An improved variable selection procedure for adaptive Lasso in high-dimensional survival analysis

Abstract

Motivated by high-dimensional genomic studies, we develop an improved procedure for adaptive Lasso in high-dimensional survival analysis. The proposed procedure effectively reduces the false discoveries while successfully maintaining the false negative proportions, which improves the existing adaptive Lasso procedures. The implementation of the proposed procedure is straightforward and it is sufficiently flexible to accommodate large-scale problems where traditional procedures are impractical. To quantify the uncertainty of variable selection and control the family-wise error rate, a multiple sample-splitting based testing algorithm is developed. The practical utility of the proposed procedure are examined through simulation studies. The methods developed are then applied to a multiple myeloma data set.

Appendix: FWER-control procedure in Sect. 3.3

• (a) Randomly split the original data multiple times (say B). Specifically, for $b=1, \ldots , B$, split the data into two disjoint sets with sample size $n_1=\lfloor n/2 \rfloor$ and $n_2=n-\lfloor n/2 \rfloor$, respectively. Here $\lfloor n/2 \rfloor$ is defined as the largest integer not greater than n / 2.

• (b) For $b=1, \ldots , B$, select variables based on the first half of the data and denote the index set of selected variables by ${\widehat{{{\mathcal {S}}}}}^{(b)}.$

• (c) Based on the second half of the data, fit conventional Cox model and assign p-values. denoted by ${\tilde{P}}_{j}$ for $j=1,\dots ,p$, using variables selected from step (b). For variables not selected from the first half of the data, assign their p-values as 1.

• (d) Compute adjusted p-values to correct for the multiplicity of the testing problem

  $$\begin{aligned} {\tilde{P}}_{corrected,j} = min({\tilde{P}}_{j} |{\widehat{{{\mathcal {S}}}}}^{(b)}|,1), \end{aligned}$$

  where $|\widehat{{\mathcal {S}}}^{(b)}|$ is the cardinality, e.g., number of variables in $\widehat{{\mathcal {S}}}^{(b)}$.

• (e) To aggregate the adjusted p-values over multiple splitting (e.g., B values for each covariate), define

  $$\begin{aligned} Q_{j}(\gamma ) = min \{q_{\gamma }( \{ {\tilde{P}}^{[b]}_{corrected,j}/\gamma ; b=1,\ldots ,B \}),1 \} \end{aligned}$$

  where $\gamma \in (0,1)$ and $q_{\gamma }$ is the emperical $\gamma$-quantile function. Define the final p-values as

  $$\begin{aligned} P_{j} = min \{(1-log \gamma _{min}) \underset{\gamma \in (\gamma _{min},1)}{\inf } Q_{j}(\gamma ),1 \}, \end{aligned}$$

  where $\gamma _{min} \in (0,1)$ is a lower bound for $\gamma$, typically 0.05 (Meinshausen et al. 2009).