A sociologist collects a dataset on friendships among $m$ Cambridge graduates. Let $y_{i, j}=1$ if persons $i$ and $j$ are friends 3 years after graduation, and $y_{i, j}=0$ otherwise. Let $z_{i}$ be a categorical variable for person $i$ 's college, taking values in the set $\{1,2, \ldots, C\}$. Consider logistic regression models,

$$\mathbb{P}\left(y_{i, j}=1\right)=\frac{e^{\theta_{i, j}}}{1+e^{\theta_{i, j}}}, \quad 1 \leqslant i<j \leqslant m$$

with parameters either

1. $\theta_{i, j}=\beta_{z_{i}, z_{j}}$; or,

2. $\theta_{i, j}=\beta_{z_{i}}+\beta_{z_{j}}$; or,

3. $\theta_{i, j}=\beta_{z_{i}}+\beta_{z_{j}}+\beta_{0} \delta_{z_{i}, z_{j}}$, where $\delta_{z_{i}, z_{j}}=1$ if $z_{i}=z_{j}$ and 0 otherwise.

(a) Write the likelihood of the models.

(b) Show that the three models are nested and specify the order. Suggest a statistic to compare models 1 and 3, give its definition and specify its asymptotic distribution under the null hypothesis, citing any necessary theorems.

(c) Suppose persons $i$ and $j$ are in the same college $k ;$ consider the number of friendships, $M_{i}$ and $M_{j}$, that each of them has with people in college $\ell \neq k$ ( $\ell$ and $k$ fixed). In each of the models above, compare the distribution of these two random variables. Explain why this might lead to a poor quality of fit.

(d) Find a minimal sufficient statistic for model 3. [You may use the following characterisation of a minimal sufficient statistic: let $f(\beta ; y)$ be the likelihood in this model, where $\beta=\left(\beta_{k}\right)_{k=0,1, \ldots, C}$ and $y=\left(y_{i, j}\right)_{i, j=1, \ldots, m} ;$ suppose $T=t(y)$ is a statistic such that $f(\beta ; y) / f\left(\beta ; y^{\prime}\right)$ is constant in $\beta$ if and only if $t(y)=t\left(y^{\prime}\right)$; then, $T$ is a minimal sufficient statistic for $\beta$.]