Table 1 Bias (MSE) for mediation effect estimates

\(n=300\)

\({\alpha }_{1}{\beta }_{1}\)

7.21E−04

(1.72E−04)

−7.07E−03

(4.23E−04)

−8.95E−03

(2.80E−04)

−8.90E−03

(1.97E−04)

−2.23E−02

(8.19E−04)

−2.25E−02

(7.14E−04)

\({\alpha }_{2}{\beta }_{2}\)

−2.44E−04

(2.80E−04)

−7.27E−03

(5.10E−04)

−1.30E−02

(4.28E−04)

−1.34E−02

(4.12E−04)

−2.71E−02

(1.22E−03)

−2.90E−02

(1.09E−03)

\({\alpha }_{3}{\beta }_{3}\)

9.45E−04

(1.00E−04)

−8.32E−03

(2.84E−04)

−7.54E−03

(1.91E−04)

−4.17E−03

(1.17E−04)

−1.43E−02

(3.63E−04)

−1.40E−02

(3.00E−04)

\({\alpha }_{4}{\beta }_{4}\)

−2.22E−03

(4.86E−04)

−1.16E−02

(7.75E−04)

−1.95E−02

(8.11E−04)

−1.87E−02

(6.97E−04)

−3.27E−02

(1.52E−03)

−3.89E−02

(1.77E−03)

\({\alpha }_{5}{\beta }_{5}\)

−4.82E−03

(5.89E−04)

−1.63E−02

(1.03E−03)

−2.66E−02

(1.23E−03)

−2.51E−02

(1.07E−03)

−4.48E−02

(2.66E−03)

−5.21E−02

(3.04E−03)

\({\alpha }_{6}{\beta }_{6}\)

6.06E−05

(1.05E−05)

−3.76E−05

(8.92E−06)

4.81E−05

(7.21E−06)

5.10E−05

(4.32E−06)

3.56E−05

(1.62E−06)

2.88E−05

(1.57E−06)

\({\alpha }_{7}{\beta }_{7}\)

2.41E−03

(3.71E−05)

3.21E−04

(5.91E−06)

3.75E−04

(8.43E−06)

1.13E−03

(1.81E−05)

5.43E−05

(1.85E−06)

6.02E−05

(1.60E−06)

\({\alpha }_{8}{\beta }_{8}\)

1.11E−04

(1.08E−06)

1.17E−05

(1.09E−07)

3.62E−05

(2.72E−07)

4.58E−05

(4.02E−07)

−1.03E−05

(4.31E−08)

−5.98E−06

(3.04E−08)

\(n=600\)

\({\alpha }_{1}{\beta }_{1}\)

1.46E−03

(1.01E−04)

2.88E−03

(1.64E−04)

−4.34E−03

(1.06E−04)

−6.78E−03

(1.18E−04)

−9.43E−03

(2.22E−04)

−1.49E−02

(2.89E−04)

\({\alpha }_{2}{\beta }_{2}\)

1.00E−03

(1.59E−04)

3.47E−03

(2.45E−04)

−7.38E−03

(2.00E−04)

−1.01E−02

(2.23E−04)

−1.32E−02

(3.40E−04)

−2.26E−02

(6.03E−04)

\({\alpha }_{3}{\beta }_{3}\)

8.87E−04

(4.72E−05)

−2.49E−04

(1.38E−04)

−3.13E−03

(6.45E−05)

−3.22E−03

(5.42E−05)

−7.51E−03

(1.67E−04)

−9.34E−03

(1.40E−04)

\({\alpha }_{4}{\beta }_{4}\)

7.51E−04

(1.97E−04)

4.32E−03

(2.97E−04)

−1.03E−02

(2.94E−04)

−1.44E−02

(4.12E−04)

−1.90E−02

(5.96E−04)

−3.22E−02

(1.18E−03)

\({\alpha }_{5}{\beta }_{5}\)

−1.03E−03

(2.80E−04)

2.39E−03

(4.03E−04)

−1.59E−02

(5.36E−04)

