Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 3

Let $Y_{it}$ be patient $i$'s QoL at visit $t$ and let $R_i$ be the last visit at which QoL is observed. The data are monotone missing-data pattern whenThus each patient contributes an observed prefix $(Y_{i0},\dots ,Y_{i{R}_i})$ followed by a missing suffix.

Let $A_i$ be treatment, $Y_i=(Y_{i0},\dots ,Y_{i,12})$, and $R_i\in \{0,\dots ,12\}$ the last observed visit. The missing at random assumption isfor every feasible $r$. Equivalently, each conditional dropout hazard may depend on treatment and observed QoL history but not on current or future unobserved QoL after conditioning on that history.

In this trial, MAR means that among patients assigned the same treatment who have the same recorded QoL trajectory up to a visit, the probability of dropping out next is unrelated to what their later QoL values would have been. Dropout may depend strongly on previous poor QoL, treatment assignment, and other observed history; MAR only rules out residual dependence on the unobserved outcomes.

MAR is plausible if clinic withdrawal is driven by recorded QoL, observed side effects, treatment, and other measured history. It is doubtful if patients leave because of an unrecorded deterioration, an imminent recovery, treatment toxicity not included in the analysis, or dissatisfaction that predicts their unseen 12-month QoL. The large dropout fraction makes such missing not at random mechanisms a serious concern, so MAR should be supported by rich predictors and sensitivity analysis rather than assumed without examination.

A missingness mechanism is ignorable for likelihood-based inference about the outcome parameter when the observed-data likelihood can be formed from the outcome model alone. The standard sufficient conditions are MAR and distinct parameters: the outcome-model parameter and missingness-model parameter have a product parameter space. The missingness indicators then carry no additional likelihood information about the outcome parameter once the observed outcomes are given.

Write the complete-data model as $p_{\theta }(y\mid a)$ and the missingness model as $q_{\psi }(r\mid y,a)$. Under MAR,Therefore the observed-data likelihood factorizes asWith distinct parameters, the first factor does not involve $\theta$. Maximizing or integrating the second factor therefore gives the same likelihood inference for $\theta$ as modeling the missingness process explicitly. Hence MAR plus distinctness makes missingness ignorable.

The monthly nurse assessments are auxiliary longitudinal outcomes observed even after clinic dropout. Investigators can include their histories, treatment, and earlier QoL in a multiple imputation model for missing 12-month QoL, or in a joint longitudinal model. Because the assessments predict disability and QoL, conditioning on them reduces residual outcome variance and makes each imputed value more informative, improving precision of the treatment-effect estimate. A prespecified covariate-adjusted or augmented inverse-probability estimator could use the same information.

The nurse measurements can also reduce bias by making MAR more credible: withdrawal may depend on unobserved clinic QoL, but that QoL is partly represented by the post-dropout disability assessments. They also permit diagnostics comparing inferred QoL trajectories with an independently recorded proxy and can support sensitivity analysis for departures from MAR. They do not by themselves prove MAR, because the rough score may omit reasons for dropout related to QoL.