e-Statistics > Medical Signal Detection

Relative Report Rate

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 multinomial model of contingency table is hierarchically described as a series of binomial distributions.

  1. $ \mathcal{L}(N_{\cdot j}\vert N_{\cdot\cdot} = n) \sim B(n,p_{\cdot j})$ for the $ j$ -th marginal count of adverse event.
  2. $ \mathcal{L}(N_{ij}\vert N_{i \cdot} = n_i) \sim B(n_i,p_{ij})$ for the $ (i,j)$ -cell count.
The relative report rate is the ratio $ \lambda_{ij} = p_{ij}/p_{\cdot j}$ , and can be estimated by

$\displaystyle \lambda_{ij}^{RR} := \hat{p}_{ij}/\hat{p}_{\cdot j}

where we use the following estimates:

$\displaystyle \hat{p}_{\cdot j} = N_{\cdot j}/N_{\cdot\cdot}$    and $\displaystyle \quad
\hat{p}_{ij} = N_{ij}/N_{i \cdot}

The estimate $ \lambda_{ij}^{RR}$ is often referred as ``relative risk (RR),'' and will be used to compare with two other estimates, Empirical Bayes Geometric Mean (EBGM) by DuMouchel, and a Stein-like estimator (STEIN). Drug-reaction combinations are ranked according to the estimate of relative report rate. The table below will display the combinations from to in the rankings.

Provided the entries for DRUG and REAC, we can display their relative report rates and rankings from another method. This enables us to compare two different rankings.