Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 32 / 3 / b

Ask the doctor whether, among patients with the same first-year drug-use status, patients who dropped out would be more or less likely to have used the drug in the second year than those who remained. The comparison is within the observed first-year categories, not just between all dropouts and all completers.

If there is no remaining relationship with second-year drug use after conditioning on first-year use, missing at random is plausible for these recorded variables. If treatment-related difficulties, new medication or deterioration lead to dropout in a way not accounted for by the recorded first-year status, missing at random may fail. A different response rate in the two first-year categories is allowed under missing at random. The unobserved second-year outcomes mean that the observed table alone cannot establish this assumption; clinical knowledge or additional follow-up information is needed.

Apply MAR standardization over a fully observed covariate. All first-year statuses are known, so the estimated probabilities of no use and use in year one are $58/102$ and $44/102$. Within those categories, missing at random lets the observed second-year probabilities represent the corresponding dropout outcomes as well. They are estimated by $7/37$ and $18/28$.

The law of total probability then givesEquivalently, impute expected drug-use counts $21(7/37)$ and $16(18/28)$ for the two dropout groups, add these to the 25 observed second-year users, and divide by 102. This uses both the complete records and the fully observed first-year information.

Under missing completely at random, the observed second-year outcomes form a random subsample, so the direct pooled complete-case analysis estimate isIt is valid under missing completely at random because dropout no longer changes the marginal distribution of second-year use. Pooling is simpler than reporting separate conditional probabilities, but MCAR alone does not make this estimate more efficient than the estimate in the previous part. The fully observed first-year variable carries useful outcome information and can improve precision even when dropout is completely random.

To make the qualification explicit, write $X=Y_1$, $m(X)=P(Y_2=1\mid X)$, $\theta =E\{m(X)\}$ and $\rho =P(R=1)$ for the constant response probability. For $N$ patients the leading variance of the pooled estimator is $\mathrm{Var}(Y_2)/(N\rho )$. The standardization estimator in the preceding part has leading varianceIndeed its first-order centered contribution is $m(X)-\theta +R\{Y_2-m(X)\}/\rho$: the two terms have zero covariance, and their variances give that expression. The law of total variance therefore yieldsThe inequality is strict when there is dropout and first-year use predicts second-year use, as the distinct conditional rates suggest here. Thus the requested universal efficiency claim needs qualification. In the unrestricted joint binary-outcome model, maximizing the observed-data likelihood under either missing at random or missing completely at random gives the same standardization estimate $0.384889$: the all-patient first-year proportion and the two observed conditional second-year proportions maximize its factored outcome likelihood. Restricting the distinct missingness parameters to a common response probability affects their factor, not this estimate. The pooled value is the simple valid MCAR answer; retaining the first-year information gives the efficient MCAR answer and does not require replacing the previous estimate.