Quantile regression of right-censored length-biased data using the Buckley–James-type method

Abstract

Length-biased data are encountered frequently due to prevalent cohort sampling in follow-up studies. Quantile regression provides great flexibility for assessing covariate effects on survival time, and is a useful alternative to Cox’s proportional hazards model and the accelerated failure time (AFT) model for survival analysis. In this paper, we develop a Buckley–James-type estimator for right-censored length-biased data under a quantile regression model. The problem of informative right-censoring of length-biased data induced by prevalent cohort sampling must be handled. Following on from the generalization of the Buckley–James-type estimator under the AFT model proposed by Ning et al. (Biometrics 67:1369–1378, 2011), we propose a Buckley–James-type estimating equation for regression coefficients in the quantile regression model and develop an iterative algorithm to obtain the estimates. The resulting estimator is consistent and asymptotically normal. We evaluate the performance of the proposed estimator on finite samples using extensive simulation studies. Analysis of real data is presented to illustrate our proposed methodology.

Appendix

To simplify the proof and rapidly utilize the arguments in Ning et al. (2011), we consider the case in which $\varepsilon _\tau$ is independent of $\mathbf{x}$, i.e. $q_\tau ( \cdot |\mathbf{x}) = q_\tau ( \cdot )$, and use an unconditional estimator $\hat{F}_{\tau 0} ( \cdot )$ (Asgharian et al. 2002) to estimate $F_{\tau 0} ( \cdot )$. However, the arguments here can be generalized to the case in which $\varepsilon _\tau$ depends on $\mathbf{x}$. It is sufficient to show that ${\hat{\varvec{\beta }}}_\tau ^{[1]}$ is consistent and asymptotically normal, and thus ${\hat{\varvec{\beta }}}_\tau ^{[m]}$ is consistent and asymptotically normal for $m = 2, \ldots ,M$ by the method of induction. For notional simplicity, we omit the “$\tau$” such that ${\varvec{\beta }}_0$, ${\hat{\varvec{\beta }}}^{[0]}$ and ${\hat{\varvec{\beta }}}^{[1]}$ represent ${\varvec{\beta }}_0 (\tau )$, ${\hat{\varvec{\beta }}}_\tau ^{[0]}$ and ${\hat{\varvec{\beta }}}_\tau ^{[1]}$, respectively. The proofs of consistency and asymptotic normality are similar to those in the work by Ning et al. (2011). To establish asymptotic properties, the following regularity conditions are required:

• C1. The matrix $\mathbf{D} = - E\{\mathbf{XX}' \delta q_{\tau } (0)/\mu _{\tau }\}$ is positive definite, where $\mu _\tau = \int {q_\tau (\log t)dt}$.

• C2. $P(V < C) > 0$.

• C3. $G_{\tau 0} ( \cdot )$ is continuous and differentiable function over $(0,\kappa )$ and $\kappa < \infty$, where $G_{\tau 0} (t) = \int _0^t {g_{\tau 0} (s)} ds$ is the cumulative function of $T_{\tau 0}$ and $\kappa = \inf \{ t:\int _0^t {g_{\tau 0} (s)} ds = 1\}$.

• C4. $\{ {\textstyle {\frac{2\kappa }{\int _0^\kappa {S_{C_{\tau 0} } (u)} du}}} - \textstyle {\frac{1}{{S_{C_{\tau 0} } (0)}}}\} \{ 1 - S_{C_{\tau 0} } (0)\} < 1$, where $S_{C_{\tau 0} } ( \cdot )$ is the survival function of $C_{\tau 0}$.

• C5. There exists a constant $\kappa _0 > 0$ such that $F_{\tau 0} (y) = 0$, $\forall y < \kappa _0$.

• C6. $\int _0^\kappa f_{\tau 0}^2 (t)/\{1 - F_{\tau 0} (t)\} dt < \infty$.

