Model diagnostics for the proportional hazards model with length-biased data

Abstract

Length-biased data are frequently encountered in prevalent cohort studies. Many statistical methods have been developed to estimate the covariate effects on the survival outcomes arising from such data while properly adjusting for length-biased sampling. Among them, regression methods based on the proportional hazards model have been widely adopted. However, little work has focused on checking the proportional hazards model assumptions with length-biased data, which is essential to ensure the validity of inference. In this article, we propose a statistical tool for testing the assumed functional form of covariates and the proportional hazards assumption graphically and analytically under the setting of length-biased sampling, through a general class of multiparameter stochastic processes. The finite sample performance is examined through simulation studies, and the proposed methods are illustrated with the data from a cohort study of dementia in Canada.

Appendices

Appendix A: regularity conditions

We assume the following regularity conditions for the large sample properties:

• (1) $(Y_i,A_i,\delta _i,\varvec{Z}_i)$ are independent and identically distributed for $i=1,\ldots ,n$.

• (2) The parameters $\varvec{\beta }_0$ belong to an interior of a known compact set.

• (3) The covariates $\varvec{Z}$ are bounded, and $\pmb {\alpha }=0$ almost surely if $\pmb {\alpha }^\top \varvec{Z}=0$ with probability one.

• (4) The differentiable baseline cumulative hazard function $\varLambda _0(\tau )<\infty$ where $\tau$ satisfies $\Pr (Y\ge \tau )>0$.

• (5) $\varGamma (\varvec{\beta })$ is positive definite.

• (6) $0<w_C(\tau )<\infty$ and $\int _0^\tau \left[ \left\{ \int _t^\tau S_C(u)\mathrm {d}u\right\} ^2/\left\{ S_C^2(t)S_V(t)\right\} \right] \mathrm {d}S_C(t)<\infty$ where $S_V(\cdot )$ is the survival function of the residual survival time.

Appendix B: Proof of Theorem 1

By applying the Taylor series expansions, we have

$$\begin{aligned}&n^{-1/2}\sum _{i=1}^n\int _0^t\widehat{E}_{\varvec{Z}}(\varvec{\beta },u,\varvec{z})\mathrm {d}N_i(u) \nonumber \\&\quad =n^{-1/2}\sum _{i=1}^n\int _0^t\widehat{E}_{\varvec{Z}} (\varvec{\beta }_0,u,\varvec{z})\mathrm {d}N_i(u)\nonumber \\&\qquad +n^{-1} \sum _{i=1}^n\frac{\partial }{\partial \varvec{\beta }}\int _0^t\widehat{E}_{\varvec{Z}}(\varvec{\beta }_0,u,\varvec{z})\mathrm {d}N_i(u)\sqrt{n}(\varvec{\beta }-\varvec{\beta }_0)+o_p(1), \nonumber \\&\quad =n^{-1/2}\sum _{i=1}^n\int _0^t\widehat{E}_{\varvec{Z}} (\varvec{\beta }_0,u,\varvec{z})\mathrm {d}N_i(u) \nonumber \\&\qquad +\,n^{-1}\sum _{i=1}^n\int _0^t\left[ \frac{\widehat{S}_{\varvec{Z}}^{(1)}(\varvec{\beta }_0,u,\varvec{z})}{\widehat{S}^{(0)}(\varvec{\beta }_0,u)}-\frac{\widehat{S}_{\varvec{Z}}^{(0)}(\varvec{\beta }_0,u,\varvec{z})\widehat{S}^{(1)}(\varvec{\beta }_0,u)}{\{\widehat{S}^{(0)}(\varvec{\beta }_0,u)\}^2}\right] \mathrm {d}N_i(u)\sqrt{n}(\varvec{\beta }-\varvec{\beta }_0)\nonumber \\&\qquad +o_p(1), \end{aligned}$$

(5)

