Analysis of growth curves with patterned correlation matrices using quasi-least squares

Abstract

Growth curve models with patterned correlation matrices for analyzing repeated measurements were studied extensively in the literature, under the assumption of multivariate normality for the response variables. This paper discusses estimation of the regression, correlation and scale parameters in those models, alleviating the normality assumption, using the method of quasi-least squares. We derive asymptotic properties of the quasi-least squares estimates and compare them with the maximum likelihood estimates obtained under the normality assumption for two important cases: the first order autoregressive (AR(1)), and the equicorrelated correlation matrices. The quasi-least squares estimate of the regression parameter is asymptotically as efficient as the maximum likelihood estimate, even though the two methods differ in how they estimate the correlation parameter. For the AR(1) correlation structure using the asymptotic relative efficiency criterion, we show that the quasi-least squares correlation and scale parameters are good competitors to the maximum likelihood estimates over the entire range of the correlation parameter. But if there is a departure from normality, for example, if the marginals are Student- t distributions, we show by simulation that the asymptotic relative efficiencies are more than one over almost the entire range. We also show that the quasi-least squares estimates are more efficient than the moment estimates. For the equicorrelated structure the quasi-least squares and the method of moment estimates coincide with the maximum likelihood estimates.