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Let $R$ indicate observation of a regression response $Y$, with fully observed predictors $X$. If $Y\perp R\mid X$ and $\mathbb{P}(R=1\mid X)>0$ on the target support, the observed conditional density equals the full conditional density. Thus a correctly specified regression model can be fitted to observed responses under conditional missing at random. This does not justify the unadjusted complete-case mean when observation rates and outcome means both vary with $X$, nor does it imply missing completely at random.

For each fixed observation pattern, the probability of that pattern may depend on observed response components and fully observed covariates, but not on the response components missing under that pattern. Conditional missing completely at random is the special case with no dependence on observed response components either.

With a fully observed covariate $X$ taking finitely many values of positive probability, response $Y$ of finite variance, and an independent response indicator with probability $\rho$, let $m(X)=E(Y\mid X)$. A covariate-standardized mean has asymptotic variance $[\mathrm{Var}(m)+E\{\mathrm{Var}(Y\mid X)\}/\rho ]/N$, whereas the complete-case mean has variance $\mathrm{Var}(Y)/(N\rho )$. The difference is $(1-\rho )\mathrm{Var}(m)/(N\rho )$. Thus missing completely at random does not justify discarding predictive observed covariates on efficiency grounds.

Data are missing completely at random when the missingness indicator is independent of both observed and missing data.

Missing data are intended measurements that were not observed. Write $R=1$ when an outcome $Y$ is observed and let $X$ denote fully observed covariates. The observation mechanism matters because observed cases can differ systematically from the target population.

Missing completely at random means $R$ is independent of both $Y$ and $X$. Missing at random allows observation to depend on $X$, but requires $R$ and $Y$ to be conditionally independent given $X$. Missing not at random permits additional dependence on the unobserved $Y$. For example, follow-up determined only by recorded age can be missing at random after conditioning on age; follow-up depending on unrecorded current symptoms can be missing not at random. The missing-at-random assumption generally cannot be established from the observed data alone.

An observed-case analysis need not estimate the population mean even under missing at random, because the distribution of $X$ among observed cases can be different. Regression, Inverse probability weighting or Multiple imputation can address this under suitable assumptions, including a positive probability of observation throughout the target population. Stating those assumptions is part of the statistical model rather than a consequence of an imputation algorithm.

The mechanism assigns probabilities to observation patterns $R$ conditional on complete measurements $D$: $p(R\mid D)$. Whether it depends on missing values distinguishes missing completely at random, missing at random and missing not at random.

For $N$ observations, let $m_X$ and $m_Y$ count missing values of two variables, and let $n_{XY}$ count observations on which both are available. The number missing at least one variable is $N-n_{XY}$. Applying the inclusion-exclusion principle to the two missing-value sets gives the displayed number missing both. The disjoint counts are consequently $m_X-m_{XY}$ missing only $X$, $m_Y-m_{XY}$ missing only $Y$, $m_{XY}$ missing both, and $n_{XY}$ complete cases. Their nonnegativity is a necessary consistency check on the supplied totals.

This identifies an aggregate missingness pattern without reconstructing individual records. It does not identify a missing-data mechanism, establish missing completely at random, or justify treating a substantive response such as an explicit unknown category as a missing observation. Counts from different cohorts cannot be inserted into the identity without first establishing a common denominator.

A frequency table gives category counts and percentages. A full-sample percentage divides by 1784, whereas a valid percentage divides by the number observed for that variable. A cumulative percentage adds categories in the displayed order. Differences between the two percentage columns therefore reflect missing values, while small discrepancies from 100% can arise from rounding error.

The age groups contain 329 respondents aged 18–24, 320 aged 25–34, 371 aged 35–44, 323 aged 45–54 and 441 aged 55+, approximately 18.44%, 17.94%, 20.80%, 18.11% and 24.72%. The oldest category is the largest, but it is open-ended and spans a different width from the younger categories, so its larger count does not establish a higher frequency per single year of age. The ordered cumulative 57.18% means that about that proportion is younger than 45. The displayed final 100.01% is compatible with rounding, not an extra respondent. There are no reported missing ages.

Blocking has valid counts 133 Not know, 1299 No and 268 Yes. Dividing by all respondents gives approximately 7.46%, 72.81% and 15.02%; dividing by the 1700 valid responses gives about 7.82%, 76.41% and 15.76%. The remaining 84 observations, 4.71% of the full sample, are missing. The distinction matters: 15.02% is not the same denominator-based statement as 15.76%. Not knowing that blocking was possible is a supplied answer category, not one of those 84 missing values. It should not be deleted or merged merely because its label contains “know”.

