testing the hypothesis that the common odds ratio in a collection of 2 x 2 tables is unity this approximation is particularly useful in evaluating the power of a study with several strata which reaches negative conclusions the corresponding formula is also provided for estimating the sample size before such a study has begun by approximating the margins of the stratum specific tables with estimates obtained from pre existing data the approximations are good for moderate odds ratios not greater than 6 and provide estimates which can differ by 10 to 30 from the traditional power estimates based on the overall table of totals obtained by collapsing the stratum specific subtables

estimation of a common odds ratio under binary cluster sampling we present results from a simulation study for the estimation of a common odds ratio in multiple 2 x 2 tables when the data are correlated within clusters we model the correlation of the data by the beta binomial distribution through a simulation study we compare the mantel haenszel estimator with rao and scott s estimator in terms of their biases observed variances relative efficiencies of their variances and 95 per cent coverage proportions we limit the simulation study to the case where there are the same number of subjects in each cluster and the same number of observations in each row of each stratum when rho 0 we recommend use of the mantel haenszel estimator gamma mh with an unadjusted variance and rao and scott s estimator gamma rsp with a pooled design effect in general when rho 0 we recommend the mantel haenszel estimator gamma mh with an adjusted variance and rao and scott s estimator gamma rsp with a pooled design effect

allocation of effort in monte carlo simulation for power of permutation tests abstract consider a randomization test in which an observed statistic is compared to a null distribution constructed by first drawing a random sample from the set of all possible permutations of the data and then calculating the statistic anew for each member of the random sample if the power of such a test is investigated by simulation the computer code must have two main loops in the outer loop which has o iterations data are generated under a specific alternative hypothesis for each outer iteration there are i inner iterations in each of which the data are permuted and the statistic calculated assume that the costs of inner and outer iterations are known and the total cost of the study is fixed so that i and o are constrained by o io for fixed this article considers what the optimal choice for i is in a mean squared sense a general expression is derived for the mean squared error of the distribution of p values if the alternative hypothesis is close to the null the optimal c

line star sets for laplacian eigenvalues abstract let g be a mixed graph with a nonzero laplacian eigenvalue of multiplicity k a line star set for in g is a set y of k edges of g such that is not a laplacian eigenvalue of g y it is shown that line star set exists for any nonzero laplacian eigenvalue of any mixed graph some basic properties for line star sets are given we also obtain some results on line star sets in undirected graphs

sample size for case control studies using cochran s statistic sample size determination for case control studies of chronic disease are often based on the simple 2 x 2 tabular cross classification of exposure and disease thereby ignoring stratification which may be considered in the analysis one consequence of this approach is that the sample size may be inadequate to attain a specified power and size when performing a statistical analysis on j 2 x 2 tables using cochran s 1954 biometrics 10 417 451 statistic or the mantel haenszel 1959 journal of the national cancer institute 22 719 748 statistic a sample size formula is derived from cochran s statistic and it is compared with the corresponding one derived when the data are treated as unstratified and also with two other formulas proposed for stratified data analysis the formula developed yields values slightly higher than one recently proposed by munoz and rosner 1984 biometrics 40 995 1004 which assumes that both margins of each 2 x 2 table are fixed while the present study considers only the case control margin to be fixed

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

mantel haenszel type inference for cumulative odds ratios with a stratified ordinal response this article proposes a mantel haenszel type estimator of an assumed common cumulative odds ratio in a proportional odds model for an ordinal response with several 2 x c contingency tables it is useful for instance for comparing two treatments on an ordinal response for data from several centers when the data are highly sparse the estimator has behavior similar to the mantel haenszel estimator of a common odds ratio for several 2 x 2 tables it is consistent under the ordinary asymptotic framework in which the number of tables is fixed and unlike the maximum likelihood ml estimator also under sparse asymptotics in which the number of tables grows with the sample size simulations reveal a considerable difference between it and the ml estimator when each table has few observations efficiency comparisons suggest that little efficiency loss occurs compared to the ml estimator when the data are not sparse tests and estimators are presented for detecting and handling heterogeneity in the odds ratios and generalizations are available for stratified r x c contingency tables

