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Exact binary-outcome bounds under unrestricted missingness

Statement

Let $Y, R \in {0, 1}$ , with $R = 1$ meaning that $Y$ is observed, and encode the observed record as $O = (R, R Y)$ . When $R = 0$ , the zero in the second recorded coordinate is a placeholder and does not assign the missing full-data value $Y = 0$ . Put $a = P (R = 1), b = P (R = 1, Y = 1), 0 \leq b \leq a \leq 1.$ Let $Θ_{a, b}$ be the set of all joint laws $P$ of $(Y, R)$ with these values of $a$ and $b$ , and let the statistical model map $P \in Θ_{a, b}$ to the law of $O$ . For each individual $P$ , use its finite conditional response kernel specified below when assessing the MAR clause; arbitrary null-state values do not affect that classification. The full-data mean $m (P) = P (Y = 1)$ is an estimand on this model. Its set of values over $Θ_{a, b}$ is exactly $[b, b + 1 - a] .$ Consequently, without a MAR restriction, the mean is not identifiable when $a < 1$ ; it is identified as $b$ when $a = 1$ . If MAR also holds, then for $a > 0$ the mean is $b / a$ , while for $a = 0$ it remains unidentified.

Facts & Assumptions

Given: The binary full-data/response setup and observed law in the Statement.

[F1]

Full and observed data under a missingness pattern: $R$ records which coordinates of the full data $W$ are observed, and $O = (R, W_{o b s} (R))$ retains the pattern label. In the binary encoding $O = (R, R Y)$ , the zero at $R = 0$ is a record placeholder and does not assign the missing full-data value $Y = 0$ .

[F2]

MCAR, MAR, and MNAR relative to an observation pattern: MAR requires each pattern probability to depend only on that pattern's observed projection; MNAR means MAR fails for at least one pattern.

[F3]

Law or distribution of a random element: the law of a random element is the pushforward probability $P_{X} (B) = P (X^{- 1} (B))$ .

[F4]

Parametric statistical models: a statistical model is a parameter map $P : Θ \to P (X, A)$ .

[F5]

Observational equivalence: two parameter values are observationally equivalent exactly when their sampling laws are equal.

[F6]

Estimands on a statistical model: an estimand is a specified function from the parameter set to target values.

[F7]

Identifiability of an estimand: an estimand is identifiable only if equality of sampling laws implies equality of its values.

[F8]

MCAR, MAR, and MNAR relative to an observation pattern and Measure kernel and probability kernel: for each finite joint law $P$ of $(Y, R)$ , a response kernel is given on states with $P (Y = y) > 0$ by $g_{P} (r ∣ y) = P (R = r, Y = y) / P (Y = y)$ ; on any null state, any probability vector may be assigned, for example $g_{P} (0 ∣ y) = 1$ and $g_{P} (1 ∣ y) = 0$ . These finite-space kernels are measurable and give the selected mechanism used to interpret MAR for each law $P$ .

Proof

technique · Directly bound the unobserved success mass and construct every admissible completion

1.1givenF1algebra

For any $P \in Θ_{a, b}$ , write $c = P (R = 0, Y = 1)$ . Since $P (R = 0) = 1 - a$ , one has $0 \leq c \leq 1 - a$ , and $m (P) = P (Y = 1, R = 1) + P (Y = 1, R = 0) = b + c$ . Hence $b \leq m (P) \leq b + 1 - a$ .

1.2givenalgebraconstruct

Conversely, fix any $m \in [b, b + 1 - a]$ and assign $P (Y = 1, R = 1) = b$ , $P (Y = 0, R = 1) = a - b$ , $P (Y = 1, R = 0) = m - b$ , and $P (Y = 0, R = 0) = 1 - a - m + b$ . Each entry is nonnegative by the bounds on $a, b, m$ ; the four entries sum to one. This law has response rate $a$ , observed success mass $b$ , and mean $m$ , so every point of the interval is attained, including both endpoints.

2.1F1F3F4F5F6F7step 1.2algebra

Under each such law, the pushforward law of $O$ has masses $b$ at $(1, 1)$ , $a - b$ at $(1, 0)$ , and $1 - a$ at $(0, 0)$ , independent of $m$ . Thus all parameters in $Θ_{a, b}$ are observationally equivalent. If $a < 1$ , the interval has positive length $1 - a$ , so it contains distinct means and the estimand is not identifiable. If $a = 1$ , it is the singleton ${b}$ and the mean is identified as $b$ .

3.1F2F8step 1.2givenalgebra∎

For each $P \in Θ_{a, b}$ use the finite conditional response kernel in [F8]. Under MAR, the probability $g_{P} (0 ∣ Y)$ depends only on the empty observed projection for pattern $R = 0$ , so it is constant $P_{Y}$ -almost surely. Hence $g_{P} (1 ∣ Y) = 1 - g_{P} (0 ∣ Y)$ is also constant $P_{Y}$ -almost surely; its constant is $P (R = 1) = a$ , by integrating over $P_{Y}$ . Therefore $b = P (R = 1, Y = 1) = a P (Y = 1)$ , and for $a > 0$ the mean is $b / a$ . If $a = 0$ , every law in the interval construction of step 1.2 has $R = 0$ almost surely; its selected kernel is constant on the support of $Y$ , so it is MCAR and hence MAR. The whole interval $[0, 1]$ therefore remains compatible with MAR.

Depends on

Used by

MCAR, MAR, MNAR, and indistinguishable observed laws Example

Dependency tree · two levels

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Sources

Lu Mao, Missing Data: Foundations and Methods, Chapter 1, §§1.6–1.7 (standard reference, not scraped)