performance of generalized estimating equations in practical situations moment methods for analyzing repeated binary responses have been proposed by liang and zeger 1986 biometrika 73 13 22 and extended by prentice 1988 biometrics 44 1033 1048 in their generalized estimating equations gee both liang and zeger 1986 and prentice 1988 estimate the parameters associated with the expected value of an individual s vector of binary responses as well as the correlations between pairs of binary responses in this paper we discuss one step estimators i e estimators obtained from one step of the generalized estimating equations and compare their performance to that of the fully iterated estimators in small samples in simulations we find the performance of the one step estimator to be qualitatively similar to that of the fully iterated estimator when the sample size is small and the association between binary responses is high we recommend using the one step estimator to circumvent convergence problems associated with the fully iterated gee algorithm furthermore we find the gee methods to be more efficient than ordinary logistic regression with variance correction for estimating the effect of a time varying covariate

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

a caveat concerning independence estimating equations with multivariate binary data clustered binary data occur commonly in both the biomedical and health sciences in this paper we consider logistic regression models for multivariate binary responses where the association between the responses is largely regarded as a nuisance characteristic of the data in particular we consider the estimator based on independence estimating equations iee which assumes that the responses are independent this estimator has been shown to be nearly efficient when compared with maximum likelihood ml and generalized estimating equations gee in a variety of settings the purpose of this paper is to highlight a circumstance where assuming independence can lead to quite substantial losses of efficiency in particular when the covariate design includes within cluster covariates assuming independence can lead to a considerable loss of efficiency in estimating the regression parameters associated with those covariates

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

channel robust classifiers a key assumption underlying traditional supervised learning algorithms is that labeled examples used to train a classifier are drawn i i d from the same distribution as test samples this assumption is violated when classifying a test sample whose statistics differ from the training samples because the test signal is the output of a noisy linear time invariant system e g from channel propagation or filtering we assume that the channel impulse response is unknown but can be modeled as a random channel with finite first and second order statistics that can be estimated from sample impulse responses we present two kernels the expected and projected rbf kernels that account for the stochastic channel compared to the strategy of virtual examples an svm trained with the proposed kernels requires dramatically less training time and may perform better in practice we also extend the joint quadratic discriminant analysis joint qda classifier which also accounts for a stochastic channel to a local version that reduces model bias results show the proposed methods achieve state of the art performance and significantly faster training times

multivariate mean parameter estimation by using a partly exponential model summary a class of partly exponential models is proposed for the regression analysis of multivariate response data the class is parameterized in terms of the response mean and a general shape parameter it includes the generalized linear error model and exponential dispersion models as special cases maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations and maximum likelihood estimates of mean and shape parameters are asymptotically independent some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix

inference for clustered data using the independence loglikelihood we use the properties of independence estimating equations to adjust the independence loglikelihood function in the presence of clustering the proposed adjustment relies on the robust sandwich estimator of the parameter covariance matrix which is easily calculated the methodology competes favourably with established techniques based on independence estimating equations we provide some insight as to why this is so the adjustment is applied to examples relating to the modelling of wind speed in europe and annual maximum temperatures in the u k

a note on implementation of decaying product correlation structures for quasi least squares this note implements an unstructured decaying product matrix via the quasi least squares approach for estimation of the correlation parameters in the framework of generalized estimating equations the structure we consider is fairly general without requiring the large number of parameters that are involved in a fully unstructured matrix it is straightforward to show that the quasi least squares estimators of the correlation parameters yield feasible values for the unstructured decaying product structure furthermore subject to conditions that are easily checked the quasi least squares estimators are valid for longitudinal bernoulli data we demonstrate implementation of the structure in a longitudinal clinical trial with both a continuous and binary outcome variable

estimating logistic regression parameters for bivariate binary data summary consider bivariate binary data with possibly different covariates for each marginal binary observation suppose that the correlation between paired observations is a nuisance and the interest lies in estimating logistic regression parameters associated with the covariates we consider the estimator obtained by using the independence estimating equation with corrected variance this estimator can be very efficient and robust compared with maximum likelihood or the estimator of liang and zeger some comments are made concerning higher dimensional multivariate data

fractional dimensions of phase space variables a tool for varying the degrees of freedom of a system in a multiscale treatment we show how the idea of fractal dimensions of phase space variables can be employed to develop a concept of adaptive resolution treatment of a molecular liquid the resulting theoretical framework allows for calculation of statistical averages of thermodynamic quantities in multiresolution computer simulation algorithms where the molecular degrees of freedom change on the fly

