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Informative censoring can hide distinct event-time laws

Statement refuted

If two studies have the same observed right-censored law of $(U, δ)$ , then they have the same event-time law of $T$ , even when censoring may depend on $T$ .

Facts & Assumptions

Given: Right-censored data are $U = \min (T, C)$ and $δ = 1 {T \leq C}$ as in Event-time observation, censoring, and truncation schemes. In Model A, $P (T = 1) = P (T = 3) = 1 / 2$ , with $C = 4$ when $T = 1$ and $C = 1 / 2$ when $T = 3$ . In Model B, $P (T = 1) = P (T = 2) = 1 / 2$ , with $C = 4$ when $T = 1$ and $C = 1 / 2$ when $T = 2$ . In both models, a separate study selects $T \sim U n i f (0, 3)$ subjects only when $T > 1$ .

[F1]

Right-censored observation records the minimum and the event indicator; left truncation selects only individuals with $T > L$ . Informative censoring requires its joint observation law to be modeled. Event-time observation, censoring, and truncation schemes

[A1]

Model A and Model B are the two finite joint laws for $(T, C)$ specified in Given.

[A2]

The separate truncation example has unconditional density $1 / 3$ on $(0, 3)$ and inclusion event $T > 1$ .

Counterexample

1.1F1A1algebra

In Model A, the outcome $T = 1$ has probability $1 / 2$ and gives $(U, δ) = (1, 1)$ because $C = 4$ . The outcome $T = 3$ has probability $1 / 2$ and gives $(U, δ) = (1 / 2, 0)$ because $C = 1 / 2 < T$ .

1.2F1A1algebra

In Model B, $T = 1$ again gives $(U, δ) = (1, 1)$ with probability $1 / 2$ , while $T = 2$ gives $(U, δ) = (1 / 2, 0)$ with probability $1 / 2$ .

1.3A2algebra

For the truncation example, $P (T > 1) = \int_{1}^{3} (1 / 3) d t = 2 / 3$ .

2.1step 1.1step 1.2algebra

Both observed laws assign probability $1 / 2$ to each of $(1, 1)$ and $(1 / 2, 0)$ , although the censoring time is a deterministic function of $T$ in each model.

2.2A2step 1.3algebra

Conditional on inclusion, the event-time density is $(1 / 3) / (2 / 3) = 1 / 2$ on $(1, 3)$ and zero outside, whereas the unselected density is $1 / 3$ on $(0, 3)$ . Thus left truncation changes the sampled event-time law.

3.1step 2.1A1algebra

The event-time laws differ: $P_{A} (T \leq 2) = 1 / 2$ and $P_{B} (T \leq 2) = 1$ . Equal observed laws with different event-time laws contradict the refuted statement.

4.1step 3.1step 2.2F1∎

The finite censoring construction refutes the claim about equal observed laws, and the uniform calculation shows that truncation changes the sampling distribution by selection. Informative censoring requires its joint observation law to be modeled.

Depends on

Event-time observation, censoring, and truncation schemes

Used by

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Sources

Qiqing Yu, Survival Analysis (Lecture Notes), §§1.2.2–1.3 and 1.7, PDF pp. 3–5 and 9 (standard reference, not scraped)
Lu Tian, STAT331 Unit 2: Likelihood Construction, Inference for Parametric Survival Distributions, §2.1, PDF pp. 1–4 (standard reference, not scraped)