TABLE 5.
Simulation scenarios and nominal size, along with the empirical type I error rates corresponding to the MAQLS z-test and t-test (DoF = I − 2) with different variance estimators. Empirical type I error rates between 4.5% and 5.5% are highlighted in boldface and considered acceptable based on the margin of error from a binomial model with 10 000 Monte Carlo replications. Pred: nominal type I error rate; MB: model-based variance; BC0: uncorrected sandwich variance of Liang and Zeger (1986); BC1: bias-corrected sandwich variance of Kauermann and Carroll (2001); BC2: bias-corrected sandwich variance of Mancl and DeRouen (2001); BC3: bias-corrected sandwich variance of Fay and Graubard (2001)
| z-test | t-test | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| τ | ρ | δ | I | N | T | Pred | MB | BC0 | BC1 | BC2 | BC3 | Pred | MB | BC1 | BC2 | BC2 | BC3 |
| 0.03 | 0.2 | 0 | 18 | 10 | 7 | 5.0 | 4.8 | 8.4 | 6.8 | 5.1 | 7.1 | 5.0 | 3.4 | 6.3 | 4.7 | 3.8 | 4.9 |
| 0.03 | 0.2 | 0 | 18 | 24 | 4 | 5.0 | 4.9 | 8.1 | 6.5 | 5.0 | 6.5 | 5.0 | 3.4 | 6.0 | 4.7 | 3.6 | 4.7 |
| 0.03 | 0.2 | 0 | 20 | 14 | 5 | 5.0 | 5.1 | 8.3 | 6.9 | 5.8 | 7.0 | 5.0 | 3.8 | 6.5 | 5.4 | 4.4 | 5.5 |
| 0.03 | 0.2 | 0 | 21 | 8 | 4 | 5.0 | 5.2 | 7.9 | 6.6 | 5.3 | 6.6 | 5.0 | 3.8 | 6.2 | 5.0 | 4.0 | 4.9 |
| 0.03 | 0.2 | 0 | 15 | 8 | 4 | 5.0 | 5.7 | 9.7 | 7.7 | 5.8 | 7.7 | 5.0 | 3.7 | 7.0 | 5.3 | 4.1 | 5.3 |
| 0.03 | 0.8 | 0 | 16 | 12 | 5 | 5.0 | 5.6 | 9.0 | 7.3 | 5.6 | 7.3 | 5.0 | 3.8 | 6.7 | 5.1 | 3.8 | 5.1 |
| 0.03 | 0.8 | 0 | 24 | 7 | 5 | 5.0 | 5.3 | 7.7 | 6.6 | 5.5 | 6.6 | 5.0 | 4.3 | 6.2 | 5.2 | 4.3 | 5.3 |
| 0.03 | 0.8 | 0 | 12 | 8 | 5 | 5.0 | 6.0 | 10.6 | 7.9 | 5.8 | 7.9 | 5.0 | 3.5 | 7.0 | 4.9 | 3.2 | 5.0 |
| 0.03 | 0.8 | 0 | 12 | 5 | 4 | 5.0 | 5.7 | 10.3 | 7.8 | 5.7 | 7.5 | 5.0 | 3.4 | 6.9 | 5.0 | 3.5 | 4.9 |
| 0.03 | 0.8 | 0 | 10 | 5 | 3 | 5.0 | 6.3 | 11.2 | 8.2 | 5.6 | 7.1 | 5.0 | 3.0 | 7.0 | 4.6 | 3.1 | 3.8 |
| 0.10 | 0.2 | 0 | 21 | 11 | 8 | 5.0 | 5.0 | 8.1 | 6.9 | 5.7 | 7.0 | 5.0 | 3.8 | 6.6 | 5.3 | 4.4 | 5.6 |
| 0.10 | 0.2 | 0 | 24 | 11 | 7 | 5.0 | 5.1 | 7.7 | 6.4 | 5.3 | 6.5 | 5.0 | 3.8 | 6.1 | 5.1 | 4.3 | 5.2 |
| 0.10 | 0.2 | 0 | 15 | 16 | 6 | 5.0 | 4.8 | 9.7 | 7.5 | 5.7 | 7.8 | 5.0 | 3.1 | 6.8 | 5.2 | 4.0 | 5.3 |
| 0.10 | 0.2 | 0 | 18 | 8 | 7 | 5.0 | 4.8 | 8.8 | 7.1 | 5.6 | 7.3 | 5.0 | 3.4 | 6.7 | 5.2 | 4.0 | 5.4 |
| 0.10 | 0.2 | 0 | 16 | 7 | 5 | 5.0 | 5.2 | 9.1 | 7.2 | 5.5 | 7.3 | 5.0 | 3.4 | 6.6 | 5.1 | 3.8 | 5.2 |
| 0.10 | 0.8 | 0 | 20 | 18 | 5 | 5.0 | 5.3 | 7.9 | 6.4 | 5.2 | 6.5 | 5.0 | 3.8 | 6.1 | 4.8 | 3.9 | 4.8 |
| 0.10 | 0.8 | 0 | 15 | 9 | 4 | 5.0 | 5.5 | 9.1 | 7.2 | 5.3 | 7.0 | 5.0 | 3.7 | 6.6 | 4.8 | 3.7 | 4.6 |
| 0.10 | 0.8 | 0 | 10 | 20 | 3 | 5.0 | 5.6 | 11.7 | 8.5 | 6.0 | 7.4 | 5.0 | 2.8 | 7.3 | 4.9 | 3.0 | 4.0 |
| 0.10 | 0.8 | 0 | 12 | 5 | 5 | 5.0 | 5.8 | 10.5 | 7.9 | 5.7 | 7.9 | 5.0 | 3.3 | 6.9 | 4.9 | 3.3 | 4.9 |
| 0.10 | 0.8 | 0 | 9 | 7 | 4 | 5.0 | 6.0 | 12.0 | 8.6 | 5.7 | 8.3 | 5.0 | 2.8 | 7.2 | 4.7 | 2.9 | 4.5 |