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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 we denote the law of probability of a random variable conditionally given for another random variable , and by the binomial distribution with parameter . Then the hierarchical binomial model of report count is formed by a series of binomial distributions.

1. for the list of adverse reactions.
2. for the pair of valid association
Then we can define the relative report rate by

Introducting Poisson distributions. The hierarchical model of binomial distribution is conditioned upon and , and related to the unconditional model via and where

It is also used to derive the model of conditional distribution with

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

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

where are hyperparameters, and is a gamma density function. The determination of hyperparameters may not be so important; 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 and the baseline are known then the posterior density given the report count is proportional to

Here we can observe that

where

with

Here represents the marginal probability distribution of the report count .

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

Then the posterior distribution of given is expressed as the mixture