Missingness weights and sensitivity

Assume AC. Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space, let $\mathcal{G}\subseteq \mathcal{F}$ be a sub-sigma-algebra, let $R$ be a random variable taking values in $\{0,1\}$, and let $Y$ be a real integrable random variable. If $R$ and $Y$ are conditionally independent given $\mathcal{G}$, then for every Borel function $\phi :\mathbb{R}\to \mathbb{R}$ such that $\phi (Y)$ is integrable, $E[R\phi (Y)\mid \mathcal{G}]=E[R\mid \mathcal{G}]E[\phi (Y)\mid \mathcal{G}]\text{almost surely}\mathrm{.}$

The cited notes define conditional independence only for bounded measurable test functions. The truncation argument above supplies the integrable extension used in this item.

Let $(\mathcal{X},\mathcal{A})$ be a measurable covariate space and let $n\ge 1$. In the one-outcome full/observed-data convention of Full and observed data under a missingness pattern, $X_i\in \mathcal{X}$ is observed for every unit, $R_i\in \{0,1\}$ indicates whether the real outcome $Y_i$ is observed, and the encoded input stores $Z_i=Y_i$ when $R_i=1$ and the placeholder $Z_i=0$ when $R_i=0$. The mask $R_i$ distinguishes the placeholder from an observed zero; it is not an imputation or a replacement of the missing full-data value. Let $\pi :\mathcal{X}\to (0,1]$ be a supplied, known, measurable response-probability function, with $\pi (x_i)>0$ for every record in the input.

Input: $n$, records $(x_i,r_i,z_i{)}_{i=1}^n$ satisfying the stated conventions, and the function $\pi$.

Procedure: For each record use weight $1\mathrm{/}\pi (x_i)$ when $r_i=1$ and weight zero when $r_i=0$. Return the unnormalized Horvitz–Thompson weighted mean ${\widehat{\mu }}_{\mathrm{H}\mathrm{T}}=\frac{1}{n}{\sum }_{i=1}^n\frac{z_i}{\pi (x_i)}\mathrm{.}$

The procedure takes the true known response probabilities as input. Substituting fitted probabilities is a different procedure; this definition makes no finite-sample stability or estimated-weight inference claim. The weighting convention is the one used for a single response indicator; no joint treatment/response weight is defined here.

Assume AC. Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space, $X$ a fully observed random element, $G=\sigma (X)$, $R\in \{0,1\}$ the response indicator ($R=1$ exactly when the real full-data outcome $Y$ is observed, under the one-outcome convention of Full and observed data under a missingness pattern), and $Y\in L^1(P)$. Suppose $R$ and $Y$ are conditionally independent given $G$, the MAR condition in the assigned one-outcome, fully observed-$X$ formulation. Let $\pi$ be a chosen version of $E[R\mid G]$ and assume $\pi >0$ almost surely on the target law. Define $RY\mathrm{/}\pi$ to be zero on the null set where $\pi =0$. Then $RY\mathrm{/}\pi \in L^1(P)$ and $E\text{\hspace{0.17em}⁣}\left[\frac{RY}{\pi }\right]=E[Y]\mathrm{.}$

Here $\pi$ is the true response probability, not a fitted estimate. This is a population identity; by itself it gives no finite-sample stability or estimated-weight inference result.

The proof uses the true conditional response probability and the explicit conditional-independence hypothesis. If $\pi$ is estimated, or if treatment and observation are weighted jointly, a separate sampling or joint/sequential law and its own positivity and proof are required. Strict positivity alone does not bound the weights; the companion example exhibits this boundary.

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.