an appraisal of methods for the analysis of longitudinal categorical data with mar drop outs a number of methods for analysing longitudinal ordinal categorical data with missing at random drop outs are considered two are maximum likelihood methods maxlik which employ marginal global odds ratios to model associations the remainder use weighted or unweighted generalized estimating equations gee two of the gee use cholesky decomposed standardized residuals to model the association structure while another three extend methods developed for longitudinal binary data in which the association structures are modelled using either gaussian estimation multivariate normal estimating equations or conditional residuals simulated data sets were used to discover differences among the methods in terms of biases variances and convergence rates when the association structure is misspecified the methods were also applied to a real medical data set two of the gee methods referred to as cond and ml norm in this paper and by their originators were found to have relatively good convergence rates and mean squared errors for all sample sizes 80 120 300 considered and one more referred to as mgee in this paper and by its originators worked fairly well for all but the smallest sample size 80

joint deconvolution and classification with applications to passive acoustic underwater multipath this paper addresses the problem of classifying signals that have been corrupted by noise and unknown linear time invariant lti filtering such as multipath given labeled uncorrupted training signals a maximum a posteriori approach to the deconvolution and classification is considered which produces estimates of the desired signal the unknown channel and the class label for cases in which only a class label is needed the classification accuracy can be improved by not committing to an estimate of the channel or signal a variant of the quadratic discriminant analysis qda classifier is proposed that probabilistically accounts for the unknown lti filtering and which avoids deconvolution the proposed qda classifier can work either directly on the signal or on features whose transformation by lti filtering can be analyzed as an example a classifier for subband power features is derived results on simulated data and real bowhead whale vocalizations show that jointly considering deconvolution with classification can dramatically improve classification performance over traditional methods over a range of signal to noise ratios

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

violations of the independent increment assumption when using generalized estimating equation in longitudinal group sequential trials in phase 3 clinical trials ethical and financial concerns motivate sequential analyses in which the data are analyzed prior to completion of the entire planned study existing group sequential software accounts for the effects of these interim analyses on the sampling density by assuming that the contribution of subsequent increments is independent of the contribution from previous data this independent increment assumption is satisfied in many common circumstances including when using the efficient estimator however certain circumstances may dictate using an inefficient estimator and the independent increment assumption may then be violated consequences of assuming independent increments in a setting where the assumption does not hold have not been previously explored one important setting in which independent increments may not hold is the setting of longitudinal clinical trials this paper considers dependent increments that arise because of heteroscedastic and correlated data in the context of longitudinal clinical trials that use a generalized estimating equation gee approach both heteroscedasticity over time and correlation of observations within subjects may lead to departures from the independent increment assumption when using gee we characterize situations leading to greater departures in this paper despite violations of the independent increment assumption simulation results suggest that operating characteristics of sequential designs are largely maintained for typically observed patterns of accrual correlation and heteroscedasticity even when using analyses that use standard software that depends on an independent increment structure more extreme scenarios may require greater care to avoid departures from the nominal type i error rate and power copyright 2014 john wiley sons ltd

robust analysis of generalized linear mixed models the method of maximum likelihood ml is widely used for analyzing generalized linear mixed models glmm s a full maximum likelihood analysis requires numerical integration techniques for calculation of the log likelihood and to avoid the computational problems involving irreducibly high dimensional integrals several maximum likelihood algorithms have been proposed in the literature to estimate the model parameters by approximating the log likelihood function although these likelihood algorithms are useful for fitting the glmm s efficiently under strict model assumptions they can be highly influenced by the presence of unusual data points in this article the author develops a technique for finding robust maximum likelihood rml estimates of the model parameters in glmm s which appears to be useful in downweighting the influential data points when estimating the parameters the asymptotic properties of the robust estimators are investigated under some regularity conditions small simulations are

estimation efficiency and tests of covariate effects with clustered binary data several approaches have been proposed to analyze clustered binary data which arise in fields such as teratology and ophthalmology these methods include mixed effects and quasi likelihood approaches as well as models that use cluster responses as covariates the three approaches measure different effects of covariates on binary responses but simple approximations relate the magnitudes of their parameters in this article we present approximations to relate the standard errors of model parameters and wald tests for covariate effects obtained from the different approaches these approximations show that wald tests involving cluster level covariates will be approximately equivalent using the different approaches however approaches that model intracluster correlation such as the mixed effects model provide more powerful tests of within cluster covariates than those that do not model the correlation simulations and example data illustrate these findings

