small sample gee estimation of regression parameters for longitudinal data longitudinal clustered response data arise in many bio statistical applications which in general cannot be assumed to be independent generalized estimating equation gee is a widely used method to estimate marginal regression parameters for correlated responses the advantage of the gee is that the estimates of the regression parameters are asymptotically unbiased even if the correlation structure is misspecified although their small sample properties are not known in this paper two bias adjusted gee estimators of the regression parameters in longitudinal data are obtained when the number of subjects is small one is based on a bias correction and the other is based on a bias reduction simulations show that the performances of both the bias corrected methods are similar in terms of bias efficiency coverage probability average coverage length impact of misspecification of correlation structure and impact of cluster size on bias correction both these methods show superior properties over the gee estimates for small samples further analysis of data involving a small number of subjects also shows improvement in bias mse standard error and length of the confidence interval of the estimates by the two bias adjusted methods over the gee estimates for small to moderate sample sizes n 50 either of the bias corrected methods geebc and geebr can be used however the method geebc should be preferred over geebr as the former is computationally easier for large sample sizes the gee method can be used copyright 2014 john wiley sons ltd

typical bad differentiable extensions abstract by a 1923 result of v jarnik every differentiable map from a closed subset p of r into r has a differentiable extension f r r it has been recently proved by the authors that among such differentiable extensions of there is always one that is nowhere monotone on r p in particular the family e 1 r of bad differentiable extensions f r r of for which the set f 0 x r f x 0 is dense in r p is nonempty we notice here that e 1 r with a natural distance is a complete metric space and prove that actually a typical function in e 1 r is nowhere monotone on r p at the same time the set m r of functions f e 1 r which are monotone on some nonempty subinterval of every nonempty open u r p is dense in e 1 r this last statement remains true when the term monotone is replaced with either of the following three terms strictly increasing strictly decreasing or constant

modelling multivariate binary data with alternating logistic regressions marginal models for multivariate binary data permit separate modelling of the relationship of the response with explanatory variables and the association between pairs of responses when the former is the scientific focus a first order generalized estimating equation method liang zeger 1986 is easy to implement and gives efficient estimates of regression coefficients although estimates of the association among the binary outcomes can be inefficient when the association model is a focus simultaneous modelling of the responses and all pairwise products prentice 1988 using second order estimating equations gives more efficient estimates of association parameters as well however this procedure can become computationally infeasible as the cluster size gets large this paper proposes an alternative approach alternating logistic regressions for simultaneously regressing the response on explanatory variables as well as modelling the association among responses in terms of pairwise odds ratios this algorithm iterates between a logistic regression using first order generalized estimating equations to estimate regression coefficients and a logistic regression of each response on others from the same cluster using an appropriate offset to update the odds ratio parameters for clusters of size n alternating logistic regression involves evaluation and inversion of matrices of order n2 rather than n4 as required for second order generalized estimating equations the alternating logistic regression estimates are shown to be reasonably efficient relative to solutions of second order equations in a few problems the new method is illustrated with an analysis of neuropsychological tests on patients with epileptic seizures

working correlation structure misspecification estimation and covariate design implications for generalised estimating equations performance the method of generalised estimating equations for regression modelling of clustered outcomes allows for specification of a working matrix that is intended to approximate the true correlation matrix of the observations we investigate the asymptotic relative efficiency of the generalised estimating equation for the mean parameters when the correlation parameters are estimated by various methods the asymptotic relative efficiency depends on three features of the analysis namely i the discrepancy between the working correlation structure and the unobservable true correlation structure ii the method by which the correlation parameters are estimated and iii the design by which we refer to both the structures of the predictor matrices within clusters and distribution of cluster sizes analytical and numerical studies of realistic data analysis scenarios show that choice of working covariance model has a substantial impact on regression estimator efficiency protection against avoidable loss of efficiency associated with covariance misspecification is obtained when a gaussian estimation pseudolikelihood procedure is used with an ar 1 structure

