Background
Methods
Statistical methods
The linear regression model
The beta regression model (and some extensions)
The fractional logit regression model
Monte Carlo simulation design
μ0
|
φ
0
| μ1
|
φ
1
| p0
| q0
| p1
| q1
| |
---|---|---|---|---|---|---|---|---|
Type-1 error experiments | 0.50 | 5 | 0.50 | 5 | 2.5 | 2.5 | 2.5 | 2.5 |
0.50 | 5 | 0.50 | 10 | 2.5 | 2.5 | 5 | 5 | |
0.50 | 100 | 0.50 | 100 | 50 | 50 | 50 | 50 | |
0.50 | 100 | 0.50 | 200 | 50 | 50 | 100 | 100 | |
0.25 | 5 | 0.25 | 5 | 1.25 | 3.75 | 1.25 | 3.75 | |
0.25 | 5 | 0.25 | 10 | 1.25 | 3.75 | 2.50 | 7.50 | |
0.25 | 100 | 0.25 | 100 | 25 | 75 | 25 | 75 | |
0.25 | 100 | 0.25 | 200 | 25 | 75 | 50 | 150 | |
0.05 | 5 | 0.05 | 5 | 0.25 | 4.75 | 0.25 | 4.75 | |
0.05 | 5 | 0.05 | 10 | 0.25 | 4.75 | 0.50 | 9.50 | |
0.05 | 100 | 0.05 | 100 | 5 | 95 | 5 | 95 | |
0.05 | 100 | 0.05 | 200 | 5 | 95 | 10 | 190 | |
Power experiments | 0.5 | 5 | 0.525 | 5 | 2.5 | 2.5 | 2.625 | 2.375 |
0.5 | 5 | 0.525 | 10 | 2.5 | 2.5 | 5.25 | 4.75 | |
0.5 | 100 | 0.525 | 100 | 50 | 50 | 52.5 | 47.5 | |
0.5 | 100 | 0.525 | 200 | 50 | 50 | 105 | 95 | |
0.25 | 5 | 0.275 | 5 | 1.25 | 3.75 | 1.375 | 3.625 | |
0.25 | 5 | 0.275 | 10 | 1.25 | 3.75 | 2.75 | 7.25 | |
0.25 | 100 | 0.275 | 100 | 25 | 75 | 27.5 | 62.5 | |
0.25 | 100 | 0.275 | 200 | 25 | 75 | 55 | 145 | |
0.05 | 5 | 0.075 | 5 | 0.25 | 4.75 | 0.375 | 4.625 | |
0.05 | 5 | 0.075 | 10 | 0.25 | 4.75 | 0.75 | 9.25 | |
0.05 | 100 | 0.075 | 100 | 5 | 95 | 7.5 | 92.5 | |
0.05 | 100 | 0.075 | 200 | 5 | 95 | 15 | 185 |
Group 0 – multinomial response probabilities | Group 1 – multinomial response probabilities | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Yi values | 0.05 | 0.15 | 0.25 | 0.35 | 0.45 | 0.55 | 0.65 | 0.75 | 0.85 | 0.95 | 0.05 | 0.15 | 0.25 | 0.35 | 0.45 | 0.55 | 0.65 | 0.75 | 0.85 | 0.95 |
Type-1 error experiments | 0.00 | 0.025 | 0.025 | 0.15 | 0.30 | 0.30 | 0.15 | 0.025 | 0.025 | 0.00 | 0.00 | 0.025 | 0.025 | 0.15 | 0.30 | 0.30 | 0.15 | 0.025 | 0.025 | 0.00 |
0.40 | 0.20 | 0.10 | 0.10 | 0.10 | 0.05 | 0.05 | 0.00 | 0.00 | 0.00 | 0.40 | 0.20 | 0.10 | 0.10 | 0.10 | 0.05 | 0.05 | 0.00 | 0.00 | 0.00 | |
Power experiments | 0.00 | 0.025 | 0.025 | 0.15 | 0.30 | 0.30 | 0.15 | 0.025 | 0.025 | 0.00 | 0.00 | 0.00 | 0.025 | 0.025 | 0.15 | 0.30 | 0.30 | 0.15 | 0.025 | 0.025 |
0.40 | 0.20 | 0.10 | 0.10 | 0.10 | 0.05 | 0.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.40 | 0.20 | 0.10 | 0.10 | 0.10 | 0.05 | 0.05 | 0.00 | 0.00 |
Statistical software
Results
Linear regression model | Beta regression model | Variable dispersion beta regression model | Fractional logit regression model | |||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
N0 = N1 | μ0
|
φ
0
| μ1
|
φ
1
| Bias | MC error bias | Variance | MC error variance | Type-1 Error | MC Error Type-1 Error | Bias | MC error bias | Variance | MC error variance | Type-1 Error | MC Error Type-1 Error | Bias | MC error bias | Variance | MC error variance | Type-1 Error | MC Error Type-1 Error | Bias | MC error bias | Variance | MC error variance | Type-1 Error | MC Error Type-1 Error |
25 | 0.5 | 5 | 0.5 | 5 | -4.18E-04 | 4.10E-04 | 3.34E-03 | 3.35E-05 | 0.050 | 0.002 | -4.52E-04 | 4.06E-04 | 3.11E-03 | 2.99E-05 | 0.063 | 0.002 | -4.57E-04 | 4.07E-04 | 3.09E-03 | 2.97E-05 | 0.064 | 0.002 | -4.18E-04 | 4.10E-04 | 3.20E-03 | 3.29E-05 | 0.061 | 0.002 |
100 | 0.5 | 5 | 0.5 | 5 | 2.07E-04 | 2.05E-04 | 8.33E-04 | 8.15E-06 | 0.050 | 0.002 | 1.65E-04 | 2.03E-04 | 8.06E-04 | 7.41E-06 | 0.052 | 0.002 | 1.61E-04 | 2.03E-04 | 8.05E-04 | 7.40E-06 | 0.053 | 0.002 | 2.07E-04 | 2.05E-04 | 8.25E-04 | 8.11E-06 | 0.053 | 0.002 |
250 | 0.5 | 5 | 0.5 | 5 | -1.72E-04 | 1.29E-04 | 3.33E-04 | 3.24E-06 | 0.050 | 0.002 | -1.57E-04 | 1.27E-04 | 3.25E-04 | 2.95E-06 | 0.052 | 0.002 | -1.61E-04 | 1.28E-04 | 3.24E-04 | 2.94E-06 | 0.052 | 0.002 | -1.72E-04 | 1.29E-04 | 3.32E-04 | 3.23E-06 | 0.051 | 0.002 |
750 | 0.5 | 5 | 0.5 | 5 | 2.88E-05 | 7.45E-05 | 1.11E-04 | 1.08E-06 | 0.050 | 0.002 | 1.20E-05 | 7.35E-05 | 1.09E-04 | 9.87E-07 | 0.049 | 0.002 | 1.15E-05 | 7.35E-05 | 1.09E-04 | 9.87E-07 | 0.049 | 0.002 | 2.88E-05 | 7.45E-05 | 1.11E-04 | 1.08E-06 | 0.050 | 0.002 |
25 | 0.5 | 100 | 0.5 | 100 | 3.16E-05 | 9.98E-05 | 1.98E-04 | 1.00E-05 | 0.051 | 0.002 | 3.25E-05 | 9.98E-05 | 1.90E-04 | 9.81E-06 | 0.062 | 0.002 | 3.24E-05 | 9.98E-05 | 1.90E-04 | 9.80E-06 | 0.062 | 0.002 | 3.16E-05 | 9.98E-05 | 1.91E-04 | 9.82E-06 | 0.061 | 0.002 |
