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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.

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.

An ignorable missingness mechanism need not be absent from the data-generating process. It means that likelihood inference about the data-model parameters can omit the missingness factor, while still integrating over missing values.

Write the complete joint data statistical probability density as $f_{\theta }(y,x)$ and the conditional missingness probability as $g_{\psi }(r\mid y,x)$. All $y$ values and only $x_{\mathrm{obs}}$ are observed. Under missing at random, $g_{\psi }(r\mid y,x)$ is constant as $x_{\mathrm{mis}}$ varies with $(y,x_{\mathrm{obs}})$ fixed. The actual observed-data likelihood is thereforeThe assumed distinctness is understood as independent variation of $\theta$ and $\psi$. Maximizing over $\psi$ multiplies $L_{\mathrm{obs}}$ by a factor independent of $\theta$; likelihood ratios, scores and likelihood curvature for $\theta$ are therefore unchanged by omitting $g_{\psi }$. This proves likelihood ignorability.

For implementation of the observed-data likelihood with a missing covariate, a linear regression model for $Y\mid X$ must be accompanied by an appropriate model for the distribution of $X$. For independent individuals, write $f_{\beta }(y\mid x)$ for the regression statistical probability density and $g_{\eta }(x)$ for the age statistical probability density. Up to the ignorable factor,Ignorability does not authorize discarding missing-age cases or assuming their ages have the distribution seen in the complete cases. It removes the need to model the observation mechanism for likelihood inference under the stated conditions, not the need to handle the missing covariates.

Write $C$ for the binary response, $P$ for the partner count, and $Z$ for fully observed background variables. Let $R=(R_C,R_P)$ indicate which of $C,P$ are observed, with 1 meaning observed. For any realised pattern $r$, partition the data into $D_{\mathrm{obs}}(r)$ and $D_{\mathrm{mis}}(r)$.

Missing completely at random means that the missing-data mechanism does not depend on any of the measurements:Missing at random allows dependence on the data observed under the particular pattern, but no further dependence on its unobserved values:Thus a response probability depending on fully observed age or sex can satisfy missing at random. Dependence on an observed partner count can also be permissible for patterns in which that count is available. Missing not at random means that this equality fails: the probability of the pattern still depends on one of its missing measurements, after conditioning on the observed ones. The distinction concerns dependence on the unseen values, not whether missingness looks haphazard. These definitions can be applied within the target population $P>1$; unknown membership of that population must itself be handled when $P$ is missing.

Missing not at random is the plausible working classification. The suggested reluctance depends on the actual partner count and actual condom response, including the values that are withheld. Among people with the same observed age, sex and other recorded information, those with an unreported high count or an unreported negative response may still be less likely to answer.

Such dependence on unobserved values violates missing at random and therefore also missing completely at random. This is a substantive judgement rather than something that can be proved from the observed data alone. If all the dependence were explained by measured predictors, the mechanism could instead satisfy missing at random.

Ordinary multiple imputation by chained equations relies on missing at random, conditional on the variables included in its models, together with adequate model specification and suitable conditional distributions. For Bayesian inference that ignores the response mechanism, the usual sufficient condition also includes distinct parameters for data and missingness models, with an appropriately separate prior.

The suspected dependence on withheld responses makes the usual missing-at-random assumption doubtful. An accurately fitted observed-data model cannot, on its own, establish what the nonresponders would have reported. Use substantive judgement and sensitivity analysis under plausible missing not at random departures, for example a pattern-mixture sensitivity analysis shifting the imputed condom log odds or partner-count distribution relative to the observed-data prediction. Report how the final association changes. Additional informative measurements may make missing at random more credible, but cannot verify it.

Treat the extra measurements as auxiliary variables for multiple imputation. Include alcohol consumption and disease history as predictors in both the $C$ and $P$ conditional distributions, together with age, sex and the other incomplete response. If these auxiliary variables themselves have missing entries, give them appropriate models and update them within the same multiple imputation by chained equations sweeps. Preserve observed entries throughout.

Useful auxiliary measurements can improve predictions, precision and the plausibility of missing at random. Their association with the missing responses reduces uncertainty in the imputed values; association with nonresponse may explain some selection that otherwise depends on unobserved values. This can reduce selection bias under the enlarged missing at random model and reduce between-imputation variance. It does not guarantee elimination of missing not at random dependence. They need not be added to the final substantive logistic regression merely because they help imputation: the final analysis should still estimate its intended association.

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.

Under missing at random, dropout before the one-month visit may depend on observed treatment $Z$ but, conditional on $Z$, not on the unseen $Y$ or $W$. Dropout between the one- and six-month visits may depend on the observed history $(Z,Y)$ but, conditional on that history, not on the unseen $W$.

Specify distributions of measurements separately by observation pattern and vary assumptions about unseen values that the observed data do not identify. For example, shift the imputed binary-response log odds for nonresponders away from its missing at random prediction and refit the substantive analysis. This investigates missing not at random departures rather than testing them from observed data.

Vary the unobserved outcome probabilities for item and unit nonresponders, and any unknown auxiliary-variable mixture, over scientifically plausible ranges. Recompute the population estimate and identify values at which substantive conclusions change. This probes untestable transport or missing at random assumptions; it is distinct from sampling uncertainty represented by a confidence interval. Letting each missing-outcome probability vary from zero to one gives worst-case identification bounds.

Item nonresponse leaves a particular measurement unobserved while retaining other measurements on that individual. Unit nonresponse leaves all survey measurements unavailable. Adjusting for an observed auxiliary variable can address item nonresponse under missing at random, but does not identify the auxiliary-variable distribution among unit nonresponders. Extrapolation to them needs further assumptions or external information.