# fit_Bayesian_FROC: Fit a model to data In BayesianFROC: FROC Analysis by Bayesian Approaches

## Description

Creates a fitted model object of class  stanfitExtended: an inherited class from the S4 class stanfit in rstan.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 fit_Bayesian_FROC( dataList, ModifiedPoisson = FALSE, prior = -1, verbose = TRUE, print_CI_of_AUC = TRUE, multinomial = TRUE, model_reparametrized = FALSE, Model_MRMC_non_hierarchical = TRUE, type_to_be_passed_into_plot = "l", ww = -11, www = 11, mm = 0.65, mmm = 11, vv = 5.31, vvv = 11, zz = 1.55, zzz = 11, prototype = FALSE, PreciseLogLikelihood = TRUE, DrawCurve = length(dataList$m) == 0, Drawcol = TRUE, summary = TRUE, make.csv.file.to.draw.curve = FALSE, mesh.for.drawing.curve = 1000, significantLevel = 0.7, new.imaging.device = TRUE, cha = 1, ite = 10000, DrawFROCcurve = TRUE, DrawAFROCcurve = FALSE, DrawCFPCTP = TRUE, dig = 5, war = floor(ite/5), see = 1234567, Null.Hypothesis = FALSE, ... )  ## Arguments dataList A list, specifying an FROC data to be fitted a model. It consists of data of numbers of TPs, FPs, lesions, images. .In addition, if in case of mutiple readers or mutiple modalities, then modaity ID and reader ID are included also. The dataList will be passed to the function rstan::sampling() of rstan. This is a variable in the function rstan::sampling() in which it is named data. For the single reader and a single modality data, the dataList is made by the following manner:  dataList.Example <- list(   h = c(41,22,14,8,1), # number of hits for each confidence level   f = c(1,2,5,11,13), # number of false alarms for each confidence level   NL = 124, # number of lesions (signals)   NI = 63, # number of images (trials)   C = 5) # number of confidence, .. the author thinks it can be calculated as the length of h or f ...? ha, why I included this. ha .. should be omitted.  Using this object dataList.Example, we can apply fit_Bayesian_FROC() such as fit_Bayesian_FROC(dataList.Example). To make this R object dataList representing FROC data, this package provides three functions:  convertFromJafroc() If data is a JAFROC xlsx formulation.  dataset_creator_new_version() Enter TP and FP data by table .  create_dataset() Enter TP and FP data by interactive manner. Before fitting a model, we can confirm our dataset is correctly formulated by using the function  viewdata(). —————————————————————————————- A Single reader and a single modality (SRSC) case. —————————————————————————————- In a single reader and a single modality case (srsc), dataList is a list consisting of f, h, NL, NI, C where f, h are numeric vectors and NL, NI, C are positive integers. f Non-negative integer vector specifying number of false alarms associated with each confidence level. The first component corresponding to the highest confidence level. h Non-negative integer vector specifying number of Hits associated with each confidence level. The first component corresponding to the highest confidence level. NL A positive integer, representing Number of Lesions. NI A positive integer, representing Number of Images. C A positive integer, representing Number of Confidence level. The detail of these dataset, see the datasets endowed with this package. 'Note that the maximal number of confidence level, denoted by C, are included, however, Note that 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 inner program and do not refer from user input data, where C is the highest number of confidence levels. So, you should write down your hits and false alarms vector so that it is compatible with this automatically created c vector. data Format: A single reader and a single modality case ——————————————————————————————————  NI=63,NL=124 confidence level No. of false alarms No. of hits In R console ->  c f  h ----------------------- ----------------------- ----------------------------- ------------- definitely present c[1] = 5 f[1] = F_5 = 1 h[1] = H_5 = 41 probably present c[2] = 4 f[2] = F_4 = 2 h[2] = H_4 = 22 equivocal c[3] = 3 f[3] = F_3 = 5 h[3] = H_3 = 14 subtle c[4] = 2 f[4] = F_2 = 11 h[4] = H_2 = 8 very subtle c[5] = 1 f[5] = F_1 = 13 h[5] = H_1 = 1 ————————————————————————————————— * false alarms = False Positives = FP * hits = True Positives = TP Note that in FROC data, all confidence level means present (diseased, lesion) case only, no confidence level indicating absent. Since each reader marks his suspicious location only if he thinks lesions are present, and marked positions generates the hits or false alarms, thus each confidence level represents that lesion is present. In the absent case, reader does not mark any locations and hence, the absent confidence level does not relate this dataset. So, if reader think it is no lesion, then in such case confidence level is not needed. Note that the first column of confidence level vector c  should not be specified. If specified, will be ignored , since it is created by  c <-c(rep(C:1)) automatically in the inner program and do not refer from user input data even if it is specified explicitly, where C is the highest number of confidence levels. So you should check the compatibility of your data and the confidence level vector  c <-c(rep(C:1)) via a table which can be displayed by the function viewdata(). ————————————————————————————— Multiple readers and multiple modalities case, i.e., MRMC case ————————————————————————————— In case of multiple readers and multiple modalities, i.e., MRMC case, in order to apply the function fit_Bayesian_FROC(), dataset represented by an R list object representing FROC data must contain components m,q,c,h,f,NL,C,M,Q. C  A positive integer, representing the highest number of confidence level, this is a scalar. M  A positive integer vector, representing the number of modalities. Q  A positive integer, representing the number of readers. m  A vector of positive integers, representing the modality ID vector. q  A vector of positive integers, representing the reader ID vector. c  A vector of positive integers, representing the confidence level. This vector must be made by rep(rep(C:1), M*Q) h  A vector of non-negative integers, representing the number of hits. f  A vector of non-negative integers, representing the number of false alarms. NL A positive integer, representing the Total number of lesions for all images, this is a scalar. Note that the maximal