100 | 0.5 | 100 | 0.5 | 100 | -1.44E-05 | 4.99E-05 | 4.95E-05 | 2.44E-06 | 0.051 | 0.002 | -1.38E-05 | 4.99E-05 | 4.90E-05 | 2.43E-06 | 0.054 | 0.002 | -1.38E-05 | 4.99E-05 | 4.90E-05 | 2.43E-06 | 0.054 | 0.002 | -1.44E-05 | 4.99E-05 | 4.90E-05 | 2.43E-06 | 0.053 | 0.002 |
250 | 0.5 | 100 | 0.5 | 100 | 6.80E-06 | 3.13E-05 | 1.98E-05 | 1.01E-06 | 0.049 | 0.002 | 6.67E-06 | 3.12E-05 | 1.97E-05 | 1.01E-06 | 0.049 | 0.002 | 6.67E-06 | 3.12E-05 | 1.97E-05 | 1.01E-06 | 0.049 | 0.002 | 6.80E-06 | 3.13E-05 | 1.97E-05 | 1.01E-06 | 0.049 | 0.002 |
750 | 0.5 | 100 | 0.5 | 100 | -1.67E-06 | 1.82E-05 | 6.66E-06 | 6.47E-07 | 0.052 | 0.002 | -1.62E-06 | 1.82E-05 | 6.65E-06 | 6.55E-07 | 0.053 | 0.002 | -1.62E-06 | 1.82E-05 | 6.65E-06 | 6.55E-07 | 0.053 | 0.002 | -1.67E-06 | 1.82E-05 | 6.65E-06 | 6.54E-07 | 0.053 | 0.002 |
25 | 0.25 | 5 | 0.25 | 5 | -2.08E-04 | 3.52E-04 | 2.50E-03 | 3.74E-05 | 0.049 | 0.002 | -7.51E-05 | 3.27E-04 | 2.05E-03 | 3.10E-05 | 0.062 | 0.002 | -1.59E-04 | 3.48E-04 | 2.33E-03 | 3.25E-05 | 0.060 | 0.002 | -2.08E-04 | 3.52E-04 | 2.40E-03 | 3.67E-05 | 0.060 | 0.002 |
100 | 0.25 | 5 | 0.25 | 5 | 2.77E-04 | 1.77E-04 | 6.26E-04 | 9.09E-06 | 0.050 | 0.002 | 2.83E-04 | 1.64E-04 | 5.28E-04 | 7.71E-06 | 0.053 | 0.002 | 3.26E-04 | 1.75E-04 | 6.06E-04 | 8.12E-06 | 0.053 | 0.002 | 2.77E-04 | 1.77E-04 | 6.20E-04 | 9.04E-06 | 0.053 | 0.002 |
250 | 0.25 | 5 | 0.25 | 5 | 1.11E-04 | 1.12E-04 | 2.50E-04 | 3.64E-06 | 0.049 | 0.002 | 1.38E-04 | 1.03E-04 | 2.12E-04 | 3.10E-06 | 0.051 | 0.002 | 1.14E-04 | 1.10E-04 | 2.44E-04 | 3.27E-06 | 0.050 | 0.002 | 1.11E-04 | 1.12E-04 | 2.49E-04 | 3.63E-06 | 0.050 | 0.002 |
750 | 0.25 | 5 | 0.25 | 5 | 4.20E-05 | 6.44E-05 | 8.33E-05 | 1.22E-06 | 0.048 | 0.002 | 2.86E-05 | 5.94E-05 | 7.09E-05 | 1.05E-06 | 0.050 | 0.002 | 4.40E-05 | 6.38E-05 | 8.15E-05 | 1.10E-06 | 0.049 | 0.002 | 4.20E-05 | 6.44E-05 | 8.32E-05 | 1.22E-06 | 0.048 | 0.002 |
25 | 0.25 | 100 | 0.25 | 100 | 2.34E-05 | 8.69E-05 | 1.48E-04 | 8.84E-06 | 0.050 | 0.002 | 1.31E-05 | 8.66E-05 | 1.41E-04 | 8.50E-06 | 0.060 | 0.002 | 2.39E-05 | 8.69E-05 | 1.42E-04 | 8.58E-06 | 0.060 | 0.002 | 2.34E-05 | 8.69E-05 | 1.42E-04 | 8.65E-06 | 0.060 | 0.002 |
100 | 0.25 | 100 | 0.25 | 100 | 9.32E-06 | 4.31E-05 | 3.71E-05 | 2.18E-06 | 0.051 | 0.002 | 1.03E-05 | 4.30E-05 | 3.65E-05 | 2.13E-06 | 0.053 | 0.002 | 9.62E-06 | 4.31E-05 | 3.67E-05 | 2.15E-06 | 0.053 | 0.002 | 9.32E-06 | 4.31E-05 | 3.67E-05 | 2.17E-06 | 0.053 | 0.002 |
250 | 0.25 | 100 | 0.25 | 100 | -7.06E-06 | 2.73E-05 | 1.49E-05 | 9.05E-07 | 0.049 | 0.002 | -8.38E-06 | 2.72E-05 | 1.47E-05 | 8.93E-07 | 0.050 | 0.002 | -7.25E-06 | 2.73E-05 | 1.48E-05 | 9.01E-07 | 0.049 | 0.002 | -7.06E-06 | 2.73E-05 | 1.48E-05 | 9.05E-07 | 0.050 | 0.002 |
750 | 0.25 | 100 | 0.25 | 100 | 1.44E-05 | 1.58E-05 | 5.00E-06 | 1.36E-07 | 0.049 | 0.002 | 1.30E-05 | 1.58E-05 | 4.99E-06 | 1.63E-07 | 0.051 | 0.002 | 1.44E-05 | 1.58E-05 | 4.99E-06 | 1.37E-07 | 0.050 | 0.002 | 1.44E-05 | 1.58E-05 | 4.99E-06 | 1.39E-07 | 0.049 | 0.002 |
25 | 0.05 | 5 | 0.05 | 5 | 3.45E-04 | 1.78E-04 | 6.35E-04 | 4.42E-05 | 0.042 | 0.001 | 1.05E-04 | 9.94E-05 | 1.96E-04 | 2.44E-05 | 0.040 | 0.001 | 3.40E-04 | 1.73E-04 | 6.24E-04 | 3.82E-05 | 0.036 | 0.001 | 3.45E-04 | 1.78E-04 | 6.09E-04 | 4.33E-05 | 0.054 | 0.002 |
100 | 0.05 | 5 | 0.05 | 5 | -7.35E-05 | 8.93E-05 | 1.59E-04 | 1.10E-05 | 0.051 | 0.002 | -3.62E-05 | 4.83E-05 | 4.68E-05 | 5.98E-06 | 0.048 | 0.002 | -6.70E-05 | 8.80E-05 | 1.56E-04 | 9.75E-06 | 0.048 | 0.002 | -7.35E-05 | 8.93E-05 | 1.57E-04 | 1.09E-05 | 0.053 | 0.002 |
250 | 0.05 | 5 | 0.05 | 5 | 2.00E-05 | 5.58E-05 | 6.32E-05 | 4.40E-06 | 0.049 | 0.002 | 4.10E-05 | 3.03E-05 | 1.84E-05 | 2.38E-06 | 0.050 | 0.002 | 1.63E-05 | 5.53E-05 | 6.21E-05 | 3.92E-06 | 0.048 | 0.002 | 2.00E-05 | 5.58E-05 | 6.29E-05 | 4.39E-06 | 0.050 | 0.002 |
750 | 0.05 | 5 | 0.05 | 5 | 6.80E-06 | 3.25E-05 | 2.11E-05 | 1.49E-06 | 0.050 | 0.002 | -6.04E-06 | 1.75E-05 | 6.13E-06 | 8.98E-07 | 0.050 | 0.002 | 6.78E-06 | 3.21E-05 | 2.07E-05 | 1.34E-06 | 0.050 | 0.002 | 6.80E-06 | 3.25E-05 | 2.11E-05 | 1.49E-06 | 0.050 | 0.002 |
25 | 0.05 | 100 | 0.05 | 100 | -4.70E-05 | 4.37E-05 | 3.76E-05 | 5.33E-06 | 0.050 | 0.002 | -5.81E-05 | 4.19E-05 | 3.32E-05 | 4.40E-06 | 0.062 | 0.002 | -4.70E-05 | 4.36E-05 | 3.64E-05 | 4.92E-06 | 0.059 | 0.002 | -4.70E-05 | 4.37E-05 | 3.61E-05 | 5.22E-06 | 0.060 | 0.002 |
100 | 0.05 | 100 | 0.05 | 100 | 2.39E-05 | 2.16E-05 | 9.41E-06 | 1.36E-06 | 0.049 | 0.002 | 2.17E-05 | 2.07E-05 | 8.54E-06 | 1.14E-06 | 0.052 | 0.002 | 2.38E-05 | 2.16E-05 | 9.32E-06 | 1.26E-06 | 0.050 | 0.002 | 2.39E-05 | 2.16E-05 | 9.31E-06 | 1.35E-06 | 0.052 | 0.002 |
