Heckman and Honoré (1989): The Identifiability of the Competing Risks Model

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 $J$ competing causes of death $\{1,2,\dots ,J\}$.

• Associated with each cause of death is a stochastic failure time $T_j$.

• We observe only the distribution of the identified minimum:

  • The time of death $T={\min }_j{T}_j$.

  • The cause of death $I=\arg {\min }_j{T}_j$.

• 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 $(T_1,T_2,\dots ,T_J)$ there exist $(S_1,S_2,\dots ,S_J)$ with $S_i⫫S_j$ for all $i\ne j$ such that $(T,I_T)$ and $(S,I_S)$ 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 $T_1,\dots ,T_J$ conditional on $X$.

• 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).

Overview

• Establish an identification theorem for a general class of competing risks models with regressors.

• This class includes models with marginal distributions that follow:

  • Proportional hazards.

  • Mixed proportional hazards.

  • Accelerated 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 $T$ from a single risk conditional on some covariates $X$.

• Conditional on $X$, $T$ has cdf $F(t\mid x)$ and pdf $f(t\mid x)$.

• Hazard function: $\lambda (t\mid x)=\frac{f(t\mid x)}{1-F(t\mid x)}$.

• Integrated hazard: $\Lambda (t\mid x)={\int }_0^t\lambda (s\mid x)\mathrm{ds}$.

• If $\lambda (t\mid x)=z(t)\varphi (x)$ then $\Lambda (t\mid x)=Z(t)\varphi (x)$ with $Z(t)={\int }_0^{t}z(s)\mathrm{ds}$.

• Equivalently, we can work with the survivor function: $$S(t\mid x)=\mathrm{Pr}(T>t\mid x)=\exp [-Z(t)\varphi (x)]$$

• It is common in practice to use $\varphi (x)=e^{x\beta }$.

• Suppose $F(t\mid x)=1-e^{-Z(t)\varphi (x)}$ where $Z(t)$ is the baseline integrated hazard and $\varphi (x)$ is a scaling term.

• If $Z$ is differentiable, then $Z\prime (t)$ 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.

• $Z(t)$ and $\varphi$ may differ across risks.

• The joint survivor function is $$S(t_1,t_2\mid x)=1-(1-\exp [-Z_1(t_1){\varphi }_1(x)])(1-\exp [-Z_2(t_2){\varphi }_2(x)]).$$

Introducing Dependence

• We could draw two independent failure times $T_1$ and $T_2$ by drawing (independently) $U_j\sim U(\mathrm{0,1})$ and solving for $T_j$:

$$S_j(t\mid x)=\exp \{-Z_j(t){\varphi }_j(x)\}.$$

• If $K(u_1,u_2)=u_1{u}_2$ is the CDF of $U_1$ and $U_2$, the joint survivor function is

$$S(t_1,t_2\mid x)=K[\exp \{-Z_1(t){\varphi }_1(x)\},\exp \{-Z_2(t){\varphi }_2(x)\}].$$

• We can introduce dependence in $T_1$ and $T_2$ by introducing dependence in $U_1$ and $U_2$ via $K$.

• Suppose $(U_1,U_2)\sim K(\cdot ,\cdot )$ on $[\mathrm{0,1}{]}^2$ and assume that $Z_1(0)=Z_2(0)=0$.

• Then the survivor function for $(T_1,T_2)$ is

Generalization: Mixed Proportional Hazards

• Suppose that the competing risks are independent, ${\varphi }_j(x)=e^{x{\beta }_j}$, and that one of the covariates, $\omega$, is not observed:

$$S(t_1,t_2\mid x)={\int }_{\Omega }\exp [-Z_1(t_1)e^{x{\beta }_1+c_1\omega }]\exp [-Z_2(t_2)e^{x{\beta }_2+c_2\omega }]\mathrm{dG}(\omega ).$$

• We can arrive at this model by choosing $K$ such that:

$$K({\eta }_1,{\eta }_2)={\int }_{\Omega }{\eta }_1^{\exp (c_1\omega )}{\eta }_2^{\exp (c_2\omega )}\mathrm{dG}(\omega ).$$

Generalization: Accelerated hazards

$$S(t\mid x)=\exp [-Z\{t\varphi (x)\}]$$

• Joint survivor with dependent competing risks:

$$S(t_1,t_2\mid x)=K(\exp [-Z_1\{t_1{\varphi }_1(x)\}],\exp [-Z_2\{t_2{\varphi }_2(x)\}]).$$

• For any $K$, the marginal distributions give rise to univariate accelerated hazard models.

