The author assume a lot of users may be radiologists or statisticians who want to compare modalities such as MRI, CT, PET,…etc.
Basic words  Truth = Positive  Truth = negative 

Reader’s positive  TP  FP 
Reader’s negative  FN  TN 
We only focus TP and FP, thus we call it hit and false alarms, respectively. That is:
Basic words  Truth = Positive  Truth = negative 

Reader’s positive  hit  False alarm 
Reader’s negative  FN  TN 
Number of hits are denoted by h
and number of false alarms are denoted by f
in the R console, respectively.
Number of hits are denoted by \(H\) in TeX and number of false alarms are denoted by \(F\) in TeX.
Suppose that there are \(\color{red}{N_I}\) images (e.g., radiograph) in which there is \(\color{red}{N_L}\) 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 multiples 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  \(H_{1}\)  \(F_{1}\) 
Note that \(H_{c},F_c \in \mathbb{N} \cup\{0\}\) for \(c=1,2,...,5\).
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  \(H_{1}=31\)  \(F_{1}=74\) 
##### Minor remark Note that the maximal number of confidence level, denoted by C
, are included, however, confidence level vector c
should not be specified. If specified, will be ignored , since it is created by c <c(rep(C:1))
in the program and do not refer from user input data, where C
is the highest number of confidence levels. Should write down your hits and false alarms vector so that it is compatible with this automatically created vector c
.
Note that the confidence level vector is not required in the above code, but we assume it is a following vector:
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:
Please use BayesianFROC::create_dataset()
to make a your own dataset.
R console  Definitions 

h 
positive integer vector, representing Number of hits for each reader, confidence level and modality. 
f 
positive integer vector, representing Number of false alarms for each reader, confidence level and modality. 
NL 
positive integer, representing Number of Lesions. 
NI 
positive integer, representing Number of Images. 
C 
A natural number. The highest confidence level, representing reader’s most highest confidence, that is Definitely lesions exist. 
It is simple to fit FROC models to data, that is run the function BayesianFROC::fit_Bayesian_FROC()
as follows:
# I do not know why, but a computer 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()
rstan::stan()
.\[\begin{eqnarray*} H_{c } & \sim &\text{Binomial} ( p_{c}, N_{L} ), \text{ for $c=1,2,...,C$.}\\ F_{c } & \sim &\text{Poisson}( (\lambda _{c} \lambda _{c+1} )\times N_{I} ), \text{ for $c=1,2,...,C1$.}\\ \lambda _{c}& =&  \log \Phi ( z_{c } ),\text{ for $c=1,2,...,C$.}\\ p_{c} &=&\Phi (\frac{z_{c +1}\mu}{\sigma})\Phi (\frac{z_{c}\mu}{\sigma}), \text{ for $c=1,2,...,C1$.}\\ p_C & =& 1\Phi (\frac{z_{C}\mu}{\sigma}),\\ F_{C} & \sim & \text{Poisson}( (\lambda _{C}  0)N_I),\\ dz_c=z_{c+1}z_{c} &\sim& \text{Uniform}(0,\infty), \text{ for $c=1,2,...,C1$.}\\ \mu &\sim& \text{Uniform}(\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( \infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( \infty,100000)\).
For the details, please see the authors paper. Note that this model is used if default value ModifiedPoisson = FALSE
is retained .
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.
This is a basic example which shows how to fit a model to data dataList
of single reader and single modality.
#0) To avoid the following error in Readme file, I have to attach the Rcpp. I do not know why such error occur withou Rcpp. This error occurs only when I run the following R scripts from readme.
#Error
#in do.call(rbind,sampler_params) :second argument must be a list Calles:<Anonymous>...get_divergent_iterations >sampler_param_vector =. do.call Execution halted
library(Rcpp) # This code can remove the above unknown error, if someone know why the error occur, please tell me.
#1) Build data for singler reader and single modality case.
dataList < list(c=c(3,2,1), # c is ignored, can omit.
h=c(97,32,31),
f=c(1,14,74),
NL=259,
NI=57,
C=3)
# where,
# c denotes confidence level, each components indicates that
# 3 = Definitely lesion,
# 2 = subtle,
# 1 = very subtle
# h denotes number of hits
# (True Positives: TP) for each confidence level,
# f denotes number of false alarms
# (False Positives: FP) for each confidence level,
# NL denotes number of lesions (signal),
# NI denotes number of images,
#2) Fit the FROC model.
fit < BayesianFROC::fit_Bayesian_FROC(
dataList,
#The number of MCMC chains
cha = 4
)
# validation of fit via alculation of p value of the chi square goodness of fit, which is
# calculated by integrating with predictive posterior measure.
BayesianFROC::p_value_of_the_Bayesian_sense_for_chi_square_goodness_of_fit(fit, head.only = TRUE)
Note that the above list object dataList
representing the following FROC data;
Number of Confidence Level  Number of Hits  Number of False alarms 