250 | 0.05 | 100 | 0.05 | 100 | -3.24E-05 | 1.37E-05 | 3.82E-06 | 7.23E-07 | 0.050 | 0.002 | -3.21E-05 | 1.31E-05 | 3.38E-06 | 9.34E-07 | 0.050 | 0.002 | -3.24E-05 | 1.37E-05 | 3.83E-06 | 6.98E-07 | 0.051 | 0.002 | -3.24E-05 | 1.37E-05 | 3.81E-06 | 7.43E-07 | 0.051 | 0.002 |
750 | 0.05 | 100 | 0.05 | 100 | -2.60E-06 | 7.93E-06 | 1.00E-06 | 0.00E + 00 | 0.052 | 0.002 | -3.87E-06 | 7.60E-06 | 1.00E-06 | 0.00E + 00 | 0.052 | 0.002 | -2.63E-06 | 7.93E-06 | 1.00E-06 | 0.00E + 00 | 0.052 | 0.002 | -2.60E-06 | 7.93E-06 | 1.00E-06 | 0.00E + 00 | 0.052 | 0.002 |
25 | 0.5 | 5 | 0.5 | 10 | -2.22E-04 | 3.59E-04 | 2.57E-03 | 3.12E-05 | 0.052 | 0.002 | -2.37E-04 | 3.62E-04 | 2.46E-03 | 3.01E-05 | 0.065 | 0.002 | -2.24E-04 | 3.58E-04 | 2.40E-03 | 2.81E-05 | 0.064 | 0.002 | -2.22E-04 | 3.59E-04 | 2.47E-03 | 3.06E-05 | 0.062 | 0.002 |
100 | 0.5 | 5 | 0.5 | 10 | -1.92E-04 | 1.80E-04 | 6.44E-04 | 7.71E-06 | 0.051 | 0.002 | -2.03E-04 | 1.82E-04 | 6.40E-04 | 7.59E-06 | 0.055 | 0.002 | -2.09E-04 | 1.79E-04 | 6.26E-04 | 7.12E-06 | 0.054 | 0.002 | -1.92E-04 | 1.80E-04 | 6.38E-04 | 7.68E-06 | 0.054 | 0.002 |
250 | 0.5 | 5 | 0.5 | 10 | -9.90E-05 | 1.13E-04 | 2.57E-04 | 3.04E-06 | 0.050 | 0.002 | -7.63E-05 | 1.14E-04 | 2.58E-04 | 3.00E-06 | 0.054 | 0.002 | -6.83E-05 | 1.12E-04 | 2.52E-04 | 2.81E-06 | 0.052 | 0.002 | -9.90E-05 | 1.13E-04 | 2.56E-04 | 3.03E-06 | 0.051 | 0.002 |
750 | 0.5 | 5 | 0.5 | 10 | 2.83E-05 | 6.59E-05 | 8.58E-05 | 1.02E-06 | 0.052 | 0.002 | 4.12E-05 | 6.65E-05 | 8.62E-05 | 1.01E-06 | 0.053 | 0.002 | 3.14E-05 | 6.54E-05 | 8.43E-05 | 9.47E-07 | 0.053 | 0.002 | 2.83E-05 | 6.59E-05 | 8.57E-05 | 1.02E-06 | 0.053 | 0.002 |
25 | 0.5 | 100 | 0.5 | 200 | -2.30E-05 | 8.62E-05 | 1.49E-04 | 9.17E-06 | 0.051 | 0.002 | -2.33E-05 | 8.63E-05 | 1.43E-04 | 9.01E-06 | 0.061 | 0.002 | -2.36E-05 | 8.62E-05 | 1.43E-04 | 8.97E-06 | 0.061 | 0.002 | -2.30E-05 | 8.62E-05 | 1.43E-04 | 8.99E-06 | 0.061 | 0.002 |
100 | 0.5 | 100 | 0.5 | 200 | 9.13E-05 | 4.29E-05 | 3.72E-05 | 2.25E-06 | 0.046 | 0.001 | 9.09E-05 | 4.29E-05 | 3.68E-05 | 2.25E-06 | 0.048 | 0.002 | 9.09E-05 | 4.29E-05 | 3.68E-05 | 2.24E-06 | 0.048 | 0.002 | 9.13E-05 | 4.29E-05 | 3.68E-05 | 2.24E-06 | 0.048 | 0.002 |
250 | 0.5 | 100 | 0.5 | 200 | -4.36E-05 | 2.74E-05 | 1.49E-05 | 9.42E-07 | 0.051 | 0.002 | -4.35E-05 | 2.74E-05 | 1.48E-05 | 9.43E-07 | 0.052 | 0.002 | -4.34E-05 | 2.74E-05 | 1.48E-05 | 9.39E-07 | 0.052 | 0.002 | -4.36E-05 | 2.74E-05 | 1.48E-05 | 9.41E-07 | 0.052 | 0.002 |
750 | 0.5 | 100 | 0.5 | 200 | 1.09E-06 | 1.58E-05 | 5.00E-06 | 1.47E-07 | 0.052 | 0.002 | 8.86E-07 | 1.58E-05 | 5.00E-06 | 1.48E-07 | 0.052 | 0.002 | 8.93E-07 | 1.58E-05 | 5.00E-06 | 1.49E-07 | 0.052 | 0.002 | 1.09E-06 | 1.58E-05 | 5.00E-06 | 1.49E-07 | 0.052 | 0.002 |
25 | 0.25 | 5 | 0.25 | 10 | -3.68E-04 | 3.11E-04 | 1.93E-03 | 3.36E-05 | 0.050 | 0.002 | 2.27E-02 | 2.96E-04 | 1.67E-03 | 2.86E-05 | 0.102 | 0.002 | -1.82E-05 | 3.09E-04 | 1.81E-03 | 2.96E-05 | 0.062 | 0.002 | -3.68E-04 | 3.11E-04 | 1.85E-03 | 3.29E-05 | 0.061 | 0.002 |
100 | 0.25 | 5 | 0.25 | 10 | -3.66E-04 | 1.55E-04 | 4.83E-04 | 8.36E-06 | 0.050 | 0.002 | 2.35E-02 | 1.47E-04 | 4.31E-04 | 7.22E-06 | 0.211 | 0.003 | -2.87E-04 | 1.54E-04 | 4.70E-04 | 7.50E-06 | 0.053 | 0.002 | -3.66E-04 | 1.55E-04 | 4.78E-04 | 8.31E-06 | 0.052 | 0.002 |
250 | 0.25 | 5 | 0.25 | 10 | 9.95E-05 | 9.78E-05 | 1.93E-04 | 3.35E-06 | 0.049 | 0.002 | 2.41E-02 | 9.22E-05 | 1.73E-04 | 2.91E-06 | 0.456 | 0.004 | 1.48E-04 | 9.69E-05 | 1.89E-04 | 3.03E-06 | 0.050 | 0.002 | 9.95E-05 | 9.78E-05 | 1.92E-04 | 3.34E-06 | 0.050 | 0.002 |
750 | 0.25 | 5 | 0.25 | 10 | -9.18E-06 | 5.71E-05 | 6.44E-05 | 1.11E-06 | 0.053 | 0.002 | 2.40E-02 | 5.38E-05 | 5.79E-05 | 9.70E-07 | 0.884 | 0.002 | 1.60E-05 | 5.66E-05 | 6.33E-05 | 1.01E-06 | 0.053 | 0.002 | -9.18E-06 | 5.71E-05 | 6.43E-05 | 1.11E-06 | 0.054 | 0.002 |
25 | 0.25 | 100 | 0.25 | 200 | 2.38E-05 | 7.50E-05 | 1.12E-04 | 8.09E-06 | 0.050 | 0.002 | 1.22E-03 | 7.48E-05 | 1.07E-04 | 7.81E-06 | 0.062 | 0.002 | 2.42E-05 | 7.50E-05 | 1.07E-04 | 7.85E-06 | 0.061 | 0.002 | 2.38E-05 | 7.50E-05 | 1.07E-04 | 7.93E-06 | 0.060 | 0.002 |
100 | 0.25 | 100 | 0.25 | 200 | -8.89E-06 | 3.76E-05 | 2.79E-05 | 1.99E-06 | 0.051 | 0.002 | 1.24E-03 | 3.75E-05 | 2.75E-05 | 1.96E-06 | 0.059 | 0.002 | -8.41E-06 | 3.76E-05 | 2.76E-05 | 1.97E-06 | 0.054 | 0.002 | -8.89E-06 | 3.76E-05 | 2.76E-05 | 1.98E-06 | 0.054 | 0.002 |
