Very Very Very Brief Description of MRMC

Issei Tsunoda

2020-01-20

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

Conventional Notation

Recall that the conventional likelihood notation;

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

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

Data \(y\)

2 readers, 2 modalities and 3 confidence levels.

Confidence Level Modality ID Reader ID Number of Hits Number of False alarms
3 = definitely present 1 1 \(H_{3,1,1}\) \(F_{3,1,1}\)
2 = equivocal 1 1 \(H_{2,1,1}\) \(F_{2,1,1}\)
1 = questionable 1 1 \(H_{1,1,1}\) \(F_{1,1,1}\)
3 = definitely present 1 2 \(H_{3,1,2}\) \(F_{3,1,2}\)
2 = equivocal 1 2 \(H_{2,1,2}\) \(F_{2,1,2}\)
1 = questionable 1 2 \(H_{1,1,2}\) \(F_{1,1,2}\)
3 = definitely present 2 1 \(H_{3,2,1}\) \(F_{3,2,1}\)
2 = equivocal 2 1 \(H_{2,2,1}\) \(F_{2,2,1}\)
1 = questionable 2 1 \(H_{1,2,1}\) \(F_{1,2,1}\)
3 = definitely present 2 2 \(H_{3,2,2}\) \(F_{3,2,2}\)
2 = equivocal 2 2 \(H_{2,2,2}\) \(F_{2,2,2}\)
1 = questionable 2 2 \(H_{1,2,2}\) \(F_{1,2,2}\)

where, each component \(H\) and \(F\) are non negative integers. By the multi-index notation, for example, \(H_{3,2,1}\) means the hit of the \(1\)-st reader over all images taken by \(2\)-nd modality with reader’s confidence level is \(3\).

So, in conventional notation we may write

\[y = (H_{c,m,r},F_{c,m,r} ;N_L,N_I).\]

Likelihood \(f(y|\theta)\)

\[ H_{c,m,r} \sim \text{Binomial}(p_{c,m,r}(\theta),N_L),\\ F_{c,m,r} \sim \text{Poisson}(q_c(\theta)).\\ \]

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

\[ A_{m,r} := \Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \\ A_{m,r} \sim \text{Normal} (A_{m},\sigma_{r}^2), \\ \]

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

Prior

\[ dz_c := z_{c+1}-z_{c},\\ dz_c, \sigma_{m,r} \sim \text{Uniform}(0,\infty),\\ z_{c} \sim \text{Uniform}( -\infty,100000),\\ A_{m} \sim \text{Uniform}(0,1).\\ \]

This is only example, and in this package I implement proper priors. The author thinks the above prior is intuitively most suitable non informative priors.

Of course I know it (uniform prior) is not suitable in some sence, Jeffrays prior is more .. but I won’t do such discussion. Only intuitive. Ha,.. I am only amateur. I am no responsiblity and no money, no position. Imagine there no position,…Oh,…I no need to imagine since I already realize it. Ha ha ha :-D but no possetion makes no money but it makes freedom, unfortunately, my body has pains from all my body by multiple chemical sensitivity. I want to d…

R script

d <- dataset_creator_new_version()



fit <- fit_Bayesian_FROC(
   ite  = 1111, 
   cha = 1,
   summary = TRUE,  
   Null.Hypothesis = F, 
   dataList = d )

Appendix:

\(\color{green}{\textit{ Why a Bayesian approach now. }}\)

In the following, the author pointed out why frequentist p value is problematic. Of course, under some condition, Bayesian p -value coincides with frequentist p value, so the scheme statistical test is problematic. We shall show the reason in the following simple example.

To tell the truth, I want to use epsilon delta manner, but to read not mathematics people, I do not use it.

I want to publish this proof, but all reviewers are against it. There is free speech here, so please enjoy my logic. I really like this because My heart is in math.

Monotonicity issues on p value

The methods of statistical testing are widely used in medical research. However, there is a well-known problem, which is that a large sample size gives a small p-value. In this section, we will provide an explicit explanation of this phenomenon with respect to simple hypothesis tests.

