Description Usage Arguments Details Value Examples

View source: R/chi_square_goodness_of_fit.R

Revised 2019 Oct 16. Revised 2019 Nov 1. Revised 2019 Nov 27. Revised 2019 Dec 1.

In the following, we explain what this function calculates.

Let *χ^2(y|θ)* be a
*chi square goodness of fit* statistic which is defined by

( Observed data - Expectation *)^2*/Exectation.

In MRMC cases, it is defined as follows.

*χ^2(D|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 *D* 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.

Note that we can rewrite the chi square as follows.

*χ^2(D|θ) := ∑_{r=1}^R ∑_{m=1}^M ∑_{c=1}^C \biggr( \frac{[ H_{c,m,r}- E_{θ}[H_{c,m,r}] ]^2}{E_{θ}[H_{c,m,r}]}+\frac{[F_{c,m,r}- E_{θ}[F_{c,m,r}] ]^2}{ E_{θ}[F_{c,m,r}] }\biggr).*

So, the chi square has two terms.

1) The first term is the difference of hit and its expectation.

2) The second term is the differences of observed false alarms and its expectatioins.

In this function, we calculate each terms, separately. So, return values retain these two terms, separately.

In this function, we calculate the following (I) and (II):

** (I) A vector ——————-**

Let us denote a collection of
posterior MCMC samples for a given dataset *D* by

* θ_1 , θ_2 , \cdots , θ_i, \cdots , θ_N,*

namely, each *θ_i* is
synthesized from posterior
*π(θ|D)*, *θ_i \sim π(θ|D).*

Substituing these MCMC samples into the above definition of the chi square, we obtain the following vector as a return value of this function.

* χ^2(D|θ_1),*

* χ^2(D|θ_2),*

* χ^2(D|θ_3),*

* : *

* : *

* χ^2(D|θ_N).*

** (II) A mean of the above vector, namely, the posterior mean of the chi square over all MCMC samples ————-**

Using the set of chi squares *(χ^2(D|θ_i);i=1,...,N)*
calculated for each posterior MCMC samples *θ_i \sim π(θ|D).*
,
the function also calculates the posterior mean of the chi square statistic, namely,

*\int χ^2(D|θ)π(θ|D) dθ,*

by approximating it as

*\frac{1}{N} ∑ _{i=1} ^N χ^2(D|θ_i),*

where *π(θ|D)* denotes the posterior probability density under the given data *D*.

Do not confuse it with the following

* χ^2(D|θ^*).*

where *θ^** denotes the posterior estimates, i.e., * θ^* := \int θ π(θ|D) dθ*.

1 2 3 4 5 | ```
Chi_square_goodness_of_fit_in_case_of_MRMC_Posterior_Mean(
StanS4class,
summary = TRUE,
dl_is_an_array_of_C_only_and_not_C_M_Q = TRUE
)
``` |

`StanS4class` |
An S4 object of class `stanfit` .
This R object is a fitted model object
as a return value of the function `fit_Bayesian_FROC()` .
To be passed to |

`summary` |
Logical: |

`dl_is_an_array_of_C_only_and_not_C_M_Q` |
A Boolean, if |

This function is implemented by vectorizations and further techinics. When the author review this, I find my past work is great,... I forget that I made this. But this function is great.

Revised 2019 Nov 1

A list, calculated by each modality reader and cofidence level, and MCMC samples.
A one the component of list contains { *χ^2(Data|θ_i)* ; i= 1,2,3,...n}, where *n* is the number of MCMC iterations.

Each component
of list isan array
whose index indicats `[MCMC, Confidence, Modality, Reader] `

.

Each component of list is
`an array whose index indicats [MCMC, C, M, Q]`

.

To be passed to the calculation of Posterior predictive p value, I need the sum of return value, that is, sum of C,M,Q and resulting quantities construct a vetor whose length is a same as the number of MCMC iterations. I love you. I need you. So, to calculate such quantites, the author .... will make a new function.

Also, it retains the posterior mean of chi square statistic for
an assumed occurrence of the data *D*:

*χ^2(Data) = \int χ^2(Data|θ) π(θ|D)d θ*

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 | ```
## Not run:
#========================================================================================
# 1) Create a fitted model object for data named dd
#========================================================================================
fit <- fit_Bayesian_FROC( ite = 1111, # Number of MCMC iterations
cha = 1,
dataList = BayesianFROC::dd # This is a MRMC dataset.
)
#========================================================================================
# 2) Calculate a chi square and meta data
#========================================================================================
a <- Chi_square_goodness_of_fit_in_case_of_MRMC_Posterior_Mean(fit)
#========================================================================================
# 3) Extract a chi square
#========================================================================================
chi.square <- a$chi.square
#========================================================================================
# A case of single reader is special in the programming perspective
# 2020 Feb 24
#========================================================================================
f <- fit_Bayesian_FROC( ite = 1111,
cha = 1,
summary = TRUE,
dataList = dddd,
see = 123)
Chi_square_goodness_of_fit_in_case_of_MRMC_Posterior_Mean(f)
# Revised 2019 August 19
# 2019 Nov 1
## End(Not run)# dontrun
``` |

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