250 | 0.25 | 100 | 0.25 | 200 | -1.15E-05 | 2.38E-05 | 1.12E-05 | 8.52E-07 | 0.051 | 0.002 | 1.23E-03 | 2.37E-05 | 1.11E-05 | 8.41E-07 | 0.069 | 0.002 | -1.16E-05 | 2.38E-05 | 1.11E-05 | 8.47E-07 | 0.052 | 0.002 | -1.15E-05 | 2.38E-05 | 1.11E-05 | 8.50E-07 | 0.052 | 0.002 |
750 | 0.25 | 100 | 0.25 | 200 | 6.24E-06 | 1.36E-05 | 3.94E-06 | 4.26E-07 | 0.050 | 0.002 | 1.26E-03 | 1.36E-05 | 3.92E-06 | 4.77E-07 | 0.099 | 0.002 | 6.41E-06 | 1.36E-05 | 3.94E-06 | 4.36E-07 | 0.050 | 0.002 | 6.24E-06 | 1.36E-05 | 3.93E-06 | 4.41E-07 | 0.050 | 0.002 |
25 | 0.05 | 5 | 0.05 | 10 | 5.41E-05 | 1.54E-04 | 4.90E-04 | 3.90E-05 | 0.051 | 0.002 | 2.16E-02 | 9.12E-05 | 1.90E-04 | 2.11E-05 | 0.374 | 0.003 | 5.03E-04 | 1.51E-04 | 4.85E-04 | 3.38E-05 | 0.047 | 0.001 | 5.41E-05 | 1.54E-04 | 4.70E-04 | 3.82E-05 | 0.061 | 0.002 |
100 | 0.05 | 5 | 0.05 | 10 | -6.01E-05 | 7.71E-05 | 1.22E-04 | 9.72E-06 | 0.049 | 0.002 | 2.19E-02 | 4.47E-05 | 4.62E-05 | 5.19E-06 | 0.920 | 0.002 | 5.21E-05 | 7.64E-05 | 1.21E-04 | 8.56E-06 | 0.048 | 0.002 | -6.01E-05 | 7.71E-05 | 1.21E-04 | 9.67E-06 | 0.051 | 0.002 |
250 | 0.05 | 5 | 0.05 | 10 | -5.42E-05 | 4.93E-05 | 4.90E-05 | 3.86E-06 | 0.049 | 0.002 | 2.22E-02 | 2.84E-05 | 1.84E-05 | 2.08E-06 | 1.000 | 0.000 | -7.90E-07 | 4.90E-05 | 4.83E-05 | 3.43E-06 | 0.049 | 0.002 | -5.42E-05 | 4.93E-05 | 4.88E-05 | 3.85E-06 | 0.050 | 0.002 |
750 | 0.05 | 5 | 0.05 | 10 | 5.39E-05 | 2.83E-05 | 1.63E-05 | 1.31E-06 | 0.049 | 0.002 | 2.22E-02 | 1.63E-05 | 6.13E-06 | 7.84E-07 | 1.000 | 0.000 | 6.76E-05 | 2.81E-05 | 1.61E-05 | 1.17E-06 | 0.049 | 0.002 | 5.39E-05 | 2.83E-05 | 1.63E-05 | 1.31E-06 | 0.049 | 0.002 |
25 | 0.05 | 100 | 0.05 | 200 | -1.58E-06 | 3.76E-05 | 2.82E-05 | 4.79E-06 | 0.054 | 0.002 | 2.09E-03 | 3.63E-05 | 2.54E-05 | 3.98E-06 | 0.082 | 0.002 | -6.31E-07 | 3.76E-05 | 2.72E-05 | 4.43E-06 | 0.062 | 0.002 | -1.58E-06 | 3.76E-05 | 2.71E-05 | 4.69E-06 | 0.064 | 0.002 |
100 | 0.05 | 100 | 0.05 | 200 | 1.51E-05 | 1.90E-05 | 7.07E-06 | 1.24E-06 | 0.052 | 0.002 | 2.19E-03 | 1.83E-05 | 6.56E-06 | 1.07E-06 | 0.144 | 0.002 | 1.55E-05 | 1.90E-05 | 7.01E-06 | 1.16E-06 | 0.054 | 0.002 | 1.51E-05 | 1.90E-05 | 6.99E-06 | 1.23E-06 | 0.056 | 0.002 |
250 | 0.05 | 100 | 0.05 | 200 | -7.23E-06 | 1.19E-05 | 2.94E-06 | 5.33E-07 | 0.051 | 0.002 | 2.19E-03 | 1.15E-05 | 2.77E-06 | 8.89E-07 | 0.272 | 0.003 | -6.93E-06 | 1.19E-05 | 2.94E-06 | 4.99E-07 | 0.052 | 0.002 | -7.23E-06 | 1.19E-05 | 2.93E-06 | 5.56E-07 | 0.052 | 0.002 |
750 | 0.05 | 100 | 0.05 | 200 | -4.05E-06 | 6.85E-06 | 1.00E-06 | 0.00E + 00 | 0.049 | 0.002 | 2.19E-03 | 6.60E-06 | 1.00E-06 | 0.00E + 00 | 0.647 | 0.003 | -4.04E-06 | 6.85E-06 | 1.00E-06 | 0.00E + 00 | 0.050 | 0.002 | -4.05E-06 | 6.85E-06 | 1.00E-06 | 0.00E + 00 | 0.050 | 0.002 |
Linear regression model | Beta regression model | Variable dispersion beta regression model | Fractional logit regression model | |||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
N0 = N1 | μ0
| φ0
| μ1
| φ1
| Bias | MC error bias | Variance | MC error variance | Power | MC error power | Bias | MC error bias | Variance | MC error variance | Power | MC error power | Bias | MC error bias | Variance | MC error variance | Power | MC error power | Bias | MC error bias | Variance | MC error variance | Power | MC error power |
25 | 0.5 | 5 | 0.5 | 5 | 4.57E-04 | 4.07E-04 | 3.32E-03 | 3.34E-05 | 0.071 | 1.82E-03 | 4.27E-04 | 4.03E-04 | 3.09E-03 | 2.98E-05 | 0.088 | 2.00E-03 | 4.80E-04 | 4.04E-04 | 3.08E-03 | 2.96E-05 | 0.090 | 2.02E-03 | 4.57E-04 | 4.07E-04 | 3.19E-03 | 3.27E-05 | 0.085 | 1.97E-03 |
100 | 0.5 | 5 | 0.5 | 5 | 1.21E-04 | 2.02E-04 | 8.33E-04 | 8.17E-06 | 0.138 | 2.44E-03 | 1.35E-04 | 2.00E-04 | 8.05E-04 | 7.41E-06 | 0.145 | 2.49E-03 | 1.41E-04 | 2.00E-04 | 8.04E-04 | 7.40E-06 | 0.146 | 2.50E-03 | 1.21E-04 | 2.02E-04 | 8.24E-04 | 8.13E-06 | 0.142 | 2.47E-03 |
250 | 0.5 | 5 | 0.5 | 5 | -3.53E-05 | 1.29E-04 | 3.33E-04 | 3.23E-06 | 0.276 | 3.16E-03 | -3.84E-05 | 1.27E-04 | 3.24E-04 | 2.94E-06 | 0.284 | 3.19E-03 | -3.46E-05 | 1.27E-04 | 3.24E-04 | 2.94E-06 | 0.284 | 3.19E-03 | -3.53E-05 | 1.29E-04 | 3.32E-04 | 3.23E-06 | 0.279 | 3.17E-03 |
750 | 0.5 | 5 | 0.5 | 5 | 2.32E-05 | 7.42E-05 | 1.11E-04 | 1.09E-06 | 0.659 | 3.35E-03 | 1.90E-05 | 7.34E-05 | 1.08E-04 | 9.95E-07 | 0.670 | 3.32E-03 | 2.07E-05 | 7.34E-05 | 1.08E-04 | 9.95E-07 | 0.671 | 3.32E-03 | 2.32E-05 | 7.42E-05 | 1.11E-04 | 1.09E-06 | 0.660 | 3.35E-03 |
