a random effects model for binary data summary this paper presents a method based on maximizing the marginal likelihood for analyzing binary data with random effects with the assumption of a parametric family that allows for a wide variety of shapes for the distribution of the random effects the marginal likelihood can be computed without numerical integrations the method uses local independence models as well as those that incorporate additional dependence among the responses two examples a panel study with binary responses and an analysis of item response data will be used to illustrate the method

interval estimation of risk difference for data sampled from clusters risk difference rd is an important measure in epidemiological studies where the probability of developing a disease for individuals in an exposed group for example is compared with that in a control group there are varying cluster sizes in each group and the binary responses within each cluster cannot be assumed independent under the cluster sampling scenario lui statistical estimation of epidemiological risk wiley ca 2004 7 27 discusses four methods for the construction of a confidence interval for the rd in this paper we introduce two very simple methods one method is based on an estimator of the variance of a ratio estimator sampling techniques 3rd edn wiley new york 1977 30 67 and the other method is based on a sandwich estimator of the variance of the regression estimator using the generalized estimating equations approach of zeger and liang biometrics 1986 42 121 130 these two methods are then compared by simulation in terms of maintaining nominal coverage probability and average coverage length with the four methods discussed by lui statistical estimation of epidemiological risk wiley ca 2004 7 27 simulations show at least as good properties of these two methods as those of the others the method based on an estimate of the variance of a ratio estimator performs best overall it involves a very simple variance expression and can be implemented with a very few computer codes therefore it can be considered as an easily implementable alternative

partial stability of catastrophe sections and its application to the cuspoids and conic umbilics the control space of a system described by an elementary catastrophe frequently has locally at least a cartesian product structure caused for example by physical distinctions among control variables or by observational necessity the effect of this structure like that of symmetries can be to stabilise singular catastrophe sections partial unfoldings the authors define partial stability to be the structural stability of a stratum of a catastrophe section under perturbations respecting the product structure these ideas are illustrated by reference to a recent observation of double cusp diffraction and then applied to previous studies of singular catastrophe sections

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

non inferiority tests for clustered matched pair data non inferiority tests for matched pair data where pairs are mutually independent may not be appropriate when pairs are clustered the tests may require an adjustment to account for the correlation within a cluster we consider the adjusted score and wald type tests and a modification of obuchowski s method for non inferiority and compare them with the non inferiority test based on a method of moments estimate in terms of type 1 error rate and power by simulations for a small cluster size under various correlation structures in general the score test adjusted by an inflation factor and the modified obuchowski s method perform as good as the test based on moments estimate in the accuracy of type 1 error rates the latter does not provide reasonably close type 1 error rates to the nominal level when the number of clusters is 25 or smaller and a positive response rate for the standard procedure is 20 per cent or lower the adjusted score test the method based on moments estimate and the modified test are comparable in power the adjusted wald type test is too anti conservative and we should caution use of the test since number of clusters is strongly related to the accuracy of empirical type 1 error rate and power it is very important to have a sufficiently large number of clusters in designing a clustered matched pair study for non inferiority

linear time representation algorithms for proper circular arc graphs and proper interval graphs our main result is a linear time that is time o m n algorithm to recognize and represent proper circular arc graphs the best previous algorithm due to a tucker has time complexity o n 2 we take advantage of the fact that among connected graphs proper circular arc graphs are precisely the graphs orientable as local tournaments and we use a new characterization of local tournaments the algorithm depends on repeated representation of portions of the input graph as proper interval graphs thus we also find it useful to give a new linear time algorithm to represent proper interval graphs this latter algorithm also depends on an orientation characterization of proper interval graphs it is conceptually simple and does not use complex data structures as a byproduct of the correctness proof of the algorithm we also obtain a new proof of a characterization of proper interval graphs by forbidden subgraphs

a logistic bivariate normal model for overdispersed two state markov processes we describe a logistic bivariate normal mixture model for a two state markov chain in which each individual makes transitions between states according to a subject specific transition probability matrix the use of the bivariate normal mixing distribution facilitates inferences regarding the correlation of the random effects and hence provides insight as to the nature of the subject to subject variability in the transition probabilities tests regarding the correlation can be based on likelihood ratio score or wald statistics estimates of the transition intensities of a latent continuous time conditionally markov process may also be computed we illustrate this methodology by application to a parasitic infection field study and contrast our findings with those previously published on this data set

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

caustics of the harmonic double cusp near an e_6 point the harmonic double cusp is the real singularity x 4 6x 2 y 2 y 4 its unfolding is eight dimensional and contains a stratum of codimension five corresponding to the e 6 singularity x 3 y 4 a complete description of the bifurcation set in a neighbourhood of the e 6 stratum is given by means of a tableau of two dimensional sections over the three dimensional base the tableau also gives information about the bifurcation set near other strata of high codimension notably d 6 x 2 y y 5 and a 7 x 2 y 8 and it shows how these strata fit together in optics the harmonic double cusp organizes a large family of caustics produced by generic thin lenses some predictions made by the tableau have already been confirmed by observations of caustics from liquid drops points in the tableau base correspond to experimental set ups and the fibre corresponds to the viewing screen

kappa statistic for clustered matched pair data kappa statistic is widely used to assess the agreement between two procedures in the independent matched pair data for matched pair data collected in clusters on the basis of the delta method and sampling techniques we propose a nonparametric variance estimator for the kappa statistic without within cluster correlation structure or distributional assumptions the results of an extensive monte carlo simulation study demonstrate that the proposed kappa statistic provides consistent estimation and the proposed variance estimator behaves reasonably well for at least a moderately large number of clusters e g k 50 compared with the variance estimator ignoring dependence within a cluster the proposed variance estimator performs better in maintaining the nominal coverage probability when the intra cluster correlation is fair 0 3 with more pronounced improvement when is further increased to illustrate the practical application of the proposed estimator we analyze two real data examples of clustered matched pair data copyright 2014 john wiley sons ltd