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

confidence limits for the common odds ratio based on the asymptotic distribution of the mantel haenszel estimator this paper proposes a new approximate confidence limit method for the common odds ratio based on the asymptotic distribution of the mantel haenszel estimator the proposed method provides easily computed limits and is applicable to both sparse data and large strata cases monte carlo experiments compare its performance with that of a method based on the asymptotic distribution of the log mantel haenszel odds ratio applications of the proposed method to the common risk and rate ratio cases are given

note the t shape tree is determined by its laplacian spectrum a t shape is a tree with exactly one of its vertices having maximal degree 3 it is proved in this paper that the t shape tree is determined by its laplacian spectrum

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

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

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

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

a jackknife estimator of a combined odds ratio a jackknife estimator of a common odds ratio is derived in the situation of stratified contingency tables it is based on pseudovalues that are constructed by omitting each single observation in each cell by means of monte carlo experiments this estimator is compared with the estimators of woolf mantel and haenszel and another jackknife estimator by breslow and liang of the common odds ratio in the situations considered this estimator has lower bias and mean squared error than the other three proposals the accompanying jackknife variance behaves in general similarly to the estimated variances of the woolf estimator and the mantel haenszel estimator

the power of the mantel haenszel test abstract the mantel haenszel mh test mantel and haenszel 1959 is a widely used method for analyzing sets of two by two tables that compare the outcome of two treatments in several strata bailar and anthony 1977 the procedure tests the null hypothesis of no partial association between treatment and outcome in any of the strata or tables against the alternative that the partial association as measured by the common odds ratio differs from 1 birch 1964 showed that for a fixed number of tables both the mh test and an earlier version due to cochran 1954 are asymptotically identical to the uniformly most powerful unbiased test of this hypothesis this article is concerned with calculating the power of the mh test as a prelude to embarking upon a study we consider versions of the test with and without corrections for continuity the spirit behind the approximations is that although the odds ratio is the appropriate parameter for computing asymptotic power under local alternatives in practice

on a laplacian spectral characterization of graphs of index less than 2 a graph is said to be determined by the adjacency respectively laplacian spectrum if there is no other non isomorphic graph with the same adjacency respectively laplacian spectrum the maximum eigenvalue of a g is called the index of g the connected graphs with index less than 2 are known and each is determined by its adjacency spectrum in this paper we show that graphs of index less than 2 are determined by their laplacian spectrum

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

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

efficient power computation for exact and mid p tests for the common odds ratio in several 2 2 tables when designing a study that may generate a set of sparse 2 x 2 tables or when confronted with negative results upon exact analysis of such tables we need to compute the power of exact tests in this paper we provide an efficient approach for computing exact unconditional power for four exact tests on the common odds ratio in a series of 2 x 2 tables these tests are the traditional exact test a test based on a probability ordering of the sample space and two tests based on ordering the sample space according to distance from the mean or median for each test we consider both a conservative version and a mid p adjusted version we explore three computational options for power determination exact power computation calculation of exact upper and lower bounds for power and monte carlo confidence bounds for power we present an interactive program implementing these options for study design the program may be run several times to arrive at a sample configuration with adequate power

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

sample size determination for case control studies and the comparison of stratified and unstratified analyses woolson bean and rojas 1986 biometrics 42 927 932 present a simple approximation of sample size for cochran s 1954 biometrics 10 417 451 test for detecting association between exposure and disease it is useful in the design of case control studies we derive a sample size formula for cochran s statistic with continuity correction which guarantees that the actual type i error rate of the test does not exceed the nominal level the corrected sample size is necessarily larger than the uncorrected one given by woolson et al and the relative difference between the two sample sizes is considerable allocation of equal number of cases and controls within each stratum is asymptotically optimal when the costs per case and control are the same when any effect of stratification is absent cochran s stratified test although valid is less efficient than the unstratified one except for the important case of a balanced design

