Time is money, and I have time but no money. Why? Nowadays, many people work in eight days a week! But, I work in zero days a week! Thus, time is no money.

Anyway, the author thinks the conventional notation would helps such no time people.

Recall that the conventional likelihood notation;

\[ f(y|\theta), \]

where \(y\) denotes data and \(\theta\) is a model parameter.

In case of a number of confidence levels \(C=5\).

Confidence Level | No. of Hits | No. of False alarms |
---|---|---|

5 = definitely present | \(H_{5}\) | \(F_{5}\) |

4 = probably present | \(H_{4}\) | \(F_{4}\) |

3 = equivocal | \(H_{3}\) | \(F_{3}\) |

2 = probably | \(H_{2}\) | \(F_{2}\) |

1 = questionable | \(H_{1}\) | \(F_{1}\) |

where, \(H_{c},F_c \in \mathbb{N}\). Moreover Number of images \(N_I\) and Number of lesions \(N_L\).

So, in conventional notation we may write

\[y = (H_1,H_2,H_3,H_4,H_5, F_1,F_2,F_3,F_4,F_5 ;N_L,N_I) \in \mathbb{N}^{2C+2}.\]

\[ H_c \sim \text{Binomial}(p_c(\theta),N_L),\\ F_c \sim \text{Poisson}(q_c(\theta)).\\ \]

\[ p_c(\theta) := \int_{\theta_c}^{\theta_{c+1}}\text{Gaussian}_{}(x|\mu,\sigma)dx,\\ q_c(\theta) := \int_{\theta_c}^{\theta_{c+1}}N_I \times \frac{d \log \Phi(z)}{dz}dz. \]

where model parameter is \(\theta = (\theta_1,\theta_2,\theta_3,...\theta_C;\mu,\sigma)\) which should be estimated and \(\Phi\) denotes the cumulative distribution functions of the canonical Gaussian. Note that \(\theta_{C+1} = \infty\)

Omitted

Or

Or

```
dat <- list(c=c(3,2,1), #Confidence level. Note that c is ignored.
h=c(97,32,31), #Number of hits for each confidence level
f=c(1,14,74), #Number of false alarms for each confidence level
NL=259, #Number of lesions
NI=57, #Number of images
C=3) #Number of confidence level
# where,
# c denotes confidence level, i.e., rating of reader.
# 3 = Definitely deseased,
# 2 = subtle,.. deseased
# 1 = very subtle
# h denotes number of hits (True Positives: TP) for each confidence level,
# f denotes number of false alarms (False Positives: FP) for each confidence level,
# NL denotes number of lesions,
# NI denotes number of images,
fit <- BayesianFROC::fit_Bayesian_FROC( dataList = d )
```