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. Author manuscript; available in PMC: 2013 Feb 14.
Published in final edited form as: Biometrics. 2008 Sep 29;65(3):919–927. doi: 10.1111/j.1541-0420.2008.01124.x

Table 2.

Comparison of the rejection rates (×10−2) of the null hypothesis H0: β2 = 0 for the t test, the adjusted ET likelihood ratio test and the adjusted EL likelihood ratio test at the 0.05 significance level. Six error distributions are N(0,1), χ2(3) − 3, heter1, heter2, 0.5N(2,1) + 0.5N(−2,1), and t(3). Moreover, heter1 and heter2 denote Gaussian distributions with heterogeneous variance σ(i). For heter1, σ (i) = exp(u) when xi2 = 0 and σ(i) = exp(u + 1) when xi2 = 1, where u ~ N(0,1); however, for heter2, σ(i) = 1 for xi2 = 0 and σ(i) = 4 for xi2 = 1. Four different (β and 10,000 simulated data sets were used.

N(0, 1) χ2(3)−3 0.5N(2, 1) + 0.5AT(−2, 1)
Methods β n=20 n=40 n=60 n=20 n=40 n=60 n=20 n=40 n=60
t 0.0 5.12 5.22 4.67 4.72 5.32 5.12 4.86 5.18 5.20
LRaET 0.0 5.65 4.63 4.85 6.63 6.02 5.52 5.93 4.93 5.15
LRaEL 0.0 7.00 5.62 5.43 8.87 6.55 5.74 6.70 5.00 4.28
t 0.5 17.41 32.32 46.58 7.25 10.12 11.37 7.14 10.56 12.77
LRaET 0.5 19.74 34.78 48.27 8.40 11.41 13.15 7.81 9.74 13.19
LRaEL 0.5 20.55 33.44 46.99 11.96 12.48 14.16 8.50 11.22 12.66
t 0.8 35.70 66.69 84.95 10.33 17.06 24.65 10.01 17.53 25.70
LRaET 0.8 39.10 69.25 84.15 13.42 19.37 25.16 12.14 19.04 27.02
LRaEL 0.8 40.94 68.97 84.11 16.85 21.34 25.32 12.88 18.68 26.88
t 1.0 50.83 85.41 96.33 14.28 25.47 35.11 12.98 26.27 38.44
LRaET 1.0 53.38 84.87 97.03 16.60 26.75 36.92 15.31 25.03 40.34
LRaEL 1.0 57.74 86.45 95.92 20.19 26.62 35.53 16.23 26.12 39.23
t(3) heter1 heter2
Methods β n=20 n=40 n=60 n=20 n=40 n=60 n=20 n=40 n=60
t 0.0 4.16 4.40 4.58 3.60 3.93 4.67 5.43 4.88 5.40
LRaET 0.0 5.63 5.68 5.50 7.80 7.46 7.19 5.45 4.97 5.33
LRaEL 0.0 8.17 6.70 5.93 11.82 9.62 9.36 7.00 6.00 5.26
t 0.5 10.71 17.71 22.64 6.11 6.23 7.05 10.51 16.40 22.13
LRaET 0.5 14.44 19.48 25.40 9.24 10.38 9.96 12.03 16.78 24.29
LRaEL 0.5 16.32 20.54 25.36 13.22 12.76 11.50 14.46 16.82 22.08
t 0.8 20.09 36.00 48.65 8.15 9.26 11.78 17.91 33.08 46.74
LRaET 0.8 24.68 37.85 51.60 11.82 13.70 16.04 20.59 34.72 48.03
LRaEL 0.8 27.29 37.95 48.27 15.90 15.84 16.08 21.81 33.02 47.33
t 1.0 29.10 50.31 64.80 9.46 12.46 15.04 25.48 47.51 65.86
LRaET 1.0 31.72 52.02 66.59 14.54 17.20 19.24 25.53 48.42 66.10
LRaEL 1.0 36.69 52.05 64.57 18.36 18.24 19.88 31.32 48.17 64.79