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Missingness and nonidentification — Examples

1 · Prerequisites

2 · Summary

Finite response mechanisms make the MCAR/MAR/MNAR distinctions concrete. Two full-data laws can agree on the response rate, observed success mass, and entire observed-record distribution while having different population means. A separate example distinguishes an unrecorded outcome from an outcome that has no value after a competing event such as death.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Death is not automatically a missing outcome

Example

Suppose the scientific target is a quality-of-life score at time $t$ for all participants enrolled at baseline, with positive probability of death before $t$ , and the score is defined only for people alive at $t$ . If death before $t$ makes the score itself undefined, those post-death values are not merely unrecorded full-data components. An unconditional mean over all enrolled participants has no value until the scientist specifies a different target, such as a composite outcome or a survivor-specific target.

Facts & Assumptions

Given: A population probability space of all baseline-enrolled participants with $P (D = 1) > 0$ ; $D = 1$ means death before $t$ ; and the proposed score $Q_{t}$ is defined only when $D = 0$ , with no post-death value stipulated.

[F1]

Random elements and real random variables: a real random variable is a measurable map from the entire probability space to $R$ .

[F2]

Full and observed data under a missingness pattern: the full-data vector contains the values that define the target, while the response pattern records which of those values are available.

[F3]

Estimands on a statistical model: an estimand is a specified function on the model's parameter set.

Verification

1.1F1given

Since $Q_{t}$ has no value stipulated on the event ${D = 1}$ , it is only a function on ${D = 0}$ and is not a map from the whole baseline probability space to $R$ . It therefore is not a real random variable on the stated population by [F1].

2.1F2F3step 1.1given∎

The observed-pattern construction [F2] can mask a coordinate of a full-data vector, but it does not define a value where the scientific outcome itself does not exist. Thus $E (Q_{t})$ over all enrolled participants is not a well-defined target function as required by [F3]. Assigning a post-death value, using a composite outcome, or restricting to survivors defines a different scientific target and must be stated separately.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

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.

Missingness and nonidentification — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Death is not automatically a missing outcome

Example

Facts & Assumptions

Verification

MCAR, MAR, MNAR, and indistinguishable observed laws

Example

Facts & Assumptions

Verification

Sources