= Solution
The <Poisson regression margin-matching score equations> from the site indicator for each $i\ge2$ directly give
$$
\sum_j e_{ij}=\sum_j y_{ij}\qquad(i=2,\ldots,8).
$$
Subtracting these seven row balances from the intercept equation $\sum_{ij}e_{ij}=\sum_{ij}y_{ij}$ gives the missing reference row balance. Therefore
$$
\boxed{\sum_j e_{ij}=\sum_j y_{ij}\quad\text{for every site }i}.
$$
The after-indicator score similarly gives $\sum_i e_{i2}=\sum_i y_{i2}$. Subtracting it from the intercept balance yields the before-column balance, so
$$
\boxed{\sum_i e_{ij}=\sum_i y_{ij}\quad(j=1,2)}.
$$
Thus the fitted before and after count totals are $114$ and $15$, and each fitted site total equals its observed total. These identities concern expected counts $e_{ij}=p_{ij}\widehat\mu_{ij}$; unequal exposures do not imply corresponding unweighted sums of fitted rates equal sums of observed rates. They follow from the unpenalized canonical Poisson <likelihood> and the available intercept/factor columns, not from balanced exposure periods.
Back to article page