testing the effect of treatment in experiments with correlated binary outcomes this paper considers the problem of testing for treatment effect in a randomized experiment with correlated binary outcomes representing success or failure for different parts of a randomized unit attention is restricted to tests that are based on a summary score for each individual randomized and thus are valid regardless of the precise nature of the correlation among parts the focus is on the efficiency of such tests under various correlation structures with special emphasis on the case in which the correlation among parts within an individual differs across treatment groups a class of summary score statistics is defined and optimal testing is discussed for some simple situations three potential general purpose tests also are described 1 the ratio estimate test discussed by henderson et al 1988 controlled clinical trials 9 189 205 2 a modified ratio estimate test with adjusted weighting based on the within individual correlation between parts 3 a test defined by applying the mantel haenszel procedure to the proportion of individuals with at least one failure stratifying by the number of parts for these general purpose tests numerical calculations of asymptotic efficiency are presented under a wide range of designs and correlation structures on the basis of these results some practical recommendations for choosing a test are made

the use of logistic models for the analysis of codon frequencies of dna sequences in terms of explanatory variables the development of the regressive logistic model applicable to the analysis of codon frequencies of dna sequences in terms of explanatory variables is presented a codon is a triplet of nucleotides that code for an amino acid and may be considered as a trivariate response b1 b2 b3 where bi i 1 2 3 is a categorical random variable with values a c g t the linear order of bases in the dna and possible statistical dependence of the bases in a given codon make the regressive logistic model a suitable tool for the analysis of codon frequencies a problem of structural zeros arises from the fact that the stopping codons terminators do not code for amino acids this is solved by normalizing the likelihood function codon frequencies may also depend on the function of the gene and they are known to differ between genes of the same genome differences also occur between synonymous codons for the same amino acid thus the use of covariates that differ between synonymous codons as well as covariates that are constant within codons of the same amino acid may be useful in explaining the frequencies as an illustration the method is applied to the human mitochondrial genome using the following as explanatory variables 1 tscore a measure of the number of single base mutations required for a given codon to become a terminator 2 aarisk an indicator of a codon s ability of changing by a single base substitution to triplets coding for amino acids with very different characteristics 3 avdist a measure of the typicality of the amino acid coded for by the triplets the results indicate that models that incorporate dependency structure and covariates are to be preferred to either the models comprising covariates alone or dependency structure alone

correlated binomial variates properties of estimator of intraclass correlation and its effect on sample size calculation in group randomized studies the sample size calculations are complicated by within group worksite community etc correlation we compare by simulation the moment method and the more standard anova method of estimating the intraclass correlation we find the former is less biased for a small to moderate number of clusters but the difference disappears when the appropriate degree of freedom is used for the anova estimator we propose a simulation approach for sample size determination and illustrate it with an example

on the use of stochastic ordering to test for trend with clustered binary data we introduce the use of stochastic ordering for defining treatment related trend in clustered exchangeable binary data for both when cluster sizes are fixed and when they vary randomly in the latter case there is a well documented tendency for such data to be sparse a problem we address by making an assumption of interpretability or equivalently marginal compatibility our procedures are based on a representation of the joint distribution of binary exchangeable random variables by a saturated model and may hence be considered nonparametric the definition of trend by stochastic ordering is proposed to ensure a flexibility that allows for various forms of monotone increases in response to the cluster as a whole to be included in the evaluation of the trend we obtain maximum likelihood estimates of probability functions under stochastic ordering using mixture likelihood based algorithms since the data are sparse we avoid the use of asymptotic results and obtain p values of the likelihood ratio procedures by permutation resampling we demonstrate how the proposed framework can be used in risk assessment and provide comparisons with existing procedures

sample size formulae for intervention studies with the cluster as unit of randomization this paper presents sample size formulae for both continuous and dichotomous endpoints obtained from intervention studies that use the cluster as the unit of randomization the formulae provide the required number of clusters or the required number of individuals per cluster when the other number is given the proposed formulae derive from student s t test with use of cluster summary measures and a variance that consists of within and between cluster components power contours are provided to help in the design of intervention studies that use cluster randomization sample size formulae for designs with and without stratification of clusters appear separately

irreversible conversion of graphs given a graph g a function f v g z and an initial 0 1 vertex labelling c 1 v g 0 1 we study an iterative 0 1 vertex labelling process on g where in each round every vertex v never changes its label from 1 to 0 and changes its label from 0 to 1 if at least f v neighbours have label 1 such processes model opinion disease spreading or fault propagation and have been studied under names such as irreversible threshold majority processes in a large variety of contexts our contributions concern computational aspects related to the minimum cardinality irr f g of sets of vertices with initial label 1 such that during the process on g all vertices eventually change their label to 1 such sets are known as irreversible conversion sets dynamic irreversible monopolies or catastrophic fault patterns answering a question posed by dreyer and roberts p a dreyer jr f s roberts irreversible k threshold processes graph theoretical threshold models of the spread of disease and of opinion discrete appl math 157 2009 1615 1627 we prove a hardness result for irr f g where f v 2 for every email protected v g furthermore we describe a general reduction principle for irr f g which leads to efficient algorithms for graphs with simply structured blocks such as trees and chordal graphs

a simple method for analysing overdispersion in clustered poisson data a simple method is proposed for analysing grouped count data exhibiting overdispersion relative to a poisson model the method is similar to the approach suggested for the analysis of clustered binary data in rao and scott 1992 it requires no specific model for the overdispersion and it can be implemented easily using standard programs designed to handle independent poisson counts after a small amount of preprocessing

