Table 3.
Empirical power and type I error rates (in parentheses) for modified Poisson regression under exchangeable working correlation when the mean cluster size is ; P0 = 0.15.
| ρ | CV | Robust | MD | KC | FG | MD/KC | MD/FG | KC/FG | |
|---|---|---|---|---|---|---|---|---|---|
| 0.01 | 0 | 11 | 91.0(7.1) | 79.7(3.4) | 86.5(5.1) | 83.3(4.0) | 83.1(4.3) | 82.2(3.7) | 84.5(4.6) |
| 0.2 | 11 | 88.8(7.7) | 78.1(3.5) | 83.9(5.2) | 81.1(4.0) | 80.7(4.1) | 79.6(3.4) | 82.2(4.5) | |
| 0.4 | 11 | 88.9(10.1) | 75.1(3.9) | 82.6(6.8) | 79.3(5.4) | 79.0(5.1) | 77.1(4.4) | 81.0(6.3) | |
| 0.6 | 11 | 88.0(10.9) | 73.1(4.0) | 82.1(7.2) | 78.4(5.8) | 78.4(5.8) | 75.4(4.8) | 81.0(6.5) | |
| 0.8 | 12 | 88.7(10.9) | 71.2(3.6) | 81.5(7.1) | 77.8(5.8) | 77.6(4.7) | 75.0(4.1) | 80.4(6.5) | |
| 0.05 | 0 | 21 | 86.9(5.0) | 82.1(3.3) | 84.6(3.9) | 83.5(3.7) | 83.6(3.8) | 82.7(3.4) | 74.1(3.8) |
| 0. 2 | 21 | 85.6(5.6) | 78.7(3.7) | 82.8(4.0) | 80.4(3.8) | 80.7(3.7) | 79.6(3.7) | 81.6(4.0) | |
| 0. 4 | 21 | 85.3(5.9) | 77.5(3.8) | 81.5(4.4) | 79.2(3.9) | 78.9(4.1) | 78.2(3.9) | 80.3(4.1) | |
| 0. 6 | 22 | 84.9(7.2) | 80.1(5.3) | 82.4(5.8) | 81.6(5.5) | 81.6(5.4) | 81.0(5.4) | 82.2(5.5) | |
| 0. 8 | 23 | 85.5(7.3) | 80.0(4.9) | 83.4(6.2) | 81.8(5.8) | 81.8(5.6) | 80.9(5.2) | 82.3(6.0) | |
| 0.10 | 0 | 33 | 84.3(6.3) | 81.7(5.4) | 82.9(5.8) | 82.3(5.6) | 82.1(5.5) | 81.8(5.4) | 82.5(5.6) |
| 0. 2 | 34 | 84.9(5.9) | 81.7(4.3) | 83.5(5.2) | 82.5(4.6) | 82.6(4.7) | 81.9(4.4) | 82.7(4.9) | |
| 0. 4 | 34 | 83.6(5.4) | 80.2(4.7) | 81.9(4.9) | 81.3(4.7) | 81.1(4.7) | 80.7(4.7) | 81.3(4.8) | |
| 0. 6 | 35 | 83.8(5.9) | 80.3(5.1) | 82.2(5.4) | 81.2(5.3) | 81.3(5.2) | 80.8(5.2) | 81.7(5.3) | |
| 0. 8 | 36 | 84.1(6.6) | 81.2(5.1) | 82.2(5.9) | 81.8(5.4) | 81.9(5.4) | 81.5(5.1) | 82.0(5.8) | |
| 0.15 | 0 | 46 | 82.5(4.8) | 79.8(3.7) | 80.8(4.2) | 80.2(3.8) | 80.4(3.9) | 80.1(3.7) | 80.5(3.9) |
| 0. 2 | 46 | 79.9(5.1) | 78.4(4.5) | 79.3(4.6) | 79.0(4.6) | 78.9(4.6) | 78.7(4.6) | 79.2(4.6) | |
| 0. 4 | 47 | 84.3(6.0) | 81.8(5.1) | 83.2(5.7) | 82.3(5.6) | 82.3(5.3) | 82.0(5.3) | 83.0(5.6) | |
| 0. 6 | 48 | 83.9(6.8) | 81.6(5.3) | 82.8(6.0) | 82.2(5.5) | 82.2(5.5) | 81.8(5.4) | 82.4(5.6) | |
| 0. 8 | 49 | 84.0(6.2) | 80.9(5.2) | 82.5(5.6) | 81.9(5.3) | 82.0(5.3) | 81.3(5.3) | 82.1(5.4) | |
| 0.20 | 0 | 59 | 83.6(6.4) | 81.6(5.9) | 82.7(6.1) | 82.0(5.9) | 82.1(6.0) | 81.7(5.9) | 82.2(6.1) |
| 0. 2 | 59 | 85.2(5.9) | 83.0(5.3) | 84.0(5.4) | 83.3(5.3) | 83.3(5.3) | 83.0(5.3) | 83.5(5.4) | |
| 0. 4 | 60 | 83.1(6.4) | 81.7(5.4) | 82.0(5.7) | 81.8(5.6) | 81.9(5.6) | 81.7(5.6) | 82.0(5.6) | |
| 0. 6 | 60 | 82.4(5.6) | 80.0(4.4) | 81.3(5.0) | 80.7(4.7) | 80.8(4.6) | 80.5(4.6) | 81.1(4.8) | |
| 0. 8 | 62 | 83.9(4.2) | 81.7(3.8) | 83.0(3.8) | 82.4(3.8) | 82.6(3.8) | 82.2(3.8) | 82.7(3.8) |
Note: Empirical type I error rate between 3.6% and 6.4% and empirical power between 77.5% and 82.5% are in bold font and considered close to nominal according to the margin of error under a binomial model with 1000 replications. Robust refers to the t-test with the uncorrected robust sandwich variance estimator; MD refers to the t-test with the bias-corrected sandwich variance estimator due to Mancl and DeRouen; KC refers to the t-test with the bias-corrected sandwich variance estimator due to Kauermann and Carroll; FG refers to the t-test with the bias-corrected sandwich variance estimator due to Fay and Graubard; MD/KC refers to the t-test with the average MD/KC standard error estimator; MD/FG refers to the t-test with the average MD/FG standard error estimator; KC/FG refers to the t-test with the average KC/FG standard error estimator. ρ refers to ICC; CV refers to the coefficient of variation of cluster sizes; refers to the estimated number of clusters.