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

Abstract

Six hypothesis testing procedures for the slope, β₁, in the simple linear regression model with one-fold nested-error structure were investigated and compared with respect to Type I error rate and power of test in a Monte Carlo simulation study. Test criteria considered were one exact F-test, four pseudo F-tests and an approximate x²-test obtained from each of the following four estimators of β₁: the 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. In the OLS case the F-statistic used a linear combination of mean squares as the denominator with degrees of freedom (d.f.) determined by Satterthwaite's approximation. In the GLS and ML cases the F-statistic was the F-statistic obtained by fitting the model using the method of Fuller and Battese (1973). Two different allocations of the denominator d.f. were used for the GLS F-statistic, namely, the GLS1 method using total error d.f. and the GLS2 method using subsampling error d.f. In the case of COV, the test was the same as the F-test obtained for the slope of regression in a one-way analysis of covariance. In the ML case an approximate x² test was also considered. GLS1 and ML F-tests behaved very similarly and were usually liberal. They became quite liberal if the number of first-stage sampling units a < 5 and the second-stage units per first-stage unit n = 2. The ML F-test became excessively liberal as the first-stage variance component decreased. GLS2 appeared to be the best choice in many situations, but in a few cases it was also liberal. Such cases tend to occur when a large proportion of the variation in the regressor variable is among first-stage unit variation and the d.f among first-stage sampling units is small relative to the d.f. within first-stage units. The ML X² test was quite liberal in all cases. The OLS test was frequently conservative. Its conservative nature was more pronounced at α (the intended significance level of the test) = .05 or .10 than at a = .01. Conservativeness of the OLS test tends to diminish as α increases, but tends to worsen as the number n of second-stage units per first-stage unit increases with a held constant.

Keywords:

• hypothesis testing
• simple linear regression
• power of test
• Monte Carlo simulation
• generalized least squares
• maximum likelihood
• mixed models
• two-stage samples
• hierarchical classification
• analysis of variance
• analysis of covariance