applications of catastrophe theory to the physical sciences abstract catastrophe theory was introduced by rene thom in the late 1960 s as an attempt to model morphogentic changes in nature using ideas from topological dynamics and the theory of singularities of mappings thom envisaged a very general approach to topological changes in the solutions to parametrized systems of equations such as differential and difference equations and in particular discussed the special case of elementary catastrophe theory singularities of smooth real valued functions popular expositions have tended to overemphasize this special case but it remains the major source of ideas and methods here we survey the applications of catastrophe theory to the physical sciences physics chemistry engineering fluid mechanics etc for brevity we confine attention to an area lying between elementary and general catastrophe theory usually known as singularity theory this is the theory of singularities of smooth vector valued functions which mathematically is a straightforward though non trivial generalization of the real valued case in the last few years it has developed into a powerful and useful technique in several areas of theoretical physics notably optics and bifurcation theory equivariant catastrophe theory taking account of symmetry is likely to prove especially interesting

estimation of a common odds ratio in paired cluster randomization designs we develop two estimators of a common odds ratio for designs in which the investigator randomly assigns each of two clusters to interventions within strata the estimators rely on an empirical adjustment for clustering to provide improved estimators of relative to the standard woolf and mantel haenszel estimators respectively the results of a simulation study show that the suggested adjustment improves the accuracy of both of these well known estimators under conditions likely to arise in practice we find the clustered woolf estimator as particularly effective in terms of mean squared error reduction we also discuss interval estimation

joint modeling and clustering paired generalized longitudinal trajectories with application to cocaine abuse treatment data in a cocaine dependence treatment study we have paired binary longitudinal trajectories that record the cocaine use patterns of each patient before and after a treatment to better understand the drug using behaviors among the patients we propose a general framework based on functional data analysis to jointly model and cluster these paired non gaussian longitudinal trajectories our approach assumes that the response variables follow distributions from the exponential family with the canonical parameters determined by some latent gaussian processes to reduce the dimensionality of the latent processes we express them by a truncated karhunen loeve kl expansion allowing the mean and covariance functions to be different across clusters we further represent the mean and eigenfunctions functions by flexible spline bases and determine the orders of the truncated kl expansions using data driven methods by treating the cluster membership as a missing value we cluster the cocaine use trajectories by a l

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

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

sample size and power for mcnemar s test with clustered data mcnemar s test is used to compare the distribution of two paired binary random variables when the data are clustered adjustment is needed to ensure that it is still a valid test this article presents two approximations for calculating the power and sample size for the adjusted mcnemar s test for clustered data working with a particular adjustment a simulation study is conducted to demonstrate the accuracy of these approximations the method is also applied to the design of a study involving positron emission tomography in detecting metastatic colorectal cancer and sensitivity of sample size computations to the design parameters are explored in this context

on the asymptotic distribution of ratio and regression estimators we give a formal derivation of the asymptotic normality of ratio and regression estimators of a finite population total for simple random sampling and for sampling with probability proportional to aggregate size the results are all well known and widely used of course but a rigorous development spelling out relatively simple sufficient conditions under which the standard results are valid does not seem to be available at present

improving small sample inference in group randomized trials with binary outcomes group randomized trials grts randomize groups of people to treatment or control arms instead of individually randomizing subjects when each subject has a binary outcome over dispersed binomial data may result quantified as an intra cluster correlation icc typically grts have a small number bin of independent clusters each of which can be quite large treating the icc as a nuisance parameter inference for a treatment effect can be done using quasi likelihood with a logistic link a wald statistic which under standard regularity conditions has an asymptotic standard normal distribution can be used to test for a marginal treatment effect however we have found in our setting that the wald statistic may have a variance less than 1 resulting in a test size smaller than its nominal value this problem is most apparent when marginal probabilities are close to 0 or 1 particularly when n is small and the icc is not negligible when the icc is known we develop a method for adjusting the estimated standard error appropriately such that the wald statistic will approximately have a standard normal distribution we also propose ways to handle non nominal test sizes when the icc is estimated we demonstrate the utility of our methods through simulation results covering a variety of realistic settings for grts

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

estimating intraclass correlation for binary data summary this paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study some of the estimators are very specific while others result from general methods such as pseudo likelihood and extended quasi likelihood estimation the simulation study identifies several useful estimators one of which does not seem to have been considered previously for binary data estimators based on extended quasi likelihood are found to have a substantial bias in some circumstances

interpretation of regressive logistic regression coefficients in analyses of familial data regressive logistic regression was introduced by bonney in 1986 and due to its easy implementation using standard logistic regression computer packages has been widely used in the analysis of binary familial disease traits associations between binary outcomes for different family members are measured by regression coefficients included in the corresponding logit models the interpretation of these coefficients is examined in detail revealing a number of features of the method that are worthy of comment

analyzing k 2 x 2 tables under cluster sampling summary in many data analytic applications such as ophthalmologic longitudinal or periodontal studies multiple observations are recorded over several sites or timepoints within the same subject bringing about dependence between measurements this correlation in turn precludes the use of standard statistical methods that assume independence between outcome measurements for example the mantel haenszel statistic used to assess association between a binary outcome and a binary exposure while adjusting for a categorical covariate does not follow the usual chi squared distribution under the null hypothesis when there is correlation between observations a modified mantel haenszel procedure which makes adjustment for dependence is proposed no particular correlation structure is assumed for responses within a cluster this closed form adjustment stems from liang and zeger s 1986 biometrika73 13 22 generalized estimating equations approach for clustered data the difference between this tabular i e noniterative technique and many earlier tabular methods is that the current method allows for consideration of site specific exposure and covariate information an example from a periodontal research study illustrates application of the method

