 Solution

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.