Fast accelerated failure time modeling for case-cohort data

Abstract

Semiparametric accelerated failure time (AFT) models directly relate the expected failure times to covariates and are a useful alternative to models that work on the hazard function or the survival function. For case-cohort data, much less development has been done with AFT models. In addition to the missing covariates outside of the sub-cohort in controls, challenges from AFT model inferences with full cohort are retained. The regression parameter estimator is hard to compute because the most widely used rank-based estimating equations are not smooth. Further, its variance depends on the unspecified error distribution, and most methods rely on computationally intensive bootstrap to estimate it. We propose fast rank-based inference procedures for AFT models, applying recent methodological advances to the context of case-cohort data. Parameters are estimated with an induced smoothing approach that smooths the estimating functions and facilitates the numerical solution. Variance estimators are obtained through efficient resampling methods for nonsmooth estimating functions that avoids full blown bootstrap. Simulation studies suggest that the recommended procedure provides fast and valid inferences among several competing procedures. Application to a tumor study demonstrates the utility of the proposed method in routine data analysis.

Appendix: Analytical details

We give the analytical form of S _i (β)’s here. Define the general rank based weighted estimating function (Jin et al. 2003)

$$ U_n(\beta)=\sum_{i=1}^n \Delta_i \varphi_{n,i}(\beta) \biggl[ X_i- \frac{W^{(1)}_{n,i}(\beta)}{W^{(0)}_{n,i}(\beta)} \biggr], $$

where φ _n,i (β) is an nonnegative weight function and

$$ W^{(k)}_{n,i}(\beta)=\frac{1}{n}\sum _{j=1}^n X_j^k I \bigl[e_j(\beta) \geq e_i(\beta)\bigr], \quad k = 0,1. $$

Equation (1) can be obtained by setting $\varphi_{n,i}(\beta) = W^{(0)}_{n,i}(\beta)$. On the other hand, the general rank based weighted estimating function for case-cohort samples has the following form:

$$ U_n^c(\beta) = \sum_{i=1}^n \Delta_i\varphi_{n,i}(\beta) \biggl[X_i- \frac{\hat{W}^{(1)}_{n, i}(\beta)}{\hat{W}^{(0)}_{n, i}(\beta)} \biggr], $$

where

$$ \hat{W}^{(k)}_{n, i}(\beta)=\frac{1}{n} \sum _{j=1}^n h_j X_j^k I\bigl[e_j(\beta)\geq e_i(\beta)\bigr], \quad k = 0,1. $$

Similarly, Eq. (2) can be obtained by setting $\varphi_{n,i}(\beta) = \hat{W}^{(0)}_{n,i}(\beta)$.

With these settings, an explicit form of S _i (β ₀) is

where

N _i (β;t)=Δ _i I(e _i (β)≤t) and λ(u) is the common hazard function of ϵ _i .

The unknown quantities in S _i (β ₀) include β ₀, w ⁽⁰⁾, w ⁽¹⁾ and λ(t). With the explicit form of S _i (β ₀), $\hat{S}_{i}(\hat{\beta})$ is obtained by replacing these unknown quantities by their sample estimators.