analysis of serially correlated data using quasi least squares summary quasi least squares qls a marginal statistical approach via generalized estimating equations that is described in the balanced data setting by chaganty 1997 journal of statistical planning and inference 63 39 54 allows for application of a wide range of working correlation structures when analyzing serially correlated data we extend the application of qls to serially correlated unequally spaced and unbalanced data using three useful working correlation models the firstorder autoregressive ar 1 the markov and the generalized markov structure described by nuniez anton and woodworth 1994 biometrics 50 445 456 we compare qls and the original formulation of the generalized estimating equation approach gee for these structures demonstrating that i infeasibility of the gee correlation parameter estimates can be a problem ii it is difficult to obtain consistent moment estimates of the correlation parameters for the generalized markov structure and iii the use of qls can lead to reduced mean square error of the estimate of the regression parameter for small samples of moderately correlated data to choose between alternative correlation models we propose a criterion that is based on the principle of generalized least squares finally data for which the generalized markov structure is appropriate are analyzed to demonstrate the use of qls in selecting a suitable working correlation structure and identifying in this paper we apply a statistical method based on the generalized estimating equation approach of liang and zeger 1986 to the analysis of longitudinal data that may be difficult to analyze using other established methods we consider repeated measures data collected by taking measurements of an outcome variable and associated covariates on each of a group of independent subjects our primary data analysis goal is to identify important covariates and to explain their effect on the marginal mean of the outcome variable while also accounting for the correlation among observations on each subject accomplishing this research objective may be difficult due to certain conditions that are typical in longitudinal studies the timing and total number of measurements taken may vary from subject to subject so that the data may be unbalanced and unequally spaced the outcome variable may not be normally distributed the intrasubject correlation may be described using a time dependent pattern for example the correlation between two measurements may decrease as they become more

use of quasi least squares to adjust for two levels of correlation this article considers data with two levels of association our motivating example is an international intervention trial with repeated observations on subjects who reside within geographically defined clusters to account for the potential correlation within clusters and within the repeated measurements that pertain to each subject we apply a method based on generalized estimating equations for a correlation structure proposed by lefkopoulou moore and ryan 1989 journal of the american statistical association 84 810 815

maximum likelihood regression methods for paired binary data we discuss maximum likelihood methods for analysing binary responses measured at two times such as in a cross over design we construct a 2 2 table for each individual with cell probabilities corresponding to the cross classification of the responses at the two times the underlying likelihood for each individual is multinomial with four cells the three dimensional parameter space of the multinomial distribution is completely specified by the two marginal probabilities of success of the 2 2 table and an association parameter between the binary responses at the two times we examine a logistic model for the marginal probabilities of the 2 2 table for individual i the association parameters we consider are either the correlation coefficient the odds ratio or the relative risk simulations show that the parameter estimates for the logistic regression model for the marginal probabilities are not very sensitive to the parameters used to describe the association between the binary responses at the two times thus we suggest choosing the measure of association for ease of interpretation

minimum entropy deconvolution and simplicity a noniterative algorithm minimum entropy deconvolution med is a technique developed by wiggins 1978 with the purpose of separating the components of a signal as the convolution model of a smooth wavelet with a series of impulses the advantage of this method as compared with traditional methods is that it obviates strong hypotheses over the components which require only the simplicity of the output the degree of simplicity is measured with the varimax norm for factor analysis an iterative algorithm for computation of the filter is derived from this norm having as an outstanding characteristic its stability in presence of noise geometrical analysis of the varimax norm suggests the definition of a new criterion for simplicity the d norm in case of multiple inputs the d norm is obtained through modification of the kurtosis norm one of the most outstanding characteristics of the new criterion by comparison with the varimax norm is that a noniterative algorithm for computation of the deconvolution filter can be deriv

bayesian quadratic discriminant analysis quadratic discriminant analysis is a common tool for classification but estimation of the gaussian parameters can be ill posed this paper contains theoretical and algorithmic contributions to bayesian estimation for quadratic discriminant analysis a distribution based bayesian classifier is derived using information geometry using a calculus of variations approach to define a functional bregman divergence for distributions it is shown that the bayesian distribution based classifier that minimizes the expected bregman divergence of each class conditional distribution also minimizes the expected misclassification cost a series approximation is used to relate regularized discriminant analysis to bayesian discriminant analysis a new bayesian quadratic discriminant analysis classifier is proposed where the prior is defined using a coarse estimate of the covariance based on the training data this classifier is termed bda7 results on benchmark data sets and simulations show that bda7 performance is competitive with and in some cases significantly better than regularized quadratic discriminant analysis and the cross validated bayesian quadratic discriminant analysis classifier quadratic bayes

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

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