(i) Suppose $Y_{1}, \ldots, Y_{n}$ are independent Poisson variables, and

$$\mathbb{E}\left(Y_{i}\right)=\mu_{i}, \log \mu_{i}=\alpha+\beta^{T} x_{i}, 1 \leqslant i \leqslant n$$

where $\alpha, \beta$ are unknown parameters, and $x_{1}, \ldots, x_{n}$ are given covariates, each of dimension $p$. Obtain the maximum-likelihood equations for $\alpha, \beta$, and explain briefly how you would check the validity of this model.

(ii) The data below show $y_{1}, \ldots, y_{33}$, which are the monthly accident counts on a major US highway for each of the 12 months of 1970 , then for each of the 12 months of 1971 , and finally for the first 9 months of 1972 . The data-set is followed by the (slightly edited) $R$ output. You may assume that the factors 'Year' and 'month' have been set up in the appropriate fashion. Give a careful interpretation of this $R$ output, and explain (a) how you would derive the corresponding standardised residuals, and (b) how you would predict the number of accidents in October 1972 .

$\begin{array}{llllllllllll}52 & 37 & 49 & 29 & 31 & 32 & 28 & 34 & 32 & 39 & 50 & 63 \\ 35 & 22 & 27 & 27 & 34 & 23 & 42 & 30 & 36 & 56 & 48 & 40 \\ 33 & 26 & 31 & 25 & 23 & 20 & 25 & 20 & 36 & & & \end{array}$

$>$ first.glm $-\operatorname{glm}(\mathrm{y} \sim$ Year $+$ month, poisson $) ;$ summary(first.glm $)$

Call:

$\operatorname{glm}($ formula $=\mathrm{y} \sim$ Year $+$ month, family $=$ poisson $)$

\begin{tabular}{lrlll} Coefficients: & & & & \ (Intercept) & Estimate & Std. Error & \multicolumn{1}{l}{ z value } & $\operatorname{Pr}(>|z|)$ \ Year1971 & $-0.81969$ & $0.09896$ & $38.600$ & $<2 e-16$ \ Year1972 & $-0.28794$ & $0.08267$ & $-3.483$ & $0.000496$ \ month2 & $-0.34484$ & $0.14176$ & $-2.433$ & $0.014994$ \ month3 & $-0.11466$ & $0.13296$ & $-0.862$ & $0.388459$ \ month4 & $-0.39304$ & $0.14380$ & $-2.733$ & $0.006271$ \ month5 & $-0.31015$ & $0.14034$ & $-2.210$ & $0.027108$ \ month6 & $-0.47000$ & $0.14719$ & $-3.193$ & $0.001408$ \ month7 & $-0.23361$ & $0.13732$ & $-1.701$ & $0.088889$ \ month8 & $-0.35667$ & $0.14226$ & $-2.507$ & $0.012168$ \ month9 & $-0.14310$ & $0.13397$ & $-1.068$ & $0.285444$ \ month10 & $0.10167$ & $0.13903$ & $0.731$ & $0.464628$ \ month11 & $0.13276$ & $0.13788$ & $0.963$ & $0.335639$ \ month12 & $0.18252$ & $0.13607$ & $1.341$ & $0.179812$ \end{tabular}

Signif. codes: 0 (, $0.001$ (, $0.01$ (, $0.05$ '.

(Dispersion parameter for poisson family taken to be 1 )

$\begin{array}{rlll}\text { Null deviance: } & 101.143 & \text { on } 32 \text { degrees of freedom } \\ \text { Residual deviance: } & 27.273 & \text { on } 19 \text { degrees of freedom }\end{array}$

Number of Fisher Scoring iterations: 3