Table 2.

Nonnull-distribution parameters and their corresponding estimates for the nonequivalence test for the factorial design. Models M1 and M3 each have an approximate noncentral $F$ nonnull distribution with noncentrality parameter $\lambda$ and degrees of freedom ${\mathrm{df}}_1=t-1=1$ and ${\mathrm{df}}_2$. Model M2 has an approximate noncentral, chi-squared nonull distribution with noncentrality parameter $\lambda$, and degrees of freedom ${\mathrm{df}}_1=t-1=1$. Notes: $d$ = effect size as defined by Eq. (26); $t$ = number of tests, with $t=2$; $r$ = number of readers; $c^{*}$ = number of cases in the pilot study from which parameter estimates were computed, or the number of cases corresponding to the error variance if parameter values are conjectured; $c$ = number of cases in planned study; ${\widehat{\sigma }}_{TR}^2$, ${\widehat{\sigma }}_{ϵ}^2$, ${\widehat{\mathrm{Cov}}}_1$, ${\widehat{\mathrm{Cov}}}_2$, and ${\widehat{\mathrm{Cov}}}_3$, are OR estimates from a factorial-design pilot study with $c^{*}$ cases, or are conjectured values; NA = not applicable.

Nonnull-distribution parameters:
Model	$\lambda$	${\mathrm{df}}_2$
M1	$\frac{r{d}^2/2}{{\sigma }_{TR}^2+{\sigma }_{ϵ}^2-{\mathrm{Cov}}_1+(r-1)({\mathrm{Cov}}_2-{\mathrm{Cov}}_3)}$	$\frac{{[{\sigma }_{TR}^2+{\sigma }_{ϵ}^2-{\mathrm{Cov}}_1+(r-1)({\mathrm{Cov}}_2-{\mathrm{Cov}}_3)]}^2}{{({\sigma }_{\tau R}^2+{\sigma }_{ϵ}^2-{\mathrm{Cov}}_1-{\mathrm{Cov}}_2+{\mathrm{Cov}}_3)}^2/[(t-1)(r-1)]}$
M2	$\frac{r{d}^2/2}{{\sigma }_{ϵ}^2-{\mathrm{Cov}}_1+(r-1)({\mathrm{Cov}}_2-{\mathrm{Cov}}_3)}$	NA
M3	$\frac{r{d}^2/2}{{\sigma }_{TR}^2+{\sigma }_w^2}$	$(t-1)(r-1)=r-1$
Nonnull-distribution parameter estimates:
Model	$\widehat{\lambda }$	${\widehat{\mathrm{df}}}_2$
M1	$\frac{r{d}^2/2}{{\widehat{\sigma }}_{TR}^2+\frac{c^{*}}{c}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1+(r-1)\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]}$	$\frac{{\{{\widehat{\sigma }}_{TR}^2+\frac{c^{*}}{c}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1+(r-1)\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]\}}^2}{{\{{\widehat{\sigma }}_{TR}^2+\frac{c^{*}}{c}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1-\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]\}}^2/[(t-1)(r-1)]}$
M2	$\frac{r{d}^2/2}{\frac{c^{*}}{c}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1+(r-1)\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]}$	NA
M3	$\frac{r{d}^2/2}{{\widehat{\sigma }}_{TR}^2+\frac{c^{*}}{c}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1-\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]}$	$(t-1)(r-1)=r-1$