and

$$\begin{aligned}&n^{-1/2}\sum _{i=1}^n\int _0^\tau \left\{ \varvec{Z}_i-\widehat{E}(\varvec{\beta },u)\right\} \mathrm {d}N_i(u) \nonumber \\&\quad =n^{-1/2}\sum _{i=1}^n\int _0^\tau \left\{ \varvec{Z}_i-\widehat{E}(\varvec{\beta }_0,u)\right\} \mathrm {d}N_i(u)+n^{-1} \nonumber \\&\quad \qquad \sum _{i=1}^n\frac{\partial }{\partial \varvec{\beta }}\int _0^\tau \left\{ \varvec{Z}_i-\widehat{E}(\varvec{\beta }_0,u)\right\} \mathrm {d}N_i(u) \sqrt{n}(\varvec{\beta }-\varvec{\beta }_0)+o_p(1) \nonumber \\&\quad =n^{-1/2}\sum _{i=1}^n\int _0^\tau \left\{ \varvec{Z}_i-\widehat{E}(\varvec{\beta }_0,u)\right\} \mathrm {d}N_i(u) \nonumber \\&\qquad -\,n^{-1}\sum _{i=1}^n\int _0^\tau \left[ \frac{\widehat{S}^{(2)}(\varvec{\beta }_0,u)}{\widehat{S}^{(0)}(\varvec{\beta }_0,u)}-\left\{ \frac{\widehat{S}^{(1)}(\varvec{\beta }_0,u)}{\widehat{S}^{(0)}(\varvec{\beta }_0,u)}\right\} ^2\right] \mathrm {d}N_i(u)\sqrt{n}(\varvec{\beta }-\varvec{\beta }_0)+o_p(1). \end{aligned}$$

(6)

The stochastic processes $\varvec{G}(t,\varvec{z})$ can be expressed in two terms, $\varvec{G}_1(t,\varvec{z})$ and $\varvec{G}_2(t,\varvec{z})$, as follows.

$$\begin{aligned}&\varvec{G}(t,\varvec{z}) \\&=\sum _{i=1}^n f(\varvec{Z}_i) I(\varvec{Z}_i\le \varvec{z})\widehat{M}_i(t) \\&=\sum _{i=1}^n f(\varvec{Z}_i)I(\varvec{Z}_i\le \varvec{z})N_i(t) \\&\qquad -f(\varvec{Z}_i)I(\varvec{Z}_i\le \varvec{z})\int _0^t \widehat{w}_C(u)R_i(u)\{\widehat{w}_C(Y_i)\}^{-1}\exp (\widehat{\varvec{\beta }}^\top \varvec{Z}_i)\mathrm {d}\widehat{\varLambda }_0(u) \\&=\sum _{i=1}^n f(\varvec{Z}_i)I(\varvec{Z}_i\le \varvec{z})N_i(t)-\int _0^t\frac{\widehat{S}_Z^{(0)}(\widehat{\varvec{\beta }},u,\varvec{z})}{\widehat{S}^{(0)}(\widehat{\varvec{\beta }},u)}\mathrm {d}N_i(u) \\&=\sum _{i=1}^n \int _0^t \left\{ f(\varvec{Z}_i)I(\varvec{Z}_i\le \varvec{z})- E_Z(\varvec{\beta }_0,u,\varvec{z})\right\} \mathrm {d}N_i(u) \\&\quad +\sum _{i=1}^n \int _0^t \left\{ E_Z(\varvec{\beta }_0,u,\varvec{z}) - \widehat{E}_Z(\widehat{\varvec{\beta }},u,\varvec{z})\right\} \mathrm {d}N_i(u) \\&=\varvec{G}_1(t,\varvec{z})+\varvec{G}_2(t,\varvec{z}) \end{aligned}$$

We exploit the Taylor expansion and empirical process approximation techniques. It is straightforward that the first term can be approximated by

$$\begin{aligned} n^{-1/2}\varvec{G}_1(t,\varvec{z})&=n^{-1/2}\sum _{i=1}^n \int _0^t \left\{ f(\varvec{Z}_i)I(\varvec{Z}_i\le \varvec{z})- E_Z(\varvec{\beta }_0,u,\varvec{z})\right\} \mathrm {d}N_i(u) \\&=n^{-1/2}\sum _{i=1}^n \int _0^t\left\{ f(\varvec{Z}_i)I(\varvec{Z}_i\le \varvec{z})-e_Z(\varvec{\beta }_0,u,\varvec{z})\right\} \mathrm {d}M_i(u)+o_p(1). \end{aligned}$$

Then, we re-express the second term based on equations (5) and (6).

