Accelerated failure time model for data from outcome-dependent sampling

Abstract

Outcome-dependent sampling designs such as the case–control or case–cohort design are widely used in epidemiological studies for their outstanding cost-effectiveness. In this article, we propose and develop a smoothed weighted Gehan estimating equation approach for inference in an accelerated failure time model under a general failure time outcome-dependent sampling scheme. The proposed estimating equation is continuously differentiable and can be solved by the standard numerical methods. In addition to developing asymptotic properties of the proposed estimator, we also propose and investigate a new optimal power-based subsamples allocation criteria in the proposed design by maximizing the power function of a significant test. Simulation results show that the proposed estimator is more efficient than other existing competing estimators and the optimal power-based subsamples allocation will provide an ODS design that yield improved power for the test of exposure effect. We illustrate the proposed method with a data set from the Norwegian Mother and Child Cohort Study to evaluate the relationship between exposure to perfluoroalkyl substances and women’s subfecundity.

Appendix

In order to establish the asymptotic properties of the proposed estimator, we need the following two lemmas.

Lemma 1

Under Conditions $(C1){-}(C4)$ and (C6), $m^{-1/2}{\tilde{U}}_{m,G}(\theta _0)$ is asymptotically normal with zero-mean and covariance matrix $\Sigma _{F}(\theta _0)+\Sigma _{O}(\theta _0)$.

Proof

Using martingales expression $M_i(\theta _0;t), i=1,\ldots , m$, we have

$$\begin{aligned} m^{-1/2}{\tilde{U}}_{m,G}(\theta _0)= & {} m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty } {\tilde{\psi }}(\theta _0;t)W_i[X_i-{\tilde{X}}(\theta _0;t)]dN_i(\theta _0;t)\nonumber \\= & {} m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }{\tilde{\psi }}(\theta _0;t)W_i {[}X_i-{\tilde{X}}(\theta _0;t)]dM_i(\theta _0;t)\nonumber \\&+\,m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }{\tilde{\psi }}(\theta _0;t)W_i {[}X_i-{\tilde{X}}(\theta _0;t)]d\Lambda _i(\theta _0;t). \end{aligned}$$

(7.1)

Obviously, the second term of (7.1) is equal to zero. Therefore, the term $m^{-1/2}{\tilde{U}}_{m,G}(\theta _0)$ can be written as

$$\begin{aligned} m^{-1/2}{\tilde{U}}_{m,G}(\theta _0)= & {} m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }{\tilde{\psi }}(\theta _0;t)W_i [X_i-{\tilde{X}}(\theta _0;t)]dM_i(\theta _0;t)\nonumber \\= & {} m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }s^{(0)}(\theta _0;t)W_i [X_i-e_X(\theta _0;t)]dM_i(\theta _0;t)\nonumber \\&+\,m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }s^{(0)}(\theta _0;t) [e_X(\theta _0;t)-{\tilde{X}}(\theta _0;t)]d\left[ W_iM_i(\theta _0;t)\right] \nonumber \\&+\, o_p(1). \end{aligned}$$

(7.2)

Next, we will show the second term of (7.2) is asymptotically negligible, which means

$$\begin{aligned} \int _{-\infty }^{\infty }s^{(0)}(\theta _0;t) {[}e_X(\theta _0;t)-{\tilde{X}}(\theta _0;t)]d\left[ \frac{1}{\sqrt{m}}\sum \nolimits _{i=1}^{m}W_iM_i(\theta _0;t) \right] =o_p(1). \end{aligned}$$

(7.3)