conditional log linear models for analyzing categorical panel data abstract conditional log linear models are developed for panel data and used to predict sequences of categorical responses the class of models considered includes conventional markov models and independence models as well as distance models in which all previous responses and present and past values of covariates are used to predict the current response the approach taken in this article has some advantages over the marginal modeling approach that has become popular for longitudinal studies quality of prediction is measured by using a logarithmic penalty function given a model conditional probabilities of responses consistent with the model are selected to provide the smallest expected penalty this minimum expected penalty provides a measure of the predictive power of a model models are compared through their predictive power as measured by the proportional reduction in expected penalty ways of incorporating the number of parameters of the competing models are discussed this emphasis on predicti

estimation of the design effect in community intervention studies this paper considers the estimation of the variance of a mortality rate in community intervention studies with little or no replication of intervention regimens our approach in estimation of this cluster sampling variance is to determine the variance for simple random sampling and multiply it by a design effect which we calculate with use of information obtained from other sources when the county is the unit of randomization and the outcome is mortality we calculate the design effect as the ratio of the age adjusted mortality rates for single stage cluster sampling relative to simple random sampling we use information from all counties in a state in the calculations we apply this approach empirically for breast cancer mortality we found that these design effects were dependent on length of time for accumulation of deaths 1 1 for one year up to 3 5 for eight years and were quite consistent for the three states and nine years considered in the investigation we present a model that accounts for the time dependence of the design effect and we show it provides a good representation of the observed relationship

logistic regression for dependent binary observations summary the likelihood of a set of binary dependent outcomes with or without explanatory variables is expressed as a product of conditional probabilities each of which is assumed to be logistic the models are called regressive logistic models they provide a simple but relatively unknown parametrization of the multivariate distribution they have the theoretical and practical advantage that they can be analyzed and fitted as in logistic regression for independent outcomes and with the same computer programs the paper is largely expository and is intended to motivate the development and usage of the regressive logistic models the discussion includes serially dependent outcomes equally predictive outcomes more specialized patterns of dependence multidimensional tables and three examples this paper seeks to promote the development and usage of a simple but relatively unknown approach to the specification of models for dependent binary outcomes that are useful in situations where it is apt to describe the dependence in terms of conditional probabilities the joint probability is decomposed into a product of successive conditional probabilities each of which is assumed to be univariate logistic since the probability of an observation is conditioned on all preceding observations the models will be referred to as regressive logistic models two broad situations are distinguished in this paper the first concerning sequences of binary outcomes possibly with measures on associated explanatory variables can usually be supported by simple patterns of dependence these will be discussed in sections 2 5 the second broad situation concerns dependent binary outcomes crossclassified into multidimensional tables that admit a fuller parametrization this is discussed in section 6 for all the regressive logistic models discussed here the parametrization is such that jointly dependent binary observations with or without explanatory variables can be fitted with existing computer programs for the standard logistic regression for independent observations some of the possibilities of the approach are illustrated with real data in section 8

sample size calculations for a split cluster beta binomial design in the assessment of toxicity mouse embryo assays are recommended to test materials used for in vitro fertilization for toxicity in such assays a number of embryos is divided in a control group which is exposed to a neutral medium and a test group which is exposed to a potentially toxic medium inferences on toxicity are based on observed differences in successful embryo development between the two groups however mouse embryo assays tend to lack power due to small group sizes this paper focuses on the sample size calculations for one such assay the nijmegen mouse embryo assay nmea in order to obtain an efficient and statistically validated design the nmea follows a stratified mouse randomized embryo balanced design also known as a split cluster design we adopted a beta binomial approach and obtained a closed sample size formula based on an estimator for the within cluster variance our approach assumes that the average success rate of the mice and the variance thereof which are breed characteristics that can be easily estimated from historical data are known to evaluate the performance of the sample size formula a simulation study was undertaken which suggested that the predicted sample size was quite accurate we confirmed that incorporating the a priori knowledge and exploiting the intra cluster correlations enable a smaller sample size also we explored some departures from the beta binomial assumption first departures from the compound beta binomial distribution to an arbitrary compound binomial distribution lead to the same formulas as long as some general assumptions hold second our sample size formula compares to the one derived from a linear mixed model for continuous outcomes in case the compound beta binomial estimator is used for the within cluster variance

simple fitting algorithms for incomplete categorical data a popular approach to estimation based on incomplete data is the em algorithm for categorical data this paper presents a simple expression of the observed data log likelihood and its derivatives in terms of the complete data for a broad class of models and missing data patterns we show that using the observed data likelihood directly is easy and has some advantages one can gain considerable computational speed over the em algorithm and a straightforward variance estimator is obtained for the parameter estimates the general formulation treats a wide range of missing data problems in a uniform way two examples are worked out in full

analysis of proportions in the presence of over under dispersion procedures for testing homogeneity of proportions in the presence of over dispersion or underdispersion occurring in several groups in toxicology teratology or mutagenicity or other similar fields are developed we consider c a neyman 1959 in probability and statistics the harold cramer volume pp 213 234 new york wiley or score type tests rao 1947 proceedings of the cambridge philosophical society 44 50 57 based on a parametric model namely the extended beta binomial model prentice 1986 journal of the american statistical association 81 321 327 and two semi parametric models using quasi likelihood wedderburn 1974 biornetrika 61 439447 and extended quasi likelihood nelder and pregibon 1987 biometrika 74 221 232 these procedures and a recent procedure by rao and scott 1992 biometrics 48 577 585 based on the concept of design effect and effective sample size are compared through simulation in terms of size power and robustness for departures from data distribution and dispersion homogeneity to study robustness in terms of departure from data distribution we simulate data from the betabinomial distribution the probit normal binomial distribution and the logit normal binomial distribution simulation shows evidence that for litter sizes and number of litters that may arise in practice a score test based on the quasi likelihood performs best in that it holds nominal level well in all data distribution situations considered here it shows some edge in power over some other statistics in some situations and also shows robustness in presence of moderate dispersion heterogeneity this statistic has a very simple form and it requires estimates of the parameters only under the null hypotheses