$$\begin{aligned} n^{-1/2}\varvec{G}_2(t,\varvec{z})&=n^{-1/2}\sum _{i=1}^n \int _0^t E_Z(\varvec{\beta }_0,u,\varvec{z}) \mathrm {d}N_i(u) \\&\quad -n^{-1/2}\sum _{i=1}^n \int _0^t \widehat{E}_Z(\widehat{\varvec{\beta }},u,\varvec{z}) \mathrm {d}N_i(u) \\&=n^{-1/2}\sum _{i=1}^n\int _0^t \left\{ E_Z(\varvec{\beta }_0,u,\varvec{z}) - \widehat{E}_Z(\varvec{\beta }_0,u,\varvec{z})\right\} \mathrm {d}N_i(u) \\&\quad -\,\varGamma _Z(\varvec{\beta }_0,t,\varvec{z})\sqrt{n}(\widehat{\varvec{\beta }}-\varvec{\beta }_0) +o_p(1) \\&=n^{-1/2}\sum _{i=1}^n \int _0^t \left\{ E_Z(\varvec{\beta }_0,u,\varvec{z})-\widehat{E}_Z(\varvec{\beta }_0,u,\varvec{z})\right\} \mathrm {d}N_i(u) \\&\quad +\,\varGamma _Z(\varvec{\beta }_0,t,\varvec{z}) \{\varGamma (\varvec{\beta }_0)\}^{-1}n^{-1/2}\widehat{U}(\varvec{\beta }_0)+o_p(1) \end{aligned}$$

where

$$\begin{aligned} \varGamma _Z(\varvec{\beta },t,\varvec{z})=\text{ E }\left\{ \int _0^t\left[ \frac{{S}_{\varvec{Z}}^{(1)}(\varvec{\beta },u,\varvec{z})}{{S}^{(0)}(\varvec{\beta },u)}-\frac{{S}_{\varvec{Z}}^{(0)}(\varvec{\beta },u,\varvec{z}){S}^{(1)}(\varvec{\beta },u)}{\{{S}^{(0)}(\varvec{\beta },u)\}^2}\right] \mathrm {d}N_i(u)\right\} \end{aligned}$$

and

$$\begin{aligned} \varGamma (\varvec{\beta })=-\text{ E }\left\{ \int _0^\tau \left[ \frac{{S}^{(2)}(\varvec{\beta },u)}{{S}^{(0)}(\varvec{\beta },u)}-\left\{ \frac{{S}^{(1)}(\varvec{\beta },u)}{{S}^{(0)}(\varvec{\beta },u)}\right\} ^2\right] \mathrm {d}N_i(u)\right\} . \end{aligned}$$

The second equation can be derived by plugging in Eq. (5). The third equation naturally follows by replacing $\sqrt{n}(\widehat{\varvec{\beta }}-\varvec{\beta }_0)$ with Eq. (6) after some algebra (Qin and Shen 2010). Note that the leading term in the last equation can be rewritten as

$$\begin{aligned}&n^{-1/2}\sum _{i=1}^n \int _0^t \left\{ E_Z(\varvec{\beta }_0,u,\varvec{z})-\widehat{E}_Z(\varvec{\beta }_0,u,\varvec{z})\right\} \mathrm {d}N_i(u) \\&=n^{-1/2}\sum _{i=1}^n \int _0^t \left\{ \frac{S_Z^{(0)}(\varvec{\beta }_0,u)-\widehat{S}_Z^{(0)}(\varvec{\beta }_0,u)}{S^{(0)}(\varvec{\beta }_0,u)}\right\} \mathrm {d}N_i(u)+o_p(1) \\&=n^{-1/2}\sum _{i=1}^n\int _0^t \frac{\sum _{k=1}^nf(\varvec{Z}_k)I(\varvec{Z}_k\le \varvec{z})R_k(u)\exp (\varvec{\beta }_0^\top \varvec{Z}_k)\left\{ \frac{1}{w_C(Y_k)}-\frac{1}{\widehat{w}_C(Y_k)}\right\} }{\sum _{k=1}^nR_k(u)\{w_C(Y_k)\}^{-1}\exp (\varvec{\beta }_0^\top \varvec{Z}_k)}\,\mathrm {d}N_i(u)\!+\!o_p(1)\\&=n^{-1/2}\sum _{i=1}^n\int _0^t \frac{\sum _{k=1}^n f(\varvec{Z}_k)I(\varvec{Z}_k\le \varvec{z})w_C(u) R_k(u) \exp (\varvec{\beta }_0^\top \varvec{Z}_k)}{nS^{(0)}(\varvec{\beta }_0,u)} \frac{\left\{ \widehat{w}_C(Y_k)\!-\!w_C(Y_k)\right\} }{\{\widehat{w}_C(Y_k)\}^{2}}\mathrm {d}N_i(u)\! +\!o_p(1) \\&=n^{-1/2}\sum _{i=1}^n \int _0^t H(\varvec{\beta }_0,u)\frac{\mathrm {d}M_{C_i}(u)}{\pi (u)}+o_p(1), \end{aligned}$$

