Table 3.

Power computation example. This table illustrates use of Table 2 equations for computing power for detecting a 0.05 difference in AUC for a study design having seven readers and 148 cases, based on the Van Dyke parameter estimates from Fig. 1, with $\text{alpha}=0.05$.

Model M1:

$$\begin{gathered} \widehat{\lambda }=\frac{r{d}^2/2}{{\widehat{\sigma }}_{TR}^2+\frac{c}{c^{*}}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1+(r-1)\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]} \\ =\frac{7{(0.05)}^2/2}{0.00020040+\frac{114}{148}[0.00080229-0.00034661+(7-1)(0.00034407-0.00023903)]}=8.439 \end{gathered}$$

$$\begin{gathered} {\widehat{\mathrm{df}}}_2=\frac{{\{{\widehat{\sigma }}_{TR}^2+\frac{c}{c^{*}}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1+(r-1)\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]\}}^2}{{\{{\widehat{\sigma }}_{TR}^2+\frac{c}{c^{*}}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1-\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]\}}^2/[(t-1)(r-1)]} \\ =\frac{{\{0.00020040+\frac{114}{148}[0.00080229-0.00034661+(7-1)(0.00034407-0.00023903)]\}}^2}{{\{0.00020040+\frac{114}{148}[0.00080229-0.00034661-(0.00034407-0.00023903)]\}}^2/(7-1)}=29.140 \end{gathered}$$

$$\begin{gathered} \text{Power}=\mathrm{Pr}(F_{{\mathrm{df}}_1,{\widehat{\mathrm{df}}}_2;\widehat{\lambda }}>F_{1-\alpha ;{\mathrm{df}}_1,{\widehat{\mathrm{df}}}_2}) \\ =\mathrm{Pr}(F_{1,29.14;8.439}>F_{0.95;1,29.14}=4.18122)=0.802 \end{gathered}$$

Model M2:

$$\begin{gathered} \widehat{\lambda }=\frac{r{d}^2/2}{\frac{c^{*}}{c}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1+(r-1)\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]} \\ =\frac{7{(0.05)}^2/2}{\frac{114}{148}[0.00080229-0.00034661+(7-1)(0.00034407-0.00023903)]}=10.461 \end{gathered}$$

$$\text{Power}=\mathrm{Pr}({\chi }_{{\mathrm{df}}_1;\widehat{\lambda }}^2>{\chi }_{1-\alpha ;{\mathrm{df}}_1}^2)=\mathrm{Pr}({\chi }_{1;10.461}^2>{\chi }_{0.95;1}^2=3.8416)=0.899$$

Model M3:

$$\begin{gathered} \widehat{\lambda }=\frac{r{d}^2/2}{{\widehat{\sigma }}_{TR}^2+\frac{c^{*}}{c}[{\widehat{\sigma }}_{ϵ}^2-{\widehat{\mathrm{Cov}}}_1-\max ({\widehat{\mathrm{Cov}}}_2-{\widehat{\mathrm{Cov}}}_3,0)]} \\ =\frac{7{(0.05)}^2/2}{0.00020040+\frac{114}{148}[0.00080229-0.00034661-(0.00034407-0.00023903)]}=18.598 \end{gathered}$$

$${\widehat{\mathrm{df}}}_2=(t-1)(r-1)=7-1=6$$

$$\begin{gathered} \text{Power}=\mathrm{Pr}(F_{{\mathrm{df}}_1,{\widehat{\mathrm{df}}}_2;\widehat{\lambda }}>F_{1-\alpha ;{\mathrm{df}}_1,{\widehat{\mathrm{df}}}_2}) \\ =\mathrm{Pr}(F_{\mathrm{1,6};18.598}>F_{0.95;\mathrm{1,6}}=5.9874)=0.945 \end{gathered}$$