ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

MCAR, MAR, MNAR, and indistinguishable observed laws

Example

Let $X$ and $Y$ be independent fair binary variables, with full data $W = (X, Y)$ ; observe $X$ under either response pattern and observe $Y$ only when $R = 1$ . Three response mechanisms are: $g_{M C A R} (1 ∣ x, y) = \frac{1}{2}, g_{M A R} (1 ∣ x, y) = \frac{1}{4} + \frac{1}{2} 1 {x = 1}, g_{M N A R} (1 ∣ x, y) = \frac{1}{4} + \frac{1}{2} 1 {y = 1} .$ In each case $g (0 ∣ x, y) = 1 - g (1 ∣ x, y)$ . The first mechanism is MCAR; the second is MAR because its probability depends only on observed $X$ ; the third is MNAR because for pattern $R = 0$ its probability varies with the unobserved $Y$ at each $X$ .

For a binary outcome with no always-observed covariate, consider two further joint laws:

Law	$P (Y = 1, R = 1)$	$P (Y = 0, R = 1)$	$P (Y = 1, R = 0)$	$P (Y = 0, R = 0)$	$P (Y = 1)$
$P^{-}$	$1 / 4$	$1 / 4$	$0$	$1 / 2$	$1 / 4$
$P^{+}$	$1 / 4$	$1 / 4$	$1 / 2$	$0$	$3 / 4$

Both yield $a = P (R = 1) = 1 / 2$ , $b = P (R = 1, Y = 1) = 1 / 4$ , and the same observed record law. Both are MNAR: their response probabilities given $Y = 1$ and $Y = 0$ differ. The respondent mean is $1 / 2$ under both laws, but the full-data mean differs. The example realizes the interval $[b, b + 1 - a] = [1 / 4, 3 / 4]$ in Exact binary-outcome bounds under unrestricted missingness.

Facts & Assumptions

Given: The full/observed-data convention and MCAR/MAR/MNAR definitions in the Statement; the finite probability models displayed in the Example.

[F1]

Full and observed data under a missingness pattern: the response pattern selects which full-data coordinates are recorded, and the pattern remains part of the observed record.

[F2]

MCAR, MAR, and MNAR relative to an observation pattern: MCAR means every pattern probability is constant in the full data almost surely.

[F3]

MCAR, MAR, and MNAR relative to an observation pattern: MAR means each pattern probability is a measurable function of that pattern's observed projection almost surely.

[F4]

MCAR, MAR, and MNAR relative to an observation pattern: MNAR means MAR fails for at least one pattern relative to the declared partition.

[F5]

Exact binary-outcome bounds under unrestricted missingness: for observed response rate $a$ and success mass $b$ , the compatible binary full means are exactly $[b, b + 1 - a]$ .

[F6]

MCAR, MAR, and MNAR relative to an observation pattern and Measure kernel and probability kernel: both displayed joint laws have $P (Y = y) > 0$ for $y = 0, 1$ , so each supplies the finite response kernel $g_{P} (r ∣ y) = P (R = r, Y = y) / P (Y = y)$ used below to classify that law.

Verification

1.1F1F2F3F4given

For the first model, $g (1 ∣ x, y) = 1 / 2$ is constant in $(x, y)$ , hence MCAR by [F2]. In the second, both pattern probabilities are functions of $x$ alone, which is recorded for both patterns, so MAR holds by [F3]; it is not MCAR because the probability differs at $x = 0$ and $x = 1$ . In the third, $P (Y = 0 ∣ X = x) = P (Y = 1 ∣ X = x) = 1 / 2$ for each $x$ , and $g (0 ∣ x, 0) = 3 / 4$ while $g (0 ∣ x, 1) = 1 / 4$ . Thus the pattern- $0$ probability cannot be a function of its observed coordinate $x$ alone, so the mechanism is MNAR by [F4].

2.1F1F4F5F6algebra∎

Each row of the table contains nonnegative entries summing to one. Under either law, $P (R = 1, Y = 1) = 1 / 4$ , $P (R = 1, Y = 0) = 1 / 4$ , and $P (R = 0) = 1 / 2$ , so the observed record laws agree and have $a = 1 / 2$ , $b = 1 / 4$ . The means are $1 / 4$ and $3 / 4$ . For the selected finite kernels in [F6], $P^{-}$ has $g_{P^{-}} (1 ∣ 1) = 1$ and $g_{P^{-}} (1 ∣ 0) = 1 / 3$ , while $P^{+}$ has $g_{P^{+}} (1 ∣ 1) = 1 / 3$ and $g_{P^{+}} (1 ∣ 0) = 1$ . In each law the pattern- $0$ response probability also differs across the two values of $Y$ , although the observed projection for that pattern is empty; hence neither law is MAR. This agrees with [F5] and shows the observed respondent mean $1 / 2$ does not identify the population mean.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lu Mao, Missing Data: Foundations and Methods, Chapter 1, §§1.3–1.7 (standard reference, not scraped)
National Research Council, The Prevention and Treatment of Missing Data in Clinical Trials, Chapter 4, pp. 50–58 (standard reference, not scraped)