fit_Null_hypothesis_model_to_: Fit the null model
In BayesianFROC: FROC Analysis by Bayesian Approaches

Fit the null model, representing the null hypothesis that all modalities are same.

fit_Null_hypothesis_model_to_(
  dataList,
  DrawCurve = FALSE,
  type_to_be_passed_into_plot = "p",
  PreciseLogLikelihood = FALSE,
  dataList.Name = "",
  ModifiedPoisson = FALSE,
  verbose = TRUE,
  summary = TRUE,
  mesh.for.drawing.curve = 10000,
  significantLevel = 0.7,
  cha = 1,
  war = floor(ite/5),
  ite = 10000,
  dig = 3,
  see = 1234569,
  ...
)

`dataList`	A list, to be fitted a model. For example, in case of a single reader and a single modality, it consists of `f, h, NL, NI, C`. The detail of these dataset, see the example data-sets. Note that the maximal number of confidence level, denoted by `C`, are included, however, should not include its each confidence level in `dataList`
`DrawCurve`	Logical: `TRUE` of `FALSE`. Whether the curve is to be drawn. TRUE or FALSE. If you want to draw the FROC and AFROC curves, then you set `DrawCurve =TRUE`, if not then `DrawCurve =FALSE`. The reason why the author make this variable `DrawCurve` is that it takes long time in MRMC case to draw curves, and thus Default value is `FALSE` in the case of MRMC data.
`type_to_be_passed_into_plot`	"l" or "p".
`PreciseLogLikelihood`	Logical, that is `TRUE` or `FALSE`. If `PreciseLogLikelihood = TRUE`(default), then Stan calculates the precise log likelihood with target formulation. If `PreciseLogLikelihood = FALSE`, then Stan calculates the log likelihood by dropping the constant terms in the likelihood function. In past, I distinct the stan file, one is target formulation and the another is not. But non-target formulation cause some Jacobian warning, thus I made all stanfile with target formulation when I uploaded to CRAN. Thus this variable is now meaningless.
`dataList.Name`	This is not for user, but the author for this package development.
`ModifiedPoisson`	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
`verbose`	A logical, if `TRUE`, then the redundant summary is printed in R console. If `FALSE`, it suppresses output from this function.
`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.
`mesh.for.drawing.curve`	A positive large integer, indicating number of dots drawing the curves, Default =10000.
`significantLevel`	This is a number between 0 and 1. The results are shown if posterior probabilities are greater than this quantity.
`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.
`war`	A variable to be passed to the function `rstan::sampling`() of rstan in which it is named `warmup`. A positive integer representing the Burn in period, which must be less than `ite`. Defaults to war = floor(ite/5)=10000/5=2000,
`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 = 1111
`dig`	A variable to be passed to the function `rstan::sampling`() of rstan in which it is named `...??`. A positive integer representing the Significant digits, used in stan Cancellation. Default = 5,
`see`	A variable to be passed to the function `rstan::sampling`() of rstan in which it is named `seed`. A positive integer representing seed used in stan, Default = 1234.
`...`	Additional arguments