Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 37 / 4

The line compares the two pooled accident rates, dividing the total number of accidents by total observation years in each period. It gives the unadjusted incidence rate ratiocorresponding to an estimated reduction of about $50.3\%$. Pooling does not adjust for differences between locations or for the unequal location composition of observation time before and after installation.

Let $Y_{it}$ be the accident count at location $i$ in period $t\in \{0,1\}$, and $e_{it}$ its exposure in years, where $t=1$ denotes after installation. The Poisson exposure model isThe offset $\log e_{it}$ has its coefficient fixed at one. The model assumes constant rates within each period, a common before rate across locations, a common treatment rate ratio, independent counts, and Poisson variance equal to the mean. The incidence rate ratio is $r=e^{\beta }$, soAn approximate 95% Wald confidence interval transforms the interval for $\beta$:

The second Poisson exposure model allows each location its own baseline accident rate:The last equality is the corner-point constraint. The common after-to-before incidence rate ratio remains $e^{\beta }$. This model is appropriate because the sites can have different background risks; treating those differences as unexplained Poisson variation made the first model fit poorly, with deviance $50.863$ on 14 residual degrees of freedom.

Adding seven location effects to the first fitted model reduces the deviance by $50.863-16.275=34.588$, which is large relative to ${\chi }_7^2$. The supplied sequential analysis of deviance reports a slightly different location contribution because it enters locations before the period indicator. The full model is much better but its deviance $16.275$ on 7 degrees of freedom still gives an approximate goodness-of-fit $p$-value $0.0227$. That suggests residual lack of fit or overdispersion, although sparse accident counts make the asymptotic deviance goodness-of-fit test approximate.

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.