Table 10.

OR-to-RMH algorithm for computing parameter values for the RMH model that correspond to specified OR parameter values.

Step 1. Solve for $x_1$ and $x_2:$

$x_1={\mathrm{\Phi }}^{-1}({\widehat{\theta }}_1)\text{and}{x}_2={\mathrm{\Phi }}^{-1}({\widehat{\theta }}_2)$

Step 2. Solve for $x_3$, using the values for $x_1$ and $x_2$ obtained in step 1:

$x_3=\{0\le x_3<1:{\widehat{\sigma }}_{R:\mathrm{OR}}^2-F_{\mathrm{BVN}}(x_1,x_2;x_3)+\mathrm{\Phi }(x_1)\mathrm{\Phi }(x_2)=0\}$

From the relationship $F_{\mathrm{BVN}}(x,y;{\rho }_1)<F_{\mathrm{BVN}}(x,y;{\rho }_2)$ if ${\rho }_1<{\rho }_2$,33 where $F(·,·;\rho )$ is the standardized bivariate normal distribution function with correlation $\rho$, it follows that ${\widehat{\sigma }}_{R:\mathrm{OR}}^2-F_{\mathrm{BVN}}(x_1,x_2;x_3)+\mathrm{\Phi }(x_1)\mathrm{\Phi }(x_2)$ is an increasing function of $x_3$ and hence $x_3$ can be easily determined numerically. Numerical solutions for $x_4,x_5,x_6$, and $x_7$ can be similarly determined in steps 3 and 6.

Step 3. Solve for $x_4$, using the values for $x_1$ and $x_2$ obtained in step 1:

$x_4=\max [x_3,\{0\le x_4<1:{\widehat{\sigma }}_{TR:\mathrm{OR}}^2+{\widehat{\sigma }}_{R:\mathrm{OR}}^2-.5\sum _{i=1}^2\{F_{\mathrm{BVN}}(x_i,x_i;x_4)-{[\mathrm{\Phi }(x_i)]}^2\}=0\left\}\right]$

Step 4. Solve for $b$ using one of the following b_method options. The resulting value of $b$ is used for the remaining steps.

b_method = unspecified: Solve for $b$, using the values for $x_1,x_2$, and $x_4$ obtained in steps 1 and 3:

$b=\{b>0:{\widehat{\sigma }}_{\epsilon :\mathrm{OR}}^2-\frac{1}{2}\sum _{i=1}^2\sum _{m=1}^4{c}_m{F}_{\mathrm{BVN}}(x_i,x_i;{\rho }_m(1-x_4)+x_4)=0\}$

where ${\rho }_1=1,{\rho }_2=\frac{1}{1+b^{-2}},{\rho }_3=\frac{1}{1+b^2},{\rho }_4=0$. With this option there can be 0, 1, or 2 possible solutions for $b$. The algorithm returns the largest solution such that $0.001\le b\le 1$ if it exists; otherwise, it returns the smallest solution such that $1\le b\le 4$ if it exists, or a missing value if it does not exist.

b_method = specified: Use the specified value of $b$.

b_method = mean_to_sigma: Solve for the value of $b$ that corresponds to a specified mean-to-sigma ratio and the minimum of the specified values for the expected test 1 and test 2 AUCs. (See Sec. B.2 for details.)

Step 5. Compute OR covariance estimates to be used in step 6.

(a) If b_method = unspecified was used in step 4, compute

${\widehat{\mathrm{Cov}}}_i={\widehat{r}}_i{\widehat{\sigma }}_{\epsilon :\mathrm{OR}}^2,i=\mathrm{1,2},3$.

(b) If one of the other two methods was used in step 4, then using the computed value of $b$ and the inputted correlations ${\widehat{r}}_1,{\widehat{r}}_2$ and ${\widehat{r}}_3$, compute a new value for the OR error variance, given by ${\tilde{\sigma }}_{\epsilon :\mathrm{OR}}^2=\frac{1}{2}\sum _{i=1}^2\sum _{m=1}^4{c}_m{F}_{\mathrm{BVN}}(x_i,x_i;{\rho }_m(1-x_4)+x_4)$, where ${\rho }_1=1,{\rho }_2=\frac{1}{1+b^{-2}},{\rho }_3=\frac{1}{1+b^2},{\rho }_4=0.$ Then compute

${\widehat{\mathrm{Cov}}}_i={\widehat{r}}_i{\tilde{\sigma }}_{\epsilon :\mathrm{OR}}^2,i=\mathrm{1,2},3.$

Step 6. Solve for $x_5,x_{6,}$ and $x_7$, using the following equations and the values for $x_1,x_2,x_4$, $b$ and ${\widehat{\mathrm{Cov}}}_i,i=\mathrm{1,2},3$, obtained in steps 1, 3, and 5:

$x_5=\{0\le x_5<1:{\widehat{\mathrm{Cov}}}_3-\sum _{m=1}^4{c}_m{F}_{\mathrm{BVN}}(x_1,x_2;{\rho }_m(1-x_4))=0\}$

where ${\rho }_1=x_5,{\rho }_2=\frac{x_5}{1+b^{-2}},{\rho }_3=\frac{x_5}{1+b^2},{\rho }_4=0$

$x_6=\max [x_5,\{0\le x_6<1:{\widehat{\mathrm{Cov}}}_2-\frac{1}{2}\sum _{i=1}^2\sum _{m=1}^4{c}_m{F}_{\mathrm{BVN}}(x_i,x_i;{\rho }_m(1-x_4))=0\left\}\right]$

where ${\rho }_1=x_6,{\rho }_2=\frac{x_6}{1+b^{-2}},{\rho }_3=\frac{x_6}{1+b^2},{\rho }_4=0$

$x_7=\max [x_5,\{0\le x_7<1:{\widehat{\mathrm{Cov}}}_1-\sum _{m=1}^4{c}_m{F}_{\mathrm{BVN}}(x_1,x_2;{\rho }_m(1-x_4)+x_3)=0\left\}\right]$

where ${\rho }_1=x_7,{\rho }_2=\frac{x_7}{1+b^{-2}},{\rho }_3=\frac{x_7}{(1+b^2)},{\rho }_4=0$

Step 7. Solve for the estimated RMH parameter values as functions of the estimated alternative RMH parameter values using the mapping given in Table 7c.

Notes: ${\widehat{\theta }}_1$ and ${\widehat{\theta }}_2$ denote specified values of the reader-averaged performance empirical AUCs for tests 1 and 2, respectively; ${\widehat{\sigma }}_{R:\mathrm{OR}}^2,{\widehat{\sigma }}_{TR:\mathrm{OR}}^2$, and ${\widehat{\sigma }}_{\epsilon :\mathrm{OR}}^2$ denote specified values of the corresponding OR parameters, and ${\widehat{r}}_1,{\widehat{r}}_2$, and ${\widehat{r}}_3$ denote specified values for the OR correlations defined by $r_i={\mathrm{Cov}}_{i:\mathrm{OR}}/{\sigma }_{\epsilon :\mathrm{OR}}^2$. These specified values can be computed from real data or conjectured. $F_{\mathrm{BVN}}(.,.;\rho )$ is the standardized bivariate normal distribution function with correlation $\rho$. Note that constraints Eq. (23) in Table 7 have been incorporated into the preceding steps.