= Solution
Use the <logit link> $g(\pi)=\log\{\pi/(1-\pi)\}$ and independent counts $R_{ij}\sim\operatorname{Bin}(n_{ij},\pi_{ij})$. This gives a <grouped-binomial logistic regression> with 24 cells and 13 free coefficients: an intercept, 11 centre contrasts, and one period contrast. Reference <treatment coding> makes the intercept the log odds for Centre 1 in the earlier of the two included periods. The two <factors> are assumed to have been coded as specified in the question.
The binomial <log-likelihood>, apart from known constants, is
$$
\ell=\sum_{i,j}\{r_{ij}\eta_{ij}-n_{ij}\log(1+e^{\eta_{ij}})\},\qquad
\eta_{ij}=\mu+\alpha_i+\beta_j.
$$
Maximize by <Fisher scoring> or <iteratively reweighted least squares>. Using response proportions $r_{ij}/n_{ij}$ with binomial trial weights $n_{ij}$ is equivalent to using the count response $(r_{ij},n_{ij}-r_{ij})$; omitting those weights would give the wrong <likelihood> and <standard errors>. The coefficient <covariance matrix> is approximately the inverse of $X^T\operatorname{diag}\{n_{ij}\widehat\pi_{ij}(1-\widehat\pi_{ij})\}X$.
The fitted reference log odds are $-0.809585$, and the period change is $-0.426758$, giving
$$
\boxed{\widehat\pi_{1,2}=0.30798,\qquad \widehat\pi_{1,3}=0.22507.}
$$
These are fitted probabilities under a common period effect, so they need not equal the separate empirical proportions at Centre 1. The estimated period <odds ratio> is
$$
\boxed{e^{-0.426758}=0.6526,\qquad 95\%\text{ CI}=e^{-0.426758\pm1.96(0.078807)}=(0.5592,0.7616).}
$$
Thus the later-period death odds are estimated to be about 35% lower at each centre, with strong evidence of a period effect: the normal <Wald statistic> is $-5.415$. The displayed “t value” is a coefficient divided by its <standard error>; with fixed binomial dispersion its usual large-sample calibration is standard normal, not a Student distribution with an estimated Gaussian-error <variance>. The <odds ratio> is not a mortality <probability> ratio.
Each centre coefficient is a log <odds ratio> relative to Centre 1, adjusted for period. All its point estimates are negative, so all other centres have lower fitted mortality probabilities in either period. For example, Centre 11's <odds ratio> is $e^{-1.079474}=0.3398$, or about $2.94$ times greater odds at Centre 1 than at Centre 11. Individual unadjusted two-sided <Wald tests> find significant differences at the 5% level for centres 2, 4, 6, 7, 8, 9, 11 and 12; the coefficients for 3, 5 and 10 do not individually reach that threshold. These separate tests are correlated comparisons against a common reference and should not be presented as 11 independent discoveries without considering <multiple testing>. They adjust only for time period, not for differing patient case mix.
For fit assessment, the residual <binomial deviance> is $15.73891$ on $24-13=11$ <statistical degrees of freedom>. Its $\chi^2_{11}$ upper tail is approximately $0.151$, so there is no appreciable evidence of lack of fit to these binomial cell counts. In particular, comparing with a saturated centre-by-period model gives this same deviance and 11 additional parameters; the data do not strongly require a centre-period <interaction>. The moderate deviance residuals are consistent with that assessment, although a residual plot by centre and period would be more informative than the five-number summary alone.
The reduction from null to fitted deviance is
$$
119.0659-15.73891=103.32699.
$$
It tests the 12 centre-and-period coefficients jointly against an intercept-only model, with reference $\chi^2_{12}$ and $p\approx1.24\times10^{-16}$. It is not a centre-only test adjusted for period. For that test, fit the period-only model and compare its deviance with the full fit: the pooled binomial probabilities in each period give deviance $90.34578$, so
$$
\boxed{D_{\text{period only}}-D_{\text{centre+period}}=74.60687,\quad 11\text{ df},\quad p\approx1.61\times10^{-11}.}
$$
Thus centre heterogeneity persists after allowing for period. Dispersion one is the stipulated binomial <variance>, and the five scoring iterations describe numerical convergence rather than statistical evidence. The fitted associations alone do not establish a causal difference in surgical quality.
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