25 | 0.5 | 100 | 0.5 | 100 | 1.51E-04 | 9.91E-05 | 1.98E-04 | 1.00E-05 | 0.416 | 3.48E-03 | 1.49E-04 | 9.91E-05 | 1.90E-04 | 9.80E-06 | 0.451 | 3.52E-03 | 1.49E-04 | 9.91E-05 | 1.90E-04 | 9.80E-06 | 0.451 | 3.52E-03 | 1.51E-04 | 9.91E-05 | 1.90E-04 | 9.82E-06 | 0.451 | 3.52E-03 |
100 | 0.5 | 100 | 0.5 | 100 | 1.54E-05 | 5.00E-05 | 4.95E-05 | 2.45E-06 | 0.942 | 1.66E-03 | 1.58E-05 | 4.99E-05 | 4.90E-05 | 2.44E-06 | 0.944 | 1.62E-03 | 1.57E-05 | 5.00E-05 | 4.90E-05 | 2.44E-06 | 0.944 | 1.62E-03 | 1.54E-05 | 5.00E-05 | 4.90E-05 | 2.44E-06 | 0.944 | 1.62E-03 |
250 | 0.5 | 100 | 0.5 | 100 | -4.13E-05 | 3.14E-05 | 1.98E-05 | 1.00E-06 | 1.000 | 8.66E-05 | -4.11E-05 | 3.14E-05 | 1.97E-05 | 9.98E-07 | 1.000 | 8.66E-05 | -4.11E-05 | 3.14E-05 | 1.97E-05 | 9.98E-07 | 1.000 | 8.66E-05 | -4.13E-05 | 3.14E-05 | 1.97E-05 | 9.97E-07 | 1.000 | 8.66E-05 |
750 | 0.5 | 100 | 0.5 | 100 | -5.02E-06 | 1.82E-05 | 6.65E-06 | 6.55E-07 | 1.000 | 0.00E + 00 | -4.95E-06 | 1.82E-05 | 6.63E-06 | 6.62E-07 | 1.000 | 0.00E + 00 | -5.02E-06 | 1.82E-05 | 6.63E-06 | 6.62E-07 | 1.000 | 0.00E + 00 | -5.02E-06 | 1.82E-05 | 6.63E-06 | 6.61E-07 | 1.000 | 0.00E + 00 |
25 | 0.25 | 5 | 0.25 | 5 | -4.33E-05 | 3.57E-04 | 2.57E-03 | 3.65E-05 | 0.076 | 1.87E-03 | -1.47E-04 | 3.34E-04 | 2.14E-03 | 3.04E-05 | 0.093 | 2.05E-03 | -1.33E-05 | 3.54E-04 | 2.40E-03 | 3.18E-05 | 0.091 | 2.03E-03 | -4.33E-05 | 3.57E-04 | 2.47E-03 | 3.58E-05 | 0.090 | 2.02E-03 |
100 | 0.25 | 5 | 0.25 | 5 | 6.89E-05 | 1.80E-04 | 6.45E-04 | 9.12E-06 | 0.166 | 2.63E-03 | 1.69E-04 | 1.68E-04 | 5.54E-04 | 7.75E-06 | 0.191 | 2.78E-03 | 1.27E-04 | 1.78E-04 | 6.24E-04 | 8.14E-06 | 0.173 | 2.68E-03 | 6.89E-05 | 1.80E-04 | 6.39E-04 | 9.08E-06 | 0.171 | 2.66E-03 |
250 | 0.25 | 5 | 0.25 | 5 | 2.85E-05 | 1.13E-04 | 2.58E-04 | 3.58E-06 | 0.340 | 3.35E-03 | -2.92E-05 | 1.05E-04 | 2.23E-04 | 3.07E-06 | 0.387 | 3.44E-03 | 2.63E-06 | 1.12E-04 | 2.51E-04 | 3.21E-06 | 0.349 | 3.37E-03 | 2.85E-05 | 1.13E-04 | 2.57E-04 | 3.57E-06 | 0.343 | 3.36E-03 |
750 | 0.25 | 5 | 0.25 | 5 | -4.56E-05 | 6.57E-05 | 8.60E-05 | 1.21E-06 | 0.768 | 2.98E-03 | -6.05E-05 | 6.13E-05 | 7.44E-05 | 1.04E-06 | 0.826 | 2.68E-03 | -5.12E-05 | 6.50E-05 | 8.41E-05 | 1.09E-06 | 0.777 | 2.94E-03 | -4.56E-05 | 6.57E-05 | 8.59E-05 | 1.21E-06 | 0.769 | 2.98E-03 |
25 | 0.25 | 100 | 0.25 | 100 | 2.72E-04 | 8.70E-05 | 1.53E-04 | 9.03E-06 | 0.516 | 3.53E-03 | 2.80E-04 | 8.67E-05 | 1.46E-04 | 8.69E-06 | 0.555 | 3.51E-03 | 2.73E-04 | 8.70E-05 | 1.47E-04 | 8.76E-06 | 0.552 | 3.52E-03 | 2.72E-04 | 8.70E-05 | 1.47E-04 | 8.85E-06 | 0.551 | 3.52E-03 |
100 | 0.25 | 100 | 0.25 | 100 | 3.28E-05 | 4.39E-05 | 3.83E-05 | 2.19E-06 | 0.981 | 9.65E-04 | 3.00E-05 | 4.38E-05 | 3.77E-05 | 2.15E-06 | 0.982 | 9.31E-04 | 3.24E-05 | 4.39E-05 | 3.80E-05 | 2.17E-06 | 0.982 | 9.43E-04 | 3.28E-05 | 4.39E-05 | 3.80E-05 | 2.18E-06 | 0.982 | 9.43E-04 |
250 | 0.25 | 100 | 0.25 | 100 | -6.18E-06 | 2.78E-05 | 1.53E-05 | 9.19E-07 | 1.000 | 0.00E + 00 | -5.28E-06 | 2.77E-05 | 1.52E-05 | 9.03E-07 | 1.000 | 0.00E + 00 | -6.03E-06 | 2.78E-05 | 1.53E-05 | 9.12E-07 | 1.000 | 0.00E + 00 | -6.18E-06 | 2.78E-05 | 1.53E-05 | 9.20E-07 | 1.000 | 0.00E + 00 |
750 | 0.25 | 100 | 0.25 | 100 | 2.44E-05 | 1.60E-05 | 5.02E-06 | 2.17E-07 | 1.000 | 0.00E + 00 | 2.49E-05 | 1.60E-05 | 5.01E-06 | 1.65E-07 | 1.000 | 0.00E + 00 | 2.43E-05 | 1.60E-05 | 5.01E-06 | 1.99E-07 | 1.000 | 0.00E + 00 | 2.44E-05 | 1.60E-05 | 5.02E-06 | 2.06E-07 | 1.000 | 0.00E + 00 |
25 | 0.05 | 5 | 0.05 | 5 | 3.18E-06 | 1.97E-04 | 7.78E-04 | 4.36E-05 | 0.147 | 2.50E-03 | -3.96E-05 | 1.22E-04 | 2.99E-04 | 2.63E-05 | 0.302 | 3.25E-03 | 5.81E-05 | 1.94E-04 | 7.58E-04 | 3.78E-05 | 0.152 | 2.54E-03 | 3.18E-06 | 1.97E-04 | 7.47E-04 | 4.27E-05 | 0.170 | 2.65E-03 |
100 | 0.05 | 5 | 0.05 | 5 | -1.26E-04 | 9.84E-05 | 1.95E-04 | 1.08E-05 | 0.438 | 3.51E-03 | -9.09E-05 | 6.01E-05 | 7.28E-05 | 6.49E-06 | 0.855 | 2.49E-03 | -1.31E-04 | 9.72E-05 | 1.91E-04 | 9.55E-06 | 0.447 | 3.52E-03 | -1.26E-04 | 9.84E-05 | 1.93E-04 | 1.08E-05 | 0.447 | 3.52E-03 |
250 | 0.05 | 5 | 0.05 | 5 | 5.34E-05 | 6.24E-05 | 7.80E-05 | 4.33E-06 | 0.812 | 2.76E-03 | 4.24E-05 | 3.81E-05 | 2.90E-05 | 2.62E-06 | 0.998 | 3.46E-04 | 5.54E-05 | 6.18E-05 | 7.65E-05 | 3.86E-06 | 0.821 | 2.71E-03 | 5.34E-05 | 6.24E-05 | 7.77E-05 | 4.33E-06 | 0.814 | 2.75E-03 |
750 | 0.05 | 5 | 0.05 | 5 | -1.36E-06 | 3.61E-05 | 2.59E-05 | 1.47E-06 | 0.999 | 2.74E-04 | -1.36E-05 | 2.19E-05 | 9.63E-06 | 9.35E-07 | 1.000 | 0.00E + 00 | -1.56E-06 | 3.57E-05 | 2.55E-05 | 1.31E-06 | 0.999 | 2.55E-04 | -1.36E-06 | 3.61E-05 | 2.59E-05 | 1.47E-06 | 0.999 | 2.69E-04 |
