Complete-case analysis

A complete-case analysis uses only observations for which every variable required by the analysis is observed. It can be biased when completeness selects observations according to their outcomes or other relevant variables.

Suppose $N>d\ge 0$ observations lie in $[a,b]$ and have sample mean $\bar{y}$. Delete any $d$ observations. The remaining sample mean obeysIndeed, the removed sum is between $da$ and $db$; subtract it from the original sum $N\bar{y}$ and divide by the remaining positive count. If only a rounded mean in $[\ell ,u]$ is known, a valid enclosing interval is $[(N\ell -db)/(N-d),(Nu-da)/(N-d)]$, optionally intersected with $[a,b]$. These deterministic bounds do not require missing at random. They constrain a complete-case summary without claiming that complete-case selection is unbiased for a target population.

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.