bivariate hoyt nakagami q distribution new exact expressions for the probability density and distribution functions of a bivariate hoyt nakagami q process with arbitrary correlation pattern in a nonstationary environment are derived a solution to a longstanding unsolved problem more specifically the following are obtained joint probability density function joint cumulative distribution function power and envelope correlation coefficients and some statistics related to the signal to noise ratio at the output of the selection combiner namely outage probability and probability density function the exact expressions are given in infinite series form but are mathematically tractable easy to evaluate and flexible enough to accommodate a myriad of correlation scenarios useful in the analysis of a more general fading environment the power correlation coefficient is found in an exact simple closed form formula whereas the envelope one also exact is more mathematically elaborate approximate simple closed form expressions for the joint distribution are also obtained that yield i excellent results for small to medium correlation coefficients and ii reasonable ones for high correlation coefficient

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

working correlation structure identification in generalized estimating equations selecting an appropriate working correlation structure is pertinent to clustered data analysis using generalized estimating equations gee because an inappropriate choice will lead to inefficient parameter estimation we investigate the well known criterion of qic for selecting a working correlation structure and have found that performance of the qic is deteriorated by a term that is theoretically independent of the correlation structures but has to be estimated with an error this leads us to propose a correlation information criterion cic that substantially improves the qic performance extensive simulation studies indicate that the cic has remarkable improvement in selecting the correct correlation structures we also illustrate our findings using a data set from the madras longitudinal schizophrenia study

a joint generalization of whitneyʼs c1 extension theorem and aversa laczkovich preissʼ extension theorem abstract we prove a joint generalization of whitney s c 1 extension theorem and aversa laczkovich preiss extension theorem it can be roughly described as a theorem on extendibility of a differentiable function defined on a closed subset of r n to a differentiable function on r n with preserving the continuity of the derivative other results on c 1 and differentiable extensions from thick subsets of r n are also proved

small sample performance of bias corrected sandwich estimators for cluster randomized trials with binary outcomes the sandwich estimator in generalized estimating equations gee approach underestimates the true variance in small samples and consequently results in inflated type i error rates in hypothesis testing this fact limits the application of the gee in cluster randomized trials crts with few clusters under various crt scenarios with correlated binary outcomes we evaluate the small sample properties of the gee wald tests using bias corrected sandwich estimators our results suggest that the gee wald z test should be avoided in the analyses of crts with few clusters even when bias corrected sandwich estimators are used with t distribution approximation the kauermann and carroll kc correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes however in cases with large variations in cluster sizes the fay and graubard fg correction should be used instead furthermore we derive a formula to calculate the power and minimum total number of clusters one needs using the t test and kc correction for the crts with binary outcomes the power levels as predicted by the proposed formula agree well with the empirical powers from the simulations the proposed methods are illustrated using real crt data we conclude that with appropriate control of type i error rates under small sample sizes we recommend the use of gee approach in crts with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure copyright 2014 john wiley sons ltd

a comparison of two bias corrected covariance estimators for generalized estimating equations summarymancl and derouen 2001 biometrics57 126 134 and kauermann and carroll 2001 jasa96 1387 1398 proposed alternative bias corrected covariance estimators for generalized estimating equations parameter estimates of regression models for marginal means the finite sample properties of these estimators are compared to those of the uncorrected sandwich estimator that underestimates variances in small samples although the formula of mancl and derouen generally overestimates variances it often leads to coverage of 95 confidence intervals near the nominal level even in some situations with as few as 10 clusters an explanation for these seemingly contradictory results is that the tendency to undercoverage resulting from the substantial variability of sandwich estimators counteracts the impact of overcorrecting the bias however these positive results do not generally hold for small cluster sizes e g 10 their estimator often results in overcoverage and the bias corrected covariance estimator of kauermann and carroll may be preferred the methods are illustrated using data from a nested cross sectional cluster intervention trial on reducing underage drinking

