Group and within-group variable selection for competing risks data

Abstract

Variable selection in the presence of grouped variables is troublesome for competing risks data: while some recent methods deal with group selection only, simultaneous selection of both groups and within-group variables remains largely unexplored. In this context, we propose an adaptive group bridge method, enabling simultaneous selection both within and between groups, for competing risks data. The adaptive group bridge is applicable to independent and clustered data. It also allows the number of variables to diverge as the sample size increases. We show that our new method possesses excellent asymptotic properties, including variable selection consistency at group and within-group levels. We also show superior performance in simulated and real data sets over several competing approaches, including group bridge, adaptive group lasso, and AIC / BIC-based methods.

Appendix

To describe the conditions to justify the asymptotics, we define

$$\begin{aligned} \begin{aligned} \ w_{ij}(t)&=\frac{I(C_{ij}\ge T_{ij}\wedge t)G(t)}{G(X_{ij}\wedge t)}\\ \mathbf{q}(u)&=-\lim _{m\rightarrow \infty }\frac{1}{m}\sum _{i=1}^m\sum _{j=1}^{n_i}\int _0^\tau \{\mathbf{Z}_{ij}(t)-\mathbf{e}({\varvec{\beta }}_0,t)\}w_{ij}(t)I(X_{ij}<u\le t)dM_{ij}({\varvec{\beta }}_0,t),\\ \pi (u)&=\lim _{m\rightarrow \infty }\frac{1}{m}\sum _{i=1}^m\sum _{j=1}^{n_i} I(X_{ij}\ge u). \end{aligned} \end{aligned}$$

where $\mathbf{a}^{\otimes 0}=\mathbf{1},\mathbf{a}^{\otimes 1}=\mathbf{a},$ and $\mathbf{a}^{\otimes 2}=\mathbf{a}{} \mathbf{a}^T$. Let $N_{ij}^c(t)=I(C_{ij}\le t)$ and $\varLambda ^c(t)$ be the cumulative hazard function by treating censored observations as events. Define $M_{ij}^c(t)=N_{ij}^c(t)-\int _0^tI(X_{ij}\ge u)d\varLambda ^c(u)$. Consider the following conditions:

• (C1) $\int _0^\tau \lambda _{10}(u)du<\infty$ and $|Z_{ijq}(0)|+\int _0^\tau |dZ_{ijq}(u)|\le M$ for all i, j, q, where M is a positive constant;

• (C2) There exists a neighborhood $\mathcal {B}$ of ${\varvec{\beta }}_0$ and $\sup _{t\in [0,\tau ],{\varvec{\beta }}\in \mathcal {B}}\Vert \mathbf{S}^{(r)}({\varvec{\beta }},\theta )-\mathbf{s}^{(r)}({\varvec{\beta }},\theta )\Vert _2$ converges in probability to 0 for $r=0,1,2$. There also exists a matrix $\varGamma =\varGamma ({\varvec{\beta }}_0)$ such that $\Vert m^{-1}\sum _{i=1}^mVar(\mathbf{D}_i)-\varGamma \Vert \rightarrow 0$, where

  $$\begin{aligned} \begin{aligned} \mathbf{D}_i&=\sum _{j=1}^{n_i}\Big [\int _0^\tau \{\mathbf{Z}_{ij}(u)-\mathbf{e}({\varvec{\beta }}_0,u)\}w_{ij}(u)dM_{ij}({\varvec{\beta }}_0,u)\\&\quad +\int _0^\tau \sum _{i=1}^m\mathbf{q}(u)/\pi (u)dM_{ij}^c(u)\Big ]. \end{aligned} \end{aligned}$$

  In addition, there exist constants $C_1$ and $C_2$ such that $0<C_1<\kappa _\text {min}(\varGamma )\le \kappa _\text {max}(\varGamma )<C_2<\infty$ for all m, where $\kappa _\text {min}(\mathbf{A})$ and $\kappa _\text {max}(\mathbf{A})$ are the minimal and maximal eigenvalues of a matrix $\mathbf{A}$, respectively;

• (C3) For $r=0,1,2$, $\mathbf{s}^{(r)}({\varvec{\beta }},t)$’s are continuous in ${\varvec{\beta }}\in \mathcal {B}$ uniformly in $t\in [0,\tau ]$ and are bounded on $\mathcal {B}\times [0,\tau ]$. ${ s}^{(0)}({\varvec{\beta }},t)$ is bounded away from 0 on $\mathcal {B}\times [0,\tau ]$. The true parameter $\beta _{j0}$ is bounded away from 0 for $j\in B_1$. Let $\varOmega =\int _0^\tau \mathbf{v}({\varvec{\beta }}_0,u){s}^{(0)}({\varvec{\beta }}_0,u)\lambda _{10}(u)du$. There exists $C_3$ and $C_4$ such that $0<C_3<\kappa _\text {min}(\varOmega )\le \kappa _\text {max}(\varOmega )<C_4<\infty$;

• (C4) There exists a constant $C_5$ such that $\sup _{1\le i\le m}E(D_{ij}^2D_{ij^{'}}^2)\le C_5<\infty$ for all $1\le j,\ j^{'}\le d_m$. $C_m^*=\max _j\sum _{k=1}^KI(j\in A_k)$ is bounded;

• (C5) $d_m^4/m\rightarrow 0$;

• (C6) $\sum _{k=1}^{K_1}c_k \left\{ \Big (\sum _{j\in A_k\cap B_1}|\beta _{j0}|^{1-\nu }\Big )^{\gamma -1} \sum _{j\in A_k\cap B_1}1/|{\beta }_{j0}|^\nu \right\} \le M_m$, where $M_m=O_p(1)$;

• (C7) $\lambda _m/\sqrt{m}\rightarrow 0$, $\sqrt{m/d_m}\tilde{\beta }_j=O_p(1)$, and $\min \big (\lambda _m m^{(\nu -1)/2}d_m^{-(1+\nu )/2}$,

  $\lambda _m m^{\gamma (\nu -1)/2}d_m^{-1+\gamma (1-\nu )/2}\big )\rightarrow \infty$.

Conditions (C1)–(C3) are standard conditions for the marginal subdistribution hazards model (Zhou et al. 2012). Conditions (C1)–(C5) are similar to the conditions of Cai et al. (2005) and Huang et al. (2014) so that they guarantee local asymptotic quadratic property of $l({\varvec{\beta }})$ and the existence of local minimizer of ${\mathcal {L}}_m({\varvec{\beta }})$. Once Conditions (C3) and (C4) are met, Condition (C6) is satisfied if the true non-zero $\beta _{j0}$’s are bounded above. Condition (C7) controls $\lambda _m$ to have the oracle property.