For each i, $W_iM_i(\theta _0;t)$ is a zero-mean process, which can be expressed as a sum of two monotone processes on the interval $[-Con_M,Con_M]$. Due to Conditions (C1) and (C2) and the follow-up time of the studies being bounded, the integrable interval $(-\infty ,+\infty )$ in formula (7.1) should be an interval of $[-Con_M,Con_M]$, which is a compact set in real space ${\mathcal {R}}$ with $Con_M$ being a positive constant and similar to the condition A of Tsiates (1990). Hence, the term $m^{-1/2}\sum _{i=1}^{m} W_iM_i(\theta _0;t)$ converges weakly to a tight Gaussian process with continuous sample paths on $[-Con_M,Con_M]$ by Example 2.11.16 of van der Vaart and Wellner (1996). We assume $X_i \ge 0$, otherwise, we decompose each $X_i(\cdot )$ into its positive and negative parts. Because ${\tilde{X}}(\theta _0;t)$ is a product of two monotone processes, which converges uniformly in probability to $e_X(\theta _0;t)$ on a compact set $[-Con_M,Con_M]$ in ${\mathcal {R}}$. Using Lemma A.1 of Kulich and Lin (2000), (7.3) holds. Hence,

$$\begin{aligned} m^{-1/2}{\tilde{U}}_{m,G}(\theta _0)=m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }s^{(0)}(\theta _0;t)W_i {[}X_i-e_X(\theta _0;t)]dM_i(\theta _0;t)+o_p(1).\nonumber \\ \end{aligned}$$

(7.4)

The first term of the right-side of (7.4) can be written as:

$$\begin{aligned}&m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }s^{(0)}(\theta _0;t) [X_i-e_{X}(\theta _0;t)]dM_i(\theta _0;t)\nonumber \\&\quad +m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }s^{(0)}(\theta _0;t)(1-\delta _i)(\xi _i/(\rho _0\rho _V)-1) [X_i-e_{X}(\theta _0;t)]dM_i(\theta _0;t)\nonumber \\&\quad +m^{-1/2}\sum \limits _{i=1}^{m}\int _{-\infty }^{\infty }s^{(0)}(\theta _0;t) \delta _i(1-\zeta _i)(\xi _i/(\rho _0\rho _V)-1) [X_i-e_{X}(\theta _0;t)]dM_i(\theta _0;t)\nonumber \\&\quad +m^{-1/2}\sum \limits _{i=1}^{m}\sum \limits _{k=\{1,3\}}\int _{-\infty }^{\infty }s^{(0)}(\theta _0;t) \delta _i(1-\xi _i)\zeta _{i,k}\left( \frac{\eta _{i,k}\pi _k(1-\rho _0\rho _V)}{\rho _k\rho _V}-1\right) \nonumber \\&\quad \times [X_i-e_{X}(\theta _0;t)]dM_i(\theta _0;t). \end{aligned}$$

(7.5)

In order to simplify the expression, we define $H_i(\theta _0)=\int _{-\infty }^{\infty }s^{(0)}(\theta _0;t) [X_i-e_{X}(\theta _0;t)]dM_i(\theta _0;t)$ and the formula (7.5) can be written as following

$$\begin{aligned}&m^{-1/2}\sum \limits _{i=1}^{m}H_i(\theta _0)+ m^{-1/2}\sum \limits _{i=1}^{m}(1-\delta _i)(\frac{\xi _i}{\rho _0\rho _V}-1)H_i(\theta _0)\nonumber \\&\quad +m^{-1/2}\sum \limits _{i=1}^{m}\delta _i(1-\zeta _i)(\frac{\xi _i}{\rho _0\rho _V}-1)H_i(\theta _0)\nonumber \\&\quad +m^{-1/2}\sum \limits _{i=1}^{m}\sum \limits _{k=\{1,3\}} \delta _i(1-\xi _i)\zeta _{i,k}(\frac{\eta _{i,k}\pi _k(1-\rho _0\rho _V)}{\rho _k\rho _V}-1)H_i(\theta _0). \end{aligned}$$

(7.6)