25 | 0.05 | 100 | 0.05 | 100 | 4.94E-05 | 4.79E-05 | 4.62E-05 | 5.71E-06 | 0.952 | 1.51E-03 | 5.38E-05 | 4.64E-05 | 4.16E-05 | 4.81E-06 | 0.970 | 1.20E-03 | 4.93E-05 | 4.79E-05 | 4.45E-05 | 5.32E-06 | 0.960 | 1.38E-03 | 4.94E-05 | 4.79E-05 | 4.43E-05 | 5.59E-06 | 0.960 | 1.39E-03 |
100 | 0.05 | 100 | 0.05 | 100 | 2.09E-06 | 2.41E-05 | 1.16E-05 | 1.45E-06 | 1.000 | 0.00E + 00 | 3.33E-06 | 2.33E-05 | 1.07E-05 | 1.25E-06 | 1.000 | 0.00E + 00 | 2.01E-06 | 2.41E-05 | 1.15E-05 | 1.36E-06 | 1.000 | 0.00E + 00 | 2.09E-06 | 2.41E-05 | 1.15E-05 | 1.44E-06 | 1.000 | 0.00E + 00 |
250 | 0.05 | 100 | 0.05 | 100 | -2.59E-05 | 1.51E-05 | 4.64E-06 | 8.17E-07 | 1.000 | 0.00E + 00 | -2.90E-05 | 1.46E-05 | 4.25E-06 | 7.46E-07 | 1.000 | 0.00E + 00 | -2.60E-05 | 1.51E-05 | 4.63E-06 | 8.10E-07 | 1.000 | 0.00E + 00 | -2.59E-05 | 1.51E-05 | 4.62E-06 | 8.25E-07 | 1.000 | 0.00E + 00 |
750 | 0.05 | 100 | 0.05 | 100 | 1.35E-06 | 8.74E-06 | 1.74E-06 | 1.18E-06 | 1.000 | 0.00E + 00 | 7.73E-07 | 8.46E-06 | 1.14E-06 | 1.16E-06 | 1.000 | 0.00E + 00 | 1.27E-06 | 8.74E-06 | 1.74E-06 | 1.18E-06 | 1.000 | 0.00E + 00 | 1.35E-06 | 8.74E-06 | 1.73E-06 | 1.20E-06 | 1.000 | 0.00E + 00 |
25 | 0.5 | 5 | 0.5 | 10 | -2.57E-04 | 3.56E-04 | 2.58E-03 | 3.14E-05 | 0.075 | 1.86E-03 | -1.42E-03 | 3.59E-04 | 2.47E-03 | 3.03E-05 | 0.089 | 2.01E-03 | -1.99E-04 | 3.54E-04 | 2.40E-03 | 2.83E-05 | 0.091 | 2.04E-03 | -2.57E-04 | 3.56E-04 | 2.47E-03 | 3.08E-05 | 0.088 | 2.01E-03 |
100 | 0.5 | 5 | 0.5 | 10 | -3.96E-05 | 1.80E-04 | 6.43E-04 | 7.72E-06 | 0.165 | 2.62E-03 | -1.34E-03 | 1.81E-04 | 6.39E-04 | 7.60E-06 | 0.161 | 2.60E-03 | -7.90E-05 | 1.78E-04 | 6.25E-04 | 7.12E-06 | 0.173 | 2.68E-03 | -3.96E-05 | 1.80E-04 | 6.37E-04 | 7.69E-06 | 0.171 | 2.66E-03 |
250 | 0.5 | 5 | 0.5 | 10 | -4.50E-05 | 1.13E-04 | 2.57E-04 | 3.05E-06 | 0.341 | 3.35E-03 | -1.29E-03 | 1.14E-04 | 2.57E-04 | 3.01E-06 | 0.313 | 3.28E-03 | -3.11E-05 | 1.12E-04 | 2.52E-04 | 2.82E-06 | 0.348 | 3.37E-03 | -4.50E-05 | 1.13E-04 | 2.56E-04 | 3.04E-06 | 0.343 | 3.36E-03 |
750 | 0.5 | 5 | 0.5 | 10 | 4.49E-05 | 6.56E-05 | 8.58E-05 | 1.02E-06 | 0.770 | 2.97E-03 | -1.22E-03 | 6.62E-05 | 8.61E-05 | 1.01E-06 | 0.725 | 3.16E-03 | 4.43E-05 | 6.52E-05 | 8.43E-05 | 9.50E-07 | 0.777 | 2.95E-03 | 4.49E-05 | 6.56E-05 | 8.57E-05 | 1.02E-06 | 0.771 | 2.97E-03 |
25 | 0.5 | 100 | 0.5 | 200 | -8.21E-06 | 8.56E-05 | 1.49E-04 | 9.09E-06 | 0.517 | 3.53E-03 | -6.91E-05 | 8.57E-05 | 1.43E-04 | 8.93E-06 | 0.551 | 3.52E-03 | -8.83E-06 | 8.56E-05 | 1.42E-04 | 8.89E-06 | 0.554 | 3.51E-03 | -8.21E-06 | 8.56E-05 | 1.43E-04 | 8.91E-06 | 0.553 | 3.52E-03 |
100 | 0.5 | 100 | 0.5 | 200 | 2.59E-06 | 4.32E-05 | 3.72E-05 | 2.26E-06 | 0.984 | 8.95E-04 | -5.88E-05 | 4.33E-05 | 3.69E-05 | 2.26E-06 | 0.984 | 8.79E-04 | 3.25E-06 | 4.32E-05 | 3.68E-05 | 2.25E-06 | 0.985 | 8.69E-04 | 2.59E-06 | 4.32E-05 | 3.68E-05 | 2.25E-06 | 0.985 | 8.73E-04 |
250 | 0.5 | 100 | 0.5 | 200 | -4.29E-06 | 2.73E-05 | 1.49E-05 | 9.32E-07 | 1.000 | 0.00E + 00 | -6.64E-05 | 2.74E-05 | 1.48E-05 | 9.32E-07 | 1.000 | 0.00E + 00 | -4.03E-06 | 2.73E-05 | 1.48E-05 | 9.30E-07 | 1.000 | 0.00E + 00 | -4.29E-06 | 2.73E-05 | 1.48E-05 | 9.30E-07 | 1.000 | 0.00E + 00 |
750 | 0.5 | 100 | 0.5 | 200 | -1.16E-05 | 1.58E-05 | 5.00E-06 | 1.54E-07 | 1.000 | 0.00E + 00 | -7.40E-05 | 1.58E-05 | 5.00E-06 | 1.56E-07 | 1.000 | 0.00E + 00 | -1.14E-05 | 1.58E-05 | 4.99E-06 | 1.55E-07 | 1.000 | 0.00E + 00 | -1.16E-05 | 1.58E-05 | 4.99E-06 | 1.56E-07 | 1.000 | 0.00E + 00 |
25 | 0.25 | 5 | 0.25 | 10 | 3.61E-04 | 3.15E-04 | 1.97E-03 | 3.36E-05 | 0.098 | 2.10E-03 | 2.22E-02 | 2.99E-04 | 1.73E-03 | 2.89E-05 | 0.223 | 2.94E-03 | 6.24E-04 | 3.12E-04 | 1.85E-03 | 2.96E-05 | 0.116 | 2.27E-03 | 3.61E-04 | 3.15E-04 | 1.89E-03 | 3.29E-05 | 0.113 | 2.24E-03 |
100 | 0.25 | 5 | 0.25 | 10 | -1.24E-04 | 1.57E-04 | 4.94E-04 | 8.31E-06 | 0.201 | 2.83E-03 | 2.26E-02 | 1.49E-04 | 4.48E-04 | 7.27E-06 | 0.613 | 3.44E-03 | -5.32E-05 | 1.56E-04 | 4.80E-04 | 7.50E-06 | 0.211 | 2.89E-03 | -1.24E-04 | 1.57E-04 | 4.89E-04 | 8.26E-06 | 0.206 | 2.86E-03 |
250 | 0.25 | 5 | 0.25 | 10 | 2.94E-05 | 9.96E-05 | 1.97E-04 | 3.33E-06 | 0.430 | 3.50E-03 | 2.29E-02 | 9.42E-05 | 1.80E-04 | 2.90E-06 | 0.945 | 1.62E-03 | 6.90E-05 | 9.89E-05 | 1.94E-04 | 3.01E-06 | 0.439 | 3.51E-03 | 2.94E-05 | 9.96E-05 | 1.97E-04 | 3.32E-06 | 0.433 | 3.50E-03 |
