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An observed-data likelihood integrates the full-data statistical probability density over unobserved values, rather than inserting arbitrary values or discarding incomplete records. When the observation pattern is random, its mechanism belongs to the joint observation model unless the conditions for an ignorable missingness mechanism permit its factor to be omitted.

The independence analysis is not justified without further assumptions. Three binary responses from the same subject generally remain correlated after conditioning on baseline covariates and treatment. The fitted independent-Bernoulli likelihood discards that correlation and gives incorrect model-based standard errors and tests if it is present. Positive within-subject correlation commonly makes those errors too small.

There is an important distinction between estimating the mean and estimating uncertainty. If the stated marginal logistic regression is correct and responses are fully observed, the independence score ${\sum }_{i,j}{w}_{ij}(Y_{ij}-m_{ij})$ still has zero expectation. With independent subjects and regularity, its root can consistently estimate marginal coefficients despite within-subject dependence; a subject-level sandwich covariance matrix is then needed. True independence is a special case in which the original standard errors would be appropriate.

Dropout creates a separate issue. Restricting that score to observed responses need not preserve its zero expectation if observation depends on the response history or unseen depression outcomes. Therefore both point estimates and uncertainty may be wrong. Neither random treatment assignment nor baseline adjustment alone establishes an ignorable missingness mechanism. We need a model matched to the scientific estimand and explicit assumptions about missing data.

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.