Single reader and Single modality

Issei Tsunoda

2019-05-04

Aims of Package

The author assume a lot of users may be radiologists or statisticians who want to compare modalities such as MRI, CT, PET.

Usage

Single reader and Single modality

Notation and Symbols for FROC task.

Suppose that there are \(N_I\) images (e.g., radiograph) in which there is \(N_L\) lesions that should be detected by radiologists. Each image may contain no lesions. Radiologist identify suspicious locations of lesions for each image if he suspects that there are lesions with his confidence level that is number \(1,2, ..., c, ..., C\). So, radiologist can answer multiple locations for a single image, this multipleness differ from ordinal ROC analysis which allows each reader only single dichotomous answer for each image. Summarizing his true positive \(H_c\) and false positive (false alarm) \(F_c\) for each confidence level generate a FROC dataset \((F_c,H_c)\). Now, we introduced the notations, \(N_L\), \(N_I\), \(H_c\), \(F_c\), \(C\). In the R console, these notations are represented by NL, NI, h, f, C.

If \(C=5\), then the dataset for FROC analysis is the follows;

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 \(H_{1}\) \(F_{1}\)

Example data.

 dat <- list(
#Confidence level.
c = c(3,2,1), 

#Number of hits for each confidence level.
h = c(97,32,31),

#Number of false alarms for each confidence level.
f = c(1,14,74),  

#Number of lesions
NL= 259,   

#Number of images
NI= 57,  

#Number of confidence level
 C= 3
)        

This code means the following data:

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 \(H_{1}=31\) \(F_{1}=74\)

Note that the confidence level vector is not required in the above code, but we assume it is a following vector:

c(3,2,1)

Do not confuse with c(1,2,3) and this order never permitted to users.

Note that the above example data is endowed in this package as the following object:

BayesianFROC::dataList.Chakra.1

Please use BayesianFROC::create_dataset() to make a your own dataset.

Fitting

It is simple to fit FROC models to data, that is run the function BayesianFROC::fit_Bayesian_FROC() as follows:

# I do not why, but Machine cannot find Rcpp function. So I have to attach the package Rcpp. This is not desired one for me.
library(Rcpp)

# Prepare dataset
dat <- BayesianFROC::dataList.Chakra.1 # data shown in the above example.

#Fitting
fit <-BayesianFROC::fit_Bayesian_FROC(dat)

The following will be done by BayesianFROC::fit_Bayesian_FROC()

Statistical model for FROC

\[H_{c } \sim \text{Binomial} ( p_{c}, N_{L} ),\] \[F_{c } \sim \text{Poisson} ( (\lambda _{c} -\lambda _{c+1} )\times N_{I} ),\] \[\lambda _{c} = - \log \Phi ( z_{c } ),\] \[p_{c} =\Phi (\frac{z_{c +1}-\mu}{\sigma})-\Phi (\frac{z_{c}-\mu}{\sigma}). \] In this model, \(z_{c},c=1,\cdots,C+1\), \(\mu\), and \(\sigma\) are the parameters to be estimated.

For the details, please see the authors paper. Note that this model is used if default value ModifiedPoisson = FALSE is retained .

Some minor change.

In the function BayesianFROC::fit_Bayesian_FROC(), if you enter ModifiedPoisson = TRUE then the above model is change into

\[F_{c } \sim \text{Poisson} ( (\lambda _{c} -\lambda _{c+1} )\times N_{L} ),\] for false alarms. Then this change the interpretation of parameters \(\lambda_c\) from false rates per image to per lesion.

The code for fitting

# I do not why, but my machine cannot find some function in Rcpp. So I have to load the package Rcpp. I think users does not need to load Rcpp.
library(Rcpp)

# Prepare dataset
dat <- BayesianFROC::dataList.Chakra.1 # data

#Fitting
fit <-BayesianFROC::fit_Bayesian_FROC(dat)

Interpretation of Outputs

The results of BayesianFROC::fit_Bayesian_FROC(dat) are as follows:

The correspondence of notations between the R console and the author’s paper:

The dictionary

R console The author’s paper(*) (LateX) Definition
A \(A\) AUC. ( the area under the AFROC curve )
z[1] \(z_1\) Threshold of the binormal assumption for confidence level 1
z[2] \(z_2\) Threshold of the binormal assumption for confidence level 2
z[3] \(z_3\) Threshold of the binormal assumption for confidence level 3
z[4] \(z_4\) Threshold of the binormal assumption for confidence level 4
m \(\mu\) Mean of the Latent Gaussian variable for signal
v \(\sigma\) Standard deviation of the Latent Gaussian variable for signal
p[1] \(p_1\) Hit rate for confidence level 1
p[2] \(p_2\) Hit rate for confidence level 2
p[3] \(p_3\) Hit rate for confidence level 3
p[4] \(p_4\) Hit rate for confidence level 4
l[1] \(\lambda_1\) False alarm rate for confidence level 1
l[2] \(\lambda_2\) False alarm rate for confidence level 2
l[3] \(\lambda_3\) False alarm rate for confidence level 3

(*) The author’s paper: Bayesian Models for Free-response Receiver Operating Characteristic Analysis.

Note that v = \(\sqrt{\sigma^2} \neq \sigma^2\).

From here, we show the case of single reader and single modality.

For multiple readers and multiple modalities case, please show the other vignette.

Questions and Supports

If user has any questions, please tell me.

tsunoda.issei1111

by

gmail.com