750 | 0.25 | 5 | 0.25 | 10 | 1.16E-04 | 5.72E-05 | 6.59E-05 | 1.12E-06 | 0.867 | 2.40E-03 | 2.31E-02 | 5.45E-05 | 6.02E-05 | 9.81E-07 | 1.000 | 0.00E + 00 | 1.25E-04 | 5.67E-05 | 6.47E-05 | 1.01E-06 | 0.874 | 2.35E-03 | 1.16E-04 | 5.72E-05 | 6.58E-05 | 1.11E-06 | 0.867 | 2.40E-03 |
25 | 0.25 | 100 | 0.25 | 200 | -1.67E-04 | 7.55E-05 | 1.14E-04 | 8.12E-06 | 0.625 | 3.42E-03 | 9.73E-04 | 7.53E-05 | 1.10E-04 | 7.94E-06 | 0.695 | 3.26E-03 | -1.66E-04 | 7.55E-05 | 1.09E-04 | 7.89E-06 | 0.659 | 3.35E-03 | -1.67E-04 | 7.55E-05 | 1.09E-04 | 7.96E-06 | 0.658 | 3.36E-03 |
100 | 0.25 | 100 | 0.25 | 200 | 2.74E-05 | 3.77E-05 | 2.85E-05 | 2.00E-06 | 0.997 | 4.12E-04 | 1.20E-03 | 3.76E-05 | 2.84E-05 | 1.99E-06 | 0.999 | 2.69E-04 | 2.76E-05 | 3.77E-05 | 2.82E-05 | 1.98E-06 | 0.997 | 3.96E-04 | 2.74E-05 | 3.77E-05 | 2.82E-05 | 2.00E-06 | 0.997 | 3.90E-04 |
250 | 0.25 | 100 | 0.25 | 200 | 1.30E-05 | 2.40E-05 | 1.14E-05 | 8.55E-07 | 1.000 | 0.00E + 00 | 1.20E-03 | 2.39E-05 | 1.14E-05 | 8.51E-07 | 1.000 | 0.00E + 00 | 1.32E-05 | 2.40E-05 | 1.14E-05 | 8.45E-07 | 1.000 | 0.00E + 00 | 1.30E-05 | 2.40E-05 | 1.14E-05 | 8.53E-07 | 1.000 | 0.00E + 00 |
750 | 0.25 | 100 | 0.25 | 200 | 1.13E-05 | 1.38E-05 | 3.98E-06 | 2.35E-07 | 1.000 | 0.00E + 00 | 1.20E-03 | 1.38E-05 | 3.99E-06 | 1.98E-07 | 1.000 | 0.00E + 00 | 1.13E-05 | 1.38E-05 | 3.98E-06 | 2.41E-07 | 1.000 | 0.00E + 00 | 1.13E-05 | 1.38E-05 | 3.98E-06 | 2.45E-07 | 1.000 | 0.00E + 00 |
25 | 0.05 | 5 | 0.05 | 10 | -3.00E-05 | 1.68E-04 | 5.69E-04 | 3.79E-05 | 0.229 | 2.97E-03 | 2.24E-02 | 1.10E-04 | 2.79E-04 | 2.19E-05 | 0.867 | 2.40E-03 | 3.93E-04 | 1.65E-04 | 5.58E-04 | 3.26E-05 | 0.242 | 3.03E-03 | -3.00E-05 | 1.68E-04 | 5.46E-04 | 3.71E-05 | 0.252 | 3.07E-03 |
100 | 0.05 | 5 | 0.05 | 10 | 6.17E-05 | 8.43E-05 | 1.42E-04 | 9.39E-06 | 0.561 | 3.51E-03 | 2.32E-02 | 5.35E-05 | 6.95E-05 | 5.41E-06 | 1.000 | 0.00E + 00 | 1.74E-04 | 8.36E-05 | 1.40E-04 | 8.27E-06 | 0.574 | 3.50E-03 | 6.17E-05 | 8.43E-05 | 1.41E-04 | 9.34E-06 | 0.569 | 3.50E-03 |
250 | 0.05 | 5 | 0.05 | 10 | -5.70E-05 | 5.35E-05 | 5.69E-05 | 3.77E-06 | 0.899 | 2.13E-03 | 2.33E-02 | 3.41E-05 | 2.78E-05 | 2.18E-06 | 1.000 | 0.00E + 00 | -1.86E-05 | 5.31E-05 | 5.61E-05 | 3.34E-06 | 0.903 | 2.09E-03 | -5.70E-05 | 5.35E-05 | 5.66E-05 | 3.76E-06 | 0.901 | 2.11E-03 |
750 | 0.05 | 5 | 0.05 | 10 | -2.69E-05 | 3.09E-05 | 1.90E-05 | 1.27E-06 | 1.000 | 1.00E-04 | 2.34E-02 | 1.96E-05 | 9.28E-06 | 7.90E-07 | 1.000 | 0.00E + 00 | -1.14E-05 | 3.06E-05 | 1.87E-05 | 1.14E-06 | 1.000 | 8.66E-05 | -2.69E-05 | 3.09E-05 | 1.90E-05 | 1.27E-06 | 1.000 | 1.00E-04 |
25 | 0.05 | 100 | 0.05 | 200 | -1.85E-05 | 4.07E-05 | 3.27E-05 | 4.82E-06 | 0.984 | 8.91E-04 | 2.03E-03 | 3.96E-05 | 3.18E-05 | 4.38E-06 | 0.997 | 4.06E-04 | -1.74E-05 | 4.06E-05 | 3.15E-05 | 4.49E-06 | 0.988 | 7.76E-04 | -1.85E-05 | 4.07E-05 | 3.14E-05 | 4.73E-06 | 0.987 | 7.92E-04 |
100 | 0.05 | 100 | 0.05 | 200 | 1.15E-05 | 2.02E-05 | 8.15E-06 | 1.23E-06 | 1.000 | 0.00E + 00 | 2.14E-03 | 1.96E-05 | 8.17E-06 | 1.14E-06 | 1.000 | 0.00E + 00 | 1.21E-05 | 2.02E-05 | 8.08E-06 | 1.16E-06 | 1.000 | 0.00E + 00 | 1.15E-05 | 2.02E-05 | 8.07E-06 | 1.23E-06 | 1.000 | 0.00E + 00 |
250 | 0.05 | 100 | 0.05 | 200 | -1.34E-05 | 1.28E-05 | 3.16E-06 | 7.35E-07 | 1.000 | 0.00E + 00 | 2.13E-03 | 1.25E-05 | 3.17E-06 | 7.53E-07 | 1.000 | 0.00E + 00 | -1.32E-05 | 1.28E-05 | 3.14E-06 | 6.89E-07 | 1.000 | 0.00E + 00 | -1.34E-05 | 1.28E-05 | 3.15E-06 | 7.11E-07 | 1.000 | 0.00E + 00 |
750 | 0.05 | 100 | 0.05 | 200 | 9.87E-06 | 7.31E-06 | 1.00E-06 | 0.00E + 00 | 1.000 | 0.00E + 00 | 2.16E-03 | 7.09E-06 | 1.00E-06 | 0.00E + 00 | 1.000 | 0.00E + 00 | 9.78E-06 | 7.31E-06 | 1.00E-06 | 0.00E + 00 | 1.000 | 0.00E + 00 | 9.87E-06 | 7.31E-06 | 1.00E-06 | 0.00E + 00 | 1.000 | 0.00E + 00 |
Linear regression model | Beta regression model | Variable dispersion beta regression model | Fractional logit regression model | |||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
N0 = N1 | E(Y0) | E(Y1) | Bias | MC error bias | Variance | MC error variance | Type-1 Error | MC Error Type-1 Error | Bias | MC error bias | Variance | MC error variance | Type-1 Error | MC Error Type-1 Error | Bias | MC error bias | Variance | MC error variance | Type-1 Error | MC Error Type-1 Error | Bias | MC error bias | Variance | MC error variance | Type-1 Error | MC Error Type-1 Error |
25 | 0.5 | 0.5 | -6.22E-04 | 2.64E-04 | 1.40E-03 | 3.04E-05 | 0.048 | 0.002 | -6.48E-04 | 2.72E-04 | 1.39E-03 | 3.04E-05 | 0.062 | 0.002 | -6.43E-04 | 2.71E-04 | 1.38E-03 | 3.02E-05 | 0.063 | 0.002 | -6.22E-04 | 2.64E-04 | 1.34E-03 | 2.98E-05 | 0.060 | 0.002 |