Identification Theorem

Assume that $(T_1,T_2)$ has joint distribution (1). Then $Z_1$, $Z_2$, ${\varphi }_1$, ${\varphi }_2$, and $K$ are identified from the minimum of $(T_1,T_2)$ under the following assumptions:

1. $K$ is continuously differentiable with partial derivatives $K_1$ and $K_2$ and for $i=1,2$, ${\lim }_{n\to \mathrm{\infty }}{K}_i({\eta }_{1n},{\eta }_{2n})$ is finite for all sequences ${\eta }_{1n}$, ${\eta }_{2n}$ for which ${\eta }_{1n}\to 1$ and ${\eta }_{2n}\to 1$ for $n\to \mathrm{\infty }$. We also assume that $K$ is strictly increasing in each of its arguments.

2. $Z_1(1)=Z_2(1)=1$ and ${\varphi }_1(x_0)={\varphi }_2(x_0)=1$ for some $x_0$.

3. The support of $\{{\varphi }_1(x),{\varphi }_2(x)\}$ is $(0,\mathrm{\infty })\times (0,\mathrm{\infty })$.

4. $Z_1$ and $Z_2$ are nonnegative, differentiable, strictly increasing functions, except that we allow them to be infinite for finite $t$.

Notes about these assumptions:

1. $K$ is already weakly increasing.

2. This is an innocuous normalization since ${\varphi }_j(x)$ and $Z_j(t)$ are not jointly identified to scale.

3. This is satisfied, for example, when ${\varphi }_j(x)=\exp (x{\beta }_j)$ and there is a common covariate with support $ℝ$ and different coefficients.

Mapping Observables to Unobservables

Observed distributions:

$$Q_1(t\mid x)=\mathrm{Pr}(T_1\ge t,T_2\ge T_1\mid x)Q_2(t\mid x)=\mathrm{Pr}(T_2\ge t,T_1\ge T_2\mid x).$$

Tsiatis (1975) establishes the following mappings:

$$\frac{\partial Q_1}{\partial t}(t\mid x)={\left[\frac{\partial S}{\partial t_1}\right]}_{t_1=t_2=t}\frac{\partial Q_2}{\partial t}(t\mid x)={\left[\frac{\partial S}{\partial t_2}\right]}_{t_1=t_2=t}.$$

We have $$\frac{\partial Q_1}{\partial t}(t\mid x)=-K_1[\exp \{-Z_1(t){\varphi }_1(x)\},\exp \{-Z_2(t){\varphi }_2(x)\}]\exp \{-Z_1(t){\varphi }_1(x)\}Z{\prime }_1(t){\varphi }_1(x).$$

Identification of ${\varphi }_j$

Taking the ratio of $\frac{\partial Q_1(t\mid x)}{\partial t}$ at $x$ and $x_0$ yields

$$\frac{K_1[\exp \{-Z_1(t){\varphi }_1(x)\},\exp \{-Z_2(t){\varphi }_2(x)\}]\exp \{-Z_1(t){\varphi }_1(x)\}Z{\prime }_1(t){\varphi }_1(x)}{K_1[\exp \{-Z_1(t){\varphi }_1(x_0)\},\exp \{-Z_2(t){\varphi }_2(x_0)\}]\exp \{-Z_1(t){\varphi }_1(x_0)\}Z{\prime }_1(t){\varphi }_1(x_0).}$$

Taking $t\to 0$ and using the normalization yields ${\varphi }_1(x)$. Our choice of $x$ was arbitrary so ${\varphi }_1(x)$ is identified on the entire support of $X$. Similarly for ${\varphi }_2(x)$.

Identification of $K$

We know $S(t,t\mid x)$ since $S(t,t\mid x)=Q_1(t\mid x)+Q_2(t\mid x)$. Furthermore, $$S(t,t\mid x)=K(\exp [-Z_1(t){\varphi }_1(x)],\exp [-Z_2(t){\varphi }_2(x)]).$$

Setting $t=1$ gives $$S(1,1\mid x)=K(\exp [-{\varphi }_1(x)],\exp [-{\varphi }_2(x)]).$$ and letting ${\varphi }_1(x)$ and ${\varphi }_2(x)$ vary over $(0,\mathrm{\infty }{)}^2$ (by Assumption 3) yields $K$.

Identification of $Z_j$

$$S(t,t\mid x_n)=K(\exp [-Z_1(t){\varphi }_1(x_n)],\exp [-Z_2(t){\varphi }_2(x_n)])$$

• Let ${\varphi }_2(x)\to 0$ while holding ${\varphi }_1(x)$ fixed.

• Then $$S(t,t\mid x)\to K(\exp [-Z_1(t){\varphi }_1(x)],1).$$

• Since $K$ and ${\varphi }_1$ are known and $K$ is strictly increasing in both arguments, we have $Z_1(1)=1$ for any $t$.

• Similarly for $Z_2(t)$.

Conclusion

Identification argument:

• Given the distribution of $(T,I)$ and exploiting multiplicative separability gives us ${\varphi }_j(x)$ for $j=\mathrm{1,2}$.

• Using the full range of ${\varphi }_j(x)$ on $(0,\mathrm{\infty })$ yields $K$.

• Using $K$, ${\varphi }_j$, and related properties gives us $Z_j(t)$.

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.