For $Z=0$, the fitted conditional success probabilities for $W$ are
$$
\mathbb P(W=1\mid Z=0,Y=0)=\frac{20}{40}=\frac12,
\qquad
\mathbb P(W=1\mid Z=0,Y=1)=\frac{75}{90}=\frac56.
$$
Among all 250 subjects with $Z=0$, <multiple imputation> asymptotically assigns $100$ to $Y=0$ and $150$ to $Y=1$. Hence
$$
\widehat\psi_0\longrightarrow
\frac{100(1/2)+150(5/6)}{250}
=\frac7{10}.
$$
Using the supplied limit for $\widehat\psi_1$ gives
$$
\widehat\beta\longrightarrow
\frac{25}{48}-\frac7{10}=-\frac{43}{240}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 2 / 2 / 7 / b / Solution

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Multiple imputation repeatedly draws missing values from fitted predictive distributions, analyzes each completed dataset, and combines the resulting estimates and uncertainty.


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