e-Statistics > Medical Signal Detection

Data Visualization

The hierarchical model of binomial distribution is conditioned upon $ N_{\cdot\cdot} = n$ and $ N_{i \cdot} = n_{i \cdot}$ , and related to the unconditional model $ N_{ij} \sim \mathrm{Poisson}(\mu_{ij})$ via $ p_{ij} = \mu_{ij}/\mu_{i \cdot}$ and $ p_{\cdot j} = \mu_{\cdot j}/\mu_{\cdot \cdot}$ where

$\displaystyle \mu_{i \cdot} = \sum_j \mu_{ij}; \quad
\mu_{\cdot j} = \sum_i \mu_{ij}; \quad
\mu_{\cdot \cdot} = \sum_{i,j} \mu_{ij}
$

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

Consider the grid of unit square spaced according to $ p_{i \cdot}$ 's and $ p_{\cdot j}$ 's determining the respective horizontal and vertical intervals. Then $ N_{ij}$ points are distributed uniformly on each $ (i,j)$ -rectangle (cell) of width $ p_{i \cdot}$ and hights $ p_{\cdot j}$ . The result is a Poisson spatial process with each cell intensity

$\displaystyle \Lambda_{ij} = \frac{\mu_{ij}}{q_{i \cdot} \times p_{\cdot j}}
\propto \lambda_{ij}
$

The Poisson intensity is proportional to the relative report rate, and can be averaged over a different (and possibly coarse) grid, which enables us to visualize the intensity in multiple resolutions. The highest intensity is represented by Red (hue value 0.0), and the lowest by the hue parameter (red=0.0, yellow=0.16, green=0.33, cyan=0.5, blue=0.66 and magenta=0.84). The horizontal grid of the intensity map indicates individual drugs ranked and spaced according to DRUG.COUNT, and the vertical grid reaction events ranked and spaced according to REAC.COUNT.