# error_MRMC: Comparison of Estimates and Truth in case of MRMC In BayesianFROC: FROC Analysis by Bayesian Approaches

## Description

In order to describe what this function calculates explicitly, let us denote a specified true model parameter by θ_0, from which fake datasets are replicated and denoted by:

D_1,D_2,...,D_k,... D_K.

We obtains estimates

θ(D_1),...,θ(D_K)

for each replicated dataset. Using these estimates, we calculate the mean of the absolute errors (= an absolute difference between estimates and a true parameter θ_0 ), namely,

\frac{1}{K}∑_{k=1}^K | θ(D_k) - θ_0 |,

or the variance of estimates:

\frac{1}{K}∑_{k=1}^K ( θ(D_k) - \frac{1}{K}∑_{k=1}^K θ(D_k) )^2.

Revised 2019 Nov 1

Revised 2020 Jan

Revised 2020 March

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 error_MRMC( replication.number = 2, initial.seed = 123, mu.truth = BayesianFROC::mu_truth, v.truth = BayesianFROC::v_truth, z.truth = BayesianFROC::z_truth, NI = 200, NL = 1142, ModifiedPoisson = FALSE, summary = FALSE, ite = 1111 ) 

## Arguments

 replication.number For fixed number of lesions, images, the dataset of hits and false alarms are replicated, and the number of replicated datasets are specified by this variable. initial.seed The variable initial.seed is used to replicate datasets. That is, if you take initial.seed = 1234, then the seed 1234, 1235, 1236, 1237, 1238, etc are for the first replication, the second replication, the third replication,etc. If the n-th model does not converge for some n, then such model has no mean and thus the non-convergent models are omitted to calculate the errors. mu.truth array of dimension (M,Q). Mean of the signal distribution of bi-normal assumption. v.truth array of dimension (M,Q). Standard Deviation of represents the signal distribution of bi-normal assumption. z.truth This is a parameter of the latent Gaussian assumption for the noise distribution. NI Number of Images. NL Number of Lesions. 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 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. 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.

## Details

2019 Sept 6 I found this program, I made this in several month ago? I forgot when this function is made. It well works, so it helps me now.

## Value

list of errors, or vaiance of estimates over all replicated datasets.

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