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The observed first time and cause do not identify the latent joint failure-time law

Statement refuted

Without restrictions on the joint law of the latent cause-specific times, the law of the observed first time and cause uniquely determines that latent joint law.

Facts & Assumptions

Given: A parametric model whose parameter is the full joint law of two latent cause-specific event times and whose observation is only the first time and its cause. Refute identifiability of the latent joint law.

[F1]

Competing-risk data, cause-specific hazard measures, and cumulative incidence: $(T, D)$ is obtained by mapping latent cause times to their unique minimum and its cause.

[F2]

Parametric statistical models: a model is a parameter map $θ \mapsto P_{θ}$ of sampling laws.

[F3]

Observational equivalence: $θ$ and $η$ are observationally equivalent exactly when $P_{θ} = P_{η}$ .

[F4]

Estimands on a statistical model: an estimand is a specified function of the model parameter.

[F5]

Identifiability of an estimand: an estimand is identifiable only if equal sampling laws imply equal target values.

Counterexample

Use two causes and parameter set $Θ = {0, 1}$ . Under $θ = 0$ , give $(T_{1}, T_{2})$ mass $1 / 2$ at $(1, 2)$ and mass $1 / 2$ at $(2, 1)$ . Under $θ = 1$ , give it mass $1 / 2$ at $(1, 3)$ and mass $1 / 2$ at $(3, 1)$ . Each law has finite positive times and no simultaneous first causes. The sampling law is the pushforward under $(T_{1}, T_{2}) \mapsto (T, D)$ , where $T = \min (T_{1}, T_{2})$ and $D$ is the unique minimizing cause.

1.1givenF4algebra

Let $ψ (θ)$ be the full joint law of $(T_{1}, T_{2})$ under parameter $θ$ . This is an estimand by [F4]. The two laws specified above differ: under $θ = 0$ , $P (T_{2} = 2) = 1 / 2$ , while under $θ = 1$ , $P (T_{2} = 2) = 0$ .

1.2givenF1F2F3algebra

Under either parameter, the first event is at time $1$ . The outcome $(T, D) = (1, 1)$ has probability $1 / 2$ from the atom with $T_{1} = 1$ , and $(T, D) = (1, 2)$ has probability $1 / 2$ from the atom with $T_{2} = 1$ ; no other observed pair occurs. Thus the two pushforward sampling laws are equal, so the parameters are observationally equivalent by [F2]–[F3].

2.1step 1.1step 1.2F5∎

Step 1.1 gives $ψ (0) \neq ψ (1)$ , while step 1.2 gives equal sampling laws. This violates the defining implication in [F5], so the latent joint law is not identifiable from the observed first time and cause.

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Sources

Lu Tian and Richard Olshen, Competing Risk and Alternatives to Cox Model, Stanford University STAT331, slide 3 (standard reference, not scraped)
Janne Pitkäniemi and Esa Läärä, Survival analysis with competing risks, Statistical Practice in Epidemiology using R, University of Tartu, slides 13–24 (standard reference, not scraped)
Peter C. Austin, D. S. Lee, and Jason P. Fine, Introduction to the Analysis of Survival Data in the Presence of Competing Risks, Circulation 133(6) (2016), pp. 601–609 (standard reference, not scraped)