Radiograph
Provides the following:
By \(\Phi()\), we means the cumulative distribution function of the Gaussian disitribution with mean 0 and variance 1.
d
Recall that the data named \(d\) appeared later in the R console has the following format:
Number of Confidence Level | Number of Hits | Number of False alarms |
---|---|---|
3 = definitely present | \(H_{3}=97\) | \(F_{3}=1\) |
2 = equivocal | \(H_{2}=32\) | \(F_{2}=14\) |
1 = questionable | \(H_{1}=31\) | \(F_{1}=74\) |
Further let \(N_L\) and \(N_I\) represent the number of Lesions and number of images, respectively. In the data d
, \(N_L=259\) and \(N_I=57\).
In case of data of multiple reader and multiple case, such hits and false alarms are calculated with each modality and reader, we omit to show it, since it is complex.
In this R script, the function rstan::sampling()
runs implicitly, thus, some stan file are used, I explain the model described in the stan file used in the above code.
First, for the simplicity, we show the definition of Bayesian Model without any explanation or proof:
\[\begin{eqnarray*} H_{c } & \sim &\text{Binomial} ( p_{c}, N_{L} ), \text{ for $c=1,2,...,C$.}\\ F_{c } & \sim &\text{Poisson}( (\lambda _{c} -\lambda _{c+1} )\times N_{I} ), \text{ for $c=1,2,...,C-1$.}\\ \lambda _{c}& =& - \log \Phi ( z_{c } ),\text{ for $c=1,2,...,C$.}\\ p_{c} &=&\Phi (\frac{z_{c +1}-\mu}{\sigma})-\Phi (\frac{z_{c}-\mu}{\sigma}), \text{ for $c=1,2,...,C-1$.}\\ p_C & =& 1-\Phi (\frac{z_{C}-\mu}{\sigma}),\\ F_{C} & \sim & \text{Poisson}( (\lambda _{C} - 0)N_I),\\ dz_c=z_{c+1}-z_{c} &\sim& \text{Uniform}(0,\infty), \text{ for $c=1,2,...,C-1$.}\\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] 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 \(( -\infty,100000)\).
Next we show the model of the following code:
\[\begin{eqnarray*} H_{c } & \sim &\text{Binomial} ( p_{c}, N_{L} ), \text{ for $c=1,2,...,C$.}\\ F_{c } & \sim &\text{Poisson}( (\lambda _{c} -\lambda _{c+1} )\times N_{L} ), \text{ for $c=1,2,...,C-1$.}\\ \lambda _{c}& =& - \log \Phi ( z_{c } ),\text{ for $c=1,2,...,C$.}\\ p_{c} &=&\Phi (\frac{z_{c +1}-\mu}{\sigma})-\Phi (\frac{z_{c}-\mu}{\sigma}), \text{ for $c=1,2,...,C-1$.}\\ p_C & =& 1-\Phi (\frac{z_{C}-\mu}{\sigma}),\\ F_{C} & \sim & \text{Poisson}( (\lambda _{C} - 0)N_I),\\ dz_c=z_{c+1}-z_{c} &\sim& \text{Uniform}(0,\infty), \text{ for $c=1,2,...,C-1$.}\\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] 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 \(( -\infty,100000)\).
The second model differs from the first model in the Poisson part.
where, dd
are data of multiple reader and multiple modality., which is not shown nor explained here for simplicity.
The R object fit
is a fitted model object.
Next, we shall show the Bayesian model using the above code.
\[\begin{eqnarray*} H_{c,m,r} & \sim &\text{Binomial }( p_{c,m,r}, N_L ),\\ F_{c,m,r} &\sim& \text{Poisson }( ( \lambda _{c} - \lambda _{c+1})N_L ),\\ \lambda _{c}& =& - \log \Phi (z_{c }),\\ p_{c,m,r} &:=&\Phi (\frac{z_{c +1}-\mu_{m,r}}{\sigma_{m,r}})-\Phi (\frac{z_{c}-\mu_{m,r}}{\sigma_{m,r}}), \\ p_C & =& 1-\Phi (\frac{z_{C}-\mu_{m,r}}{\sigma_{m,r}}),\\ F_{C,m,r} & \sim &\text{Poisson } ( (\lambda _{C} - 0)N_I),\\ A_{m,r}&:=&\Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \\ A_{m,r}&\sim&\text{Normal} (A_{m},\sigma_{r}^2), \\ dz_c&:=&z_{c+1}-z_{c},\\ dz_c, \sigma_{m,r} &\sim& \text{Uniform}(0,\infty),\\ z_{c} &\sim& \text{Uniform}( -\infty,100000),\\ A_{m} &\sim& \text{Uniform}(0,1).\\ \end{eqnarray*}\] Our new model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C}\), \(A_{m}\), \(\sigma_{r}\), \(\mu_{m,r}\), and \(\sigma_{m,r}\).
In the above model, author write down the highest confidence level separately. Because if not, then it cause the divergent transition issue. In the past, I wrote the model with the assumption that the highest threshold \(z_{C+1}\) is the very large number (theoretically it is infinity) and whose prior is the uniform distribution with very large support, then it cause the divergent transitions almost all iterations in the MCMC. Also, I use the target formulation to avoid Jacobin warnings, such warnings also difficult for me to solve or understand what it say or how overcome.
One may thinks my models are very complex. In fact many people will hate my explanation. One reason why my model is so complex is the FROC statistical model is very complex and without the reader’s effort, cannot understand. Further more, the description of my model is changed to avoid the divergent transition issues, it make my model more complex one. Furhter I had to overcome Jacobian issues, which formulation is not intuitive for me.
I am very tired to explain my model and also disappeared my ability :’-D
These model are made in two years ago, 2017.May ~ 2017. August.
I briefly explain the divergent transition in my model. In theoretical perspective, it natural to introduce the highest decision threshold as an very large number, which is a parameter of model. Unfortunately, it is distributed by the uniform distributions with very large support. Then once we introduce such theoretically infinity parameter in the numerical program, it case the divergent transition issues.
This vignettes are most important one, and if reader wish to understand these theory, then please read my paper.
My paper cannot upload to Arxiv, since I cannot find some who endorse my paper.
include in this package other hierarchical models, but I do not exhibit all of them here, my health is very bad, so, I am tired, sorry.
In FROC model. we need to … my aches stop me to write … good bye !!