These notes are based on the following article:
Heckman, James J. and Bo E. Honoré (1989). The identifiability of the competing risks model. Biometrika, 76: 325–330.
The Classical Competing Risks Model
Suppose there are competing causes of death .
Associated with each cause of death is a stochastic failure time .
We observe only the distribution of the identified minimum:
The time of death .
The cause of death .
Goal: Identify the joint distribution of the latent failure times given that we only observe the distribution of the identified minimum.
Note that we aren’t considering regressors yet.
Cox and Tsiatis Nonidentification Theorem
For any joint distribution of latent failure times, there exists another such distribution with independent failure times that yields the same distribution of the minimum (Cox, 1959, 1962; Tsiatis, 1975).
That is, given r.v.’s there exist with for all such that and are observationally equivalent.
In light of this result, any empirical work needed to proceed by placing some structure on the form of dependence across risks, for example, by assuming independence.
Importance of Dependence
We are concerned with conditional independence—independence of the risks conditional on .
Even conditional independence may not hold if, for example, we are studying an individual whose behavior may affect all of the risks.
Yashin, Manton, and Stallard (1986): How do smoking, blood pressure, and body weight (regressors) affect time of death from cancer, heart disease, etc. (risks).
Establish an identification theorem for a general class of competing risks models with regressors.
This class includes models with marginal distributions that follow:
Mixed proportional hazards.
Results are presented for only two competing risks but generalize to any arbitrary finite number of risks.
Proportional Hazards Model
We want to model the time of death from a single risk conditional on some covariates .
Conditional on , has cdf and pdf .
Hazard function: .
Integrated hazard: .
If then with .
Equivalently, we can work with the survivor function:
It is common in practice to use .
Suppose where is the baseline integrated hazard and is a scaling term.
If is differentiable, then is the baseline hazard.
Proportional Hazards and Competing Risks
Assuming for the moment that failure times are independent, we can easily generalize this to model competing risks.
The distribution of each failure time has a proportional hazard specification.
and may differ across risks.
The joint survivor function is
- We could draw two independent failure times and by drawing (independently) and solving for :
- If is the CDF of and , the joint survivor function is
We can introduce dependence in and by introducing dependence in and via .
Suppose on and assume that .
Then the survivor function for is
Generalization: Mixed Proportional Hazards
- Suppose that the competing risks are independent, , and that one of the covariates, , is not observed:
- We can arrive at this model by choosing such that:
Generalization: Accelerated hazards
- Joint survivor with dependent competing risks:
- For any , the marginal distributions give rise to univariate accelerated hazard models.
Assume that has joint distribution (eq:prop_survivor). Then , , , , and are identified from the minimum of under the following assumptions:
is continuously differentiable with partial derivatives and and for , is finite for all sequences , for which and for . We also assume that is strictly increasing in each of its arguments.
and for some .
The support of is .
and are nonnegative, differentiable, strictly increasing functions, except that we allow them to be infinite for finite .
Notes about these assumptions:
is already weakly increasing.
This is an innocuous normalization since and are not jointly identified to scale.
This is satisfied, for example, when and there is a common covariate with support and different coefficients.
Mapping Observables to Unobservables
Tsiatis (1975) establishes the following mappings:
Taking the ratio of at and yields
Taking and using the normalization yields . Our choice of was arbitrary so is identified on the entire support of . Similarly for .
We know since . Furthermore,
Setting gives and letting and vary over (by Assumption 3) yields .
Let while holding fixed.
Since and are known and is strictly increasing in both arguments, we have for any .
Similarly for .
Given the distribution of and exploiting multiplicative separability gives us for .
Using the full range of on yields .
Using , , and related properties gives us .
Implications of Nonparametric Identification:
Identification does not depend on parametric functional forms or assumed forms of risk dependence (modulo separability of the hazard).
Highlights the role of regressors in identification in contrast to the Cox-Tsiatis nonidentification result.
Suggests the possibility of a nonparametric estimator.