Competing risks regression with dependent multiple spells: Monte Carlo evidence and an application to maternity leave

Abstract

Copulas are a convenient tool for modelling dependencies in competing risks models with multiple spells. This paper introduces several practical extensions to the nested copula model and focuses on the choice of the hazard model and copula. A simulation study looks at the relevance of the assumed parametric or semiparametric model for hazard functions, copula and whether a full or partial maximum likelihood approach is chosen. The results show that the researcher must be careful which hazard is being specified as similar functional form assumptions for the subdistribution and cause-specific hazard will lead to differences in estimated cumulative incidences. Model selection tests for the choice of the hazard model and copula are found to provide some guidance for setting up the model. The nice practical properties and flexibility of the copula model are demonstrated with an application to a large set of maternity leave periods of mothers for up to three maternity leave periods.

Appendix

1.1 Proof of Proposition 2

To simplify the notation, we omit index k and ${\varvec{\beta }}_k$. For the same reason, we treat all relevant functionals to be sufficiently smooth such that they are differentiable and integrable in t and differentiable in ${\varvec{x}}$. For example the PWCON model has a discontinuous cause-specific hazard in t and would require a combination of summation and integration (see Kyrrä, 2009). Without that this changes the nature of the results. Let

$$\begin{aligned} f_{j}(t;\,{\varvec{x}})= & {} \lambda _{j}(s;\,{\varvec{x}}) S(s;\,{\varvec{x}}) \end{aligned}$$

and therefore $Q_{j}(t;\,{\varvec{x}}) = \int _0^{t} f_{j}(s;\,{\varvec{x}})\mathrm{d}s$. We assume $f'_{jl}(t;\,{\varvec{x}})=\partial f_{j}(t;\,{\varvec{x}})/\partial x_l$ is bounded and there exists an interval for t such that $f'_{jl}(t;\,{\varvec{x}})$ has the same sign for all $t \in [t_1, t_2)$.

We have

$$\begin{aligned} Q'_{jl}(t_2;\,{\varvec{x}})= & {} \int _0^{t_2}f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s\\= & {} \int _0^{t_1} f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s + \int _{t_1}^{t_2} f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s. \end{aligned}$$

The sign of $Q'_{jl}(t_1;\,{\varvec{x}})$ is different from $Q'_{jl}(t_2;\,{\varvec{x}})$ if

$$\begin{aligned} (C1)&\left| \int _0^{t_1} f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s \right| < \left| \int _{t_1}^{t_2}f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s \right| , \text {and} \\ (C2)&{\mathrm{{sign}}}\left( \int _0^{t_1}f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s \right) \ne {\mathrm{{sign}}}\left( \int _{t_1}^{t_2}f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s \right) . \end{aligned}$$

Without loss of generality, we assume that

$$\begin{aligned} \int _0^{t_1}f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s >0 \end{aligned}$$

and

$$\begin{aligned} \int _{t_1}^{t_2} f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s <0. \end{aligned}$$

The above conditions C1 and C2 can be combined as

$$\begin{aligned} (C3)&\int _0^{t_1} f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s < -\int _{t_1}^{t_2} f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s. \end{aligned}$$

Let $s_1 = \arg \sup _{t\in [0,t_1)} f'_{jl}(s;\,{\varvec{x}})$ and $s_2 = \arg \sup _{t\in [t_1,t_2)} f'_{jl}(s;\,{\varvec{x}})$. Then, $f'_{jl}(s_2;\,{\varvec{x}})<0$ and takes the least negative value of $f'_{jl}(t;\,{\varvec{x}})$ for $t\in [t_1,t_2)$. We therefore have

$$\begin{aligned} 0<\int _0^{t_1} f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s\le & {} \int _0^{t_1} f'_{jl}(s_1;\,{\varvec{x}}) \ \mathrm{d}s \equiv t_1 f'_{jl}(s_1;\,{\varvec{x}}) \text {, and}\\ 0<-(t_2-t_1) f'_{jl}(s_2;\,{\varvec{x}})\equiv & {} -\int _{t_1}^{t_2} f'_{jl}(s_2;\,{\varvec{x}}) \ \mathrm{d}s \le -\int _{t_1}^{t_2} f'_{jl}(s;\,{\varvec{x}}) \ \mathrm{d}s. \end{aligned}$$

Hence, C3 holds if there existed $t_1$ and $\epsilon = t_2 - t_1 >0$ such that :

$$\begin{aligned} 0< & {} t_1 f'_{jl}(s_1;\,{\varvec{x}})< - \epsilon f'_{jl}(s_2;\,{\varvec{x}}),\nonumber \\ 0< & {} \frac{t_1}{\epsilon } f'_{jl}(s_1;\,{\varvec{x}}) < -f'_{jl}(s_2;\,{\varvec{x}}). \end{aligned}$$

(21)

Equation (21) holds if (i) $f'_{jl}(s_2;\,{\varvec{x}}) <0$ and (ii) $|f'_{jl}(s_2;\,{\varvec{x}})|>c f'_{jl}(s_1;\,{\varvec{x}}) >0$ for some $c>0$. c is small if $t_1$ is small or $t_2-t_1$ is large. It depends on the model and covariate $x_l$, whether these conditions are satisfied or not. This is illustrated and discussed in more detail in what follows.