3 = definitely present  97  1 
2 = equivocal  32  14 
1 = questionable  31  74 
#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:
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 bi normal 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 Freeresponse 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.
\({}^{\dagger}\) traditionally, case means modality in this context. # Modality Comparison Which methods are more useful to detect lesions in radiographs from MRI, CT, PET,…
This package provides the solution of this Modality comparison issue by Bayesian approaches.
dataset_creator_new_version()
or create_dataset()
convertFromJafroc()
fit_Bayesian_FROC()
DrawCurves()
Prepare data
#An example dataset for the case of Multiple readers and Multiple Modalities.
dat < BayesianFROC::dataList.Chakra.Web
Fitting
# Fitting for your data with respect to the hierarchical Bayesian model.
fit<BayesianFROC::fit_Bayesian_FROC(dat)
Draw Curves
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  \(H_{1,2,2}\)  \(F_{1,2,2}\) 
where, each component \(H\) or \(F\) is non negative integers.
This package has example data, for example, the following object in this package is an MRMC dataset:
R console  Definitions 

m 
positive integer vector, representing Modality ID, 
q 
positive integer vector, representing Reader ID, 
c 
positive integer vector, representing Confidence level, 
h 
positive integer vector, representing Number of hits for each reader, confidence level and modality. 
f 
positive integer vector, representing Number of false alarms for each reader, confidence level and modality. 
NL 
positive integer, representing Number of Lesions. 
NI 
positive integer, representing Number of Images. 
M 
positive integer, representing Number of modalities. 
Q 
positive integer, Number of Readers. 
Note the confidence level should be the above order, i.e., 5,4,3,2,1,5,4,3,… and not 1,2,3,4,5,1,2,3,… . If you make by your hand, please be careful !!
This data can be shown more intuitive manner by the following code: Note that what we need to run the function, the above data is only needed to run other procedures. The following code merely show the data.
Compatibility for Jafroc:
Note that this dataset is transformed from an Rjafroc dataset made by Chakraborty.
By the function BayesianFROC:convertFromJafroc()
, users can transform the Jafroc datasets to this package dataset.
Help assistance for user’s making data:
Please use BayesianFROC::create_dataset()
to make a dataset.
If you have some Jafroc data, then please use BayesianFROC::convertFromJafroc()
.
In this package, the author use the sheet names and column names in programs, so user have to obey the strict rules. For details, please use help("convertFromJafroc")
.
To fit the model, it is same as the case of single reader and single modality as follows:
# I do not why, but Machine cannot find some function in Rcpp. So I have to load the package Rcpp.
library(Rcpp)
# Prepare dataset
dat < BayesianFROC::dataList.Chakra.Web
#Fitting
fit <BayesianFROC::fit_Bayesian_FROC(dat)
It is different to the single reader and single modality case user should run the another function DrawCurves_MRMC_pairwise()
to draw the curves such as FROC, AFROC, or cumulative hits and cumulative false alarms. Since drawing the curve is very heavy procedures, so the author separate this drawing process from fitting procedures.
