Monte Carlo Method for Obtaining the Cutoff Value

Monte Carlo simulation method for obtaining the cutoff value \(c_{univ,j}^+\) in Step 2 of the MDDC method.

1. Obtain the marginals \(n_{1\bullet}, \ldots, n_{I\bullet}, n_{\bullet 1}, \ldots, n_{\bullet J}\) from the original \(I \times J\) contingency table.

2. Under the assumption of no association between drugs and AEs (i.e., independence of rows and columns), compute cell probabilities

\[p_{ij} = \left(\frac{n_{i\bullet}}{n_{\bullet \bullet}}\right)\left(\frac{n_{\bullet j}}{n_{\bullet \bullet}}\right),\]

where \(i = 1, \ldots, I\) and \(j = 1, \ldots, J\).

3. Generate 10,000 \(I \times J\) contingency tables with the above specified marginals and cell probabilities \(\{p_{ij}\}\) through multinomial distribution

\[(n_{11}, n_{12}, \ldots, n_{IJ}) \sim \text{Multinomial}(n_{\bullet \bullet}, p),\]

where \(p = (p_{11}, p_{12}, \ldots, p_{IJ})^T\).

4. For the \(r\)-th simulated table, \(r = 1, \ldots, 10000\), compute \(e_{ij}\) for all the cells in the table, and obtain

\[m_{j,r} = \max_{1 \leq i \leq I} e_{ij} \times \mathbf{1}\{n_{ij}>5\}\]

for \(j = 1, \ldots, J\). For each drug \(j\), this will provide \(m_{j,1}, m_{j,2}, \ldots, m_{j,10000}\).

5. For each drug \(j\), obtain the cutoff value \(c_{univ,j}^+\) as the 95-th quantile by ordering \(m_{j,1}, m_{j,2}, \ldots, m_{j,10000}\) from smallest to largest.