two sample comparison for large groups of correlated binary responses although there is much literature on testing the treatment effect in experiments with correlated binary outcomes the existing methodologies only work well when the cluster size is small in this paper i propose a simple method to test the treatment effect in experiments with large groups of correlated binary outcomes i use the weighted estimating equations approach to estimate the treatment effect the proposal is appropriate under any correlation structure power comparisons show the advantage of the new procedure

on the existence of maximum likelihood estimates in logistic regression models summary the problems of existence uniqueness and location of maximum likelihood estimates in log linear models have received special attention in the literature haberman 1974 chapter 2 wedderburn 1976 silvapulle 1981 for multinomial logistic regression models we prove existence theorems by considering the possible patterns of data points which fall into three mutually exclusive and exhaustive categories complete separation quasicomplete separation and overlap our results suggest general rules for identifying infinite parameter estimates in log linear models for frequency tables

comparison of tests for categorical data from a stratified cluster randomized trial two features commonly exhibited by randomized trials of health promotion interventions are cluster randomization and stratification ignoring correlations between individuals within clusters can lead to an inflated type i error rate and hence a p value which overstates the significance of the result this paper compares several methods for analysing categorical data from a stratified cluster randomized trial we propose an extension of a method from survey sampling that uses the design effect to reduce the effective sample size we compare this with three methods from zhang and boos that extend the standard cochran mantel haenszel cmh statistic by using appropriate covariance matrices and with a bootstrap method the comparison is based on empirical type i error rates from a simulation study in which the number of clusters randomized is small as in most public health intervention studies the method that performs consistently well is one of the zhang and boos extensions of the standard cmh statistic

sample size calculations for clustered binary data in this paper we propose a sample size calculation method for testing on a binomial proportion when binary observations are dependent within clusters in estimating the binomial proportion in clustered binary data two weighting systems have been popular equal weights to clusters and equal weights to units within clusters when the number of units varies cluster by cluster performance of these two weighting systems depends on the extent of correlation among units within each cluster in addition to them we will also use an optimal weighting method that minimizes the variance of the estimator a sample size formula is derived for each of the estimators with different weighting schemes we apply these methods to the sample size calculation for the sensitivity of a periodontal diagnostic test simulation studies are conducted to evaluate a finite sample performance of the three estimators we also assess the influence of misspecified input parameter values on the calculated sample size the optimal estimator requires equal or smaller sample sizes and is more robust to the misspecification of an input parameter than those assigning equal weights to units or clusters

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

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

multivariate methods for clustered binary data with more than one level of nesting abstract clustered data occur frequently in statistical practice in some areas of application such as ophthalmology clustered data are the rule rather than the exception in this setting standard multivariate methods such as logistic regression are invalid because of the lack of independence among outcomes for individual sample points within a cluster in rosner 1984 a polychotomous logistic regression model was presented to control for the effect of both cluster and individual specific covariates while accounting for the correlation among units within a cluster this model reduces to a beta binomial model in the absence of covariates and to an ordinary logistic model for clusters of size 1 and for larger clusters when no correlation is present a 1 level of nesting correlation structure is assumed as is typically found in ophthalmology where the person is the cluster and the individual eyes are the units within a cluster in this article this model is generalized to a 2 level of nesting struc

sample size for a two group comparison of repeated binary measurements using gee controlled clinical trials often randomize subjects to two treatment groups and repeatedly evaluate them at baseline and intervals across a treatment period of fixed duration a popular primary objective in these trials is to compare the change rates in the repeated measurements between treatment groups repeated measurements usually involve missing data and a serial correlation within each subject the generalized estimating equation gee method has been widely used to fit the time trend in repeated measurements because of its robustness to random missing and misspecification of the true correlation structure in this paper we propose a closed form sample size formula for comparing the change rates of binary repeated measurements using gee for a two group comparison the sample size formula is derived incorporating missing patterns such as independent missing and monotone missing and correlation structures such as ar 1 model we also propose an algorithm to generate correlated binary data with arbitrary marginal means and a markov dependency and use it in simulation studies copyright 2005 john wiley sons ltd

on the caratheodory number of interval and graph convexities inspired by a result of caratheodory uber den variabilitatsbereich der fourierschen konstanten von positiven harmonischen funktionen rend circ mat palermo 32 1911 193 217 the caratheodory number of a convexity space is defined as the smallest integer k such that for every subset u of the ground set v and every element u in the convex hull of u there is a subset f of u with at most k elements such that u in the convex hull of f we study the caratheodory number for generalized interval convexities and for convexity spaces derived from finite graphs we establish structural properties bounds and hardness results

an evaluation of methods for the stratified analysis of clustered binary data in community intervention trials a simulation study is conducted in a community intervention setting several methods of stratified analysis of clustered binary data are compared in terms of empirical significance and empirical power levels they are the mantel haenszel test statistic chi 2 mh the adjusted mantel haenszel test statistic of donald donner chi 2 dd rao scott chi 2 rsn and chi 2 rsp and zhang boos chi 2 zbn and chi 2 zbp wald chi 2 w robust wald chi 2 rw score chi 2 s robust score chi 2 rs and the test statistic based on generalized linear mixed model glmm chi 2 glmm when rho not equal 0 chi 2 mh has inflated type i error and it should not be used when observations are correlated the results also warn of the use of chi 2 rsn and chi 2 rw due to their poor performance in terms of empirical significance level chi 2 zbp and chi 2 glmm have better empirical significance levels as compared to other statistics however chi 2 zbp tends to have lower empirical powers than other statistics when the number of clusters n is less than 24 chi 2 rsp provides the highest empirical powers when rho or 0 1 and n or 12 when rho or 0 01 we recommend the use of chi 2 rs and chi 2 glmm since they have better overall performance in terms of empirical significance levels and empirical power levels