propensity score methods for estimating relative risks in cluster randomized trials with low incidence binary outcomes and selection bias despite randomization selection bias may occur in cluster randomized trials classical multivariable regression usually allows for adjusting treatment effect estimates with unbalanced covariates however for binary outcomes with low incidence such a method may fail because of separation problems this simulation study focused on the performance of propensity score ps based methods to estimate relative risks from cluster randomized trials with binary outcomes with low incidence the results suggested that among the different approaches used multivariable regression direct adjustment on ps inverse weighting on ps and stratification on ps only direct adjustment on the ps fully corrected the bias and moreover had the best statistical properties

differentiable extension theorem a lost proof of v jarnik abstract we present a beautiful but relatively unknown theorem that every differentiable function f p r with p r being closed admits differentiable extension f r r we present an elementary proof of this result based on a construction sketched in a hard to access 1923 paper 4 of v jarnik using this construction we also obtain an elegant version of whitney extension theorem characterizing when such an f admits continuously differentiable extension

extensions of vector valued baire one functions with preservation of points of continuity abstract we prove an extension theorem with non tangential limits for vector valued baire one functions moreover at every point where the function is continuous or bounded the continuity or boundedness is preserved more precisely let h be a closed subset of a metric space x and let z be a normed vector space let f h z be a baire one function we show that there is a continuous function g x h z such that for every a h the non tangential limit of g at a equals f a and moreover if f is continuous at a h respectively bounded in a neighborhood of a h then the extension f f g is continuous at a respectively bounded in a neighborhood of a we also prove a result on pointwise approximation of vector valued baire one functions by a sequence of locally lipschitz functions that converges uniformly or continuously at points where the approximated function is continuous in an accompanying paper extensions of vector valued functions with preservation of derivatives the main result is applied to extensions of vector valued functions defined on a closed subset of euclidean or banach space with preservation of differentiability continuity and pointwise lipschitz property

extensions of vector valued functions with preservation of derivatives abstract let x and y be banach or normed linear spaces and f x a closed set we apply our recent extension theorem for vector valued baire one functions to obtain an extension theorem for vector valued functions f f y with pre assigned derivatives with preservation of differentiability at every point where the pre assigned derivative is actually a derivative preservation of continuity preservation of point wise lipschitz property and for finite dimensional domain x preservation of strict differentiability and global eventually local lipschitz continuity this work depends on the paper extensions of vector valued baire one functions with preservation of points of continuity koc and kola 2016 20

bayesian methods of analysis for cluster randomized trials with binary outcome data we explore the potential of bayesian hierarchical modelling for the analysis of cluster randomized trials with binary outcome data and apply the methods to a trial randomized by general practice an approximate relationship is derived between the intracluster correlation coefficient icc and the between cluster variance used in a hierarchical logistic regression model by constructing an informative prior for the icc on the basis of available information we are thus able implicitly to specify an informative prior for the between cluster variance the approach also provides us with a credible interval for the icc for binary outcome data several approaches to constructing informative priors from empirical icc values are described we investigate the sensitivity of results to the prior specified and find that the estimate of intervention effect changes very little in this data set while its interval estimate is more sensitive the bayesian approach allows us to assume distributions other than normality for the random effects used to model the clustering this enables us to gain insight into the robustness of our parameter estimates to the classical normality assumption in a model with a more complex variance structure bayesian methods can provide credible intervals for a difference between two variance components in order for example to investigate whether the effect of intervention varies across clusters we compare our results with those obtained from classical estimation discuss the relative merits of the bayesian framework and conclude that the flexibility of the bayesian approach offers some substantial advantages although selection of prior distributions is not straightforward

intra cluster correlation selection for cluster randomized trials in this paper we give focus to cluster randomized trials also known as group randomized trials which randomize clusters or groups of subjects to different trial arms such as intervention or control outcomes from subjects within the same cluster tend to exhibit an exchangeable correlation measured by the intra cluster correlation coefficient icc our primary interest is to test if the intervention has an impact on the marginal mean of an outcome using recently developed methods we propose how to select a working icc structure with the goal of choosing the structure that results in the smallest standard errors for regression parameter estimates and thus the greatest power for this test specifically we utilize small sample corrections for the estimation of the covariance matrix of regression parameter estimates this matrix is incorporated within correlation selection criteria proposed in the generalized estimating equations literature to choose one of multiple working icc structures under consideration we demonstrate the potential power and utility of this approach when used in cluster randomized trial settings via a simulation study and application example and we discuss practical considerations for its use in practice copyright 2016 john wiley sons ltd

