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Relative Report Rate

Relative report rate (also called relative risk in statistics literature) is used for ``interesting'' association between drugs and adverse events.

Hierarchical multinomial model. By $ \mathcal{L}(X\vert Y=y)$ we denote the law of probability of a random variable $ X$ conditionally given $ Y = y$ for another random variable $ Y$, and by $ B(n,p)$ the binomial distribution with parameter $ (n,p)$. Then the hierarchical binomial model of report count is formed by a series of binomial distributions.

  1. $ \mathcal{L}(C_{\cdot B}\vert C_{\cdot\cdot} = n) \sim B(n,p_{\cdot B})$ for the list $ B$ of adverse reactions.
  2. $ \mathcal{L}(C_{A,B}\vert C_{A \cdot} = n_A) \sim B(n_A,p_{A,B})$ for the pair $ (A,B)$ of valid association
Then we can define the relative report rate by

$\displaystyle \lambda_{A,B} = p_{A,B}/p_{\cdot B}
$

Introducting Poisson distributions. The hierarchical model of binomial distribution is conditioned upon $ C_{\cdot\cdot} = n$ and $ C_{A \cdot} = n_{A \cdot}$, and related to the unconditional model $ C_{A,B} \sim \mathrm{Poisson}(\mu_{A,B})$ via $ p_{A,B} = \mu_{A,B}/\mu_{A \cdot}$ and $ p_{\cdot B} = \mu_{\cdot B}/\mu_{\cdot \cdot}$ where

$\displaystyle \mu_{A \cdot} = \sum_B \mu_{A,B}; \quad
\mu_{\cdot B} = \sum_A \mu_{A,B}; \quad
\mu_{\cdot \cdot} = \sum_{(A,B)\in\mathcal{V}} \mu_{A,B}
$

It is also used to derive the model $ \mathcal{L}(C_{A \cdot}\vert C_{\cdot\cdot} = n) \sim B(n,p_{A \cdot})$ of conditional distribution with $ p_{A \cdot} = \mu_{A \cdot}/\mu_{\cdot \cdot}$

Model parameters. Empirical Bayes approach can achieve the interpretability of the relative report model. Assume that each report count $ C$ is a draw from a Poisson distribution with unknown mean $ \mu$. Here the values

$\displaystyle \lambda=\mu/E
$

is treated as parameters, drawn from a common prior distribution.

Prior density. The prior distribution of relative report rate is assumed to be the mixture of two gamma distributions

$\displaystyle \pi(\lambda)
=p g(\lambda;\alpha_1,\beta_1)+(1-p)g(\lambda;\alpha_2,\beta_2)
$

where $ \alpha_1, \beta_1, \alpha_2, \beta_2, p$ are hyperparameters, and $ g(\lambda;\alpha,\beta)
=\beta^{\alpha}\lambda^{\alpha-1}e^{-\beta\lambda}/\Gamma(\alpha)$ is a gamma density function. The determination of hyperparameters may not be so important; $ \alpha_1 = 0.2, \beta_1 = 0.1, \alpha_2 = 2, \beta_2 = 4, p = 1/3$ can be a good choice, suggested by the fact that the majority of relative report rates are well below one.

Posterior density. If the prior density $ \pi(\lambda)$ and the baseline $ E$ are known then the posterior density $ \pi(\lambda \vert n)$ given the report count $ C = n$ is proportional to

$\displaystyle \phi(\lambda; n, E) = e^{-E \lambda + E} \lambda^n \pi(\lambda)
$

Here we can observe that

$\displaystyle \Phi(n, E)
= \int_0^\infty \rho(\lambda; n) d\lambda
= \pi(n)\left/\left(e^{-E} \frac{E^n}{n!}\right)\right.
$

where

$\displaystyle \pi(n) = p f(n;\alpha_1, \beta_1, E)+(1-p) f(n;\alpha_2, \beta_2, E)
$

with

$\displaystyle f(n;\alpha,\beta,E) = (1+\beta/E)^{-n}(1+E/\beta)^{-\alpha}
\Gamma(\alpha+n)/\Gamma(\alpha)n!
$

Here $ \pi(n)$ represents the marginal probability distribution of the report count $ C = n$.

Gamma-Poisson shrinker. The posterior probability $ q$ of the first component can be derived as

$\displaystyle q=\frac{p f(n;\alpha_1, \beta_1, E)}{\pi(n)}
$

Then the posterior distribution of $ \lambda$ given $ C = n$ is expressed as the mixture

$\displaystyle f(\lambda\vert n,E)
=\pi(\lambda;\alpha_1+n,\beta_1+E,\alpha_2+n,\beta_2+E,q)
$


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