−2.12E−02

(7.05E−04)

−2.89E−02

(1.13E−03)

−4.57E−02

(2.27E−03)

\({\alpha }_{6}{\beta }_{6}\)

1.15E−05

(4.16E−06)

−9.97E−05

(4.28E−06)

−3.24E−05

(3.21E−06)

1.59E−04

(2.27E−06)

1.43E−04

(2.06E−06)

1.37E−04

(1.45E−06)

\({\alpha }_{7}{\beta }_{7}\)

2.20E−03

(2.09E−05)

1.29E−04

(1.32E−06)

2.99E−04

(7.03E−06)

6.68E−04

(1.07E−05)

4.24E−05

(1.28E−06)

1.01E−04

(1.29E−06)

\({\alpha }_{8}{\beta }_{8}\)

−4.17E−06

(4.54E−07)

3.83E−06

(7.98E−08)

−4.66E−06

(2.06E−07)

−6.36E−06

(1.62E−07)

−4.77E−06

(9.88E−09)

5.34E−06

(4.52E−08)

\(n=300\)

\({\alpha }_{1}{\beta }_{1}\)

4.46E−04

(1.82E−04)

−7.65E−03

(4.95E−04)

−7.79E−03

(2.49E−04)

−6.92E−03

(1.95E−04)

−1.85E−02

(6.84E−04)

−1.84E−02

(5.13E−04)

\({\alpha }_{2}{\beta }_{2}\)

3.22E−04

(3.08E−04)

−6.70E−03

(5.90E−04)

−1.07E−02

(3.88E−04)

−1.07E−02

(3.65E−04)

−2.17E−02

(9.31E−04)

−2.58E−02

(8.66E−04)

\({\alpha }_{3}{\beta }_{3}\)

5.90E−04

(1.09E−04)

−7.05E−03

(2.83E−04)

−4.87E−03

(1.40E−04)

−3.934E−03

(1.01E−04)

−1.33E−02

(3.34E−04)

−1.22E−02

(2.38E−04)

\({\alpha }_{4}{\beta }_{4}\)

−2.44E−03

(4.26E−04)

−1.04E−02

(7.13E−04)

−1.74E−02

(7.17E−04)

−1.81E−02

(6.81E−04)

−3.33E−02

(1.69E−03)

−3.79E−02

(1.74E−03)

\({\alpha }_{5}{\beta }_{5}\)

−3.57E−03

(5.93E−04)

−1.53E−02

(1.02E−03)

−2.34E−02

(1.15E−03)

−2.25E−02

(9.91E−04)

−4.21E−02

(2.39E−03)

−5.08E−02

(2.94E−03)

\({\alpha }_{6}{\beta }_{6}\)

1.77E−04

(9.38E−06)

1.56E−04

(7.85E−06)

2.24E−04

(7.31E−06)

−4.44E−05

(4.70E−06)

2.45E−05

(2.84E−06)

−1.12E−05

(2.35E−06)

\({\alpha }_{7}{\beta }_{7}\)

3.63E−03

(4.74E−05)

4.27E−04

(1.09E−05)

7.49E−04

(1.07E−05)

1.89E−03

(2.29E−05)

1.84E−04

(3.29E−06)

2.20E−04

(3.13E−06)

\({\alpha }_{8}{\beta }_{8}\)

7.33E−05

(9.05E−07)

−2.12E−05

(1.81E−07)

−9.48E−06

(2.56E−07)

3.63E−05

(4.97E−07)

−1.24E−05

(3.90E−08)

6.43E−07

(2.70E−08)

\(n=600\)

\({\alpha }_{1}{\beta }_{1}\)

8.96E−04

(9.43E−05)

2.08E−03

(1.75E−04)

−3.94E−03

(1.07E−04)

−6.44E−03

(1.24E−04)

−8.81E−03

(1.98E−04)

−1.44E−02

(2.69E−04)

\({\alpha }_{2}{\beta }_{2}\)

3.74E−04

(1.50E−04)