Blocking is a nominal categorical variable in this form. Yes and No identify a binary behaviour, while Not know additionally records awareness. There is no automatic substantive ordinal scale Not know $<$ No $<$ Yes. Consequently its cumulative percentage is tied to the printing order and should not be read as a universal low-to-high blocking scale. This point will limit interpretation of the later ordinal coefficients.

There are 478 observed Not-populist classifications and 1214 Populist classifications. These represent 26.79% and 68.05% of the full sample, or 28.25% and 71.75% of the 1692 classified respondents. The 92 missing classifications account for 5.16%. These percentages describe the particular classification and sample; the output does not supply the instrument used to define populism or establish prevalence in a wider population. Adding missing cases to either substantive group would invent their classifications.

The later table uses only 1629 observations with both blocking and populism observed. It is smaller than either marginal valid total, as expected under complete-case analysis. The missingness overlap from marginal and complete-case counts givesThus 63 have blocking missing but populism observed, 71 have populism missing but blocking observed, and 21 have both missing. These totals reconcile the denominators without reconstructing individual records. They do not establish a missing-data mechanism such as missing completely at random; selection into complete cases could affect later associations.

For a population question use a population-averaged logistic model for repeated binary outcomes. Code $z_i=1$ for the new treatment and $0$ for the control, with visit times $t_j=2j$ months for $j=1,2,3$. Let $w_{ij}=(1,z_i,x_i^T,t_j{)}^T$, ${\gamma }_M=({\alpha }_M,{\varphi }_M,{\beta }_M^T,{\delta }_M{)}^T$, andDifferent subjects are independent. Each response has a Bernoulli distribution with this marginal mean, but no independence of a subject's three responses is assumed. The linear time effect and absence of treatment-by-time interaction are substantive mean assumptions; if inappropriate they can be replaced by time indicators and interaction terms.

For complete data, set $m_i=(m_{i1},m_{i2},m_{i3}{)}^T$, $A_i=\mathrm{diag}(m_{ij}(1-m_{ij}))$, and $D_i=\partial m_i/\partial {\gamma }_M^T=A_i{W}_i$, where $W_i$ has rows $w_{ij}^T$. A working exchangeable correlation matrix has unit diagonal and off-diagonal value $\rho \in (-1/2,1)$; set $V_i=A_i^{1/2}C(\rho )A_i^{1/2}$. The generalized estimating equation isWith correct means and independent subjects, a positive-definite working covariance matrix and regularity give consistency even if the working correlation is wrong. Write $H={\sum }_i{D}_i^T{V}_i^{-1}{D}_i$ and $u_i=D_i^T{V}_i^{-1}(Y_i-m_i)$. The fitted subject-level sandwich covariance matrix isevaluated at the estimates. This inference is asymptotic in the number of independent subjects, not the number of visits.

For the actual incomplete data, define $R_{ij}=1$ if visit $j$ is observed, with $R_{i0}=1$ and monotone dropout. Let ${\mathcal{H}}_{i,j-1}$ contain treatment, baseline covariates and responses observed before visit $j$. Under sequential missing at random, specifyThe assumption is that the retention decision, conditional on this history, does not additionally depend on unseen current or future outcomes. Fit the $q_{ij}$, for example by visit-specific logistic regressions. A simple valid choice uses working independence and the inverse-observation-weighted estimating equations for longitudinal dropout:Terms with $R_{ij}=0$ are zero and require no missing response. Given the full response vector, sequential missing at random makes $\mathbb{E}(R_{ij}/{\rho }_{ij}\mid Y_i,z_i,x_i)=1$. Consequently the weighted score has the same expectation as the complete-data score, proving its mean-zero property. Correct retention probabilities, positivity and a correctly specified marginal mean are required. Use cluster-robust uncertainty, accounting for fitted retention-model coefficients, or resample entire subjects and refit both models. Under missing completely at random, the simpler unweighted observed-response GEE can suffice; ordinary unweighted GEE is not valid under arbitrary history-dependent missing at random.

The treatment coefficient gives a population conditional-on-baseline odds ratio $e^{{\varphi }_M}$ at each visit. To communicate an absolute population treatment benefit, standardize predicted probabilities over the full baseline population, rather than only completers:A causal reading additionally uses the randomized controlled trial assignment, consistency of the treatment definition, no interference and the dropout assumptions; the statistical model alone does not supply those conditions.

