Quadratic Pearson approximation to the Poisson deviance ($D\simeq {\sum }_i(y_i-e_i{)}^2/e_i$)

Write $y_i=e_i+r_i$. For $|r_i|/e_i$ small, the cell contribution to the Poisson deviance expands asSumming gives the Pearson chi-squared statistic as the quadratic approximation. The approximation can be inaccurate for sparse expected counts or cells with relative residuals near minus one.