2.77E−03

(2.21E−04)

−6.33E−03

(1.95E−04)

−9.28E−03

(2.15E−04)

−1.32E−02

(3.50E−04)

−2.20E−02

(5.88E−04)

\({\alpha }_{3}{\beta }_{3}\)

1.41E−03

(5.39E−05)

1.33E−04

(1.31E−04)

−1.94E−03

(5.17E−05)

−3.45E−03

(5.67E−05)

−6.69E−03

(1.41E−04)

−8.19E−03

(1.06E−04)

\({\alpha }_{4}{\beta }_{4}\)

3.66E−04

(2.13E−04)

3.32E−03

(2.95E−04)

−8.58E−03

(2.91E−04)

−1.35E−02

(3.75E−04)

−1.93E−02

(5.98E−04)

−3.14E−02

(1.13E−03)

\({\alpha }_{5}{\beta }_{5}\)

4.79E−04

(3.29E−04)

3.09E−03

(4.39E−04)

−1.15E−02

(4.74E−04)

−1.95E−02

(6.63E−04)

−2.81E−02

(1.11E−03)

−4.37E−02

(2.12E−03)

\({\alpha }_{6}{\beta }_{6}\)

−2.74E−05

(3.95E−06)

−1.16E−04

(4.39E−06)

−2.32E−05

(3.27E−06)

−1.33E−04

(2.97E−06)

−1.96E−04

(2.85E−06)

−6.82E−05

(1.90E−06)

\({\alpha }_{7}{\beta }_{7}\)

3.17E−03

(2.65E−05)

3.37E−04

(4.72E−06)

1.13E−03

(8.65E−06)

2.24E−03

(1.90E−05)

1.23E−04

(1.92E−06)

2.37E−04

(2.55E−06)

\({\alpha }_{8}{\beta }_{8}\)

1.62E−05

(3.35E−07)

−5.68E−07

(4.13E−08)

−6.73E−07

(2.26E−07)

2.72E−05

(1.54E−07)

1.69E−05

(6.54E−08)

1.30E−05

(2.72E−08)

\(n=300\)

\({\alpha }_{1}{\beta }_{1}\)

3.76E−03

(2.34E−04)

−3.46E−03

(4.90E−04)

−1.78E−03

(2.13E−04)

−5.14E−03

(1.98E−04)

−1.63E−02

(6.36E−04)

−1.46E−02

(3.74E−04)

\({\alpha }_{2}{\beta }_{2}\)

2.82E−03

(3.66E−04)

−5.79E−03

(8.16E−04)

−4.11E−03

(3.79E−04)

−7.97E−03

(3.72E−04)

−2.14E−02

(1.13E−03)

−2.18E−02

(7.77E−04)

\({\alpha }_{3}{\beta }_{3}\)

2.25E−03

(1.42E−04)

−6.91E−03

(3.36E−04)

−9.13E−04

(1.19E−04)

−3.27E−03

(1.19E−04)

−1.32E−02

(3.33E−04)

−8.64E−03

(1.59E−04)

\({\alpha }_{4}{\beta }_{4}\)

1.03E−03

(5.02E−04)

−8.62E−03

(9.89E−04)

−7.04E−03

(5.47E−04)

−1.15E−02

(5.83E−04)

−2.90E−02

(1.65E−03)

−2.97E−02

(1.26E−03)

\({\alpha }_{5}{\beta }_{5}\)

6.66E−03

(7.53E−04)

−5.38E−03

(1.01E−03)

−7.22E−03

(7.94E−04)

−1.27E−02

(7.87E−04)

−3.52E−02

(2.06E−03)

−4.02E−02

(2.24E−03)

\({\alpha }_{6}{\beta }_{6}\)

−3.47E−05

(8.66E−06)

−2.58E−04

(7.30E−06)

9.43E−05

(7.46E−06)

6.55E−05

(6.88E−06)

6.48E−05

(6.34E−06)

7.46E−05

(5.69E−06)

\({\alpha }_{7}{\beta }_{7}\)

