A tutorial on rank-based coefficient estimation for censored data in small- and large-scale problems

Abstract

The analysis of survival endpoints subject to right-censoring is an important research area in statistics, particularly among econometricians and biostatisticians. The two most popular semiparametric models are the proportional hazards model and the accelerated failure time (AFT) model. Rank-based estimation in the AFT model is computationally challenging due to optimization of a non-smooth loss function. Previous work has shown that rank-based estimators may be written as solutions to linear programming (LP) problems. However, the size of the LP problem is O(n ²+p) subject to n ² linear constraints, where n denotes sample size and p denotes the dimension of parameters. As n and/or p increases, the feasibility of such solution in practice becomes questionable. Among data mining and statistical learning enthusiasts, there is interest in extending ordinary regression coefficient estimators for low-dimensions into high-dimensional data mining tools through regularization. Applying this recipe to rank-based coefficient estimators leads to formidable optimization problems which may be avoided through smooth approximations to non-smooth functions. We review smooth approximations and quasi-Newton methods for rank-based estimation in AFT models. The computational cost of our method is substantially smaller than the corresponding LP problem and can be applied to small- or large-scale problems similarly. The algorithm described here allows one to couple rank-based estimation for censored data with virtually any regularization and is exemplified through four case studies.

Appendix

Operating characteristics of polynomial-smoothed Gehan estimator

In this section, we outline the large sample properties of the estimator $\widehat{{\boldsymbol {\beta }}}_{G,\varepsilon }$. Let the parameter β belong to a parameter space $\mathbb{B}$, a compact subset of ℜ^p and let f ₀(β) be a convex function for ${\boldsymbol {\beta }}\in\mathbb{B}$. The proof of Theorem 1 relies on the following two facts regarding the loss functions f _G (β) and f _G,ε (β).

Lemma 1

Under Conditions A1–A3 in Johnson and Strawderman (2009, p. 586),

$$\sup_{{\boldsymbol {\beta }}\in\mathbb{B}} \bigl \vert f_{G}({\boldsymbol {\beta }}) - f_0( {\boldsymbol {\beta }}) \bigr \vert \rightarrow0 \quad \mbox{\textit{almost surely}}. $$

Lemma 2

Under Conditions A1–A3 in Johnson and Strawderman (2009, p. 586),

$$\sup_{{\boldsymbol {\beta }}\in\mathbb{B}} \bigl \vert f_{G,\varepsilon }({\boldsymbol {\beta }}) - f_0( {\boldsymbol {\beta }}) \bigr \vert \rightarrow0 \quad \mbox{\textit{almost surely}}. $$

Lemma 1 is also Lemma 1 in Johnson and Strawderman (2009) under exactly the same conditions and stated without proof.

Outline proof of Lemma 2

By the triangle inequality, we have

(19)

By Lemma 1, the second term in (19) can be made arbitrarily small, uniformly for all ${\boldsymbol {\beta }}\in\mathbb{B}$, except on a set of probability measure zero. The first term in (19) is

Hence, the absolute difference between the Gehan loss and its smooth approximation can be made arbitrarily small, for every ${\boldsymbol {\beta }}\in\mathbb{B}$. The conclusion then follows. □

Proof of Theorem 1

Under Conditions A1–A3 of Johnson and Strawderman, f _G (β) and f _G,ε (β) converge uniformly to the convex function f ₀(β) by Lemmas 1 and 2, respectively. By Condition A4, f ₀(β) is strictly convex at its unique minimizer, β ₀. Thus, the minimizers of the random convex functions f _G,ε (β) and f _G (β) converge almost surely to β ₀. □

Asymptotic distribution

The polynomial-smoothed Gehan estimator bears a close similarity to Heller’s (2007) estimator and one expects the asymptotic distribution theory follows similarly. A straightforward calculation confirms that K _ε (z) in Ψ _G,ε (β) in (14) is a survivor function and k _ε (z)=(d/dz)K _ε (z) is symmetric about zero with finite second moment (that is, Heller’s 2007, Condition C3, p. 553). Define the asymptotic slope matrix A _ε (β) and asymptotic covariance B _ε (β),

Then, assuming the covariate matrix has finite second moment and the non-singularity of A _ε (β) in a neighborhood of the true value β ₀, one can show $\sqrt{n}(\widehat{{\boldsymbol {\beta }}}_{G,\varepsilon } - {\boldsymbol {\beta }}_{0})$ converges in distribution to a mean-zero normal random vector with asymptotic covariance

$$\bigl\{\mathbf {A}_{\varepsilon }({\boldsymbol {\beta }}_0)\bigr\}^{-1} \mathbf {B}_{\varepsilon }({\boldsymbol {\beta }}_0)\bigl\{\mathbf {A}_{\varepsilon }( {\boldsymbol {\beta }}_0)\bigr\}^{-1}, $$

(see Heller 2007, Appendix). As with Heller’s estimator, both A _ε (β) and B _ε (β) are directly estimable from the data, the latter derived from a theory of U-statistics.