(i) Assume that independent observations $Y_{1}, \ldots, Y_{n}$ are such that

$$Y_{i} \sim \operatorname{Binomial}\left(t_{i}, \pi_{i}\right), \log \frac{\pi_{i}}{1-\pi_{i}}=\beta^{T} x_{i} \quad \text { for } 1 \leqslant i \leqslant n$$

where $x_{1}, \ldots, x_{n}$ are given covariates. Discuss carefully how to estimate $\beta$, and how to test that the model fits.

(ii) Carmichael et al. (1989) collected data on the numbers of 5 -year old children with "dmft", i.e. with 5 or more decayed, missing or filled teeth, classified by social class, and by whether or not their tap water was fluoridated or non-fluoridated. The numbers of such children with dmft, and the total numbers, are given in the table below:

\begin{tabular}{l|ll} Social Class & Fluoridated & Non-fluoridated \ \hline I & $12 / 117$ & $12 / 56$ \ II & $26 / 170$ & $48 / 146$ \ III & $11 / 52$ & $29 / 64$ \ Unclassified & $24 / 118$ & $49 / 104$ \end{tabular}

A (slightly edited) version of the $R$ output is given below. Explain carefully what model is being fitted, whether it does actually fit, and what the parameter estimates and Std. Errors are telling you. (You may assume that the factors SClass (social class) and Fl (with/without) have been correctly set up.)

$\begin{array}{lrrrr} & \text { Estimate } & \text { Std. } & \text { Error } & \text { z value } \\ \text { (Intercept) } & -2.2716 & 0.2396 & -9.480 \\ \text { SClassII } & 0.5099 & 0.2628 & 1.940 \\ \text { SClassIII } & 0.9857 & 0.3021 & 3.262 \\ \text { SClassUnc } & 1.0020 & 0.2684 & 3.734 \\ \text { Flwithout } & 1.0813 & 0.1694 & 6.383\end{array}$

Here 'Yes' is the vector of numbers with dmft, taking values $12,12, \ldots, 24,49$, 'Total' is the vector of Total in each category, taking values $117,56, \ldots, 118,104$, and SClass, Fl are the factors corresponding to Social class and Fluoride status, defined in the obvious way.