number of confidence level (denoted by C) are included in the above R object. However, each confidence level vector is not included in the data, because it is created automatically from C. To confirm false positives and hits are correctly ordered with respect to the automatically generated confidence vector, the function viewdata() shows the table. Revised 2019 Nov 27 Revised 2019 Dec 5 Example data. Multiple readers and multiple modalities ( i.e., MRMC) —————————————————————————————————  Modality ID Reader ID Confidence levels No. of false alarms No. of hits. m  q c  f  h -------------- ------------- ------------------------ ------------------- ---------------- 1 1 3 20 111 1 1 2 29 55 1 1 1 21 22 1 2 3 6 100 1 2 2 15 44 1 2 1 22 11 2 1 3 6 66 2 1 2 24 55 2 1 1 23 1 2 2 3 5 66 2 2 2 30 55 2 2 1 40 44 ————————————————————————————————— * false alarms = False Positives = FP * hits = True Positives = TP ModifiedPoisson Logical, that is TRUE or FALSE. If ModifiedPoisson = TRUE, then Poisson rate of false alarm is calculated per lesion, and model is fitted so that the FROC curve is an expected curve of points consisting of the pairs of TPF per lesion and FPF per lesion. Similarly, If ModifiedPoisson = TRUE, then Poisson rate of false alarm is calculated per image, and model is fitted so that the FROC curve is an expected curve of points consisting of the pair of TPF per lesion and FPF per image. For more details, see the author's paper in which I explained per image and per lesion. (for details of models, see vignettes , now, it is omiited from this package, because the size of vignettes are large.) If ModifiedPoisson = TRUE, then the False Positive Fraction (FPF) is defined as follows (F_c denotes the number of false alarms with confidence level c ) \frac{F_1+F_2+F_3+F_4+F_5}{N_L}, \frac{F_2+F_3+F_4+F_5}{N_L}, \frac{F_3+F_4+F_5}{N_L}, \frac{F_4+F_5}{N_L}, \frac{F_5}{N_L}, where N_L is a number of lesions (signal). To emphasize its denominator N_L, we also call it the False Positive Fraction (FPF) per lesion. On the other hand, if ModifiedPoisson = FALSE (Default), then False Positive Fraction (FPF) is given by \frac{F_1+F_2+F_3+F_4+F_5}{N_I}, \frac{F_2+F_3+F_4+F_5}{N_I}, \frac{F_3+F_4+F_5}{N_I}, \frac{F_4+F_5}{N_I}, \frac{F_5}{N_I}, where N_I is the number of images (trial). To emphasize its denominator N_I, we also call it the False Positive Fraction (FPF) per image. The model is fitted so that the estimated FROC curve can be ragraded as the expected pairs of FPF per image and TPF per lesion (ModifiedPoisson = FALSE ) or as the expected pairs of FPF per image and TPF per lesion (ModifiedPoisson = TRUE) If ModifiedPoisson = TRUE, then FROC curve means the expected pair of FPF per lesion and TPF. On the other hand, if ModifiedPoisson = FALSE, then FROC curve means the expected pair of FPF per image and TPF. So,data of FPF and TPF are changed thus, a fitted model is also changed whether ModifiedPoisson = TRUE or FALSE. In traditional FROC analysis, it uses only per images (trial). Since we can divide one image into two images or more images, number of trial is not important. And more important is per signal. So, the author also developed FROC theory to consider FROC analysis under per signal. One can see that the FROC curve is rigid with respect to change of a number of images, so, it does not matter whether ModifiedPoisson = TRUE or FALSE. This rigidity of curves means that the number of images is redundant parameter for the FROC trial and thus the author try to exclude it. Revised 2019 Dec 8 Revised 2019 Nov 25 Revised 2019 August 28 prior positive integer, to select the prior verbose A logical, if TRUE, then the redundant summary is printed in R console. If FALSE, it suppresses output from this function. print_CI_of_AUC Logical, if TRUE then Credible intervals of AUCs for each modality are plotted. multinomial A logical, if TRUE then model is the most classical one using multinomial distribution. model_reparametrized A logical, if TRUE, then a model under construction is used. Model_MRMC_non_hierarchical A logical. If TRUE, then the model of multiple readers and multiple modalities consits of no hyper parameters. The reason why the author made this parameter is that the hyper parameter make the MCMC posterior samples be unstable. And also, my hierarachical model is not so good in theoretical perspective. Thus, I made this. The Default is TRUE. type_to_be_passed_into_plot "l" or "p". zz, zzz, ww, www, mm, mmm, vv, vvv Each of which is a real number specifying one of the parameter of prior prototype A logical, if TRUE then the model is no longer a generative model. Namely, in generally speaking, a dataset drawn from the model cannot satisfy the condition that the sum of the numbers of hits over all confidence levels is bounded from the above by the number of lesions, namely, Σ_c H_c ≤ N_L However, this model (TRUE ) is good in the sense that it admits various initial values of MCMC sampling. if FALSE, then the model is precisely statistical model in the sense that any dataset drawn from the model satisfies that the sum of the number of hits is not greater than the number of lesions, namely, Σ_c H_c ≤ N_L. This model is theoretically perfect. However, in the practically, the calculation will generates some undesired results which caused by the so-called floo .... I forget English :'-D. The flood point??? I forgeeeeeeeeeeeeet!! Ha. So, prior synthesizes very small hit rates such as 0.0000000000000001234 and it cause the non accurate calculation such as 0.00000,,,00000123/0.000.....000012345= 0.0012 which becomes hit rate and thus OH No!. Then it synthesizes Bernoulli success rate which is not less than 1 !! To avoid this, the author should develop the theory of prior to avoid this very small numbers, however the author has idea but now it does not success. If prototype = TRUE, then the model for hits is the following: H_5 \sim Binomial(p_5,N_L) H_4 \sim Binomial(p_4,N_L) H_3 \sim Binomial(p_3,N_L) H_2 \sim Binomial(p_2,N_L) H_1 \sim Binomial(p_1,N_L) On the other hand, if prototype = FALSE, then the model for hits is the following: H_5 \sim Binomial( p_5,N_L ) H_4 \sim Binomial( \frac{p_4}{1-p_5},N_L - H_5) H_3 \sim Binomial( \frac{p_3}{1-p_5-p_4},N_L - H_5-H_4) H_2 \sim Binomial( \frac{p_2}{1-p_5-p_4-p_3},N_L - H_5-H_4-H_3) H_1 \sim Binomial( \frac{p_1}{1-p_5-p_4-p_3-p_2},N_L - H_5-H_4-H_3-H_2) Each number of lesions is adjusted so that the sum of hits Σ_c H_c is less than the number of lesions (signals, targets) N_L. And hence the model in case of prototype = FALSE is a generative model in the sense that it can replicate datasets of FROC arises. Note that the adjustment of the number of lesions in the above manner leads us the adjustment of hit rates. The reason why we use the hit rates such as \frac{p_2}{1-p_5-p_4-p_3} instead of p_c is that it ensures the equality E[H_c/N_L] = p_c. This equality is very important. To establish Bayesian FROC theory so that it is compatible to the classical FROC theory, we need the following two equations, E[H_c/N_L] = p_c, E[F_c/N_X] = q_c, where E denotes the expectation and N_X is the number of lesion or the number of images and q_c is a false alarm rate, namely, F_c \sim Poisson( q_c N_X). Using the above two equations, we can establish the alternative Bayesian FROC theory preserving classical notions and formulas. For the details, please see the author's pre print: Bayesian Models for ,,, for?? I forget my paper title .... :'-D. What the hell!? I forget,... My health is so bad to forget , .... I forget. The author did not notice that the prototype is not a generative model. And hence the author revised the model so that the model is exactly generative model. But the reason why the author remains the prototype model(prototype = TRUE) is that the convergence of MCMC sampling in case of MRMC is not good in the current model (prototype = FALSE) . Because it uses fractions \frac{p_1}{1-p_5-p_4-p_3-p_2} and which is very dangerous to numerical perspective. For example, if p_1 is very small, then the numerator and denominator of \frac{p_1}{1-p_5-p_4-p_3-p_2} is very small. Both of them is like 0.000000000000000123.... and such small number causes the non accurate results. So, sometimes, it occurs that \frac{p_1}{1-p_5-p_4-p_3-p_2} >1 which never occur in the theoretical perspective but unfortunately, in numerically occurs. SO, now, the author try to avoid such phenomenon by using priors but it now does not success. Here of course we interpret the terms such as N_L - H_5-H_4-H_3 as the remained targets after reader get hits. The author thinks it is another manner to do so like N_L -H_1-H2-H_3, but it does not be employed. Since the author thinks that the reader will assign his suspicious lesion location from high confidence level and in this view point the author thinks it should be considered that targets are found from the highest confidence suspicious location. PreciseLogLikelihood Logical, that is TRUE or FALSE. If PreciseLogLikelihood = TRUE(default), then Stan calculates the precise log likelihood with target formulation. If PreciseLogLikelihood = FALSE, then Stan calculates the log likelihood by dropping the constant terms in the likelihood function. In past, I distinct the stan file, one is target formulation and the another is not. But non-target formulation cause some Jacobian warning, thus I made all stanfile with target formulation when I uploaded to CRAN. Thus this variable is now meaningless. DrawCurve Logical: TRUE of FALSE. Whether the curve is to be drawn. TRUE or FALSE. If you want to draw the FROC and AFROC curves, then you set DrawCurve =TRUE, if not then DrawCurve =FALSE. The reason why the author make this variable DrawCurve is that it takes long time in MRMC case to draw curves, and thus Default value is FALSE in the case of MRMC data. Drawcol Logical: TRUE of FALSE. Whether the (A)FROC curve is to be drawn by using color of dark theme. The Default value is a TRUE. summary Logical: TRUE of FALSE. Whether to print the verbose summary. If TRUE then verbose summary is printed in the R console. If FALSE, the output is minimal. I regret, this variable name should be verbose. make.csv.file.to.draw.curve Logical: TRUE of FALSE. Whether to create a csv file. If TRUE then csv file is created in your desktop to draw an FROC curve and cumulative hits and false alarms by scatter plot. Default is FALSE since it took times to create csv files. mesh.for.drawing.curve A positive large integer, indicating number of dots drawing the curves, Default =10000. significantLevel This is a number between 0 and 1. The results are shown if posterior probabilities are greater than this quantity. new.imaging.device Logical: TRUE of FALSE. If TRUE (default), then open a new device to draw curve. Using this we can draw curves in same plain by new.imaging.device=FALSE. cha A variable to be passed to the function rstan::sampling() of rstan in which it is named chains. A positive integer representing the number of chains generated by Hamiltonian Monte Carlo method, and, Default = 1. ite A variable to be passed to the function rstan::sampling() of rstan in which it is named iter. A positive integer representing the number of samples synthesized by Hamiltonian Monte Carlo method, and, Default = 10000. DrawFROCcurve Logical: TRUE of FALSE. Whether the FROC curve is to be drawn. DrawAFROCcurve Logical: TRUE of FALSE. Whether the AFROC curve is to be drawn. DrawCFPCTP Logical: TRUE of FALSE. Whether the CFP and CTP points are to be drawn. CFP: Cumulative false positive per lesion (or image) which is also called False Positive Fraction (FPF). CTP Cumulative True Positive per lesion which is also called True Positive Fraction (TPF).. dig A variable to be passed to the function rstan::sampling() of rstan in which it is named ...??. A positive integer representing the Significant digits, used in stan Cancellation. Default = 5, war A variable to be passed to the function rstan::sampling() of rstan in which it is named warmup. A positive integer representing the Burn in period, which must be less than ite. Defaults to war = floor(ite/5)=10000/5=2000, see A variable to be passed to the function rstan::sampling() of rstan in which it is named seed. A positive integer representing seed used in stan, Default = 1234567. Null.Hypothesis Logical, that is TRUE or FALSE. If Null.or.Alternative.Hypothesis = FALSE(default), then fit the alternative model to dataList (for details of models, see vignettes ). If Null.or.Alternative.Hypothesis = TRUE, then fit the null model to dataList.