$\sum_{i < j} \frac{\pi_i \pi_j - \pi_{ij}}{\pi_{ij}} (\frac{y_i}{\pi_i} - \frac{y_j}{\pi_j})^2$

$\sum_{i < j} \frac{\pi_{ij} - \pi_i \pi_j}{\pi_{ij}} (\frac{y_i}{\pi_i} - \frac{y_j}{\pi_j})^2$

$\sum_{i < j} (\pi_i \pi_j - \pi_{ij}) (\frac{y_i}{\pi_i} + \frac{y_j}{\pi_j})^2$

$\sum_{i < j} \pi_{ij} (\frac{y_i}{\pi_i} - \frac{y_j}{\pi_j})^2$