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If a regression response $Y$ is observed but covariate $X$ is sometimes missing, likelihood inference requires a joint model: for example $f_{\beta }(y\mid x)g_{\eta }(x)$. An incomplete record contributes $\int f_{\beta }(y\mid x)g_{\eta }(x)dx$ to the observed-data likelihood. Missing at random with distinct parameters allows the missingness mechanism to be omitted; it does not remove the need to integrate over $X$.

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.

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.