100 | 0.5 | 0.5 | 8.00E-06 | 1.32E-04 | 3.50E-04 | 7.51E-06 | 0.049 | 0.002 | -2.97E-06 | 1.37E-04 | 3.61E-04 | 7.68E-06 | 0.055 | 0.002 | -4.54E-07 | 1.37E-04 | 3.61E-04 | 7.66E-06 | 0.055 | 0.002 | 8.00E-06 | 1.32E-04 | 3.46E-04 | 7.47E-06 | 0.052 | 0.002 |
250 | 0.5 | 0.5 | 1.07E-04 | 8.33E-05 | 1.40E-04 | 3.02E-06 | 0.051 | 0.002 | 1.09E-04 | 8.60E-05 | 1.46E-04 | 3.10E-06 | 0.053 | 0.002 | 1.09E-04 | 8.59E-05 | 1.45E-04 | 3.10E-06 | 0.053 | 0.002 | 1.07E-04 | 8.33E-05 | 1.39E-04 | 3.01E-06 | 0.052 | 0.002 |
750 | 0.5 | 0.5 | -3.01E-06 | 4.84E-05 | 4.67E-05 | 1.01E-06 | 0.051 | 0.002 | -2.56E-06 | 4.99E-05 | 4.87E-05 | 1.03E-06 | 0.054 | 0.002 | -2.57E-06 | 4.99E-05 | 4.87E-05 | 1.03E-06 | 0.054 | 0.002 | -3.01E-06 | 4.84E-05 | 4.66E-05 | 1.01E-06 | 0.052 | 0.002 |
25 | 0.215 | 0.215 | -4.46E-04 | 3.72E-04 | 2.74E-03 | 3.41E-05 | 0.051 | 0.002 | -3.06E-04 | 2.80E-04 | 1.82E-03 | 3.12E-05 | 0.037 | 0.001 | -3.94E-04 | 3.51E-04 | 2.19E-03 | 3.13E-05 | 0.072 | 0.002 | -4.46E-04 | 3.72E-04 | 2.63E-03 | 3.34E-05 | 0.062 | 0.002 |
100 | 0.215 | 0.215 | 5.06E-05 | 1.85E-04 | 6.86E-04 | 8.38E-06 | 0.050 | 0.002 | -2.96E-05 | 1.38E-04 | 4.64E-04 | 7.95E-06 | 0.030 | 0.001 | 5.99E-05 | 1.74E-04 | 5.63E-04 | 7.95E-06 | 0.061 | 0.002 | 5.06E-05 | 1.85E-04 | 6.79E-04 | 8.33E-06 | 0.053 | 0.002 |
250 | 0.215 | 0.215 | 1.18E-04 | 1.17E-04 | 2.74E-04 | 3.31E-06 | 0.051 | 0.002 | 1.19E-04 | 8.69E-05 | 1.86E-04 | 3.17E-06 | 0.030 | 0.001 | 1.08E-04 | 1.10E-04 | 2.26E-04 | 3.16E-06 | 0.060 | 0.002 | 1.18E-04 | 1.17E-04 | 2.73E-04 | 3.30E-06 | 0.053 | 0.002 |
750 | 0.215 | 0.215 | -1.10E-05 | 6.78E-05 | 9.13E-05 | 1.11E-06 | 0.050 | 0.002 | -1.93E-05 | 5.02E-05 | 6.20E-05 | 1.06E-06 | 0.029 | 0.001 | -1.64E-06 | 6.37E-05 | 7.55E-05 | 1.06E-06 | 0.059 | 0.002 | -1.10E-05 | 6.78E-05 | 9.12E-05 | 1.11E-06 | 0.050 | 0.002 |
Linear regression model | Beta regression model | Variable dispersion beta regression model | Fractional logit regression model | |||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
N0 = N1 | E(Y0) | E(Y1) | Bias | MC error bias | Variance | MC error variance | Power | MC error power | Bias | MC error bias | Variance | MC error variance | Power | MC error power | Bias | MC error bias | Variance | MC error variance | Power | MC error power | Bias | MC error bias | Variance | MC error variance | Power | MC error power |
25 | 0.5 | 0.6 | -1.36E-04 | 2.63E-04 | 1.40E-03 | 3.04E-05 | 0.745 | 0.003 | 2.32E-03 | 2.76E-04 | 1.44E-03 | 3.22E-05 | 0.769 | 0.003 | 1.77E-03 | 2.73E-04 | 1.43E-03 | 3.20E-05 | 0.767 | 0.003 | -1.36E-04 | 2.63E-04 | 1.35E-03 | 2.98E-05 | 0.774 | 0.003 |
100 | 0.5 | 0.6 | -2.17E-04 | 1.33E-04 | 3.50E-04 | 7.54E-06 | 1.000 | 0.000 | 2.37E-03 | 1.39E-04 | 3.72E-04 | 8.18E-06 | 1.000 | 0.000 | 1.86E-03 | 1.38E-04 | 3.72E-04 | 8.16E-06 | 1.000 | 0.000 | -2.17E-04 | 1.33E-04 | 3.46E-04 | 7.50E-06 | 1.000 | 0.000 |
250 | 0.5 | 0.6 | -5.06E-05 | 8.32E-05 | 1.40E-04 | 3.02E-06 | 1.000 | 0.000 | 2.55E-03 | 8.73E-05 | 1.50E-04 | 3.29E-06 | 1.000 | 0.000 | 2.05E-03 | 8.64E-05 | 1.50E-04 | 3.28E-06 | 1.000 | 0.000 | -5.06E-05 | 8.32E-05 | 1.39E-04 | 3.01E-06 | 1.000 | 0.000 |
750 | 0.5 | 0.6 | -8.70E-05 | 4.85E-05 | 4.66E-05 | 1.01E-06 | 1.000 | 0.000 | 2.55E-03 | 5.08E-05 | 5.02E-05 | 1.10E-06 | 1.000 | 0.000 | 2.05E-03 | 5.04E-05 | 5.02E-05 | 1.10E-06 | 1.000 | 0.000 | -8.70E-05 | 4.85E-05 | 4.66E-05 | 1.01E-06 | 1.000 | 0.000 |
25 | 0.215 | 0.315 | 2.82E-04 | 3.70E-04 | 2.75E-03 | 3.42E-05 | 0.466 | 0.004 | 1.35E-02 | 2.96E-04 | 2.02E-03 | 3.04E-05 | 0.725 | 0.003 | 3.33E-03 | 3.55E-04 | 2.21E-03 | 3.07E-05 | 0.589 | 0.003 | 2.82E-04 | 3.70E-04 | 2.64E-03 | 3.35E-05 | 0.501 | 0.004 |
100 | 0.215 | 0.315 | 6.30E-06 | 1.84E-04 | 6.85E-04 | 8.36E-06 | 0.966 | 0.001 | 1.36E-02 | 1.46E-04 | 5.17E-04 | 7.65E-06 | 1.000 | 0.000 | 3.10E-03 | 1.77E-04 | 5.68E-04 | 7.71E-06 | 0.987 | 0.001 | 6.30E-06 | 1.84E-04 | 6.78E-04 | 8.32E-06 | 0.967 | 0.001 |
250 | 0.215 | 0.315 | 1.87E-05 | 1.17E-04 | 2.74E-04 | 3.31E-06 | 1.000 | 0.000 | 1.37E-02 | 9.28E-05 | 2.08E-04 | 3.04E-06 | 1.000 | 0.000 | 3.14E-03 | 1.12E-04 | 2.28E-04 | 3.07E-06 | 1.000 | 0.000 | 1.87E-05 | 1.17E-04 | 2.73E-04 | 3.30E-06 | 1.000 | 0.000 |
750 | 0.215 | 0.315 | 1.02E-04 | 6.74E-05 | 9.14E-05 | 1.10E-06 | 1.000 | 0.000 | 1.38E-02 | 5.33E-05 | 6.94E-05 | 1.02E-06 | 1.000 | 0.000 | 3.22E-03 | 6.45E-05 | 7.64E-05 | 1.03E-06 | 1.000 | 0.000 | 1.02E-04 | 6.74E-05 | 9.13E-05 | 1.11E-06 | 1.000 | 0.000 |