Skip to main content

Advertisement

Table 4 Mean estimates from 20000 replicate simulations of bias (MC error of bias), variance (MC error of variance) and power (MC error power), from the fitted linear, beta, variable-dispersion beta and fractional logit regression models estimated on the beta distributed response data (Power experiments)

From: A Monte Carlo simulation study comparing linear regression, beta regression, variable-dispersion beta regression and fractional logit regression at recovering average difference measures in a two sample design

      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
  1. The first 24 rows of the Table describe experiments where the dispersion parameter does not vary as a function of group membership; whereas, in the bottom 24 rows of the Table the dispersion parameter varies as a function of group membership. Rows 1–8 (and 25–32) correspond to simulated experiments where the mass of the distribution is near 0.50, rows 9–16 (and 33–40) correspond to simulated experiments where the mass of the distribution is centered near 0.25, and finally rows 17–24 (and 41–48) correspond to simulated experiments where the mass of the distribution is near 0.05.
  2. ∆ = 0.025 (power experiments).
  3. Power refers to the proportion of null hypothesis rejected.