(for details of models, see vignettes ). Note that the null model is constructed under the null hypothesis that all modality are same observer performance ability. The alternative model is made under the assumption that all modality are not same. The reason why author creates this parameter is to test the null hypothesis by the Bayes factor. But the result of test is not desired one for me. Thus the test is under construction. ... Additional arguments ## Details For details, see vignettes P value calculation is improved by using generated quatinties block in Stan files. P value is the following. Appendix: p value In order to evaluate the goodness of fit of our model to the data, we used the so-called the posterior predictive p value. In the following, we use general conventional notations. Let y_{obs} be an observed dataset and f(y|θ) be a model (likelihood) for future dataset y. We denote a prior and a posterior distribution by π(θ) and π(θ|y) \propto f(y|θ)π(θ), respectively. In our case, the data y is a pair of hits and false alarms; that is, y=(H_1,H_2, … H_C; F_1,F_2, … F_C) and θ = (z_1,dz_1,dz_2,…,dz_{C-1},μ, σ) . We define the χ^2 discrepancy (goodness of fit statistics) to validate that our model fit the data. T(y,θ) := ∑_{c=1,.......,C} \biggr( \frac{\bigl(H_c-N_L\times p_c(θ) \bigr)^2}{N_L\times p_c(θ)}+ \frac{\bigl(F_c- q_{c}(θ) \times N_{X}\bigr)^2}{ q_{c}(θ) \times N_{X} }\biggr). for a single reader and a single modality. T(y,θ) := ∑_{r=1}^R ∑_{m=1}^M ∑_{c=1}^C \biggr( \frac{(H_{c,m,r}-N_L\times p_{c,m,r}(θ))^2}{N_L\times p_{c,m,r}(θ)}+ \frac{\bigl(F_c- q_{c}(θ) \times N_{X}\bigr)^2}{ q_{c}(θ) \times N_{X} }\biggr). for multiple readers and multiple modalities. Note that p_c and λ _{c} depend on θ. In classical frequentist methods, the parameter θ is a fixed estimate, e.g., the maximal likelihood estimator. However, in a Bayesian context, the parameter is not deterministic. In the following, we show the p value in the Bayesian sense. Let y_{obs} be an observed dataset (in an FROC context, it is hits and false alarms). Then, the so-called posterior predictive p value is defined by p_value = \int \int \, dy\, dθ\, I( T(y,θ) > T(y_{obs},θ) )f(y|θ)π(θ|y_{obs}) In order to calculate the above integral, let θ_1,θ _2, ......., θ_i,.......,θ_I be samples from the posterior distribution of y_{obs} , namely, θ_1 \sim π(....|y_{obs} ), ......., θ_i \sim π(....|y_{obs} ), ......., θ_I \sim π(....|y_{obs} ). we obtain a sequence of models (likelihoods), i.e., f(....|θ_1),f(....|θ_2),......., f(....|θ_n). We then draw the samples y^1_1,....,y^i_j,.......,y^I_J , such that each y^i_j is a sample from the distribution whose density function is f(....|θ_i), namely, y^1_1,.......,y^1_j,.......,y^1_J \sim f(....|θ_1), ......., y^i_1,.......,y^i_j,.......,y^i_J \sim f(....|θ_i), ......., y^I_1,.......,y^I_j,.......,y^I_J \sim f(....|θ_I). Using the Monte Carlo integral twice, we calculate the integral of any function φ(y,θ). \int \int \, dy\, dθ\, φ(y,θ)f(y|θ)π(θ|y_{obs}) \approx \int \, \frac{1}{I}∑_{i=1}^I φ(y,θ_i)f(y|θ_i)\,dy \frac{1}{IJ}∑_{i=1}^I ∑_{j=1}^J φ(y^i_j,θ_i) In particular, substituting φ(y,θ):= I( T(y,θ) > T(y_{obs},θ) ) into the above equation, we can approximate the posterior predictive p value. p_value \approx \frac{1}{IJ}∑_i ∑_j I( T(y^i_j,θ_i) > T(y_{obs},θ_i) ) ## Value An object of class  stanfitExtended which is an inherited S4 class from the S4 class stanfit By rstan::sampling, the function fit the author's FROC Bayesian models to user data. Use this fitted model object for sequential analysis, such as drawing the FROC curve and alternative FROC (AFROC) curves. ———————————————————————————————————— Notations and symbols for the Outputs of a single reader and a single modality case —————————————————————————————————————- In the following, the notations for estimated parameters are shown.  w  A real number representing the lowest threshold of the Gaussian assumption (bi-normal assumption). so w=z[1]. dz[1]  A real number representing the difference of the first and second threshold of the Gaussian assumption: dz[1] := z[2] - z[1]. dz[2]  A real number representing the difference of the second and third threshold of the Gaussian assumption: dz[2] := z[3] - z[2]. dz[3]  A real number representing the difference of the third and fourth threshold of the Gaussian assumption: dz[3] := z[4] - z[3]. ... m A real number representing the The mean of the Latent Gaussian distribution for diseased images. In TeX, it denoted by μ v A positive real number representing the standard deviation of the Latent Gaussian distribution for diseased images.In TeX, it will be denoted by σ, not the square of σ. p[1] A real number representing the Hit rate with confidence level 1. p[2]A real number representing the Hit rate with confidence level 2. p[3]A real number representing the Hit rate with confidence level 3. ... l[1]A positive real number representing the (Cumulative) False positive rate with confidence level 1. In TeX, it will be denoted by λ_1. l[2]A positive real number representing the (Cumulative) False positive rate with confidence level 2. In TeX, it will be denoted by λ_2. l[3]A positive real number representing the (Cumulative) False positive rate with confidence level 3. In TeX, it will be denoted by λ_3. l[4]A positive real number representing the (Cumulative) False positive rate with confidence level 4. In TeX, it will be denoted by λ_4. ... dl[1]A positive real number representing the difference  l[1] - l[2]. dl[2]A positive real number representing the difference  l[2] - l[3]. dl[3]A positive real number representing the difference  l[3] - l[4]. ... z[1] A real number representing the lowest threshold of the (Gaussian) bi-normal assumption. z[2] A real number representing the 2nd threshold of the (Gaussian) bi normal assumption. z[3] A real number representing the 3rd threshold of the (Gaussian) bi normal assumption. z[4] A real number representing the fourth threshold of the (Gaussian) bi-normal assumption. a A real number defined by m/v, please contact the author's paper for detail. b A real number representing defined by 1/v, please contact the author's paper for detail. A A positive real number between 0 and 1, representing AUC, i.e., the area under the alternative ROC curve. lp__ The logarithmic likelihood of our model for your data. ————————————————————————————————————— —- Notations and symbols: Outputs of Multiple Reader and Multiple Modality case —— ——————————————————————————————————————  w  The lowest threshold of