Regression modelling of weighted κ by using generalized estimating equations

In many clinical studies more than one observer may be rating a characteristic measured on an ordinal scale. For example, a study may involve a group of physicians rating a feature seen on a pathology specimen or a computer tomography scan. In clinical studies of this kind, the weighted κ coefficient is a popular measure of agreement for ordinally scaled ratings. Our research stems from a study in which the severity of inflammatory skin disease was rated. The investigators wished to determine and evaluate the strength of agreement between a variable number of observers taking into account patient-specific (age and gender) as well as rater-specific (whether board certified in dermatology) characteristics. This suggested modelling κ as a function of these covariates. We propose the use of generalized estimating equations to estimate the weighted κ coefficient. This approach also accommodates unbalanced data which arise when some subjects are not judged by the same set of observers. Currently an estimate of overall κ for a simple unbalanced data set without covariates involving more than two observers is unavailable. In the inflammatory skin disease study none of the covariates were significantly associated with κ, thus enabling the calculation of an overall weighted κ for this unbalanced data set. In the second motivating example (multiple sclerosis), geographic location was significantly associated with κ. In addition we also compared the results of our method with current methods of testing for heterogeneity of weighted κ coefficients across strata (geographic location) that are available for balanced data sets.     

Calculating power for the comparison of dependent κ-coefficients

Summary. In the psychosocial and medical sciences, some studies are designed to assess the agreement between different raters and/or different instruments. Often the same sample will be used to compare the agreement between two or more assessment methods for simplicity and to take advantage of the positive correlation of the ratings. Although sample size calculations have become an important element in the design of research projects, such methods for agreement studies are scarce. We adapt the generalized estimating equations approach for modelling dependent κ -statistics to estimate the sample size that is required for dependent agreement studies. We calculate the power based on a Wald test for the equality of two dependent κ -statistics. The Wald test statistic has a non-central χ ²-distribution with non-centrality parameter that can be estimated with minimal assumptions. The method proposed is useful for agreement studies with two raters and two instruments, and is easily extendable to multiple raters and multiple instruments. Furthermore, the method proposed allows for rater bias. Power calculations for binary ratings under various scenarios are presented. Analyses of two biomedical studies are used for illustration.     

A generalized estimating equation method for fitting autocorrelated ordinal score data with an application in horticultural research

Summary.  Generalized estimating equations for correlated repeated ordinal score data are developed assuming a proportional odds model and a working correlation structure based on a first-order autoregressive process. Repeated ordinal scores on the same experimental units, not necessarily with equally spaced time intervals, are assumed and a new algorithm for the joint estimation of the model regression parameters and the correlation coefficient is developed. Approximate standard errors for the estimated correlation coefficient are developed and a simulation study is used to compare the new methodology with existing methodology. The work was part of a project on post-harvest quality of pot-plants and the generalized estimating equation model is used to analyse data on poinsettia and begonia pot-plant quality deterioration over time. The relationship between the key attributes of plant quality and the quality and longevity of ornamental pot-plants during shelf and after-sales life is explored.     

A mixed poisson–inverse-gaussian regression model

The mixed Poisson–inverse-Gaussian distribution has been used by Holla, Sankaran, Sichel, and others in univariate problems involving counts. We propose a Poisson–inverse-Gaussian regression model which can be used for regression analysis of counts. The model provides an attractive framework for incorporating random effects in Poisson regression models and in handling extra-Poisson variation. Maximum-likelihood and quasilikelihood-moment estimation is investigated and illustrated with an example involving motor-insurance claims.     

Deletion diagnostics for marginal mean and correlation model parameters in estimating equations

Regression diagnostics are introduced for parameters in marginal association models for clustered binary outcomes in an implementation of generalized estimating equations. Estimating equations for intracluster correlations facilitate computational formulae for one-step deletion diagnostics in an extension of earlier work on diagnostics for parameters in the marginal mean model. The proposed diagnostics measure the influence of an observation or a cluster of observations on the estimated regression parameters and on the overall fit of the model. The diagnostics are applied to data from four research studies from public health and medicine.     