4.69E−03

(5.74E−05)

9.18E−04

(1.63E−05)

2.09E−03

(2.36E−05)

3.81E−03

(4.57E−05)

5.83E−04

(9.04E−06)

1.06E−03

(1.21E−05)

\({\alpha }_{8}{\beta }_{8}\)

1.82E−05

(1.91E−06)

−3.56E−05

(5.92E−07)

−1.17E−05

(7.80E−07)

−1.28E−05

(4.19E−07)

−2.13E−05

(1.67E−07)

1.08E−05

(1.89E−07)

\(n=600\)

\({\alpha }_{1}{\beta }_{1}\)

2.44E−03

(1.08E−04)

2.14E−03

(1.74E−04)

4.14E−04

(1.01E−04)

−3.50E−03

(1.01E−04)

−6.87E−03

(1.99E−04)

−1.07E−02

(1.95E−04)

\({\alpha }_{2}{\beta }_{2}\)

3.41E−03

(2.02E−04)

4.48E−03

(2.75E−04)

7.19E−04

(1.71E−04)

−5.51E−03

(2.26E−04)

−9.64E−03

(3.32E−04)

−1.68E−02

(4.46E−04)

\({\alpha }_{3}{\beta }_{3}\)

1.76E−03

(7.22E−05)

−3.66E−03

(1.91E−04)

6.19E−04

(6.90E−05)

−1.24E−03

(6.31E−05)

−6.00E−03

(1.74E−04)

−5.34E−03

(7.98E−05)

\({\alpha }_{4}{\beta }_{4}\)

4.34E−03

(2.65E−04)

4.77E−03

(3.36E−04)

2.98E−04

(2.42E−04)

−9.92E−03

(3.48E−04)

−1.59E−02

(5.71E−04)

−2.48E−02

(8.05E−04)

\({\alpha }_{5}{\beta }_{5}\)

8.04E−03

(4.34E−04)

9.29E−03

(5.06E−04)

1.81E−03

(3.96E−04)

−9.00E−03

(4.37E−04)

−1.82E−02

(7.82E−04)

−3.04E−02

(1.23E−03)

\({\alpha }_{6}{\beta }_{6}\)

1.10E−04

(4.60E−06)

5.09E−05

(3.85E−06)

9.57E−05

(4.21E−06)

−1.66E−04

(3.28E−06)

−1.12E−04

(2.79E−06)

−7.62E−05

(2.46E−06)

\({\alpha }_{7}{\beta }_{7}\)

2.91E−03

(3.12E−05)

8.64E−04

(1.22E−05)

2.17E−03

(2.10E−05)

3.23E−03

(2.66E−05)

6.05E−04

(6.67E−06)

1.23E−03

(9.06E−06)

\({\alpha }_{8}{\beta }_{8}\)

6.18E−06

(4.08E−07)

−1.44E−06

(1.23E−07)

−2.07E−06

(4.32E−07)

2.24E−05

(2.45E−07)

1.13E−05

(5.31E−08)

5.00E−06

(1.28E−07)

\(n=300\)

\({\alpha }_{1}{\beta }_{1}\)

7.39E−03

(3.47E−04)

−2.75E−03

(7.68E−04)

4.38E−03

(3.12E−04)

2.65E−03

(2.98E−04)

−8.93E−03

(7.81E−04)

−4.19E−03

(2.82E−04)

\({\alpha }_{2}{\beta }_{2}\)

8.32E−03

(7.65E−04)

−1.43E−02

(2.29E−03)

5.21E−03

(6.63E−04)

2.17E−03

(5.13E−04)

−2.20E−02

(2.15E−03)

−8.00E−03

(6.11E−04)

\({\alpha }_{3}{\beta }_{3}\)

4.03E−03

(2.70E−04)

−1.05E−02

(5.41E−04)

2.83E−03

(2.53E−04)

8.24E−05

(2.15E−04)

−1.32E−02

(5.05E−04)

−3.20E−03

(2.06E−04)

