Very Brief description

Nowadays many people have no time, so, it is for such busy person.

The author entrust complement of the logic to reader.

Before reading this, execute the following R script:

Graphical User Interface: GUI

library(BayesianFROC)
          BayesianFROC::fit_GUI() #   Enjoy fitting!

Or

library(BayesianFROC)
          fit_GUI_dashboard() #   Enjoy fitting!

Or

library(BayesianFROC)
           fit_GUI_simple() #   Enjoy fitting!

Then reader will understand what this package is.

\(\color{green}{\textit{Single reader and Single modality }}\)

Data
Confidence Level No. of Hits No. of False alarms
5 = definitely present \(H_{5}\) \(F_{5}\)
4 = probably present \(H_{4}\) \(F_{4}\)
3 = equivocal \(H_{3}\) \(F_{3}\)
2 = probably absent \(H_{2}\) \(F_{2}\)
1 = questionable \(H_{1}\) \(F_{1}\)

Note that \(H_{c},F_c \in \mathbb{N} \cup\{0\}\) for \(c=1,2,...,5\).

Model
false positive fraction per image

\[\begin{eqnarray*} H_{c } & \sim &\text{Binomial} ( p_{c}, N_{L} ), \text{ for $c=1,2,...,C$.}\\ F_{c } & \sim &\text{Poisson}( (\lambda _{c} -\lambda _{c+1} )\times N_{I} ), \text{ for $c=1,2,...,C-1$.}\\ \lambda _{c}& =& - \log \Phi ( z_{c } ),\text{ for $c=1,2,...,C$.}\\ p_{c} &=&\Phi (\frac{z_{c +1}-\mu}{\sigma})-\Phi (\frac{z_{c}-\mu}{\sigma}), \text{ for $c=1,2,...,C-1$.}\\ p_C & =& 1-\Phi (\frac{z_{C}-\mu}{\sigma}),\\ F_{C} & \sim & \text{Poisson}( (\lambda _{C} - 0)N_I),\\ dz_c=z_{c+1}-z_{c} &\sim& \text{Uniform}(0,\infty), \text{ for $c=1,2,...,C-1$.}\\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C-1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( -\infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( -\infty,100000)\).

Or equivalently, if \(C=5\)

\[\begin{eqnarray*} H_{1 } & \sim &\text{Binomial} ( p_{1}, N_{L} ) \\ H_{2 } & \sim &\text{Binomial} ( p_{2}, N_{L} ) \\ H_{3 } & \sim &\text{Binomial} ( p_{3}, N_{L} ) \\ H_{4 } & \sim &\text{Binomial} ( p_{4}, N_{L} ) \\ H_{5 } & \sim &\text{Binomial} ( p_{4}, N_{L} ) \\ F_{1 } & \sim &\text{Poisson}( (\lambda _{1} -\lambda _{2} )\times N_{I} ), \\ F_{2 } & \sim &\text{Poisson}( (\lambda _{2} -\lambda _{3} )\times N_{I} ), \\ F_{3 } & \sim &\text{Poisson}( (\lambda _{3} -\lambda _{4} )\times N_{I} ), \\ F_{4 } & \sim &\text{Poisson}( (\lambda _{4} -\lambda _{5} )\times N_{I} ), \\ F_{5 } & \sim &\text{Poisson}( (\lambda _{5} -0 )\times N_{I} ), \text{Be careful !!:'-D}\\ \lambda _{1}& =& - \log \Phi ( z_{1 } ),\\ \lambda _{2}& =& - \log \Phi ( z_{2 } ),\\ \lambda _{3}& =& - \log \Phi ( z_{3 } ),\\ \lambda _{4}& =& - \log \Phi ( z_{4 } ),\\ \lambda _{5}& =& - \log \Phi ( z_{5 } ),\\ p_{1} &:=&\Phi (\frac{z_{2}-\mu}{\sigma})-\Phi (\frac{z_{1}-\mu}{\sigma}), \\ p_{2} &:=&\Phi (\frac{z_{3}-\mu}{\sigma})-\Phi (\frac{z_{2}-\mu}{\sigma}), \\ p_{3} &:=&\Phi (\frac{z_{c4}-\mu}{\sigma})-\Phi (\frac{z_{3}-\mu}{\sigma}), \\ p_{4} &:=&\Phi (\frac{z_{5}-\mu}{\sigma})-\Phi (\frac{z_{4}-\mu}{\sigma}), \\ p_5 &:=& 1-\Phi (\frac{z_{5}-\mu}{\sigma}),\text{Be careful !!:'-D}\\ dz_c=z_{2}-z_{1} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{3}-z_{2} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{4}-z_{3} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{5}-z_{4} &\sim& \text{Uniform}(0,\infty), \\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C-1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( -\infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( -\infty,100000)\).