the Gaussian assumption (bi-normal assumption). so w=z[1]. dz[1]  The difference of the first and second threshold of the Gaussian assumption. dz[2]  The difference of the second and third threshold of the Gaussian assumption. dz[3]  The difference of the third and fourth threshold of the Gaussian assumption. ... mu The mean of the Latent Gaussian distribution for diseased images. v The variance of the Latent Gaussian distribution for diseased images. ppp[1,1,1] Hit rate with confidence level 1, modality 1, reader 1. ppp[2,1,1] Hit rate with confidence level 2, modality 1, reader 1. ppp[3,1,1] Hit rate with confidence level 3, modality 1, reader 1. ... l[1] (Cumulative) False positive rate with confidence level 1. l[2] (Cumulative) False positive rate with confidence level 2. l[3] (Cumulative) False positive rate with confidence level 3. l[4] (Cumulative) False positive rate with confidence level 4. ... dl[1] This is defined by the difference  l[1] - l[2]. dl[2] This is defined by the difference  l[2] - l[3]. dl[3] This is defined by the difference  l[3] - l[4]. ... z[1] The lowest threshold of the (Gaussian) bi-normal assumption. z[2] The 2nd threshold of the (Gaussian) bi normal assumption. z[3] The 3rd threshold of the (Gaussian) bi normal assumption. z[4] The fourth threshold of the (Gaussian) bi-normal assumption. aa This is defined by m/v, please see the author's paper for more detail. bb This is defined by 1/v, please see the author's paper for more detail. AA The area under alternative FROC curve associated to reader and modality. A The area under alternative FROC curve associated to modality. hyper_v Standard deviation of AA around A. lp__ The logarithmic likelihood of our model for your data. ## References Bayesian Models for Free-response Receiver Operating Characteristic Analysis; Pre-print See vignettes ## See Also ——— Before fitting: create a dataset  convertFromJafroc Convert from JAFROC format xlsx file to the author's format  dataset_creator_new_version Create an R object which represent user data.  create_dataset Create an R object which represent user data. ——— Further sequential analysis: Plot curves Using the result of fitting a Bayesian FROC model, we can go sequential analysis.  DrawCurves for drawing free response ROC curves. ——— Further sequential analysis: Validation of the Model  ppp Calculation of a p-value in the Bayesian paradigm. ——— R objects of example datasets from real world or fictitious: dataList.Chakra.1 A list for an example dataset of a single reader and a single modality data. The word Chakra in the dataset name means that it appears in the paper of Chakraborty. dataList.Chakra.2 A list for an example dataset of a single reader and a single modality data. The word Chakra in the dataset name means that it appears in the paper of Chakraborty. dataList.Chakra.3 A list for an example dataset of a single reader and a single modality data. The word Chakra in the dataset name means that it appears in the paper of Chakraborty. dataList.Chakra.4 A list for an example dataset of a single reader and a single modality data. The word Chakra in the dataset name means that it appears in the paper of Chakraborty. dataList.high.ability A list for an example dataset of a single reader and a single modality data dataList.low.ability A list for an example dataset of a single reader and a single modality data dataList.Chakra.Web A list for an example dataset of multiple readers and multiple modalities data. The word Chakra in the dataset name means that it appears in the paper of Chakraborty. data.hier.ficitious A list for an example dataset of multiple readers and multiple modalities data dataList.High A list for an example dataset of a single reader and a single modality data whose AUC is high. dataList.Low A list for an example dataset of a single reader and a single modality data whose AUC is low. data.bad.fit A list for an example dataset of a single reader and a single modality data whose fitting is bad, that is chi square is very large. However the MCMC convergence criterion is satisfied with very high quality. Thus the good MCMC convergence does not mean the model is correct. So, to fit a model to this data, we should change the latent Gaussian and differential logarithmic Gaussian to more appropriate distributions for hit and false alarm rate. In theoretically perspective, there is no a a prior distribution for hit and false alarm rate. So, if we encounter not good fitting data, then we should change the model, and such change will occur in the latent distributions. The reason why the author saved this data is to show that our model is not unique nor good and gives a future research directions. To tell the truth the author is not interested the FROC theory. My background is mathematics, geometry, pure mathematics. So, I want to go back to my home ground. This program are made to show my skill for programming or my ability. But, now, I do not think to get job. I want to go back mathematics. Soon, my paper is published which is related Gromov Hausdorff topology. Of course, I will publish this package's theory soon. Please wait. d,dd ,ddd ,dddd ,ddddd,dddddd,ddddddd The other datasets, the author like these datasets because name is very simple. ## Examples   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 ## Not run: #======================================================================================== # The 1-st example #======================================================================================== # # # Making FROC Data and Fitting a Model to the data # # Notations # # h = hits = TP = True Positives # f = False alarms = FP = False Positives # # #======================================================================================== # 1) Build a data-set #======================================================================================== BayesianFROC:::clearWorkspace() # For a single reader and a single modality case. dat <- list(c=c(3,2,1), # Confidence level. Note that c is ignored. h=c(97,32,31), # Number of hits for each confidence level f=c(1,14,74), # Number of false alarms for each confidence level NL=259, # Number of lesions NI=57, # Number of images C=3) # Number of confidence level if (interactive()){ viewdata(dat)} # where, # c denotes confidence level, i.e., rating of reader. # 3 = Definitely diseased, # 2 = subtle,.. diseased # 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, # NI denotes number of images, # For example, in the above example data, # the number of hits with confidence level 3 is 97, # the number of hits with confidence level 2 is 32, # the number of hits with confidence level 1 is 31, # the number of false alarms with confidence level 3 is 1, # the number of false alarms with confidence level 2 is 14, # the number of false alarms with confidence level 1 is 74, #======================================================================================== # 2) Fit an FROC model to the above dataset. #======================================================================================== fit <- fit_Bayesian_FROC( dat, # dataset ite = 111, #To run in time <5s. cha = 1, # number of chains, it is better more large. summary = FALSE ) # The return value "fit" is an S4 object of class "stanfitExtended" which is inherited # from the S4 class "stanfit". #======================================================================================== # 3) Change the S4 class of fitted model object # Change the S4 class from "stanfitExtended" to "stanfit" to apply other packages. # The fitted model object of class "stanfit" is widely available. # For example the package ggmcmc, rstan, shinystan::launch_shinystan(stanfit_object) # Thus, to use such packages, we get back the inherited class into "stanfit" as follows: # Changing the class from stanfitExtended to stanfit, # we can apply other pakcage's functions to the resulting object. #======================================================================================== fit.stan <- methods::as(fit,"stanfit") # Then, return value "fit.stan" is no longer an S4 object of class "stanfitExtended" but # the S4 object of class "stanfit" which is widely adequate for many packages. #======================================================================================== # 3.1) Apply the functions for the class stanfit #======================================================================================== grDevices::dev.new();rstan::stan_hist(fit.stan, bins=33,pars = c("A")) grDevices::dev.new();rstan::stan_hist(fit.stan, bins=22,pars = c("A")) grDevices::dev.new();rstan::stan_hist(fit.stan, bins=11,pars = c("A")) grDevices::dev.off() # I am not sure why the above stan_hist also works for the new S4 class "stanfitExtended" # Get pipe operator # %>% <- utils::getFromNamespace("%>%", "magrittr") # Plot about MCMC samples of parameter name "A", representing AUC # The author does not think the inherited class "stanfitExtended" is good, # cuz the size of object is very redundant and large, # which caused by the fact that inherited class contains plot data for FROC curve. # To show the difference of size for the fitted model object of class # stanfitExtended and stanfit, we execute the following code; size_of_return_value(fit) - size_of_return_value(methods::as(fit,"stanfit")) #4) Using the S4 object fit, we can go further step, such as calculation of the # Chisquare and the p value as the Bayesian sense for testing the goodness of fit. # I think p value has problems that it relies on the sample size monotonically. # But it is widely used, thus I hate it but I implement the p value. #======================================================================================== # REMARK #======================================================================================== # # Should not write the above data as follows: # MANNER (A) dat <- list(c=c(1,2,3),h=c(31,32,97),f=c(74,14,1),NL=259,NI=57,C=3) # Even if user writes data in the above MANNER (A), # the program interprets it as the following MANNER (B); # MANNER (B) dat <- list(c=c(3,2,1),h=c(31,32,97),f=c(74,14,1),NL=259,NI=57,C=3) # Because the vector c is ignored in the program, # and it is generated by the code rep(C:1) automatically in the internal of the function. # So, we can omit the vector c from the list. #This package is very rigid format, so please be sure that data-format is #exactly same to the format in this package. #More precisely, the confidence level vector should be denoted rep(C:1) (Not rep(1:C)). # Note that 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 confidence level vector, # where C is the highest number of confidence levels. # I regret this order, this order is made when I start, so I was very beginner, # but it is too late to fix,...tooooooo late. #======================================================================================== # The 2-nd example #======================================================================================== # # (1)First, we prepare the data from this package. dat <- BayesianFROC::dataList.Chakra.1 # (2)Second, we run fit_Bayesian_FROC() in which the rstan::stan() is implemented. # with data named "dat" and the author's Bayesian model. fit <- fit_Bayesian_FROC(dat, ite = 111 #To run in time <5s. ) # Now, we get the object named "fit" which is an S4 object of class stanfitExtended. # << Minor Comments>> # More precisely, this is an S4 object of some inherited class (named stanfitExtended) # which is extended using stan's S4 class named "stanfit". fit.stan <- methods::as(fit,"stanfit") # Using the output "fit.stan", # we can use the functions in the "rstan" package, for example, as follows; grDevices::dev.new(); rstan::stan_trace(fit.stan, pars = c("A"))# stochastic process of a posterior estimate rstan::stan_hist(fit.stan, pars = c("A")) # Histogram of a posterior estimate rstan::stan_rhat(fit.stan, pars = c("A")) # Histogram of rhat for all parameters rstan::summary(fit.stan, pars = c("A")) # summary of fit.stan by rstan grDevices::dev.off() #======================================================================================== # The 3-rd example #======================================================================================== # Fit a model to a hand made data # 1) Build the data for a single reader and a single modality case. dat <- list( c=c(3,2,1), # Confidence level, which is ignored. h=c(97,32,31), # Number of hits for each confidence level f=c(1,14,74), # Number of false alarms for each confidence level NL=259, # Number of lesions NI=57, # Number of images C=3) # Number of confidence level # where, # c denotes confidence level, , each components indicates that # 3 = Definitely lesion, # 2 = subtle, # 1 = very