Table 2 Common multinomial logistic regression estimates for three grid models and the corresponding centralized model in Study 3 and Study 4

\( {\widehat{\alpha}}_1 \)

2

1.01e-1

4.12e-1

1.29e-1

6.01e-1

\( {\widehat{\beta}}_{1,1} \)

0.5

3.43e-2

2.51e-1

3.76e-2

3.69e-1

\( {\widehat{\beta}}_{2,1} \)

0.5

2.58e-2

2.51e-1

3.35e-2

3.69e-1

\( {\widehat{\beta}}_{3,1} \)

0.5

1.39e-2

4.75e-1

1.13e-1

7.11e-1

\( {\widehat{\beta}}_{4,1} \)

0.5

5.55e-2

4.78e-1

8.62e-2

7.08e-1

\( {\widehat{\alpha}}_2 \)

3

1.11e-1

4.09e-1

1.47e-1

5.95e-1

\( {\widehat{\beta}}_{1,2} \)

2

5.43e-2

2.70e-1

8.40e-2

3.96e-1

\( {\widehat{\beta}}_{2,2} \)

2

4.60e-2

2.70e-1

8.35e-2

3.95e-1

\( {\widehat{\beta}}_{3,2} \)

2

2.55e-2

4.86e-1

1.56e-1

7.26e-1

\( {\widehat{\beta}}_{4,2} \)

2

7.86e-2

4.89e-1

1.30e-1

7.22e-1

\( {\widehat{\alpha}}_3 \)

1

6.06e-2

4.64e-1

5.53e-2

6.80e-1

\( {\widehat{\beta}}_{1,3} \)

1

3.23e-2

3.02e-1

5.55e-2

4.43e-1

\( {\widehat{\beta}}_{2,3} \)

1

2.61e-2

3.02e-1

4.65e-2

4.42e-1

\( {\widehat{\beta}}_{3,3} \)

1

1.78e-2

5.55e-1

1.18e-1

8.30e-1

\( {\widehat{\beta}}_{4,3} \)

1

6.04e-2

5.58e-1

9.09e-2

8.25e-1