On eliminating the asymptotic bias in the quasi-least squares estimate of the correlation parameter

In a recent paper, Chaganty (1997, J. Statist. Plann. Inference 63, 39-54) introduced the method of quasi-least squares (QLS) for estimating the regression, correlation and scale parameters in longitudinal data analysis problems. The QLS estimates of the regression and scale parameters are consistent even if the working correlation structure is misspecified. The estimate of the correlation parameter, however, is asymptotically biased. In this paper, we present modified (C-QLS) estimates of the correlation parameter for the following working correlation structures that are appropriate for the analysis of balanced and equally spaced longitudinal data: the unstructured matrix, for which the C-QLS estimate is a positive definite, consistent correlation matrix; and the exchangeable, tridiagonal, and autoregressive structures, for which the C-QLS estimates are feasible, consistent and robust against misspecification. We also present feasible and consistent C-QLS estimates for two structures appropriate for the analysis of unbalanced and unequally spaced longitudinal data: the Markov and generalized Markov working correlation structures that were discussed by Nunez-Anton and Woodworth (1994, Biometrics 50, 445-456) and Shults and Chaganty (1998, Biometrics 54, 1622-1630). We then present an improved consistent estimate of the scale parameter. Finally, examples are given to contrast the C-QLS estimates with estimates obtained using the widely used generalized estimating equation (GEE) approach. (C) 1999 Elsevier Science B.V. All rights reserved.