subtle # That is the high number indicates the high confidence level. # 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, # NI denotes number of images, # 2) Fit and draw FROC and AFROC curves. fit <- fit_Bayesian_FROC(dat, DrawCurve = TRUE) # (( REMARK )) # Changing the hits and false alarms denoted by h and f # in the above dataset denoted by dat, # user can fit a model to various datasets and draw corresponding FROC curves. # Enjoy drawing the curves for various datasets in case of # a single reader and a single modality data #======================================================================================== # For Prior and Bayesian Update: # Calculates a posterior mean and variance # for each parameter #======================================================================================== # Mean values of posterior samples are used as a point estimates, and # Although the variance of posteriors receives less attention, # but to make a prior, we will need the it. # For, example, if we assume that model parameter m has prior distributed by # Gaussian, then we have to know the mean and variance to characterize prior. e <- rstan::extract(fit) # model parameter m and v is a number, # indicating the mean and variance of signal distribution, respectively. stats::var(e$m) mean(e$m) stats::var(e$v) mean(e$v) # The model parameter z or dz is a vector, and thus we execute the following; # z = ( z[1], z[2], z[3] ) # dz = ( z[2]-z[1], z[3]-z[2] ) # Posterior mean of posterior MCMC samples for parameter z and dz apply(e$dz, 2, mean) apply(e$z, 2, mean) # Posterior variance of posterior MCMC samples for parameter z and dz apply(e$dz, 2, var) apply(e$z, 2, var) apply(e$dl, 2, mean) apply(e$l, 2, mean) apply(e$p, 2, mean) apply(e$p, 2, var) # Revised 2019 Sept 6 #======================================================================================== # The 4-th example #======================================================================================== # ## Only run examples in interactive R sessions if (interactive()) { # 1) Build the data interactively, dataList <- create_dataset() #Now, as as a return value of create_dataset(), we get the FROC data (list) named dataList. # 2) Fit an MRMC or srsc FROC model. fit <- fit_Bayesian_FROC(dataList) }## Only run examples in interactive R sessions #======================================================================================== # The 5-th example #======================================================================================== # Comparison of the posterior probability for AUC # In the following, we calculate the probability of the events that # the AUC of some modality is greater than the AUC of another modality. #======================================================================================== # Posterior Probability for some events of AUCs by using posterior MCMC samples #======================================================================================== # This example shows how to use the stanfit (stanfit.Extended) object. # Using stanfit object, we can extract posterior samples and using these samples, # we can calculate the posterior probability of research questions. fit <- fit_Bayesian_FROC(dataList.Chakra.Web.orderd,ite = 111,summary =FALSE) # For example, we shall show the code to compute the posterior probability of the ever # that the AUC of modality 1 is larger than that of modality 2: e <- extract(fit) # Then, the MCMC samples are extracted in the object "e" for all parameters. # From this, e.g., AUC can be extracted by the code e$A that is a two dimensional array. # The first component of e$A indicates the ID of MCMC samples and # the second component indicates the modality ID. # For example, the code e$A[,1] means the vector of MCMC samples of the 1 st modality. # For example, the code e$A[,2] means the vector of MCMC samples of the 2 nd modality. # For example, the code e$A[,3] means the vector of MCMC samples of the 3 rd modality. # To calculate the posterior probability of the event # that the AUC of modality 1 is larger than that of modality 2, # we execute the following R script: mean(e$A[,1] > e$A[,2]) # Similarly, to compute the posterior probability of the event that # the AUC of modality 1 is larger than that of modality 3: mean(e$A[,1] > e$A[,3]) # Similarly, to compute the posterior probability of the event that # the AUC of modality 1 is larger than that of modality 4: mean(e$A[,1] > e$A[,4]) # Similarly, to compute the posterior probability of the event that # the AUC of modality 1 is larger than that of modality 5: mean(e$A[,1] > e$A[,5]) # Similarly, to compute the posterior probability of the event that # the AUC of modality 1 is larger than that of modality 5 at least 0.01 mean(e$A[,1] > e$A[,5]+0.01) # Similarly, mean( e$A[,1] > e$A[,5] + 0.01 ) mean( e$A[,1] > e$A[,5] + 0.02 ) mean( e$A[,1] > e$A[,5] + 0.03 ) mean( e$A[,1] > e$A[,5] + 0.04 ) mean( e$A[,1] > e$A[,5] + 0.05 ) mean( e$A[,1] > e$A[,5] + 0.06 ) mean( e$A[,1] > e$A[,5] + 0.07 ) mean( e$A[,1] > e$A[,5] + 0.08 ) # Since any posterior distribution tends to the Dirac measure whose center is # true parameter under the assumption that the model is correct in the sense that the # true distribution is belongs to a family of models. # Thus using this procedure, we will get # the true parameter if any more large sample size we can take. # Close the graphic device to avoid errors in R CMD check. Close_all_graphic_devices() #======================================================================================== # The 6-th Example for MRMC data #======================================================================================== # To draw FROC curves for each modality and each reader, the author provides codes. # First, we make a fitted object of class stanfitExtended as following manner. fit <- fit_Bayesian_FROC( ite = 1111, cha = 1, summary = FALSE, Null.Hypothesis = FALSE, dataList = dd # This is a MRMC dataset. ) # Using this fitted model object called fit, we can draw FROC curves for the # 1-st modality as following manner: DrawCurves( # This is a fitted model object fit, # Here, the modality is specified modalityID = 1, # Reader is specified as 1,2,3,4 readerID = 1:4, # If TRUE, the new imaging device is created and curves are drawn on it. new.imaging.device = TRUE ) # The next codes are quite same, except