Location-shift pattern-mixture sensitivity function

Assume AC. Let $X$ take values in a standard-Borel space $\mathcal{X}$, let $Y$ be real-valued, and let $R\in \{0,1\}$ indicate response in the one-outcome full/observed-data setup of Full and observed data under a missingness pattern. Suppose $P(R=1)>0$, $P(R=0)>0$, and the covariate laws overlap in this direction: for every measurable $A\subseteq \mathcal{X}$, $P(X\in A\mid R=1)=0$ implies $P(X\in A\mid R=0)=0$. Under the respondent law $P(\cdot \mid R=1)$, choose a conditional law $K_1(x,\cdot )$ of $Y$ given $X=x$; standard-Borel disintegration supplies such a probability kernel under AC (Disintegration of a joint law on standard borel spaces, Conditional law given a random element). The overlap condition makes the respondent version relevant almost surely under the nonrespondent covariate law.

A location-shift pattern-mixture sensitivity function is a measurable $\mathrm{\Delta }:\mathcal{X}\to \mathbb{R}$ for which the nonrespondent conditional law is specified by $K_0^{\mathrm{\Delta }}(x,B)={\int }_{\mathbb{R}}{\mathbf{1}}_B(y+\mathrm{\Delta }(x))K_1(x,dy)$ for every Borel set $B\subseteq \mathbb{R}$, on the covariate support of nonrespondents. This is the full conditional distribution restriction for $Y_1\sim K_1(x,\cdot )$, not merely a restriction on conditional means. For each $x$, $K_0^{\mathrm{\Delta }}(x,\cdot )$ is the pushforward of $K_1(x,\cdot )$ under the measurable translation $y\mapsto y+\mathrm{\Delta }(x)$, hence is a probability measure; its evaluations are measurable in $x$ by kernel integration because $(x,y)\mapsto {\mathbf{1}}_B(y+\mathrm{\Delta }(x))$ is measurable (Measure kernel and probability kernel, Measurability of integration against a kernel). The relation specifies the missing conditional law while leaving the respondent conditional law fixed. A finite mean target additionally requires integrability under the resulting full-data law; the definition alone does not assert that moment condition.

In this specified family, $\mathrm{\Delta }=0$ gives equality of the respondent and nonrespondent conditional laws. This does not make equality of conditional means alone equivalent to MAR in a general full-data model; equality of means need not imply equality of distributions. Values where no respondents occur are outside the definition because $K_1$ is unavailable there, and a pattern with no nonrespondents requires no sensitivity function.

The formal one-outcome MCAR/MAR/MNAR taxonomy for this pair is supplied by MCAR, MAR, and MNAR relative to an observation pattern. The location-shift relation above is the selected sensitivity model within that taxonomy; it does not claim that all MNAR laws are location shifts.

National Research Council Chapter 5 introduces sensitivity parameters through differences between respondent and nonrespondent distributions and illustrates a mean-shift parameter. This item deliberately specifies the stronger full-distribution location-shift model shown above; the cited source supports the modeling framework, not a theorem that this particular translation restriction is empirically true.