Ignorable missingness mechanism

A missingness mechanism is ignorable for likelihood or Bayesian inference about outcome-model parameters when the missingness-model factor can be omitted without changing that inference. Missing at random together with distinct outcome and missingness parameters is a standard sufficient condition.

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 $X$.

Outcome and missingness parameters are distinct when their joint parameter space is the Cartesian product of their separate parameter spaces, so variation-independent choices can be made for the two models.