Table 1 Common ordinal logistic regression estimates for three grid models with various local site sample sizes and the corresponding centralized model in Study 1 and Study 2

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

−1

−4.75e-3

1.27e-1

−1.46e-2

1.82e-1

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

0

−1.17e-3

1.23e-1

−3.47e-3

1.75e-1

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

1

1.93e-3

1.30e-1

1.02e-2

1.84e-1

\( {\widehat{\beta}}_1 \)

1

7.22e-3

8.02e-2

5.53e-3

1.14e-1

\( {\widehat{\beta}}_2 \)

1

6.60e-3

8.02e-2

3.79e-3

1.14e-1

\( {\widehat{\beta}}_3 \)

1

1.04e-2

1.42e-1

1.64e-3

2.01e-1

\( {\widehat{\beta}}_4 \)

1

7.34e-3

1.42e-1

1.36e-2

2.02e-1