We use \(H_{c,m,r}\) instead of \(H_c\), with \(H_{c,m,r}\) indicating the number of hits by the \(r^{\text{th}}\) reader using \(m^{\text{th}}\) modality with his or her confidence level \(c\), and we will use \(F_{c,m,r}\) similarly. Other quantities are also extended in this manner by adding subscripts \(m,r\), indicating that calculations are performed, respectively, for each \(r^{\text{th}}\) reader and \(m^{\text{th}}\) modality.
In the function BayesianFROC::fit_Bayesian_FROC()
of this package for MRMC data, we implement the following statistical models which will be used to explain how the data \((H_{c,m,r},F_{c,m,r})\) arise for number of lesions \(N_L\).
\[H_{c,m,r} \sim \text{Binomial}( p_{c,m,r}, N_L ), \] \[F_{c,m,r} \sim \text{Poisson} ( ( \lambda _{c}  \lambda _{c+1})N_L ), \] \[\lambda _{c} =  \log \Phi (z_{c }), \] \[p_{c,m,r}
:= \Phi (\frac{z_{c +1}\mu_{m,r}}{\sigma_{m,r}})\Phi (\frac{z_{c}\mu_{m,r}}{\sigma_{m,r}}), \]
\[A_{m,r} := \Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \] \[A_{m,r} \sim \text{Normal} (A_{m},\sigma_{r}^2), \] \[dz_c := z_{c+1}z_{c}, \] \[dz_c, \sigma_{m,r} \sim \text{Uniform}(0,\infty), \] \[z_{c,m,r} \sim \text{Uniform}( \infty,100000), \] \[A_{m} \sim \text{Uniform}(0,1). \]
Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C}\), \(A_{m}\), \(\sigma_{r}\), \(\mu_{m,r}\), and \(\sigma_{m,r}\).
library(Rcpp)
# Prepare a dataset
dat < BayesianFROC::dataList.Chakra.Web
# Fitting
fit < BayesianFROC::fit_Bayesian_FROC(dat)
# Draw curves for the 1st modality and 2nd reader
DrawCurves(
# This is the above fitted model object
fit,
# Specify Modatity ID by vector whose curve are drawn.
modalityID =1,
# Specify readerID ID by vector whose curve are drawn.
readerID =2)
Note that you need to input your dataset dat
and estimates in fit
to draw the curves. Of course you should specify reader and modality whose curves you want to write.
To identify reader and modality ID you can use vector, for example, modalityID = c(1,3)
for the first modality and the second modalities. readerID = c(1,2)
for the first and the second readers.
If user has any questions, send me a mail.
tsunoda.issei1111 at gmail.com
If user has any questions, please tell me.
If I am wrong, then please let me know. My background is mathematics, especially Differential geometry. So I can understand any mathematical materials,
tsunoda.issei1111 at gmail.com
abbreviation  word  meaning  TeX  R console 

Reader  Radiologist or doctor or etc  reader try to find lesion (nodule) from  Subscript \(r\) for reader ID and \(R\) for the number of readers  qd for reader ID and Q for the number of readers 
SRSC  single reader and single case  Data type  srsc 

MRMC  multiple reader and multiple case  Data type  
H , TP  Hit  Number of True Positive  \(H\)  h 
F , FP  Number of False Positive  \(F\)  f 

AUC  Area under the curve  curve indicating AFROC curve  Single reader and single modality case it is denoted by \(A\). In MRMC case, \(A_m\) for mth modality or \(A_{m,r}\) for the \(m\)th modality and \(r\)th reader  A for one indexed array or AA for array having two subscripts, \(A_m\)=A[md] , \(A_{m,r}\) = AA[md,qd] where \(m\)=md indicating modality ID, and \(r\) = rd indicating reader ID. 
Signal  nodule or lesion  Nonhealthy case  