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

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- Solution c

Solution

For

m_{j}

exchangeable Bernoulli responses of mean

p_{j}

and pairwise correlation

ρ

,

E p_{j} = p_{j}, Var (p_{j}) = \frac{p _{j} ( 1 - p _{j} )}{m _{j}} {1 + (m_{j} - 1) ρ} .

A standard quasi-binomial model uses a common dispersion multiplier

ϕ

. It cannot represent this variance simultaneously when the offer counts

m_{j}

vary, because the multiplier

1 + (m_{j} - 1) ρ

then varies by user.

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