A generalized linear mixed model for longitudinal binary data with a marginal logit link function

Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. In general, a feature of random effects logistic regression models for longitudinal binary data is that the marginal functional form, when integrated over the distribution of the random effects, is no longer of logistic form. Recently, Wang and Louis [Biometrika 90 (2003) 765--775] proposed a random intercept model in the clustered binary data setting where the marginal model has a logistic form. An acknowledged limitation of their model is that it allows only a single random effect that varies from cluster to cluster. In this paper we propose a modification of their model to handle longitudinal data, allowing separate, but correlated, random intercepts at each measurement occasion. The proposed model allows for a flexible correlation structure among the random intercepts, where the correlations can be interpreted in terms of Kendall's $\tau$. For example, the marginal correlations among the repeated binary outcomes can decline with increasing time separation, while the model retains the property of having matching conditional and marginal logit link functions. Finally, the proposed method is used to analyze data from a longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS390 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

QUEM: An Achievement Test for Knowledge-Based Systems

Abstract-This paper describes the QUality and Experience Metric (QUEM), a method for estimating the skill level of a knowledgebased system based on the quality of the solutions it produces. It allows one to assess how many years of experience the system would be judged to have if it were a human by providing a quantitative measure of the system's overall competence. QUEM can be viewed as a type of achievement or job-placement test administered to knowledge-based systems to help system designers determine how the system should be used and by what level of user. To apply QUEM, a set of subjects, experienced judges, and problems must be identified. The subjects should have a broad range of experience levels. Subjects and the knowledge-based system are asked to solve the problems; and judges are asked to rank order all solutions}from worst quality to best. The data from the subjects is used to construct a skill-function relating experience to solution quality, and confidence bands showing the variability in performance. The system's quality ranking is then plugged into the skill function to produce an estimate of the system's experience level. QUEM can be used to gauge the experience level of an individual system, to compare two systems, or to compare a system to its intended users. This represents an important advance in providing quantitative measures of overall performance that can be applied to a broad range of systems

A measure of partial association for generalized estimating equations

Abstract: In a regression setting, the partial correlation coefficient is often used as a measure of 'standardized' partial association between the outcome y and each of the covariates in In a linear regression model estimated using ordinary least squares, with y as the response, the estimated partial correlation coefficient between y and x k can be shown to be a monotone function, denoted f (z), of the Z-statistic for testing if the regression coefficient of x k is 0. When y is non-normal and the data are clustered so that y and x are obtained from each member of a cluster, generalized estimating equations are often used to estimate the regression parameters of the model for y given x. In this paper, when using generalized estimating equations, we propose using the above transformation f (z) of the GEE Z-statistic as a measure of partial association. Further, we also propose a coefficient of determination to measure the strength of association between the outcome variable and all of the covariates. To illustrate the method, we use a longitudinal study of the binary outcome heart toxicity from chemotherapy in children with leukaemia or sarcoma

Joint generalized estimating equations for multivariate longitudinal binary outcomes with missing data: an application to acquired immune deficiency syndrome data

In a large, prospective longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women, instead of a single outcome variable, there are multiple binary outcomes (e.g., abnormal heart rate, abnormal blood pressure, abnormal heart wall thickness) considered as joint measures of heart function over time. In the presence of missing responses at some time points, longitudinal marginal models for these multiple outcomes can be estimated using generalized estimating equations (GEE) (Liang and Zeger, 1986), and consistent estimates can be obtained under the assumption of a missing completely at random (MCAR) mechanism. When the missing data mechanism is missing at random (MAR), that is the probability of missing a particular outcome at a time-point depends on observed values of that outcome and the remaining outcomes at other time points, we propose joint estimation of the marginal models using a single modified GEE based on an EM-type algorithm. The proposed method is motivated by the longitudinal study of cardiac abnormalities in children born to HIV-infected women and analyses of these data are presented to illustrate the application of the method. Further, in an asymptotic study of bias, we show that under an MAR mechanism in which missingness depends on all observed outcome variables, our joint estimation via the modified GEE produces almost unbiased estimates, provided the correlation model has been correctly specified, whereas estimates from standard GEE can lead to substantial bias

Spatial theory and human behavior

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45971/1/10110_2005_Article_BF01952731.pd

Perturbing the minimand resampling with Gamma(1,1) random variables as an extension of the Bayesian Bootstrap

Jin et al. (2001) proposed a clever resampling method useful for calculating a variance estimate of the solution to an estimating equation. The method multiplies each independent subject's contribution to the estimating equation by a randomly sampled random variable with mean and variance 1. They showed that this resampling technique gives consistent variance estimates under mild conditions. Rubin (1981. The Bayesian Bootstrap. Ann. Statist. 9, 130-134) proposed the Bayesian Bootstrap as a modification of the usual bootstrap. In this note, we show that the Bayesian Bootstrap is a special case of Jin et al.'s resampling approach.Generalized linear models Missing at random Missing data mechanism Riemann summation

Approximate median regression via the Box-Cox transformation

Median regression is used increasingly in many different areas of applications. The usual median regression estimating equations are derived from minimizing the least absolute deviations (LAD). Because they are not a smooth function of the regression parameters, a solution is best obtained using a linear programming algorithm. As an alternative, we propose estimating the median regression parameters via Gaussian estimating equations after applying a Box-Cox transformation to both the outcome and the linear predictor. The proposed estimator is notably more efficient than the standard LAD estimator, albeit with an acknowledged loss of robustness