Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 13J (Statistical Modelling)

Conditional on the observed colleges, the stochastic block model assumes that the friendship indicators are independent random variables. For arbitrary linear predictors ${\theta }_{ij}$, their Bernoulli distribution likelihood isThus the three requested likelihoods are

Model 2 is obtained from model 3 by setting ${\beta }_0=0$. Model 3 is contained in model 1 by choosing the symmetric block parametersHence

Assuming every college-pair cell is observed and the models are identifiable, model 1 has $C(C+1)/2$ parameters and model 3 has $C+1$. The likelihood-ratio test statistic for model 3 against model 1 isUnder model 3 and the regularity conditions of Wilks theorem,

Let $n_{\ell }$ be the number of graduates in college $\ell$. In model 1, every edge from a person in college $k$ to a person in college $\ell$ has probabilitysoIn model 2 the corresponding probability isBecause $k\ne \ell$, the within-college indicator in model 3 vanishes, so $p_{k\ell }^{(3)}=p_{k\ell }^{(2)}$. Consequently models 2 and 3 both give

The distinct edge indicators used in $M_i$ and $M_j$ are independent under all three models, so every model also forces $M_i$ and $M_j$ to be independent. Thus all members of one college have the same cross-college friendship propensity, with no person-level heterogeneity or dependence between their friendship counts. Real friendship networks can have unusually sociable individuals and correlated choices of friends, producing overdispersion and dependence that these models cannot fit.

For $k=1,\dots ,C$, define the class degree sumand define the number of within-college friendships byEach between-college edge contributes once to the degree sum of each endpoint's college, while each within-college edge contributes twice to its college's degree sum. The model-3 likelihood can therefore be writtenThe Fisher-Neyman factorization theorem shows that $T=(T_0,T_1,\dots ,T_C)$ is sufficient.

For two networks $y,y^{\prime }$, the likelihood ratio isBecause the parameters range independently over ${\mathbb{R}}^{C+1}$, this ratio is constant in $\beta$ exactly when every displayed difference vanishes. The likelihood-ratio criterion for minimal sufficiency therefore gives the answerThis is the degree-sum sufficient statistic for an additive logistic network model.