Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 2

For user $j$ and offer $r$, both fits useThe first model treats each click as Bernoulli$(p_j)$. The grouped model treats the click count $C_j=m_j{\mathrm{prop}}_j$ as $\mathrm{Bin}(m_j,p_j)$. Their likelihoods differ only by binomial coefficients independent of the parameters, so their fitted coefficients agree.

The grouped binomial likelihood assumes conditional independence of repeated offers to one user. This is doubtful: persistent unmeasured user preferences and temporal feedback make clicks from the same user positively correlated.

For $m_j$ exchangeable Bernoulli responses of mean $p_j$ and pairwise correlation $\rho$,A standard quasi-binomial model uses a common dispersion multiplier $\varphi$. It cannot represent this variance simultaneously when the offer counts $m_j$ vary, because the multiplier $1+(m_j-1)\rho$ then varies by user.

The beta-binomial regression iswhere the beta distribution is parameterized by mean ${\mu }_j$ and variance parameter $\theta$. Marginally,It matches part c when $\rho =\theta /(1+\theta )$, so it is appropriate at the mean-variance level for a common nonnegative intraclass correlation.

The beta-binomial model mixes binomials over a beta-distributed success probability. The random-intercept generalized linear mixed model instead takesBoth create within-user dependence and overdispersion, but use different mixing distributions. The mixed-model coefficients are conditional on the random effect, while beta-binomial regression is naturally phrased through a marginal mean.