Studies on the estimation of the slope parameter in the simple linear regression model with one-fold nested error structure

Abstract

Four estimators of the slope, β ₁, in the simple linear regression model with one-fold nested error structure were compared with respect, to their mean squared error in a Monte Cario simulation study. Estimators considered were ordinary least squares (OLS), maximum likelihood (ML), estimated generalized least squares (GLS) using analysis of variance estimates of variance components, and the “covariance” estimator (COV) which uses only within-first-stage-unit information. GLS and ML behave quite well if the number of first-stage sampling units a>5 with n≥2 second-stage units per first-stage unit or if a=5 and n>2. When the first-stage variance component is large, GLS is better than ML, but the reverse is true when is small. Some approximate formulas for and derived by regression methods are given. Kackar-Harville approximations for and are satisfactory if a≥11 and may be “good enough” if a≥7.

Keywords:

• estimation
• simple linear regression
• mean squared error
• Monte Carlo simulation
• generalized least squares
• maximum likelihood
• mixed models
• two-stage samples
• hierarchical classification