the nonparametric behrens fisher problem with dependent replicates purely nonparametric methods are developed for general two sample problems in which each experimental unit may have an individual number of possibly correlated replicates in particular equality of the variances or higher moments of the distributions of the data is not assumed even under the null hypothesis of no treatment effect thus a solution for the so called nonparametric behrens fisher problem is proposed for such models the methods are valid for metric count ordered categorical and even dichotomous data in a unified way point estimators of the treatment effects as well as their asymptotic distributions will be studied in detail for small sample sizes the distributions of the proposed test statistics are approximated using satterthwaite welch type t approximations extensive simulation studies show favorable performance of the new methods in particular in small sample size situations a real data set illustrates the application of the proposed methods

a comparison of methods for testing homogeneity of proportions in teratologic studies summary we consider teratologic studies in which the aim is to compare the survival rate of animals in a treatment group to the corresponding rate in a control group the design of such studies often involves the allocation of intact litters of animals to treatment invalidating the application of standard statistical methods we review the strengths and weaknesses of several approaches for dealing with this problem including methodology recently developed for the analysis of clustered binary data a simulation study is conducted in which litter sizes are generated from a distribution having specified mean and degree of imbalance it is recommended on the basis of this study and on theoretical considerations that the choice of method should depend on whether the comparison of interest is experimental or observational for experimental comparisons involving the random assignment of litters to different treatment groups methods based on the adjustment of standard chi square statistics are recommended unless the number of litters in each group is very large

imperfection sensitivity of elastically supported beams and its relation to the double cusp instability model this paper deals with the two mode initial postbuckling analysis of geometrically imperfect beams on elastic foundation under compression by adjusting the beam length to the characteristic length ratio it is found that the resulting two mode stability problem may be classified as the double cusp model in catastrophe theory the appropriate postbuckling coefficients are computed with the use of koiter s theory of elastic stability the equilibrium paths and the critical sets are plotted and analysed eight independent control parameters are suggested the paper is the first in the open literature to apply the double cusp model to a two mode stability problem which displays imperfection sensitivity

singular coordinate sections of the conic umbilic catastrophes studies the intersection of a normal form for the conic umbilic catastrophe of general codimension k with planes in control space having all but two control variables set to zero and display in a set of figures and a table the geometry in which strata of the bifurcation set intersect these planes and the singularity types occurring on them five distinct curve forms arise the authors compare these sections with those of the cuspoids and discuss their occurrence as optical caustics

on the caratheodory and exchange numbers of geodetic convexity in graphs abstract a set of vertices s of a graph g is geodesically convex if for every pair of vertices of s all vertices of all shortest paths joining them also lie in s the geodetic convex hull of s denoted by s is the minimum geodesically convex set of g containing s we say that s is caratheodory independent if s a s s a and that s is exchange independent if s 1 or there is p s such that s p a s p s a the caratheodory number exchange number of g is the cardinality of a maximum caratheodory exchange independent set in this paper we show that deciding whether a given graph has exchange number at most k is an np complete problem we also present characterizations of the caratheodory and exchange numbers for some graph classes like unit interval graphs and powers of cycles which allow us to determine these parameters in polynomial time

design and analysis of trials with a partially nested design and a binary outcome measure where treatments are administered to groups of patients or delivered by therapists outcomes for patients in the same group or treated by the same therapist may be more similar leading to clustering trials of such treatments should take account of this effect where such a treatment is compared with an un clustered treatment the trial has a partially nested design this paper compares statistical methods for this design where the outcome is binary investigation of consistency reveals that a random coefficient model with a random effect for group or therapist is not consistent with other methods for a null treatment effect and so this model is not recommended for this design small sample performance of a cluster adjusted test of proportions a summary measures test and logistic generalised estimating equations and random intercept models are investigated through simulation the expected treatment effect is biased for the logistic models empirical test size of two sided tests is raised only slightly but there are substantial biases for one sided tests three formulae are proposed for calculating sample size and power based on i the difference of proportions ii the log odds ratio or iii the arc sine transformation of proportions calculated power from these formulae is compared with empirical power from a simulations study logistic models appeared to perform better than those based on proportions with the likelihood ratio test performing best in the range of scenarios considered for these analyses the log odds ratio method of calculation of power gave an approximate lower limit for empirical power

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

modelling the covariance structure of repeated measurements models for repeated measurements used by wilson hebel and sherwin 1981 biometrics 37 553565 and mansour nordheim and rutledge 1985 biometrics 41 287 294 are considered these two models are generalized by the a p e k model obtained by introducing an extra parameter k a more general class of models described by kenward unpublished ph d thesis university of reading 1981 is briefly discussed simulation results are presented in which model parameters are estimated by maximum likelihood and likelihood ratios are used as criteria for choosing between models a set of blood pressure data is used to illustrate model fitting

evaluation of an adjusted chi square statistic as applied to observational studies involving clustered binary data a simple adjustment to the pearson chi square test has been proposed for comparing proportions estimated from clustered binary observations however the assumptions needed to assure the validity of this test have not yet been thoroughly addressed these assumptions will hold for experimental comparisons but could be violated for some observational comparisons in this paper we investigate the conditions under which the adjusted chi square statistic is valid and examine its performance when these assumptions are violated we also introduce some alternative test statistics that do not require these assumptions the test statistics considered are then compared through simulation and an example presented based on real data the simulation study shows that the adjusted chi square statistic generally produces empirical type i errors close to nominal under the assumption of a common intracluster correlation coefficient even if the intracluster correlations are different the adjusted chi square statistic performs well when the groups have equal numbers of clusters