Condition C1 is to ensure that the difference between ${\hat{\varvec{\beta }}}^{[1]}$ and ${\varvec{\beta }}_0$ is asymptotically linear in ${\varvec{\beta }}_0$. Conditions C2 to C5 ensure uniform consistency and asymptotic normality of the unconditional estimator $\hat{F}_{\tau 0} (t)$ for $0 < t \le \kappa$ and the properties of $\hat{\mu }_\tau$ (Asgharian et al. 2002; Asgharian and Wolfson 2005). Condition C6 ensures that the various expectations are finite.

Proof of consistency of ${\hat{\varvec{\beta }}}^{[1]}$. First, consider the equality

$$\begin{aligned}&\mathbf{U}({\hat{\varvec{\beta }}}^{[1]} ;{\hat{\varvec{\beta }}}^{[0]}) - \mathbf{U}({\hat{\varvec{\beta }}}^{[0]};{\hat{\varvec{\beta }}}^{[0]} )\nonumber \\&\quad = \sum \limits _{i = 1}^n {\mathbf{x}_i \delta _i \left\{ {\frac{{I(\log y_i \le \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[1]} ) - I(\log y_i \le \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} )}}{{y_i e^{ - \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} } }}} \right\} }\nonumber \\&\quad = \sum \limits _{i = 1}^n {E\!\left[ \!{\mathbf{X}_i \delta _i \left\{ {\frac{{I(T_i e^{ - \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} } \le e^{\mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[1]} - \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} } ) - I(T_i e^{ - \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} } \le 1)}}{{T_i e^{ - \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} } }}} \!\right\} } \!\right] } \!+\! o_p (\sqrt{n} )\nonumber \\&\quad = \sum \limits _{i = 1}^n {E\left[ {\mathbf{X}_i \delta _i \left\{ {\int \limits _0^{e^{\mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[1]} - \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} } } {\frac{1}{t}\frac{{q_\tau (\log t)}}{{\mu _\tau }}dt} - \int \limits _0^1 {\frac{1}{t}\frac{{q_\tau (\log t)}}{{\mu _\tau }}dt} } \right\} } \right] } + o_p (\sqrt{n} )\nonumber \\&\quad = \sum \limits _{i = 1}^{n} {E\left[ {\mathbf{X}_i \delta _i \left\{ {\int \limits _{ - \infty }^{\mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[1]} - \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} } {\frac{{q_\tau (s)}}{{\mu _{\tau }}}ds} - \int \limits _{ - \infty }^0 {\frac{{q_\tau (s)}}{{\mu _{\tau }}}ds}} \right\} } \right] } + o_p (\sqrt{n} )\nonumber \\&\quad = nE\{\mathbf{XX}'\delta q_{\tau } (0)/\mu {_\tau }\} ({\hat{\varvec{\beta }}}^{[1]} - {\hat{\varvec{\beta }}}^{[0]} ) + o_p (\sqrt{n} ), \end{aligned}$$

(4)

due to using the condition of the consistent initial estimator ${\hat{\varvec{\beta }}}^{[0]}$ and applying the Taylor expansion. Similarly to the proof of Ning et al. (2011) in their Appendix, we can show that $n^{-1/2} \mathbf{U}({\varvec{\beta }}_0 ;{\varvec{\beta }}_0 ) - n^{-1/2} \mathbf{U} ({\hat{\varvec{\beta }}}^{[0]} ;{\hat{\varvec{\beta }}}^{[0]} )$ is asymptotically linear in ${\hat{\varvec{\beta }}}^{[0]}$

