Table 2.

Denominator degrees of freedom summary.

a) In terms of the OR mean squares and covariances estimates:

ddfO = (t-1)(r-1)

$${\mathrm{ddf}}_{\mathrm{D}}=\{\begin{array}{cc}\frac{{[\mathrm{MS}\left({\mathrm{T}}^{\ast }\mathrm{R}\right)+r({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3)]}^2}{\frac{\mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{R}\right)}^2}{(t-1)(r-1)}+\frac{{[{\widehat{\sigma }}_{∊}^2-{\widehat{\mathrm{Cov}}}_1+(r-1)({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3)]}^2}{(t-1)(c-1)}+\frac{{[{\widehat{\sigma }}_{∊}^2-{\widehat{\mathrm{Cov}}}_1-{\widehat{\mathrm{Cov}}}_2+{\widehat{\mathrm{Cov}}}_3]}^2}{(t-1)(r-1)(c-1)}},\hfill & {\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3>0\hfill \\ (t-1)(r-1),\hfill & {\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3\le 0\hfill \end{array}\phantom{\}}$$

$${\mathrm{ddf}}_{\mathrm{H}}=\frac{{[\mathrm{MS}\left({\mathrm{T}}^{\ast }\mathrm{R}\right)+\max [r({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3),0]]}^2}{\frac{{\left[\mathrm{MS}\left({\mathrm{T}}^{\ast }\mathrm{R}\right)\right]}^2}{(t-1)(r-1)}}$$

b) Relationships:

ddfD ≤ddfH

$${\mathrm{ddf}}_{\mathrm{o}}=(t-1)(r-1)\{\begin{array}{cc}\le {\mathrm{ddf}}_{\mathrm{H}},\hfill & {\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3>0\hfill \\ ={\mathrm{ddf}}_{\mathrm{H}},\hfill & {\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3=0\hfill \end{array}\phantom{\}}$$

ddfD can be larger or smaller than ddfO

c) In terms of the DBM mean squares:

ddfO = (t-1)(r-1)

$${\mathrm{ddf}}_{\mathrm{D}}=\{\begin{array}{cc}\frac{{[\mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{R}\right)}_{\text{pseudo}}+\mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}-\mathrm{MS}{\left({\mathrm{T}}^{\ast }{\mathrm{R}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}]}^2}{\frac{\mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{R}\right)}_{\text{pseudo}}^2}{(t-1)(r-1)}+\frac{\mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}^2}{(t-1)(c-1)}+\frac{\mathrm{MS}{\left({\mathrm{T}}^{\ast }{\mathrm{R}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}^2}{(t-1)(r-1)(c-1)}},\hfill & \mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}-\mathrm{MS}{\left({\mathrm{T}}^{\ast }{\mathrm{R}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}>0\hfill \\ (t-1)(r-1),\hfill & \mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}-\mathrm{MS}{\left({\mathrm{T}}^{\ast }{\mathrm{R}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}\le 0\hfill \end{array}\phantom{\}}$$

$${\mathrm{ddf}}_{\mathrm{H}}=\frac{{\{\mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{R}\right)}_{\text{pseudo}}+\max [\mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}}-\mathrm{MS}{\left({\mathrm{T}}^{\ast }{\mathrm{R}}^{\ast }\mathrm{C}\right)}_{\text{pseudo}},0]\}}^2}{\mathrm{MS}{\left({\mathrm{T}}^{\ast }\mathrm{R}\right)}_{\text{pseudo}}^2∕\left[(t-1)(r-1)\right]}$$