The population-averaged logistic regression describes $P(Y_{ij}=1\mid z_i,x_i)$ after averaging over unmeasured individual heterogeneity. Its intercept is a marginal log odds at reference baseline values and time zero, its treatment coefficient is a population conditional-on-baseline log odds ratio, and its time coefficient is the change in that marginal log odds per month. Time zero is an extrapolation here because the recorded visits begin later; centring time at the first visit would give a more interpretable intercept.

In the random-intercept logistic model, the intercept is instead the conditional log odds for a subject with $B_i=0$. The treatment and time coefficients apply at fixed $B_i$; the subject's actual intercept is ${\alpha }_C+B_i$. Even with a treatment-independent random intercept, the conditional and marginal coefficients generally differ through noncollapsibility of the odds ratio. Indeed, if $\eta ={\alpha }_C+{\varphi }_{C}z+{\beta }_C^{T}x+{\delta }_{C}t$, marginalization giveswhich is not generally a logistic curve with the same linear coefficients. To quantify random-intercept attenuation of marginal logistic slopes, let $p_B={\mathrm{logit}}^{-1}(\eta +B)$. ThenFor a nondegenerate finite random intercept this derivative lies strictly between zero and one and usually varies with $\eta$. Thus the marginal treatment contrast is an integrated, attenuated version of the conditional contrast, and the marginal time slope need not be constant. One should not equate the two sets of coefficients, even in a randomized controlled trial without treatment confounding.

For missing completely at random, an unweighted observed-response GEE with cluster-robust standard errors can consistently fit the marginal mean, and a correctly specified integrated GLMM likelihood is valid as well. Under missing at random depending on previous observed outcomes, ordinary unweighted GEE is generally biased because the remaining responses are selected by informative observed history. Correct inverse-observation-weighted estimating equations for longitudinal dropout recover the marginal target under the sequential MAR and positivity assumptions already stated. Covariate-only observation mechanisms are a simpler special case in which correctly conditioned unweighted mean equations can remain valid.

A correctly specified joint response GLMM, fitted by observed-data likelihood, is valid under ignorable missing at random with distinct parameters, including dependence of missingness on observed response history. This is a likelihood property, not a claim that every mixed model automatically fixes missingness. Under missing not at random, neither ordinary GEE, MAR-based weights nor a response-only GLMM is generally valid; the unseen-outcome dependence requires further modelling and sensitivity assumptions.

Interpret each mechanism for a fixed missingness pattern $r$: the observed and missing response components are the coordinates selected by that pattern, and the covariates $X$ are assumed fully recorded.

For missing completely at random, missingness depends on neither response values nor covariates:For covariate-dependent missing completely at random, response observation may depend on $X$ but not on the response after conditioning on $X$:This is conditional independence $R\perp Y\mid X$ and can still produce a marginal association between response and missingness.

For covariate-dependent missing at random, the observation probability may use observed response components and covariates, but not the values missing under that pattern:For missing not at random, there remains dependence on $Y^m$ after conditioning on $Y^o,X$; the displayed simplification is not valid. For example, refusal could depend on the unreported response itself, even among individuals sharing all recorded covariates. These mechanisms concern the reason for nonobservation, not whether the incomplete records merely look irregular. Their distinction is not generally identifiable from the observed responses alone.

Let $R_i=1$ indicate an observed outcome. Fitting the conditional outcome model to responders requireswith a positive observation probability over the relevant predictor support. This is conditional missing at random for the predictors used in the regression: after conditioning on them, observation must not further select participants by their unobserved outcome. It gives $f(Y_i\mid x_i,Z_i,R_i=1)=f(Y_i\mid x_i,Z_i)$, the complete-case regression under conditional missing at random condition. Correct specification of the conditional mean model is also required. Missing completely at random is sufficient but stronger; simply assuming missingness depends only on a post-randomization variable excluded from this model would not suffice.

Let $Y$ denote the full data and $R$ the pattern of indicators, with $R_j=1$ for an observed component and $R_j=0$ for a missing one. For each fixed pattern $r$, split the data as $y=(y_{\mathrm{obs}}(r),y_{\mathrm{mis}}(r))$. The missing at random condition isHere $\psi$ parameterizes the missingness mechanism. Equivalently its conditional probability, with the observed data fixed, is constant over all possible completions of the missing data. The restriction is pattern-specific because the observed components depend on $r$. Missing at random allows missingness to depend on observed values; missing completely at random imposes independence from all the data.

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.

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.