$$\begin{aligned}&\frac{1}{{\sqrt{n} }}\mathbf{U}({\varvec{\beta }}_0 ;{\varvec{\beta }}_0 ) - \frac{1}{{\sqrt{n} }}\mathbf{U}({\hat{\varvec{\beta }}}^{[0]} ;{\hat{\varvec{\beta }}}^{[0]} )\\&\quad = \frac{1}{{\sqrt{n} }}\sum \limits _{i = 1}^n {\mathbf{x}_i \delta _i \left\{ {\frac{{I(\log y_i \le \mathbf{x}_{i}^{\prime } {\varvec{\beta }}_0 ) - \tau }}{{y_i e^{ - \mathbf{x}_{i}^{\prime } {\varvec{\beta }}_0 } }} - \frac{{I(\log y_i \le \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} ) - \tau }}{{y_i e^{ - \mathbf{x}_{i}^{\prime } {\hat{\varvec{\beta }}}^{[0]} } }}} \right\} }\\&\quad \quad + \frac{1}{{\sqrt{n} }}\sum \limits _{i = 1}^n \mathbf{x}_i (1 - \delta _i )\left\{ {\frac{{\int _{y_{\tau 0,i} }^\infty {t^{ - 1} \{ I(t \le 1) - \tau \} } d\hat{F}_{\tau 0} (t;{\varvec{\beta }}_0 )}}{{1 - \hat{F}_{\tau 0} (y_{\tau 0,i} ;{\varvec{\beta }}_0 )}}}\right. \\&\quad \quad \left. - \frac{{\int _{\hat{y}_{\tau 0,i}^{[0]} }^\infty {t^{ - 1} \{ I(t \le 1) - \tau \} } d\hat{F}_{\tau 0} (t;{\hat{\varvec{\beta }}}^{[0]} )}}{{1 - \hat{F}_{\tau 0} (\hat{y}_{\tau 0,i}^{[0]} ;{\hat{\varvec{\beta }}}^{[0]} )}} \right\} \\&\quad = - \mathbf{B}\sqrt{n} ( {{\hat{\varvec{\beta }}}^{[0]} - {\varvec{\beta }}_0 } ) + o_p (1), \end{aligned}$$

where $\mathbf{B}$ is a matrix, in which the form is similar to Ning et al. (2011) in the Web Appendices. Through the definition of ${\hat{\varvec{\beta }}}^{[1]}$, $n^{-1/2} \mathbf{U}({\hat{\varvec{\beta }}}^{[1]} ;{\hat{\varvec{\beta }}}^{[0]} ) = 0$, the above linear approximation, combined with (4), implies that

$$\begin{aligned} {\hat{\varvec{\beta }}}^{[1]} - {\varvec{\beta }}_0 = (\mathbf{I}_p + \mathbf{D}^{ - 1} \mathbf{B})({\hat{\varvec{\beta }}}^{[0]} - {\varvec{\beta }}_0 ) + n^{ - 1} \mathbf{D}^{ - 1} \mathbf{U}({\varvec{\beta }}_0 ;{\varvec{\beta }}_0 ) + o_p (n^{-1/2}), \end{aligned}$$

(5)

where $\mathbf{I}_p$ is an identity matrix. Hence, ${\hat{\varvec{\beta }}}^{[1]}$ is consistent.

Proof of weak convergence of ${\hat{\varvec{\beta }}}^{[1]}$. Following the argument in Ning et al. (2011), the estimating equation with true regression coefficients can be represented by independent and identically distributed forms derived by the asymptotic properties of Vardi’s estimator,

$$\begin{aligned} U({\varvec{\beta }}_0 ,{\varvec{\beta }}_0 )&= \sum \limits _{i = 1}^n \mathbf{x}_i \left\{ \delta _i \frac{{I(\log y_i \le \mathbf{x}_{i}^{\prime } {\varvec{\beta }}_0 ) - \tau }}{{y_i e^{ - \mathbf{x}_{i}^{\prime } {\varvec{\beta }}_0 } }}+ (1 - \delta _i )\frac{{\int _{y_{\tau 0,i} }^\infty {t^{ - 1} \{ I(t \le 1) - \tau \} dF_{\tau 0} (t)} }}{{F_{\tau 0} (y_{\tau 0,i} )}} \right\} \nonumber \\&+ \sum \limits _{i = 1}^n {\left\{ {\int {\mathbf{L}(t)t^{ - 1} \{ I(t \le 1) - \tau \} d\mathcal G(V_{{\varvec{\beta }}_0 ,i} )(t)} + \mathbf{L}_i } \right\} }+ o_p(1), \end{aligned}$$