R script for the model with FPF per image
false positive fraction per image

\[\begin{eqnarray*} H_{c } & \sim &\text{Binomial} ( p_{c}, N_{L} ), \text{ for $c=1,2,...,C$.}\\ F_{c } & \sim &\text{Poisson}( (\lambda _{c} -\lambda _{c+1} )\times N_{L} ), \text{ for $c=1,2,...,C-1$.}\\ \lambda _{c}& =& - \log \Phi ( z_{c } ),\text{ for $c=1,2,...,C$.}\\ p_{c} &=&\Phi (\frac{z_{c +1}-\mu}{\sigma})-\Phi (\frac{z_{c}-\mu}{\sigma}), \text{ for $c=1,2,...,C-1$.}\\ p_C & =& 1-\Phi (\frac{z_{C}-\mu}{\sigma}),\\ F_{C} & \sim & \text{Poisson}( (\lambda _{C} - 0)N_L),\\ dz_c=z_{c+1}-z_{c} &\sim& \text{Uniform}(0,\infty), \text{ for $c=1,2,...,C-1$.}\\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C-1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( -\infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( -\infty,100000)\).

Or equivalently, if \(C=5\)

\[\begin{eqnarray*} H_{1 } & \sim &\text{Binomial} ( p_{1}, N_{L} ) \\ H_{2 } & \sim &\text{Binomial} ( p_{2}, N_{L} ) \\ H_{3 } & \sim &\text{Binomial} ( p_{3}, N_{L} ) \\ H_{4 } & \sim &\text{Binomial} ( p_{4}, N_{L} ) \\ H_{5 } & \sim &\text{Binomial} ( p_{4}, N_{L} ) \\ F_{1 } & \sim &\text{Poisson}( (\lambda _{1} -\lambda _{2} )\times N_{L} ), \\ F_{2 } & \sim &\text{Poisson}( (\lambda _{2} -\lambda _{3} )\times N_{L} ), \\ F_{3 } & \sim &\text{Poisson}( (\lambda _{3} -\lambda _{4} )\times N_{L} ), \\ F_{4 } & \sim &\text{Poisson}( (\lambda _{4} -\lambda _{5} )\times N_{L} ), \\ F_{5 } & \sim &\text{Poisson}( (\lambda _{5} -0 )\times N_{L} ), \text{Be careful !!:'-D}\\ \lambda _{1}& =& - \log \Phi ( z_{1 } ),\\ \lambda _{2}& =& - \log \Phi ( z_{2 } ),\\ \lambda _{3}& =& - \log \Phi ( z_{3 } ),\\ \lambda _{4}& =& - \log \Phi ( z_{4 } ),\\ \lambda _{5}& =& - \log \Phi ( z_{5 } ),\\ p_{1} &:=&\Phi (\frac{z_{2}-\mu}{\sigma})-\Phi (\frac{z_{1}-\mu}{\sigma}), \\ p_{2} &:=&\Phi (\frac{z_{3}-\mu}{\sigma})-\Phi (\frac{z_{2}-\mu}{\sigma}), \\ p_{3} &:=&\Phi (\frac{z_{c4}-\mu}{\sigma})-\Phi (\frac{z_{3}-\mu}{\sigma}), \\ p_{4} &:=&\Phi (\frac{z_{5}-\mu}{\sigma})-\Phi (\frac{z_{4}-\mu}{\sigma}), \\ p_5 &:=& 1-\Phi (\frac{z_{5}-\mu}{\sigma}),\text{Be careful !!:'-D}\\ dz_c=z_{2}-z_{1} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{3}-z_{2} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{4}-z_{3} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{5}-z_{4} &\sim& \text{Uniform}(0,\infty), \\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C-1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( -\infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( -\infty,100000)\).

R script for the model with FPF per lesion

\(\color{green}{\textit{Multiple reader and Multiple case}{}^{\dagger} }\)

\({}^{\dagger}\) traditionally, case means modality in this context.