A unified approach to estimating association measures via a joint generalized linear model for paired binary data

Paired binary data arise frequently in biomedical studies with unique features of their own. For instance, in clinical studies involving pairs such as ears, eyes etc., often both the intrapair association parameter and the event probability are of interest. In addition, we may be interested in the dependence of the association parameter on certain covariates as well. Although various methods have been proposed to model paired binary data, this paper proposes a unified approach for estimating various intrapair measures under a generalized linear model with simultaneous maximum likelihood estimates of the marginal probabilities and the intrapair association. The methods are illustrated with a twin morbidity study.     

Estimated estimating equations: semiparametric inference for clustered and longitudinal data

Summary.  We introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions. This estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link, and variance–covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance–covariance functions via smoothing methods, whereas the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed estimated estimating equations approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model, with either a specified or an unspecified distribution of random effects and link function, the model proposed emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model, without the need to specify any other components of this generalized linear mixed model. Among marginal models, the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations. Applications of estimated estimating equations include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.     

Stratified analysis of correlated binary outcome data: a comparison of model dependent and robust tests of significance

Model dependent and robust test statistics constructed using a generalized estimating equations extension of logistic regression applicable to the analysis of correlated binary outcome data are shown to have relatively simple algebraic expressions in stratified analyses where all variables are measured at the cluster level These expressions are used to demonstrate the close relationship to standard procedures which assume that subjects responses are independent, to prove that the asymptotic validity of model dependent test statistics is assured if the average correlation between cluster members is constant, and that this assumption can be relaxed when there are the same number of subjects in each cluster.     

A practical extension of the generalized estimating equation approach for longitudinal data

Liang and Zeger (1986) proposed an extension of generalized linear models to the analysis of longitudinal data. In their formulation, a common dispersion parameter assumption across observation times is required. However, this assumption is not expected to hold in most situations. Park (1993) proposed a simple extension of Liang and Zeger's formulation to allow for different dispersion parameters for each time point. The proposed model is easy to apply without heavy computations and useful to handle the cases when variations in over-dispersion over time exist. In this paper, we focus on evaluating the effect of additional dispersion parameters on the estimators of model parameters. Through a Monte Carlo simulation study, efficiency of Park's method is compared with the Liang and Zeger's method.     

Estimation for generalized partially functional linear additive regression model

In practice, it is not uncommon to encounter the situation that a discrete response is related to both a functional random variable and multiple real-value random variables whose impact on the response is nonlinear. In this paper, we consider the generalized partial functional linear additive models (GPFLAM) and present the estimation procedure. In GPFLAM, the nonparametric functions are approximated by polynomial splines and the infinite slope function is estimated based on the principal component basis function approximations. We obtain the estimator by maximizing the quasi-likelihood function. We investigate the finite sample properties of the estimation procedure via Monte Carlo simulation studies and illustrate our proposed model by a real data analysis.     

Weighted empirical adaptive variance estimators for correlated data regression

Estimating equations based on marginal generalized linear models are useful for regression modelling of correlated data, but inference and testing require reliable estimates of standard errors. We introduce a class of variance estimators based on the weighted empirical variance of the estimating functions and show that an adaptive choice of weights allows reliable estimation both asymptotically and by simulation in finite samples. Connections with previous bootstrap and jackknife methods are explored. The effect of reliable variance estimation is illustrated in data on health effects of air pollution in King County, Washington.     

Efficiency of generalized estimating equations for binary responses

Summary.  Using standard correlation bounds, we show that in generalized estimation equations (GEEs) the so-called 'working correlation matrix' R ( α ) for analysing binary data cannot in general be the true correlation matrix of the data. Methods for estimating the correlation param-eter in current GEE software for binary responses disregard these bounds. To show that the GEE applied on binary data has high efficiency, we use a multivariate binary model so that the covariance matrix from estimating equation theory can be compared with the inverse Fisher information matrix. But R ( α ) should be viewed as the weight matrix, and it should not be confused with the correlation matrix of the binary responses. We also do a comparison with more general weighted estimating equations by using a matrix Cauchy–Schwarz inequality. Our analysis leads to simple rules for the choice of α in an exchangeable or autoregressive AR(1) weight matrix R ( α ), based on the strength of dependence between the binary variables. An example is given to illustrate the assessment of dependence and choice of α .     

