= Quadratic Pearson approximation to the Poisson deviance
{title2=$D\simeq\sum_i(y_i-e_i)^2/e_i$}
Write $y_i=e_i+r_i$. For $|r_i|/e_i$ small, the cell contribution to the <Poisson deviance> expands as
$$
2[(e_i+r_i)\log(1+r_i/e_i)-r_i]
=\frac{r_i^2}{e_i}-\frac{r_i^3}{3e_i^2}+O(r_i^4/e_i^3).
$$
Summing 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.
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