where

$$\begin{aligned} H(\varvec{\beta },t)&=\lim _{n\rightarrow \infty } \frac{1}{n^2}\sum _{i=1}^n\sum _{k=1}^n \\&\quad \frac{f(\varvec{Z}_k)I(\varvec{Z}_k\le \varvec{z})w_C(Y_i) R_k(Y_i) \exp (\varvec{\beta }^\top \varvec{Z}_k)\{{w}_C(Y_k)\}^{-2}h_k(t)}{S^{(0)}(\varvec{\beta },Y_i)} \\ M_{C_i}(t)&=I(V_i\le t,\delta _i=0)-\int _0^tI(V_i\ge u)\mathrm {d}\varLambda _C(u) \\ h_k(t)&=I(Y_k\ge t)\int _t^{Y_k}S_C(u)\mathrm {d}u \\ \pi (t)&=S_C(t)S_V(t), \end{aligned}$$

in which $\varLambda _C(t)$ is the cumulative hazard function of the residual censoring time and $S_V(t)$ is the survival function of the residual survival time. The last equation can be obtained by expressing $\{\widehat{w}_C(y)-w_C(y)\}$ as an i.i.d. sum of martingales (Pepe and Fleming 1991). Finally, the general class of stochastic processes $G(t,\varvec{z})$ can be asymptotically represented by

$$\begin{aligned} n^{-1/2}\varvec{G}(t,\varvec{z})&=n^{-1/2}\sum _{i=1}^n \int _0^t\left\{ f(\varvec{Z}_i)I(\varvec{Z}_i\le \varvec{z})-e_Z(\varvec{\beta }_0,u)\right\} \mathrm {d}M_i(u) \nonumber \\&+\,n^{-1/2}\sum _{i=1}^n \int _0^t H(\varvec{\beta }_0,u)\frac{\mathrm {d}M_{C_i}(u)}{\pi (u)} \nonumber \\&+\,\varGamma _Z(\varvec{\beta }_0,t,\varvec{z}) \{\varGamma (\varvec{\beta }_0)\}^{-1}n^{-1/2} \sum _{i=1}^n\int _0^\infty \left\{ \varvec{Z}_i-e(\varvec{\beta }_0,u)\right\} \mathrm {d}M_i(u)+o_p(1) \end{aligned}$$

(7)

$$\begin{aligned}&=n^{-1/2}\sum _{i=1}^n\varvec{G}_i^*(t,\varvec{z})+o_p(1). \end{aligned}$$

(8)

Under the regularity conditions, for any given $\varvec{z}$, $\varvec{G}_i^*(t,\varvec{z})$ is a mean zero process bounded on $[0,\tau ]$. This process can be classified as a Donsker class (Kosorok 2008). Thus, as $n\rightarrow \infty$, the summation of $\varvec{G}_i^*(t,\varvec{z})$ in (8) converges weakly to a mean zero Gaussian process, for which the asymptotic covariance function is $\text{ E }\left\{ \varvec{G}_i^*(t_1,\varvec{z}_1)\varvec{G}_i^*(t_2,\varvec{z}_2)^\top \right\}$.

Appendix C: Proof of Theorem 2

We note that $\varGamma _Z(\varvec{\beta }_0,t,\varvec{z}) \{\varGamma (\varvec{\beta }_0)\}^{-1}$ in the third term of (7) converges in probability to a non-random function. Conditional on the observed data, the process (7) is a linear combination of independent normally distributed processes with mean zero. Thus, given that $\widehat{\varvec{\beta }}$ is a consistent estimator for $\varvec{\beta }_0$, we can show that $n^{-1}\sum _{i=1}^n\widehat{\varvec{G}}_i^*(t_1,\varvec{z}_1)\widehat{\varvec{G}}_i^*(t_2,\varvec{z}_2)^\top$ converges in probability to the asymptotic covariance function $\text{ E }\left\{ \varvec{G}_i^*(t_1,\varvec{z}_1)\varvec{G}_i^*(t_2,\varvec{z}_2)^\top \right\}$ as $n\rightarrow \infty$. By applying the multiplier central limit theorem (Kosorok 2008), it follows that conditional on the observed data and $\varvec{z}$, $n^{-1/2}\widetilde{\varvec{G}}_m(t,\varvec{z})$ and $n^{-1/2}\sum _{i=1}^n{\varvec{G}}_i^*(t,\varvec{z})$ converge to the same mean zero Gaussian process.