Robust estimation of covariance parameters in partial linear model for longitudinal data

For longitudinal data, the within-subject dependence structure and covariance parameters may be of practical and theoretical interests. The estimation of covariance parameters has received much attention and been studied mainly in the framework of generalized estimating equations (GEEs). The GEEs method, however, is sensitive to outliers. In this paper, an alternative set of robust generalized estimating equations for both the mean and covariance parameters are proposed in the partial linear model for longitudinal data. The asymptotic properties of the proposed estimators of regression parameters, non-parametric function and covariance parameters are obtained. Simulation studies are conducted to evaluate the performance of the proposed estimators under different contaminations. The proposed method is illustrated with a real data analysis.     

Estimation of a two-equation panel model with mixed continuous and ordered categorical outcomes and missing data

Summary.  The paper proposes an estimation approach for panel models with mixed continuous and ordered categorical outcomes based on generalized estimating equations for the mean and pseudoscore equations for the covariance parameters. A numerical study suggests that efficiency can be gained in the mean parameter estimators by using individual covariance matrices in the estimating equations for the mean parameters. The approach is applied to estimate the returns to occupational qualification in terms of income and perceived job security in a 9-year period based on the German Socio-Economic Panel. To compensate for missing data, a combined multiple imputation–weighting approach is adopted.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Building adaptive estimating equations when inverse of covariance estimation is difficult

Summary. To construct an optimal estimating function by weighting a set of score functions, we must either know or estimate consistently the covariance matrix for the individual scores. In problems with high dimensional correlated data the estimated covariance matrix could be unreliable. The smallest eigenvalues of the covariance matrix will be the most important for weighting the estimating equations, but in high dimensions these will be poorly determined. Generalized estimating equations introduced the idea of a working correlation to minimize such problems. However, it can be difficult to specify the working correlation model correctly. We develop an adaptive estimating equation method which requires no working correlation assumptions. This methodology relies on finding a reliable approximation to the inverse of the variance matrix in the quasi-likelihood equations. We apply a multivariate generalization of the conjugate gradient method to find estimating equations that preserve the information well at fixed low dimensions. This approach is particularly useful when the estimator of the covariance matrix is singular or close to singular, or impossible to invert owing to its large size.     

Consistent model selection and data-driven smooth tests for longitudinal data in the estimating equations approach

Summary.  Model selection for marginal regression analysis of longitudinal data is challenging owing to the presence of correlation and the difficulty of specifying the full likelihood, particularly for correlated categorical data. The paper introduces a novel Bayesian information criterion type model selection procedure based on the quadratic inference function, which does not require the full likelihood or quasi-likelihood. With probability approaching 1, the criterion selects the most parsimonious correct model. Although a working correlation matrix is assumed, there is no need to estimate the nuisance parameters in the working correlation matrix; moreover, the model selection procedure is robust against the misspecification of the working correlation matrix. The criterion proposed can also be used to construct a data-driven Neyman smooth test for checking the goodness of fit of a postulated model. This test is especially useful and often yields much higher power in situations where the classical directional test behaves poorly. The finite sample performance of the model selection and model checking procedures is demonstrated through Monte Carlo studies and analysis of a clinical trial data set.     

Slope modified confidence bands for a simple linear regression model

Considerable attention has been directed in the statistical literature towards the construction of confidence bands for a simple linear regression model. These confidence bands allow the experimenter to make inferences about the model over a particular region of interest. However, in practice an experimenter will usually first check the significance of the regression line before proceeding with any further inferences such as those provided by the confidence bands. From a theoretical point of view, this raises the question of what the conditional confidence level of the confidence bands might be, and from a practical point of view it is unsatisfactory if the confidence bands contain lines that are inconsistent with the directional decision on the slope. In this paper it is shown how confidence bands can be modified to alleviate these two problems.     

On the application of extended quasi‐likelihood to the clustered data case

The author describes the relationship between the extended generalized estimating equations (EGEEs) of Hall & Severini (1998) and various similar methods. He proposes a true extended quasi‐likelihood approach for the clustered data case and explores restricted maximum likelihood‐like versions of the EGEE and extended quasi‐likelihood estimating equations. He also presents simulation results comparing the various estimators in terms of mean squared error of estimation based on three moderate sample size, discrete data situations.