\({\alpha }_{4}{\beta }_{4}\)

1.07E−02

(1.07E−03)

−6.93E−03

(2.81E−03)

5.83E−03

(9.77E−04)

1.03E−03

(7.32E−04)

−2.05E−02

(2.85E−03)

−1.21E−02

(9.51E−04)

\({\alpha }_{5}{\beta }_{5}\)

1.74E−02

(1.50E−03)

−1.48E−02

(4.46E−03)

1.08E−02

(1.33E−03)

7.51E−03

(1.09E−03)

−3.15E−02

(4.61E−03)

−1.28E−02

(1.26E−03)

\({\alpha }_{6}{\beta }_{6}\)

2.06E−05

(8.13E−06)

−1.05E−04

(6.97E−06)

4.44E−05

(8.01E−06)

9.89E−06

(7.72E−06)

−1.27E−04

(6.67E−06)

6.18E−05

(7.25E−06)

\({\alpha }_{7}{\beta }_{7}\)

6.19E−03

(1.20E−04)

1.66E−03

(4.07E−05)

4.97E−03

(1.08E−04)

4.63E−03

(8.72E−05)

9.38E−04

(1.99E−05)

2.55E−03

(7.09E−05)

\({\alpha }_{8}{\beta }_{8}\)

−1.15E−04

(5.76E−06)

−8.81E−06

(8.67E−07)

−1.04E−04

(5.11E−06)

−1.78E−05

(2.51E−06)

−5.95E−06

(1.02E−06)

−6.78E−05

(2.77E−06)

\(n=600\)

\({\alpha }_{1}{\beta }_{1}\)

4.81E−03

(1.64E−04)

8.99E−04

(4.05E−04)

3.88E−03

(1.57E−04)

1.75E−03

(1.52E−04)

−2.42E−03

(3.86E−04)

−3.55E−03

(1.46E−04)

\({\alpha }_{2}{\beta }_{2}\)

8.29E−03

(4.00E−04)

−2.07E−03

(1.30E−03)

6.75E−03

(3.79E−04)

3.43E−03

(3.20E−04)

−8.50E−03

(1.30E−03)

−4.32E−03

(3.05E−04)

\({\alpha }_{3}{\beta }_{3}\)

2.88E−03

(1.17E−04)

−1.15E−02

(4.25E−04)

2.38E−03

(1.15E−04)

6.43E−04

(1.17E−04)

−1.38E−02

(4.22E−04)

−2.27E−03

(1.18E−04)

\({\alpha }_{4}{\beta }_{4}\)

1.06E−02

(5.68E−04)

6.82E−03

(1.20E−03)

9.09E−03

(5.32E−04)

2.99E−03

(4.38E−04)

−4.25E−03

(1.26E−03)

−7.83E−03

(4.99E−04)

\({\alpha }_{5}{\beta }_{5}\)

1.57E−02

(8.59E−04)

6.22E−03

(1.70E−03)

1.26E−02

(7.55E−04)

6.31E−03

(6.40E−04)

−8.25E−03

(1.96E−03)

−9.78E−03

(7.13E−04)

\({\alpha }_{6}{\beta }_{6}\)

−9.35E−05

(3.98E−06)

−9.84E−06

(2.76E−06)

−1.16E−04

(3.92E−06)

−1.13E−04

(3.75E−06)

−6.49E−05

(2.45E−06)

−9.40E−05

(3.77E−06)

\({\alpha }_{7}{\beta }_{7}\)

4.63E−03

(6.31E−05)

1.03E−03

(1.79E−05)

3.94E−03

(5.85E−05)

4.57E−03

(5.64E−05)

1.13E−03

(1.85E−05)

2.41E−03

(3.99E−05)

\({\alpha }_{8}{\beta }_{8}\)

2.92E−05

(1.56E−06)

−1.46E−05

(2.31E−07)

−2.62E−05

(1.51E−06)

9.12E−07

(1.19E−06)

−1.37E−05

(7.91E−08)

−2.86E−05

(1.11E−06)