Note that

$$\begin{aligned} f'_{jl}(s;\,{\varvec{x}})= & {} \lambda _j(s;\,{\varvec{x}})S'(s;\,{\varvec{x}}) + \lambda '_{jl}(s;\,{\varvec{x}})S(s;\,{\varvec{x}})\\= & {} -f_{j}(s;\,{\varvec{x}})(\Lambda '_{1l}(s;\,{\varvec{x}}) +\Lambda '_{2l}(s;\,{\varvec{x}})) +\lambda '_{jl}(s;\,{\varvec{x}}) f_j(s;\,{\varvec{x}})/\lambda _j(s;\,{\varvec{x}})\\= & {} -f_j(s;\,{\varvec{x}})[\Lambda '_{1l}(s;\,{\varvec{x}}) +\Lambda '_{2l}(s;\,{\varvec{x}}) - \lambda '_{jl}(s;\,{\varvec{x}}) /\lambda _j(s;\,{\varvec{x}})]. \end{aligned}$$

Condition (i) $f'_{jl}(s_2;\,{\varvec{x}}) <0$ requires

$$\begin{aligned} \Lambda '_{1l}(s_2;\,{\varvec{x}}) +\Lambda '_{2l}(s_2;\,{\varvec{x}}) > \Lambda '_{jl}(s_2;\,{\varvec{x}}) /\lambda _j(s_2;\,{\varvec{x}}), \end{aligned}$$

(22)

which may be the case or not. Take as an example a proportional hazard model $\Lambda _j(t;\,{\varvec{x}})=\Lambda _{j0}(t)\phi _j({\varvec{x}})$, $\lambda _j(t;\,{\varvec{x}})=\lambda _{j0}(t)\phi _j({\varvec{x}})$ and $\phi _j({\varvec{x}})>0$. In this case (22) becomes

$$\begin{aligned} \Lambda _{10}(s_2)\phi '_{1l}({\varvec{x}}) + \Lambda _{20}(s_2)\phi '_{2l}({\varvec{x}})> & {} \phi '_{jl}({\varvec{x}})/\phi _{j}({\varvec{x}}). \end{aligned}$$

This likely holds for $s_2$ large as $\Lambda _{j0}(s)$ grows to $\infty$ as t increases, whenever $\phi '_{jl}({\varvec{x}})>0$ for $j=1,2$. (i) therefore cannot be ruled out even under strong restrictions on the role of the covariates. For condition (ii) to hold, we need for some $c>0$

$$\begin{aligned}&|f_j(s_2;\,{\varvec{x}})[\Lambda '_{1l}(s_2;\,{\varvec{x}}) +\Lambda '_{2l}(s_2;\,{\varvec{x}}) - \Lambda '_{jl}(s_2;\,{\varvec{x}}) /\lambda _j(s_2;\,{\varvec{x}})]|\nonumber \\&\quad > c|f_j(s_1;\,{\varvec{x}})[\Lambda '_{1l}(s_1;\,{\varvec{x}}) +\Lambda '_{2l}(s_1;\,{\varvec{x}}) - \lambda '_{jl}(s_1;\,{\varvec{x}}) /\lambda _j(s_1;\,{\varvec{x}})]|. \end{aligned}$$

Again, this may be true or not. Take again as an example $\Lambda _j(t;\,{\varvec{x}})=\Lambda _{j0}(t)\phi _j({\varvec{x}})$, the above condition becomes

$$\begin{aligned}&|f_j(s_2;\,{\varvec{x}})[\Lambda _{10}(s_2)\phi '_{1l}({\varvec{x}}) +\Lambda _{20}(s_2)\phi '_{2l}({\varvec{x}}) - \phi '_{jl}({\varvec{x}})/\phi _j({\varvec{x}})]|\nonumber \\&\quad > c|f_j(s_1;\,{\varvec{x}})[\Lambda _{10}(s_1)\phi '_{1l}({\varvec{x}}) +\Lambda _{20}(s_1)\phi '_{2l}({\varvec{x}}) - \phi '_{jl}({\varvec{x}})/\phi _j({\varvec{x}})]|. \end{aligned}$$

This generally holds if $f_j(s_2;\,{\varvec{x}})>cf_j(s_1;\,{\varvec{x}})$ since by definition $\Lambda _{j0}(s_2) >\Lambda _{j0}(s_1)$ for all $s_2>s_1$. Condition (ii) therefore holds for c small enough. We have therefore illustrated that even for the proportional hazards model, where the direction of the partial effect on the hazard is unique for all t, there is no analogous result for $Q'_{jl}(t;\,{\varvec{x}})$ as the direction of partial covariate effect is not restricted.