Bounds on the covariate-time transformation for competing-risks survival analysis

Abstract

A fundamental problem with the latent-time framework in competing risks is the lack of identifiability of the joint distribution. Given observed covariates along with assumptions as to the form of their effect, then identifiability may obtain. However it is difficult to check any assumptions about form since a more general model may lose identifiability. This paper considers a general framework for modelling the effect of covariates, with the single assumption that the copula dependency structure of the latent times is invariant to the covariates. This framework consists of a set of functions: the covariate-time transformations. The main result produces bounds on these functions, which are derived solely from the crude incidence functions. These bounds are a useful model checking tool when considering the covariate-time transformation resulting from any particular set of further assumptions. An example is given where the widely-used assumption of independent competing risks is checked.

Appendices

Appendices

A copula invariance

Given a multivariate survival function, S(t), its survival copula (Nelsen 1998, pp. 28–29) is a function $C:[0,1]^k\mapsto [0,1]$ with the property that C(1,...,1,u,1,...,1) = u and that S(t) = C{S ₁(t ₁), ..., S _k (t _k )}, where S _i (t _i ) is the marginal survival function for T _i .

If the effect of a covariate, Z, can be represented in the form

$$ S({\bf t}|z)=S\{\phi_{1}(t_{1},z), \ldots, \phi_{k}(t_{k},z)|Z=\mathcal{Z} \} $$

then it follows that $\phi_{i}(t,z)=S_{i}^{-1}\{S_{i}(t|Z=z)|Z=\mathcal{Z}\}$. Hence if we denote C _z to be the survival copula of the joint survival function conditional on Z = z, it follows that

$$ \begin{aligned} C_{z}\left[S_{1}(t_{1}|z), \ldots, S_{k}(t_{k}|z)\right]=& S\left[ \phi_{1}(t_{1},z), \ldots, \phi_{k}(t_{k},z)|Z={\mathcal Z} \right] \\ =& C_{\mathcal Z}\left[S_{1}\{\phi_{1}(t_{1},z)|{\mathcal Z}\}, \ldots, S_{k}\{\phi_{k}(t_{k},z)|{\mathcal Z}\}\right] \\ =& C_{\mathcal Z}\left[S_{1}\circ S_{1}^{-1}\{ S_{1}(t|z)|{\mathcal Z}\}, \ldots, S_{k}\circ S_{k}^{-1}\{ S_{k}(t|z)|{\mathcal Z}\}\right] \\ =& C_{\mathcal Z}\left[ S_{1}(t_{1}|z), \ldots, S_{k}(t_{k}|z)\right] \\ \Rightarrow C_{z}=&C_{\mathcal Z} \end{aligned} $$

Hence our assumption about the form of the CTT implies that the survival copula is invariant to the value of the covariate. Given that the copula describes the dependency structure of a multivariate distribution (Nelsen 1998, chapter 5) this gives a useful insight into the implications of assumption (4).

B Lemmas

Lemma B.1.

For all doubly indexed sets  A _ij ,

$$ \bigcup_{i} \bigcap_{j} A_{ij} \subseteq \bigcap_{j} \bigcup_{i} A_{ij}. $$

Proof

Consider

$$ \begin{aligned} \left\{\bigcup_{i} \bigcap_{j} A_{ij} \right\} \cap \left\{ \bigcap_{j} \bigcup_{i} A_{ij}\right\}^{c} =& \left\{ \bigcup_{i} \bigcap_{j} A_{ij} \right\} \cap \left\{ \bigcup_{j} \bigcap_{i} A_{ij}^{c} \right\} \\ =& \bigcup_{i,j} \left\{\bigcap_{m} A_{im} \cap \bigcap_{n} A_{nj}^{c} \right\} \end{aligned} $$

Since $A_{im} \cap A_{nj}^{c}=\emptyset$ for m = j, n = i

$$=\bigcup_{i,j} \emptyset = \emptyset$$

□

Lemma B.2.

For any subset, $\mathcal{C}$, of the indices, i, over which the random variables  T _i  are defined, define  $A= \bigcup_{i \in \mathcal{C}} \bigcap_{j \neq i} \{ T_{i} < T_{j} \}$, and, $B= \bigcup_{i \in \mathcal{C}} \bigcap_{j \not\in \mathcal{C}} \{ T_{i} < T_{j} \}$, then  A = B.

Proof

Since

$$ \bigcap_{j \neq i} \{T_{i}< T_{j}\} \subseteq \bigcap_{j \not\in \mathcal{C}}\{T_{i} < T_{j} \}, $$

we have that $A \subseteq B$.

Consider

$$ B^{c}= \bigcap_{i \in \mathcal{C}} \bigcup_{j \not\in\mathcal{C}} \{ T_{i}> T_{j}\}. $$

Swapping the indices i and j, and using lemma B.1 we have,

$$ B^{c}=\bigcap_{j \in \mathcal{C}} \bigcup_{i \not\in\mathcal{C}} \{ T_{i}< T_{j}\} \supseteq \bigcup_{i \not\in \mathcal{C}} \bigcap_{j \in \mathcal{C}}\{ T_{i} < T_{j} \}. $$

Clearly $\bigcap_{j \in \mathcal{C}} \{T_{i}< T_{j}\} \supseteq \bigcap_{j \neq i} \{ T_{i}< T_{j}\}$, hence

$$ B^{c} \supseteq\bigcup_{i\not\in \mathcal{C}} \bigcap_{j \neq i} \{ T_{i}< T_{j}\} = A^{c}, $$

where the last equality arises from the fact that $\bigcap_{j\neq i}\{ T_{i}< T_{j}\}$ partitions Ω, the sample space.

So we have $A\subseteq B$ and $A^{c}\subseteq B^{c}$ hence A = B. □

Lemma B.3.

For any subset, $\mathcal{C}$, of the indices, i, over which the random variables  T _i  are defined, define  $A= \bigcup_{i \in \mathcal{C}} \left( \{T_{i}< t\} \cap \bigcap_{j \neq i} \{ T_{i} < T_{j} \}\right)$, and, $B= \left(\bigcup_{i \in \mathcal{C} } \{T_{i}< t\} \right) \cap \left( \bigcup_{k \in \mathcal{C}} \bigcap_{j \neq k} \{ T_{k} < T_{j} \} \right)$ then  A = B.

Proof

Define X _i  =  {T _i  < t} and $Y_{i}= \bigcap_{j \neq i} \{T_{i}< T_{j}\}$, then

$$ A = \bigcup_{i \in \mathcal{C}} X_{i} \cap Y_{i} $$

and

$$ B= \bigcup_{i, k \in \mathcal{C}} X_{i} \cap Y_{k}. $$

By inspection A ⊂ B.

Since, $\{ T_{k} < T_{i}\} \cap \{T_{i} < t\} \Rightarrow \{T_{k}< t\}$, for a fixed $k\in\mathcal{C}$, and for all i

$$ \begin{array}{lll} Y_{k} \cap X_{i} & \subset & Y_{k} \cap X_{k} \\ \Rightarrow Y_{k} \cap \bigcup\limits_{i \in \mathcal{C} } X_{i}& \subset & Y_{k} \cap X_{k} \\ \Rightarrow \bigcup\limits_{k \in \mathcal{C}} \left( Y_{k} \cap \bigcup\limits_{i \in \mathcal{C}} X_{i} \right)&\subset& \bigcup\limits_{k\in\mathcal{C}} Y_{k} \cap X_{k} \\ \Rightarrow B & \subset & A \end{array} $$

Since A ⊂ B and B ⊂ A, then A = B. □