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

Table of contents
- Solution d

Solution

The beta-binomial regression is

P_{j} \sim Beta (μ_{j}, θ), C_{j} ∣ P_{j} \sim Bin (m_{j}, P_{j}), logit (μ_{j}) = z_{j}^{T} β,

where the beta distribution is parameterized by mean

μ_{j}

and variance parameter

θ

. Marginally,

Var (C_{j} / m_{j}) = \frac{μ _{j} ( 1 - μ _{j} )}{m _{j}} {1 + (m_{j} - 1) \frac{θ}{1 + θ}} .

It matches part c when

ρ = θ / (1 + θ)

, so it is appropriate at the mean-variance level for a common nonnegative intraclass correlation.

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