Variance function estimation by replicate analysis and generalized least squares: A Monte Carlo comparison

Abstract

The two methods normally used to estimate variance functions from data — replicate analysis and residuals analysis (via generalized least squares, GLS) — are compared quantitatively through Monte Carlo simulations on simple models having a linear response function (RF) and two different two-parameter variance functions (VFs) — the power model and a components of variance model. The computations emphasize the small numbers of total data points (~ 25) practicable in many data analysis applications, under moderately strong heteroscedasticity (variance range a factor of ~ 100). Results are obtained for weighted LS fitting of replicate-based variance estimates ( s i 2) and squared residuals ( r ij 2), and unweighted LS fitting of their logarithms. For residuals analysis, the effects of the studentization correction are examined. The computations cover the accuracy and precision of estimation of both the VF and the RF, and the reliability of prediction of their standard errors. The results show that the s i 2 and r ij 2 analyses are best, with an efficiency edge for the latter that is much smaller than predicted in the asymptotic limit. The logarithmic methods give significantly biased estimates of the VF, worst for ln( r ij 2) analysis. However, all methods yield at most 5% loss in precision in the estimation of the RF, and < 20% bias in the estimation of that precision, for just 24 data points. Thus, in applications that target the RF, all of these methods would be deemed acceptable under many circumstances. By contrast, use of unweighted or OLS gives precision losses and precision estimation biases more than an order of magnitude greater. These results support the choice of replicate analysis in the common situation where the functional forms of the RF or VF or both are unknown, since VF estimation and RF estimation are largely decoupled in replicate analysis.