statistical inference for familial disease clusters in many epidemiologic studies the first indication of an environmental or genetic contribution to the disease is the way in which the diseased cases cluster within the same family units the concept of clustering is contrasted with incidence we assume that all individuals are exchangeable except for their disease status this assumption is used to provide an exact test of the initial hypothesis of no familial link with the disease conditional on the number of diseased cases and the distribution of the sizes of the various family units new parametric generalizations of binomial sampling models are described to provide measures of the effect size of the disease clustering we consider models and an example that takes covariates into account ascertainment bias is described and the appropriate sampling distribution is demonstrated four numerical examples with real data illustrate these methods

statistical methods in ophthalmology an adjusted chi square approach ophthalmologic studies often compare several groups of subjects for the presence or absence of some ocular finding where each subject may contribute two eyes to the analysis the values from the two eyes being highly correlated rosner 1982 biometrics 38 105 114 and dallal 1988 biometrics 44 253 257 proposed procedures for testing whether the proportion of affected eyes is the same among the different groups while accounting for the intrasubject correlation in this paper we propose an alternative approach based on a simple adjustment of the standard pearson chi square test for the equality of proportions the suggested approach utilizes information on subjects who supply only one eye to the analysis and readily generalizes to studies in which more than two units of analysis are provided by each subject

analysis of proportions of affected foetuses in teratological experiments this paper is concerned with the analysis of proportions affected when an increasing dose of a compound is applied to a group of laboratory animals several sets of data including a set from a teratological experiment at the shell toxicology laboratory sittingbourne are analysed and some simulations are performed among the distributional models the beta binomial model is in general found to be the most sensitive to departures from the binomial for testing the equality of two proportions a comparison is made between the pseudo t test based on the jackknife method and the likelihood ratio test based on the beta binomial model from the limited comparison no definite advantage of one approach over the other has been found at the 5 level of significance both approaches lead to similar conclusions

multivariate methods for binary longitudinal data with heterogeneous correlation over time clustered binary data occur frequently in biostatistical work one particular application is in binary longitudinal data where several visits are available for the same individual several approaches have been proposed for the analysis of clustered binary data in rosner a polychotomous logistic regression model was proposed which is a generalization of the beta binomial distribution and allows for person and visit specific covariates while controlling for clustering effects one assumption of this model is that all pairs of visits within an individual are equally correlated which may be inappropriate if several visits are available over a long follow up period in this paper this approach is extended to allow for heterogeneous correlation over time the total time period is divided into subintervals and a beta binomial mixture model is introduced to estimate odds ratios relating outcomes for pairs of visits both within a subinterval as well as in different subintervals to include covariates an extension of the polychotomous logistic regression model is proposed which allows one to estimate effects of person subinterval and visit specific covariates while controlling for clustering using the beta binomial mixture model this model is applied to the analysis of respiratory symptom data in children collected over a 14 year period in east boston ma in relation to maternal and child smoking where the unit is the child and symptom history is divided into early adolescent and late adolescent symptom experience

comparison of operational characteristics for binary tests with clustered data although statistical methodology is well developed for comparing diagnostic tests in terms of their sensitivity and specificity comparative inference about predictive values is not in this paper we consider the analysis of studies comparing operating characteristics of two diagnostic tests that are measured on all subjects and have test outcomes from multiple sites with varying number of sites among subjects we have developed a new approach for comparing sensitivity specificity positive predictive value and negative predictive value with simple variance calculation and in particular focus on comparing tests using difference of positive and negative predictive values simulation studies are conducted to show the performance of our approach we analyze real data on patients with lung cancer diagnostic tests to illustrate the methodology

on model selection and the arc sine laws generalizations of the arc sine laws are shown to provide insight into the operating characteristics of certain techniques for selecting models to fit a given data set when the available models are nested as a corollary one sees that a popular technique may be expected to include about one superfluous parameter even if the sample size is large

bias correction in maximum likelihood logistic regression this paper provides an expression for bias of the maximum likelihood logistic regression estimates for use with small sample sizes this bias correction is based on an expansion of the maximum likelihood equation simulation results show these corrections to be highly effective in small samples

a simple method for the analysis of clustered binary data a simple method for comparing independent groups of clustered binary data with group specific covariates is proposed it is based on the concepts of design effect and effective sample size widely used in sample surveys and assumes no specific models for the intracluster correlations it can be implemented using any standard computer program for the analysis of independent binary data after a small amount of preprocessing the method is applied to a variety of problems involving clustered binary data testing homogeneity of proportions estimating dose response models and testing for trend in proportions and performing the mantel haenszel chi squared test for independence in a series of 2 x 2 tables and estimating the common odds ratio and its variance illustrative applications of the method are also presented

tests of independence treatment heterogeneity dose related trend with exchangeable binary data abstract existing methods of testing for treatment effects for clustered binary data include the beta binomial quasilikelihood gee procedures all of these methods revolve around the mean response and the second order correlation however these two parameters alone do not fully determine the effect of treatment this article develops nonparametric likelihood ratio procedures to test for independence heterogeneity dose related trend in dose response studies involving exchangeable binary data the hypotheses of independence heterogeneity trend are expressed in terms of joint probabilities of similar responses among cluster mates constrained maximum likelihood estimates of these probabilities are computed and used to construct test statistics unlike the test statistics for independence and heterogeneity the asymptotic distribution of the likelihood ratio test for trend is not exactly a chi square however an upper bound of its p value is obtained by using a chi squared distribution a set of clustere

