chi_square_at_replicated_data_and_MCMC_samples_MRMC: chi square at replicated data drawn (only one time) from...
In BayesianFROC: FROC Analysis by Bayesian Approaches

Description Usage Arguments Details Value Examples

View source: R/chi_square_goodness_of_fit.R

To pass the return value to the calculator of the posterior predictive p value.

chi_square_at_replicated_data_and_MCMC_samples_MRMC(
  StanS4class,
  summary = TRUE,
  seed = NA,
  serial.number = NA
)

`StanS4class`	An S4 object of class `stanfitExtended` which is an inherited class from the S4 class `stanfit`. This R object is a fitted model object as a return value of the function `fit_Bayesian_FROC()`. To be passed to `DrawCurves()` ... etc
`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.
`seed`	This is used only in programming phase. If seed is passed, then, in procedure indicator the seed is printed. This parameter is only for package development.
`serial.number`	A positive integer or Character. This is for programming perspective. The author use this to print the serial numbre of validation. This will be used in the validation function.

For a given dataset D_0, let us denote by π(|D_0) a posterior distribution of the given data D_0.

Then, we draw poterior samples.

θ_1 \sim π(.| D_0),

θ_2 \sim π(.| D_0),

θ_3 \sim π(.| D_0),

....,

θ_n \sim π(.| D_0).

We let L(|θ) be a likelihood function or probability law of data, which is also denoted by L(y|θ) for a given data y. But, the specification of data y is somehow conversome, thus, to denote the function sending each y into L(y|θ), we use the notation L(|θ).

Now, we synthesize data-samples (y_i;i=1,2,...,n) in only one time drawing from the collection of likelihoods L(|θ_1),L(|θ_2),...,L(|θ_n).

y_1 \sim L(.| θ_1),

y_2 \sim L(.| θ_2),

y_3 \sim L(.| θ_3),

....,

y_n \sim L(.| θ_n).

Altogether, using these pair of samples (y_i, θ_i), i= 1,2,...,n, we calculate the chi squares as the return value of this function. That is,

χ(y_1|θ_1),

χ(y_2|θ_2),

χ(y_3|θ_3),

....,

χ(y_n|θ_n).

This is contained as a vector in the return value,

so the return value is a vector whose length is the number of MCMC iterations except the burn-in period.

Note that in MRMC cases,

χ(y|θ).

is defined as follows.

χ^2(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{[F_{c,m,r}-(λ _{c} -λ _{c+1} )\times N_{L}]^2}{(λ_{c}(θ) -λ_{c+1}(θ) )\times N_{L} }\biggr).

where a dataset y consists of the pairs of the number of False Positives and the number of True Positives (F_{c,m,r}, H_{c,m,r}) together with the number of lesions N_L and the number of images N_I and θ denotes the model parameter.

Application of this return value to calculate the so-called Posterior Predictive P value.

As will be demonstrated in the other function, chaning seed, we can obtain

y_{1,1},y_{1,2},y_{1,3},...,y_{1,j},....,y_{1,J} \sim L ( . |θ_1),

y_{2,1},y_{2,2},y_{2,3},...,y_{2,j},....,y_{2,J} \sim L ( . |θ_2),

y_{3,1},y_{3,2},y_{3,3},...,y_{3,j},....,y_{3,J} \sim L ( . |θ_3),

...,

y_{i,1},y_{i,2},y_{i,3},...,y_{i,j},....,y_{I,J} \sim L ( . |θ_i),

...,

y_{I,1},y_{I,2},y_{I,3},...,y_{I,j},....,y_{I,J} \sim L ( . |θ_I).

where L ( . |θ_i) is a likelihood function for a model parameter θ_i. And thus, we calculate the chi square statistics.

χ(y_{1,1}|θ_1), χ(y_{1,2}|θ_1), χ(y_{1,3}|θ_1),..., χ(y_{1,j}|θ_1),...., χ(y_{1,J}|θ_1),

χ(y_{2,1}|θ_2), χ(y_{2,2}|θ_2), χ(y_{2,3}|θ_2),..., χ(y_{2,j}|θ_2),...., χ(y_{2,J}|θ_2),

χ(y_{3,1}|θ_3), χ(y_{3,2}|θ_3), χ(y_{3,3}|θ_3),..., χ(y_{3,j}|θ_3),...., χ(y_{3,J}|θ_3),

...,

χ(y_{i,1}|θ_i), χ(y_{i,2}|θ_i), χ(y_{i,3}|θ_i),..., χ(y_{i,j}|θ_i),...., χ(y_{I,J}|θ_i),

...,

χ(y_{I,1}|θ_I), χ(y_{I,2}|θ_I), χ(y_{I,3}|θ_I),..., χ(y_{I,j}|θ_I),...., χ(y_{I,J}|θ_I).

whih are used when we calculate the so-called Posterior Predictive P value to test the null hypothesis that our model is fitted a data well.

Revised 2019 Sept. 8

Revised 2019 Dec. 2

Revised 2020 March

Revised 2020 Jul

A list.

From any given posterior MCMC samples θ_1,θ_2,...,θ_i,....,θ_n (provided by stanfitExtended object), it calculates a return value as a vector of the form χ(y_i|θ_i),i=1,2,...., where each dataset y_i is drawn from the corresponding likelihood likelihood(.|θ_i),i=1,2,..., namely,

y_i \sim likelihood(.| θ_i).

The return value also retains these y_i, i=1,2,...

Revised 2019 Dec. 2

## Not run: 

  fit <- fit_Bayesian_FROC( ite  = 1111,  dataList = ddd )
 a <- chi_square_at_replicated_data_and_MCMC_samples_MRMC(fit)

 b<-a$List_of_dataList
 lapply(b, plot_FPF_and_TPF_from_a_dataset)




## End(Not run)