# error_srsc: Validation via replicated datasets from a model at a given... In BayesianFROC: FROC Analysis by Bayesian Approaches

## Description

Print for a given true parameter, a errors of estimates from replicated dataset.

Also print a standard error which is the variance of estimates.

Suppose that θ_0 is a given true model parameter with a given number of images N_I and a given number of lesions N_L, specified by user.

(I)
(I.1) Synthesize a collection of dataset D_k (k=1,2,...,K) from a likelihood (model) at a given parameter θ_0, namely

D_k \sim likelihood( θ_0).

(I.2) Replicates K models fitted to each dataset D_k (k=1,2,...,K), namely, draw MCMC samples \{ θ_i (D_k);i=1,...,I\} from each posterior of the dataset D_k, namely

θ _i(D_k) \sim π(|D_k).

(I.3) Calculate posterior means for the set of data D_k (k=1,2,...,K), namely

\bar{θ}(D_k) := \frac{1}{I} ∑_i θ_i(D_k) .

(I.4) Calculates error for each dataset D_k

ε_k:= Truth - estimates = θ_0 - \bar{θ}(D_k).

(II) Calculates mean of errors over all datasets D_k (k=1,2,...,K)

mean of errors \bar{ε}(θ_0,N_I,N_L)= \frac{1}{K} ∑ ε_k .

NOTE

We note that if a fitted model does not converge,( namely R hat is far from one), then it is omiited from this calculation.

(III) Calculates mean of errors for various number of lesions and images

mean of errors \bar{ε}(θ_0,N_I,N_L)

For example, if (N_I^1,N_L^1),(N_I^2,N_L^2),(N_I^3,N_L^3),...,(N_I^m,N_L^m), then \bar{ε}(θ_0,N_I^1,N_L^1), \bar{ε}(θ_0,N_I^2,N_L^2), \bar{ε}(θ_0,N_I^3,N_L^3),..., \bar{ε}(θ_0,N_I^m,N_L^m) are calculated.

To obtain precise error, The number of replicated fitted models (denoted by K) should be large enough. If K is small, then it causes a bias. K = replicate.datset: a variable of the function error_srsc.

Running this function, we can see that the error \bar{ε}(θ_0,N_I,N_L) decreases monotonically as a given number of images N_I or a given number of lesions N_L increases.

Also, the scale of error also will be found. Thus this function can show how our estimates are correct. Scale of error differs for each componenet of model parameters.

Revised 2019 August 28

## Usage

  1 2 3 4 5 6 7 8 9 10 11 error_srsc( NLvector = c(100L, 10000L, 1000000L), ratio = 2, replicate.datset = 3, ModifiedPoisson = FALSE, mean.truth = 0.6, sd.truth = 5.3, z.truth = c(-0.8, 0.7, 2.38), ite = 2222, cha = 1 ) 

## Arguments

 NLvector A vector of positive integers, indicating a collection of numbers of Lesions. ratio A positive rational number, with which Number of Images is determined by the formula: (number of images) = ratio times (numbser of lesions). Note that in calculation, it rounds ratio * NLvector  to an integer. replicate.datset A Number indicate that how many you replicate dataset from user's specified dataset. 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 mean.truth This is a parameter of the latent Gaussian assumption for the noise distribution. sd.truth This is a parameter of the latent Gaussian assumption for the noise distribution. z.truth This is a parameter of the latent Gaussian assumption for the noise distribution. 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. 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.

## Details

In Bayesian inference, if sample size is large, then posterior tends to the Dirac measure. So, the error and variance of estimates should be tends to zero as sample size tends to infinity.

This function check this phenomenen.

If model has problem, then it contains some non-decreasing vias with respect to sample size.

Revised 2019 Nov 1

Provides a reliability of our posterior mean estimates. Using this function, we can find what digit makes sence.

In the real world, the data for modality comparison or observer performan evaluation is 100 images or 200 images. In such scale data, any estimate of AUC will contain error at most 0.0113.... So, the value of AUC should round in 0.XXX and not 0.XXXX or 0.XXXXX or more. Since error is 0.00113... and hence 4 digit or more digit is meaningless. In such manner, we can analyize the errors.

We note that if we increase the number of images or lesinons, the errors decrease.

For example, if we use 20000 images in FROC trial, then the error of AUC will be 0.0005... and thus, and so on. Thus large number of images gives us more reliable AUC. However the radiologist cannot read such large (20000) images.

Thus, the error will be 0.00113...

If the number of images are given before hand and moreover if we obtains the estimates, then we can run this function using these two, we can find the estimated errors by simulation. Of course, the esimates is not the truth, but roughly speaking, if we assume that the estimates is not so far from truth, and the error analysis is rigid with respect to changing the truth, then we can say using estimates as truth, the result of this error analysis can be regarded as an actual error.