Example Data format

Two readers and two modalities and three kind of confidence levels.

Confidence Level Modality ID Reader ID Number of Hits Number of False alarms
3 = definitely present 1 1 \(H_{3,1,1}\) \(F_{3,1,1}\)
2 = equivocal 1 1 \(H_{2,1,1}\) \(F_{2,1,1}\)
1 = questionable 1 1 \(H_{1,1,1}\) \(F_{1,1,1}\)
3 = definitely present 1 2 \(H_{3,1,2}\) \(F_{3,1,2}\)
2 = equivocal 1 2 \(H_{2,1,2}\) \(F_{2,1,2}\)
1 = questionable 1 2 \(H_{1,1,2}\) \(F_{1,1,2}\)
3 = definitely present 2 1 \(H_{3,2,1}\) \(F_{3,2,1}\)
2 = equivocal 2 1 \(H_{2,2,1}\) \(F_{2,2,1}\)
1 = questionable 2 1 \(H_{1,2,1}\) \(F_{1,2,1}\)
3 = definitely present 2 2 \(H_{3,2,2}\) \(F_{3,2,2}\)
2 = equivocal 2 2 \(H_{2,2,2}\) \(F_{2,2,2}\)
1 = questionable 2 2 \(H_{1,2,2}\) \(F_{1,2,2}\)

Alternative model

\[\begin{eqnarray*} H_{c,m,r} & \sim &\text{Binomial }( p_{c,m,r}, N_L ),\\ F_{c,m,r} &\sim& \text{Poisson }( ( \lambda _{c} - \lambda _{c+1})N_L ),\\ \lambda _{c}& =& - \log \Phi (z_{c }),\\ p_{c,m,r} &:=&\Phi (\frac{z_{c +1}-\mu_{m,r}}{\sigma_{m,r}})-\Phi (\frac{z_{c}-\mu_{m,r}}{\sigma_{m,r}}), \\ p_C & =& 1-\Phi (\frac{z_{C}-\mu_{m,r}}{\sigma_{m,r}}),\\ F_{C,m,r} & \sim &\text{Poisson } ( (\lambda _{C} - 0)N_I),\\ A_{m,r}&:=&\Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \\ A_{m,r}&\sim&\text{Normal} (A_{m},\sigma_{r}^2), \\ dz_c&:=&z_{c+1}-z_{c},\\ dz_c, \sigma_{m,r} &\sim& \text{Uniform}(0,\infty),\\ z_{c} &\sim& \text{Uniform}( -\infty,100000),\\ A_{m} &\sim& \text{Uniform}(0,1).\\ \end{eqnarray*}\] Our new model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C}\), \(A_{m}\), \(\sigma_{r}\), \(\mu_{m,r}\), and \(\sigma_{m,r}\).

R script for fitting the alternative model.
Null Hypothesis model

\[\begin{eqnarray*} H_{c,m,r} & \sim &\text{Binomial }( p_{c,m,r}, N_L ),\\ F_{c,m,r} &\sim& \text{Poisson }( ( \lambda _{c} - \lambda _{c+1})N_L ),\\ \lambda _{c}& =& - \log \Phi (z_{c }),\\ p_{c,m,r} &:=&\Phi (\frac{z_{c +1}-\mu_{m,r}}{\sigma_{m,r}})-\Phi (\frac{z_{c}-\mu_{m,r}}{\sigma_{m,r}}), \\ p_C & =& 1-\Phi (\frac{z_{C}-\mu_{m,r}}{\sigma_{m,r}}),\\ F_{C,m,r} & \sim &\text{Poisson } ( (\lambda _{C} - 0)N_I),\\ A_{m,r}&:=&\Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \\ A_{m,r}&\sim&\text{Normal} ( \color{red}{A},\sigma_{r}^2), \\ dz_c&:=&z_{c+1}-z_{c},\\ dz_c, \sigma_{m,r} &\sim& \text{Uniform}(0,\infty),\\ z_{c} &\sim& \text{Uniform}( -\infty,100000),\\ \color{red}{A} &\sim& \text{Uniform}(0,1).\\ \end{eqnarray*}\] Our new model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C}\), \(A\), \(\sigma_{r}\), \(\mu_{m,r}\), and \(\sigma_{m,r}\).