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

the concept of residual confounding in regression models and some applications in this paper the concept of residual confounding is generalized to various types of regression models such as logistic regression or cox regression residual confounding and a newly suggested parameter the relative residual confounding are defined on the regression parameters of the models the estimator gives the proportion of confounding which has been removed by incomplete adjustment the concept quantifies the effects of categorizing continuous covariables and of model misspecification these are investigated by a simulation study and with data from an epidemiological investigation a case control study of laryngeal cancer is used to illustrate the residual confounding effect of arbitrary transformation of a continuous confounder smoking on the effect of alcohol consumption on laryngeal cancer risk the data also showed that categorization into two levels can yield high residual confounding the parameters described in this paper are of some use in quantifying the effect of inadequate adjustment for confounding variables

power calculations for likelihood ratio tests in generalized linear models summary recently self and mauritsen 1988 biometrics 44 79 86 described an approach for sample size power calculations within the framework of generalized linear models that is based on a noncentral chi square approximation to the distribution of score statistics in this work we present another approach to this same problem that is based on a noncentral chi square approximation to the distribution of the likelihood ratio statistic this approach is easier to implement than the previous one and simulation studies indicate that it is accurate over a much wider range of parameter values and data configurations than the earlier method the new method is also applied to the problem of power calculations for 1 k matched case control studies which is related to but formally outside of the generalized linear model framework

on a signless laplacian spectral characterization of t shape trees let m be an associated matrix of a graph g the adjacency laplacian and signless laplacian matrix two graphs are said to be cospectral with respect to m if they have the same m spectrum a graph is said to be determined by m spectrum if there is no other non isomorphic graph with the same spectrum with respect to m it is shown that t shape trees are determined by their laplacian spectra moreover among them those are determined by their adjacency spectra are characterized in this paper we identify graphs which are cospectral to a given t shape tree with respect to the signless laplacian matrix subsequently t shape trees which are determined by their signless laplacian spectra are identified

power and sample size for matched case control studies the problem of sample size for matched case control studies is approached without any advance information in current practice the probability of exposure 0 of matched controls and the correlation coefficient 0 between outcomes of a matched pair have to be available as in dupont 1988 biometrics 43 1157 1168 the probability of discordant pairs is required in schlesselman 1982 case control studies design conduct analysis and parker and bregman 1986 biometrics 42 919 926 the measure of association w between a case s and the matched control s exposure status and 0 are also needed in fleiss 1988 proceedings of the xivth international biometrics conference 29 44 for a test of ho 7p 1 against the alternative hypothesis h1 if 7o 1 sample sizes are determined without requiring any advance information both for pair matching and for multiple matching using the model of ejigou and mchugh 1977 biometrics 33 552 556 1981 biometrika 68 85 91 and the results in ejigou 1990 biometricss 46 61 69 the weakness of ury s rule 1975 biometrics 31 643 649 is reviewed and alternatives are proposed as in current procedures our method enables one to determine power for a given hi test size and sample size in addition it also provides a solution to the problem of the number of controls to be matched to each case to achieve a desired level of power this is also a new feature of our method

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

the spectral characterization of graphs of index less than 2 with no path as a component a graph is said to be determined by the adjacency and laplacian spectrum or to be a ds graph for short if there is no other non isomorphic graph with the same adjacency and laplacian spectrum respectively it is known that connected graphs of index less than 2 are determined by their adjacency spectrum in this paper we focus on the problem of characterization of ds graphs of index less than 2 first we give various infinite families of cospectral graphs with respect to the adjacency matrix subsequently the results will be used to characterize all ds graphs with respect to the adjacency matrix of index less than 2 with no path as a component moreover we show that most of these graphs are ds with respect to the laplacian matrix