(6)

where

$$\begin{aligned} \mathbf{L}(t)&= E\left\{ {\frac{{\mathbf{X}(1 - \delta )I(Y_{\tau 0} < t)}}{{1 - F_{\tau 0} (Y_{\tau 0} )}}} \right\} ,\\ \mathbf{L}_i&= E\left\{ {\frac{{\mathbf{X}(1 - \delta )\int _{Y_{\tau 0} }^\kappa {t^{ - 1} \{ I(t \le 1) - \tau \} dF_{\tau 0} (t)\mathcal G(V_{{\varvec{\beta }}_0 ,i} )(Y_{\tau 0} )} }}{{\{ 1 - F_{\tau 0} (Y_{\tau 0} )\} ^2 }}} \right\} ,\\ V_{{\varvec{\beta }}_0 ,i} (t)&= p^{1/2} \{ I(y_{\tau 0,i} \le t,\delta _i = 1) - G_\tau ^* (t)\} \!+\! \left( {\frac{p}{{1\! -\! p}}} \right) \{ G_\tau ^* (t) - G_{\tau 0} (t)\} (\delta _i - p)\\&\quad + \,(1 - p)^{1/2} h_\tau (t)\int \limits _{0 < z \le t} {\{ I(y_{\tau 0,i} \le z,\delta _i = 0) - F_\tau ^* (z)\} d\frac{1}{{h_\tau (z)}}}, \end{aligned}$$

$G_\tau ^* (t) = P(Y_{\tau 0} \le t|\delta = 1)$, $F_\tau ^* (t) = P(Y_{\tau 0} \le t|\delta = 0)$, $h_\tau (t) = \int _t^\infty {z^{ - 1} dG_{\tau 0} (z)}$, $p = P(\delta = 1)$, $\mathcal G$ and $\mathcal F$ are linear operators defined by

$$\begin{aligned} \mathcal G(\nu )(t)&= \frac{{\int _0^\infty {s^{ - 1} dG_{\tau 0} (s)} \int _0^t {s^{ - 1} d \mathcal F^{ - 1} (\nu )(s)} \! -\! \int _0^t {s^{ - 1} dG_{\tau 0} (s)} \int _0^\infty {s^{ - 1} d \mathcal F^{ - 1} (\nu )(s)} }}{{\left\{ {\int _0^\infty {s^{ - 1} dG_{\tau 0} (s)} } \right\} ^2 }}\!,\\ \mathcal F(u)(t)&= p\int \limits _{0 < x \le t} {\frac{{g_\tau ^* (x)}}{{g_{\tau 0} (x)}}du(x)}\\&+\, (1 - p) \int \limits _{0 < y \le t} {y\left( \,\, {\int \limits _{y \le z} {\frac{{u(z)}}{{z^2 }}dz} } \right) d\left\{ {\left( {\frac{{h_\tau (t)}}{{h_\tau (y)}} - 1} \right) \frac{{f_\tau ^* (y)}}{{h_\tau (y)}}} \right\} }, \end{aligned}$$

and $g_\tau ^* (x)$ and $f_\tau ^* (x)$ are the conditional density functions of $G_\tau ^* (x)$ and $F_\tau ^* (x)$, respectively. Hence, equality (5) combined with the asymptotic independent and identically distributed form of an initial estimator ${\hat{\varvec{\beta }}}^{[0]}$ and representation (6), implies that $n^{1/2} ({\hat{\varvec{\beta }}}^{[1]} - {\varvec{\beta }}_0 )$ converges in distribution to a zero-mean Gaussian process.