### Abstract

Six hypothesis testing procedures for the slope, β_{1}, 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 χ^{2}-test obtained from each of the following four estimators of β_{1}: 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 Satterth waite'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 χ^{2}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 σ_{1}^{2} 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 χ^{2} 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 α =. 01. Conservativeness of the OLS test tends to diminish as a increases, but tends to worsen as the number n of second-stage units per first-stage unit increases with a held constant.

Original language | English |
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Pages (from-to) | 2023-2043 |

Number of pages | 21 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 20 |

Issue number | 7 |

DOIs | |

State | Published - 1 Jan 1991 |

### Keywords

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

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## Cite this

*Communications in Statistics - Theory and Methods*,

*20*(7), 2023-2043. https://doi.org/10.1080/03610929108830619