likelihood inference in a correlated probit regression model correlated binary observations arise in a variety of applications for example in animal studies the term litter effect is used to describe the greater alikeness of responses within a litter as compared to that between litters at a given set of experimental conditions in a setting that motivated this work cells on individual atomic bomb survivors were scored as to the presence or absence of chromosomal aberrations the number of aberrant cells varied among subjects at a specified radiation exposure estimate age at exposure city of exposure and sex in a manner that substantially exceeds that consistent with a binomial error structure otake prentice 1984 this extra binomial variation or overdispersion reflects the fact that the binary response of cells from a given survivor tend to be more alike than are the responses of cells from distinct survivors having the same estimated radiation exposure level and other covariate values such overdispersion may result from individual differences in susceptibility to radiation damage from omitted covariates or most plausibly in this setting from substantial random errors in the estimated radiation exposure levels failure to acknowledge overdispersion may lead to serious underestimation of the standard errors associated with regression parameter estimates and to unduly precise inferences

a simulation study of the analysis of sets of 2 2 contingency tables under cluster sampling estimation of a common odds ratio abstract this article presents the results of a simulation study of the estimation of a common odds ratio in multiple contingency tables when the data are correlated within clusters the correlation is modeled by the beta binomial distribution the three objectives of the study were to determine a the variances biases and relative efficiencies of the beta binomial maximum likelihood estimator mle the binomial mle and the mantel haenszel estimator of the common odds ratio b the variances and biases of two estimators of the intracluster correlation coefficient and c the effectiveness of adjusting variance estimates of the binomial and mantel haenszel estimators using the two estimators of the simulation shows that increasing the size of the intracluster correlation coefficient increases the variance and the positive bias of all three estimators of the odds ratio the mantel haenszel estimator displays less variance and bias than the mle except for large values of and an unexpected b

analysis of data arising from a stratified design with the cluster as unit of randomization this paper discusses statistical techniques for the analysis of dichotomous data arising from a design in which the investigator randomly assigns each of two clusters of possibly varying size to interventions within strata the problem addressed is that of assessing the statistical significance of the intervention effect over all strata we propose a weighted paired t test based on the empirical logistic transform for designs that randomize large aggregate clusters in each of several strata

multivariate methods for clustered binary data with multiple subclasses with application to binary longitudinal data clustered binary data occur frequently in biostatistical work several approaches have been proposed for the analysis of clustered binary data in rosner 1984 biometrics 40 1025 1035 a polychotomous logistic regression model was proposed that is a generalization of the beta binomial distribution and allows for unit and subunit specific covariates while controlling for clustering effects one assumption of this model is that all pairs of subunits within a cluster are equally correlated this is appropriate for ophthalmologic work where clusters are generally of size 2 but may be inappropriate for larger cluster sizes a beta binomial mixture model is introduced to allow for multiple subclasses within a cluster and to estimate odds ratios relating outcomes for pairs of subunits within a subclass as well as in different subclasses to include covariates an extension of the polychotomous logistic regression model is proposed which allows one to estimate effects of unit class and subunit specific covariates while controlling for clustering using the beta binomial mixture model this model is applied to the analysis of respiratory symptom data in children collected over a 14 year period in east boston massachusetts in relation to maternal and child smoking where the unit is the child and symptom history is divided into early adolescent and late adolescent symptom experience

estimation of a common odds ratio in case control studies of familial aggregation a new estimator of a common odds ratio is proposed for case control studies of familial aggregation the proposed estimator is a modification to the usual mantel haenszel estimator that relies on an empirical adjustment for the within family clustering which is typical of such designs a simulation study shows that the estimator tends to have smaller mean squared error than the unmodified mantel haenszel estimator under conditions likely to arise in practice the construction of confidence intervals is also discussed

cluster sampling techniques in quantal response teratology and developmental toxicity studies summary this paper presents a model free approach for evaluating teratology and developmental toxicity data involving clustered binary responses in teratology studies a major statistical problem arises from the effect of intralitter correlation or the potential for littermates to respond similarly some statistical methods impose strict distributional assumptions to account for extra binomial variation while others rely on nonparametric resampling and empirical variance estimation techniques quasi likelihood methods and generalized estimating equations gee which model the marginal mean variance relationship also avoid strict distributional assumptions the proposed approach often used to analyze complex sample survey data is based on a first order taylor series approximation and a between cluster variance estimation procedure yielding consistent variance estimates for binomial based proportions and regression coefficients from dose response models the cluster sample technique presented here in the context of a logistic dose response model incorporates many of the advantages of quasi likelihood methods are valid for any underlying nested correlation structure and are adaptable to a variety of analytical settings the results of a simulation study show the cluster sample technique to be a viable competitor to gee methods currently receiving attention

incorporation of clustering effects for the wilcoxon rank sum test a large sample approach the wilcoxon rank sum test is frequently used in statistical practice for the comparison of measures of location when the underlying distributions are far from normal or not known in advance an assumption of the ordinary rank sum test is that individual sampling units are independent in many ophthalmologic clinical trials the early treatment for diabetic retinopathy scale etdrs is a principal endpoint used for measuring the level of diabetic retinopathy this is an ordinal scale and it is natural to consider the wilcoxon rank sum test for the comparison of the level of diabetic retinopathy between treatment groups however under this design unlike the usual wilcoxon rank sum test the subject is the unit of randomization but the eye is the unit of analysis furthermore a person will tend to have different but correlated etdrs scores for fellow eyes thus we propose a correction to the variance of the wilcoxon rank sum statistic that accounts for clustering effects and that can be used for both balanced same number of subunits per cluster or unbalanced different number of subunits per cluster data both in the presence or absence of ties with p value adjusted accordingly in this article we present large sample theory and simulation results for this test procedure and apply it to diabetic retinopathy data from type i diabetics in the sorbinil retinopathy trial

a note on generating correlated binary variables summary it is important to be able to generate correlated binary data in an efficient easily programmed manner for among other things the generation of large bootstrap samples in this note nothing more than the basic ingredient of the uniform random number generator is required for simulating binary data from the most commonly occurring correlation structures