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

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

using second order generalized estimating equations to model heterogeneous intraclass correlation in cluster randomized trials in cluster randomized trials it is commonly assumed that the magnitude of the correlation among subjects within a cluster is constant across clusters however the correlation may in fact be heterogeneous and depend on cluster characteristics accurate modeling of the correlation has the potential to improve inference we use second order generalized estimating equations to model heterogeneous correlation in cluster randomized trials using simulation studies we show that accurate modeling of heterogeneous correlation can improve inference when the correlation is high or varies by cluster size we apply the methods to a cluster randomized trial of an intervention to promote breast cancer screening

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

smooth extension theorems for one variable maps abstract we characterize the real valued functions f defined on perfect subsets p of r which admit n times differentiable extensions f r r in this characterization no continuity of f n is imposed in particular it generalizes jarnik s extension theorem according to which f admits differentiable extension f r r if and only if f is differentiable the new characterization is also closely related to the whitney s extension theorem which characterizes partial maps f admitting n times differentiable extensions f r r with continuous nth derivative f n we also provide an elegant description of a linear extension operator t n c p c r such that t n f d n r for every d n r extendable f c p and t n f c n r whenever f c p is c n r extendable

a bias correction for covariance estimators to improve inference with generalized estimating equations that use an unstructured correlation matrix generalized estimating equations gees are routinely used for the marginal analysis of correlated data the efficiency of gee depends on how closely the working covariance structure resembles the true structure and therefore accurate modeling of the working correlation of the data is important a popular approach is the use of an unstructured working correlation matrix as it is not as restrictive as simpler structures such as exchangeable and ar 1 and thus can theoretically improve efficiency however because of the potential for having to estimate a large number of correlation parameters variances of regression parameter estimates can be larger than theoretically expected when utilizing the unstructured working correlation matrix therefore standard error estimates can be negatively biased to account for this additional finite sample variability we derive a bias correction that can be applied to typical estimators of the covariance matrix of parameter estimates via simulation and in application to a longitudinal study we show that our proposed correction improves standard error estimation and statistical inference

a covariance estimator for gee with improved small sample properties summary in this paper we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations gee hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small resampling methods such as the jackknife and bootstrap have been suggested for covariance estimation when the number of clusters is small a drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near zero cell counts caused by resampling we propose a bias corrected covariance estimator that avoids this problem in a small simulation study we compare the bias corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10 40 subjects with equal and unequal cluster sizes of 16 64 observations the bias corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2 3 times the nominal level the methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease

approximations for standard errors of estimators of fixed and random effects in mixed linear models abstract best linear unbiased estimators of the fixed and random effects of mixed linear models are available when the true values of the variance ratios are known if the true values are replaced by estimated values the mean squared errors of the estimators of the fixed and random effects increase in size the magnitude of this increase is investigated and a general approximation is proposed the performance of this approximation is investigated in the context of a the estimation of the effects of the balanced one way random model and b the estimation of treatment contrasts for balanced incomplete block designs

an auto homeomorphism of a cantor set with derivative zero everywhere abstract we construct a closed bounded subset x of r with no isolated points which admits a differentiable bijection f x x such that f x 0 for all x x we also show that any such function admits a restriction f p to an uncountable closed p x forming a minimal dynamical system the existence of such a map f seems to contradict several well know results the map f marks a limit beyond which banach fixed point theorem cannot be generalized