Consider the following null hypothesis \(H_0\) and its alternative hypothesis \(H_1\); \[\begin{eqnarray*} H_0: \mathbb E[ X_i] &=&m_0, \\ H_1: \mathbb E[ X_i] &>&m_0, \\ \end{eqnarray*}\] where \(\mathbb E[ X_i]\) means the expectation of random samples \(X_i\) from a normal distribution whose variance \(\sigma _0 ^2\) is known. In this situation, the test statistic is given by \[ Z^{\text{test}} := \frac{\overline{X_{n}} -m_0 }{\sqrt{\sigma _0 ^2/n}}, \] where \(\overline{X_{n}} := \sum_{i=1,\cdots,n} X_i /n\) is normally distributed with mean \(m_0\) and standard deviation \(\sigma_0/\sqrt{n}\). Under the null hypothesis, \(Z^{\text{test}}\) is normally distributed with mean \(0\) and a standard deviation \(1\) (standard normal distribution). The null hypothesis is rejected if \(Z^{\text{test}} >z_{2\alpha}\) , where \(z_{2\alpha}\) is a percentile point of the normal distribution, e.g., \(z_{0.025}=1.96 .\)

Suppose that the true distribution of \(X_1, \cdots, X_n\) is a normal distribution with mean \(m_0 + \epsilon\) and variance \(\sigma _0 ^2\), where \(\epsilon\) is an arbitrary fixed positive number. Then \[\begin{eqnarray*} Z^{\text{test}} &=&\frac{\overline{X_{n}} -(m_0+\epsilon -\epsilon) }{\sqrt{\sigma _0 ^2/n}}\\ &=& Z^{\text{Truth}} + \frac{\epsilon}{\sqrt{\sigma _0 ^2/n}}\\ \end{eqnarray*}\] where \(Z^{\text{Truth} }:=(\overline{X_{n}} -(m_0+\epsilon ))/\sqrt{\sigma _0 ^2/n}\).

In the following, we calculate the probability with which we reject the null hypothesis \(H_0\) with confidence level \(\alpha\). \[\begin{eqnarray*} \text{Prob}(Z^{\text{test}} >z_{2\alpha} ) &=&\text{Prob} (Z^{\text{Truth} } + \frac{\epsilon}{\sqrt{\sigma _0 ^2/n}} >z_{2\alpha})\\ &=&\text{Prob} (Z^{\text{Truth} } >z_{2\alpha} - \frac{\epsilon}{\sqrt{\sigma _0 ^2/n}})\\ &=&\text{Prob} (Z^{\text{Truth} } >z_{2\alpha} - \frac{\epsilon}{\sigma _0 }\sqrt{n} )\\ \end{eqnarray*}\] Note that \(\epsilon /\sigma _0\) is called the effect size.

Thus, if \(z_{2\alpha} - \epsilon \sqrt{n} /\sigma _0 < z_{2(1-\beta)}\), i.e., if \(n > ( z_{2\alpha}- z_{2(1-\beta)})^2 \sigma _0 ^2 \epsilon ^{-2}\), then the probability that the null hypothesis is rejected is greater than \(1 - \beta\).

For example, consider the case \(\sigma _0 =1\), \(\alpha =0.05\), and \((1-\beta) =\alpha\), then \(z_{2\alpha}=1.28\) and in this case, for all \(\epsilon>0\), if \(n > 7 \epsilon ^{-2}\) then the probability in which above hypothesis test concludes that the difference of the observed mean from the hypothesized mean is significant is greater than \(0.95\). This means that almost always the p-value is less than 0.05. Thus a large sample size induces a small p-value.

For example,

  • if \(\epsilon =1\) then by taking a sample size such that \(n > 7\), then almost always the conclusion of the test will be that the observed difference is statistically significant. Similarly,

  • if \(\epsilon =0.1\) then by taking a sample size such that \(n > 700\), then almost all tests will reach the conclusion that the difference is significant; and

  • if \(\epsilon =0.01\) then by taking sample size so that \(n > 70000\), then the same problem will arise.

This phenomenon also means that in large samples statistical tests will detect very small differences between populations.

By above consideration we can get the result ``significance difference’’ with respect to any tiny difference \(\epsilon\) by collecting a large enough sample \(n\), and thus we must not use the statistical test.