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Use $e_{ij}$ for the expected accident count, not for the accident rate. With a design row $z_{ij}$ consisting of an intercept, seven non-reference site indicators and the after-treatment indicator,Here $\log p_{ij}$ is a known generalized linear model offset, with coefficient fixed at one. Differentiating the Poisson regression log-likelihood gives the score function equationsThese nine equations estimate the intercept, seven site contrasts and one treatment contrast, with the stated reference-level constraints.

The Fisher information is $Z^{\mathsf{T}}WZ$, where $W=\mathrm{diag}(e_{ij})$. A Fisher scoring update adds $(Z^{\mathsf{T}}WZ{)}^{-1}{Z}^{\mathsf{T}}(y-e)$ to the current coefficient vector. In iteratively reweighted least squares, the working response is $z_{ij}^{\ast }=\log e_{ij}+(y_{ij}-e_{ij})/e_{ij}$; regress $z_{ij}^{\ast }-\log p_{ij}$ on the design with weights $e_{ij}$. Thus the reported four iterations solve the likelihood equations while accounting for unequal exposure periods.

The common adjusted rate ratio is obtained by exponentiating the after coefficient:This estimates a $54.2\%$ lower accident rate after the intervention, conditional on the additive site model. The standard error $0.2751810$ gives normal Wald statistic $-2.8369$, two-sided $p\simeq 0.0046$, and an approximate $95\%$ confidence interval for the rate ratioThe label t value is again a normal-reference Wald ratio under fixed Poisson dispersion, rather than an exact Student statistic.

The intercept gives the before rate at reference site one: $e^{0.2707792}\simeq 1.311$ accidents per year. The seven site coefficients are log baseline-rate ratios relative to that site. In site order two through eight, their exponentials are about $0.615,2.767,1.711,0.769,1.797,0.615,1.221$. The third site has the largest estimated rate and a nominal Wald ratio about $3.118$ against the reference. These are adjusted baseline comparisons, not differences in absolute accident counts. The overall unadjusted before/after rate ratio is $(15/18)/(114/68)\simeq 0.497$, and it differs from $0.458$ because site risks and their relative exposure weights differ.

An independent way to solve the fit is the profile likelihood for a common Poisson rate ratio with unequal exposures. Put $t_i=y_{i1}+y_{i2}$ and ${\lambda }_i=\exp (\mu +{\alpha }_i)$. For fixed $r$, the site score gives ${\hat{\lambda }}_i=t_i/(p_{i1}+r{p}_{i2})$. The remaining equation isIts positive root reproduces the reported treatment estimate. Comparing the same site model with $r=1$ to the fitted model gives a deviance improvement about $9.750$ on one degree of freedom, with nominal likelihood-ratio $p\simeq 0.0018$.

There are $16$ count observations and $9$ fitted mean parameters, leaving $7$ residual degrees of freedom. The null deviance $132.9485$ on $15$ degrees of freedom is for the intercept-only rate model, still including the exposure offset. The residual Poisson deviance is $16.27524$ on $7$ degrees of freedom. A chi-squared comparison gives approximately $p=0.023$, suggesting potential lack of fit of a single common treatment ratio; the extreme deviance residuals near $-2.03$ and $2.14$ also deserve inspection. Several after-cell expected counts are below one or near two, so this absolute-fit calibration is only approximate. Site-specific treatment effects, overdispersion or temporal dependence are possible explanations to investigate, with a parametric bootstrap available to calibrate the sparse-count fit statistic. A before/after association alone does not isolate a causal effect from secular changes or regression to the mean at selected sites.

Use the original PDF's value $\hat{p}=0.386$; the TeX aid misreads its last digit. In the fitted conditional mixture, $\hat{\theta }=0.575$ means that about 57.5% are assigned to the structural-zero, or modelled cured, component. Its asymptotic 95% confidence interval of $(0.30,0.81)$ shows substantial uncertainty about that fraction. An observed post-treatment zero alone is not proof of cure, since the non-cured Poisson distribution can also produce zero.

The parameter $p$ is a conditional allocation probability, not the ratio of post-treatment to pre-treatment means. That ratio among non-cured patients is $\beta =p/(1-p)$. HenceThe map $p\mapsto p/(1-p)$ is strictly increasing, so transform the endpoints of the supplied 95% confidence interval:No change in the non-cured mean is $\beta =1$, equivalently $p=1/2$, which lies in this interval. Thus the point estimate suggests a reduction, but this interval does not firmly establish one among the non-cured patients at the corresponding two-sided 5% level. It is not correct to describe $p=0.386$ itself as a 61.4% reduction in their mean count.

Under a full pre-conditioning mixture with cure independent of the baseline rate, the population mean ratio would be $(1-\theta )\beta$, whose plug-in value is about $0.425(0.629)=0.267$. This combines disappearance of counts in one component with a change of rate in the other; it is distinct from the conditional estimate's direct interpretation. The two marginal confidence intervals cannot simply be multiplied to obtain a calibrated joint 95% interval for this derived quantity.

Only 12 patients were observed, so the asymptotic intervals, particularly for a mixture parameter near a boundary, should be interpreted cautiously. The data describe one-minute counts and do not establish permanent cure. Nor does an uncontrolled before-and-after comparison alone prove a causal effect of the drug: natural count variability, secular changes and regression to the mean remain possible. The structural-zero interpretation and the reduction among non-cured patients are conclusions within the specified statistical model.

After adjusting for location, adding the period effect reduces deviance by $9.750$ on one degree of freedom, giving the supplied likelihood-ratio test $p\simeq 0.002$. The adjusted incidence rate ratio isUnder this Poisson exposure model, the accident rate after installation is estimated to be about $54.2\%$ lower, with a statistically significant reduction. The remaining lack of fit merits residual checks and possibly a Quasi-Poisson model or a model allowing location-specific changes. This before-and-after observational study establishes an association under the fitted model; changes in traffic volume, broader trends, or regression to the mean at sites chosen for high prior counts could also contribute, so the estimated reduction is not by itself a causal treatment effect.