modality ID and new.imaging.device # The code that "new.imaging.device = F" means that the curves are drawn using # the previous imaging device to plot the 1-st and 2-nd modality curves draw in the same # Plot plain. Drawing in different curves in same plain, we can compare the curve # of modality. Of course, the interpretation of FROC curve is the ordinal ROC curve, # that is, # if curve is upper then the observer performance with its modality is more greater. # So, please enjoy drawing curves. DrawCurves(fit,modalityID = 2,readerID = 1:4, new.imaging.device = FALSE) DrawCurves(fit,modalityID = 3,readerID = 1:4, new.imaging.device = FALSE) DrawCurves(fit,modalityID = 4,readerID = 1:4, new.imaging.device = FALSE) DrawCurves(fit,modalityID = 5,readerID = 1:4, new.imaging.device = FALSE) Close_all_graphic_devices() #======================================================================================== # The 7-th example NON-CONVERGENT CASE 2019 OCT. #======================================================================================== ff <- fit_Bayesian_FROC( ite = 1111, cha = 1, summary = TRUE, dataList = ddd ) #' dat <- list( c=c(3,2,1), #Confidence level h=c(73703933,15661264,12360003), #Number of hits for each confidence level f=c(1738825,53666125 , 254965774), #Number of false alarms for each confidence level NL=100000000, #Number of lesions NI=200000000, #Number of images C=3) #Number of confidence level # From the examples of the function mu_truth_creator_for_many_readers_MRMC_data() #======================================================================================== # Large number of readers cause non-convergence #======================================================================================== v <- v_truth_creator_for_many_readers_MRMC_data(M=4,Q=6) m <- mu_truth_creator_for_many_readers_MRMC_data(M=4,Q=6) d <-create_dataList_MRMC(mu.truth = m,v.truth = v) #fit <- fit_Bayesian_FROC( ite = 111, cha = 1, summary = TRUE, dataList = d ) plot_FPF_and_TPF_from_a_dataset(d) #======================================================================================== # convergence #======================================================================================== v <- v_truth_creator_for_many_readers_MRMC_data(M=2,Q=21) m <- mu_truth_creator_for_many_readers_MRMC_data(M=2,Q=21) d <- create_dataList_MRMC(mu.truth = m,v.truth = v) fit <- fit_Bayesian_FROC( ite = 200, cha = 1, summary = TRUE, dataList = d) plot_FPF_TPF_via_dataframe_with_split_factor(d) plot_empirical_FROC_curves(d,readerID = 1:21) #======================================================================================== # non-convergence #======================================================================================== v <- v_truth_creator_for_many_readers_MRMC_data(M=5,Q=6) m <- mu_truth_creator_for_many_readers_MRMC_data(M=5,Q=6) d <- create_dataList_MRMC(mu.truth = m,v.truth = v) #fit <- fit_Bayesian_FROC( ite = 111, cha = 1, summary = TRUE, dataList = d) #======================================================================================== # convergence #======================================================================================== v <- v_truth_creator_for_many_readers_MRMC_data(M=1,Q=36) m <- mu_truth_creator_for_many_readers_MRMC_data(M=1,Q=36) d <- create_dataList_MRMC(mu.truth = m,v.truth = v) #fit <- fit_Bayesian_FROC(ite = 111, cha = 1,summary = TRUE, dataList = d, see = 123) #======================================================================================== # non-convergence #======================================================================================== v <- v_truth_creator_for_many_readers_MRMC_data(M=1,Q=37) m <- mu_truth_creator_for_many_readers_MRMC_data(M=1,Q=37) d <- create_dataList_MRMC(mu.truth = m,v.truth = v) #fit <- fit_Bayesian_FROC( ite = 111, cha = 1, summary = TRUE, dataList = d) #======================================================================================== # convergence A single modality and 11 readers #======================================================================================== v <- v_truth_creator_for_many_readers_MRMC_data(M=1,Q=11) m <- mu_truth_creator_for_many_readers_MRMC_data(M=1,Q=11) d <- create_dataList_MRMC(mu.truth = m,v.truth = v) fit <- fit_Bayesian_FROC( ite = 111, cha = 1, summary = TRUE, dataList = d, see = 123455) DrawCurves( summary = FALSE, modalityID = c(1:fit@dataList$M), readerID = c(1:fit@dataList$Q), StanS4class = fit ) #======================================================================================== # convergence A single modality and 17 readers #======================================================================================== v <- v_truth_creator_for_many_readers_MRMC_data(M=1,Q=17) m <- mu_truth_creator_for_many_readers_MRMC_data(M=1,Q=17) d <- create_dataList_MRMC(mu.truth = m,v.truth = v) fit <- fit_Bayesian_FROC( ite = 1111, cha = 1, summary = TRUE, dataList = d,see = 123455) DrawCurves( summary = FALSE, modalityID = c(1:fit@dataList$M), readerID = c(1:fit@dataList$Q),fit ) DrawCurves( summary = FALSE, modalityID = 1, readerID = c(8,9),fit ) # ## For readerID 8,9, this model is bad # Close_all_graphic_devices() #======================================================================================== # convergence 37 readers, 1 modality #======================================================================================== v <- v_truth_creator_for_many_readers_MRMC_data(M=1,Q=37) m <- mu_truth_creator_for_many_readers_MRMC_data(M=1,Q=37) d <- create_dataList_MRMC(mu.truth = m,v.truth = v) fit <- fit_Bayesian_FROC(see = 2345678, ite = 1111, cha = 1, summary = TRUE, dataList = d) DrawCurves( summary = FALSE, modalityID = c(1:fit@dataList$M), readerID = c(1:fit@dataList$Q),fit ) DrawCurves( summary = FALSE, modalityID = 1, readerID = c(8,9),fit ) # In the following, consider two readers whose ID are 8 and 15, respectively. # Obviously, one of them will have high performamce than the other, # however, # Sometimes, the FROC curve does not reflect it, # Namely, one of the FROC curve is upper than the other # even if the FPF and TPF are not.... WHY??? DrawCurves( summary = FALSE, modalityID = 1, readerID = c(8,15),fit ) Close_all_graphic_devices() Close_all_graphic_devices() ## End(Not run)# dontrun 

BayesianFROC documentation built on Jan. 13, 2021, 5:22 a.m.