 Solution

The Poisson regression margin-matching score equations from the site indicator for each $i\ge 2$ directly giveSubtracting these seven row balances from the intercept equation ${\sum }_{ij}{e}_{ij}={\sum }_{ij}{y}_{ij}$ gives the missing reference row balance. ThereforeThe after-indicator score similarly gives ${\sum }_i{e}_{i2}={\sum }_i{y}_{i2}$. Subtracting it from the intercept balance yields the before-column balance, soThus 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 $e_{ij}=p_{ij}{\hat{\mu }}_{ij}$; 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.