In order to describe what this function calculates explicitly, let us denote a specified true model parameter by \(\theta_0\), from which fake datasets are replicated and denoted by:
$$D_1,D_2,...,D_k,... D_K.$$
We obtains estimates
$$ \theta(D_1),...,\theta(D_K)$$
for each replicated dataset. Using these estimates, we calculate the mean of the absolute errors (= an absolute difference between estimates and a true parameter \(\theta_0\) ), namely,
$$ \frac{1}{K}\sum_{k=1}^K | \theta(D_k) - \theta_0 |, $$
or the variance of estimates:
$$ \frac{1}{K}\sum_{k=1}^K ( \theta(D_k) - \frac{1}{K}\sum_{k=1}^K \theta(D_k) )^2. $$
Revised 2019 Nov 1
Revised 2020 Jan
Revised 2020 March
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
)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.
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.
array of dimension (M,Q). Mean of the signal distribution of bi-normal assumption.
array of dimension (M,Q). Standard Deviation of represents the signal distribution of bi-normal assumption.
This is a parameter of the latent Gaussian assumption for the noise distribution.
Number of Images.
Number of Lesions.
Logical, that is TRUE or FALSE.
If ModifiedPoisson = TRUE,
then Poisson rate of false alarm is calculated per lesion,
and a 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 a 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
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.
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 = 1111
list of errors, or vaiance of estimates over all replicated datasets.
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.