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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) $

    © TTU Mathematics