I want to go home. Unfortunatly, my house is ...

## Value

Replicated datasets, estimates, errors,...etc I made this program 1 years ago? and now I forget ... the precise return values. When I see today, 2019 August. It retains too many return values to explain all of them.

## 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 ## Not run: #======================================================================================== # 0) 0-th example #======================================================================================== datasets <-error_srsc( NLvector = c(100,10000,1000000), ite = 2222 ) # By the following, we can extract only datasets whose # model has converged. datasets$convergent.dataList.as.dataframe #======================================================================================== # 1) 1-st example #======================================================================================== # Long width is required in R console. datasets <-error_srsc(NLvector = c( 50L, 111L, 11111L ), # NIvector, ratio=2, replicate.datset =3, ModifiedPoisson = FALSE, mean.truth=0.6, sd.truth=5.3, z.truth =c(-0.8,0.7,2.38), ite =2222 ) #======================================================================================== # 2) Plot the error of AUC with respect to NI #======================================================================================== a <-error_srsc(NLvector = c( 33L, 50L, 111L, 11111L ), # NIvector, ratio=2, replicate.datset =3, ModifiedPoisson = FALSE, mean.truth=0.6, sd.truth=5.3, z.truth =c(-0.8,0.7,2.38), ite =2222 ) aa <- a$Bias.for.various.NL error.of.AUC <- aa[8,] y <- subset(aa[8,], select = 2:length(aa[8,])) y <- as.numeric(y) y <- abs(y) upper_y <- max(y) lower_y <- min(y) x <- 1:length(y) plot(x,y, ylim=c(lower_y, upper_y)) # From this plot, we cannot see whether the error has decreased or not. # Thus, we replot with the log y-axis, the we will see that the error # has decreased with respect to number of images and lesions. ggplot(data.frame(x=x,y=y), aes(x = x, y = y)) + geom_line() + geom_point() + scale_y_log10() # Revised 2019 Sept 25 # General print of log scale df<-data.frame(x=c(10,100,1000,10,100,1000), y=c(1100,220000,33000000,1300,240000,36000000), group=c("1","1","1","2","2","2") ) ggplot2::ggplot(df, aes(x = x, y = y, shape = group)) + ggplot2::geom_line(position = position_dodge(0.2)) + # Dodge lines by 0.2 ggplot2::geom_point(position = position_dodge(0.2), size = 4)+ # Dodge points by 0.2 ggplot2::scale_y_log10()+ ggplot2::scale_x_log10() #======================================================================================== # 2) Add other param into plot plain of the error of AUC with respect to NI #======================================================================================== a <-error_srsc(NLvector = c( 111L, 11111L ), # NIvector, ratio=2, replicate.datset =3, ModifiedPoisson = FALSE, mean.truth=0.6, sd.truth=5.3, z.truth =c(-0.8,0.7,2.38), ite =2222 ) aa <- a\$Bias.for.various.NL error.of.AUC <- aa[8,] y1 <- subset(aa[8,], select = 2:length(aa[8,])) y1 <- as.numeric(y1) y1 <- abs(y1) LLL <-length(y1) y2 <- subset(aa[7,], select = 2:length(aa[7,])) y2 <- as.numeric(y2) y2 <- abs(y2) y <- c(y1,y2) upper_y <- max(y) lower_y <- min(y) group <- rep(seq(1,2,1),1 , each=LLL) x <- rep(seq(1,LLL,1),2 , each=1) group <- as.character(group) df <- data.frame(x=x,y=y,group=group) ggplot2::ggplot(df, aes(x = x, y = y, shape = group)) + ggplot2::geom_line(position = position_dodge(0.2)) + # Dodge lines by 0.2 ggplot2::geom_point(position = position_dodge(0.2), size = 4)+ # Dodge points by 0.2 ggplot2::scale_y_log10() # ggplot2::scale_x_log10() #======================================================================================== # Confidence level = 4 #======================================================================================== datasets <-error_srsc(NLvector = c( 111L, 11111L ), # NIvector, ratio=2, replicate.datset =3, ModifiedPoisson = FALSE, mean.truth=-0.22, sd.truth=5.72, z.truth =c(-0.46,-0.20,0.30,1.16), ite =2222 ) error_srsc_variance_visualization(datasets) # The parameter of model is 7 in which the ggplot2 fails with the following warning: # The shape palette can deal with a maximum of 6 discrete values because more than 6 # becomes difficult to # discriminate; you have 7. Consider specifying shapes manually if you must have them. ## End(Not run)# dontrun 

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