The five terms on the right-hand side of (7.6) have mean zero. Because $E[\xi /(\rho _0\rho _V)]=1$, the covariance matrix between the first term and the second term is

$$\begin{aligned} E\left[ (1-\delta )(\frac{\xi }{\rho _0\rho _V}-1)H(\theta _0)^{\otimes 2}\right]= & {} E\left\{ E\left[ (1-\delta )(\frac{\xi }{\rho _0\rho _V}-1)H(\theta _0)^{\otimes 2} |X,\delta ,T\right] \right\} \\= & {} E\left\{ (1-\delta )H(\theta _0)^{\otimes 2}E\left[ (\frac{\xi }{\rho _0\rho _V}-1)|X,\delta ,T\right] \right\} \\= & {} E\left\{ (1-\delta )H(\theta _0)^{\otimes 2}E\left[ (\frac{\xi }{\rho _0\rho _V}-1)\right] \right\} \\= & {} 0. \end{aligned}$$

By similar arguments, we can obtain the five terms on the right hand side of (7.6) are uncorrelated with each other. Besides, each term is a sum of independent and identically distributed zero-mean random vectors. Using a slight extension of H$\acute{a}$jek’s (1960) central limit theorem, $m^{-1/2}{\tilde{U}}_{m,G}(\theta _{0})$ can be shown to converge in distribution to a zero-mean normal vector with covariance matrix being $\Sigma _{F}(\theta _0)+\Sigma _{O}(\theta _0)$, where $\Sigma _F(\theta _0)=E[H_1(\theta _0)^{\otimes 2}]$, $\Sigma _O(\theta _0)=\frac{1-\rho _0\rho _V}{\rho _0\rho _V}E[(1-\delta _1)H_1(\theta _0)^{\otimes 2}]+ \frac{1-\rho _0\rho _V}{\rho _0\rho _V}E[\delta _1(1-\zeta _1)H_1(\theta _0)^{\otimes 2}]+ \sum \limits _{k=\{1,3\}}\frac{(1-\rho _0\rho _V)(\pi _k(1-\rho _0\rho _V)-\rho _k\rho _V)}{\rho _k\rho _V} E[\delta _1\zeta _{1,k}H_1(\theta _0)^{\otimes 2}]$, with $a^{\otimes 2}=aa^{'}$ for a vector a. Therefore, Lemma 1 holds. $\square$

Lemma 2

Under Conditions $(C1){-}(C4)$, the weighted Gehan estimating function and the smoothed weighted Gehan estimating function are asymptotically equivalent:

$$\begin{aligned} m^{-1/2}{\tilde{U}}_{m,G}(\theta _0)=m^{-1/2}{\bar{U}}_{m,G}(\theta _0)+o_p(1). \end{aligned}$$

Proof

Due to the induced smoothness method, we have

$$\begin{aligned} m^{-1/2}\left( {\tilde{U}}_{m,G}(\theta _0)-{\bar{U}}_{m,G}(\theta _0)\right)= & {} \frac{1}{m^{3/2}}\sum \limits _{i=1}^{m}\sum \limits _{j=1}^{m}\delta _iW_iW_j(X_i-X_j)\\&\times \left[ I(e_j(\theta _0)-e_i(\theta _0)\ge 0)-\Phi (\frac{e_j(\theta _0) -e_i(\theta _0)}{r_{ij}})\right] . \end{aligned}$$

Due to the inequality $I(e_j(\theta _0)-e_i(\theta _0)\ge 0)-\Phi (\frac{e_j(\theta _0) -e_i(\theta _0)}{r_{ij}})\le \Phi (-|\frac{e_j(\theta _0) -e_i(\theta _0)}{r_{ij}}|)$, we can obtain

$$\begin{aligned} \Vert m^{-1/2}({\tilde{U}}_{m,G}(\theta _0)-{\bar{U}}_{m,G}(\theta _0))\Vert\le & {} \left\| \frac{1}{m^2}\sum \limits _{i=1}^{m}\sum \limits _{j=1}^{m}\frac{\delta _iW_iW_j (X_{i}-X_{j})\sqrt{m}r_{ij}}{|e_j(\theta _0) -e_i(\theta _0)|}\right\| \\&\times \left| \frac{e_j(\theta _0) -e_i(\theta _0)}{r_{ij}}\right| \Phi \left( -\left| \frac{e_j(\theta _0) -e_i(\theta _0)}{r_{ij}}\right| \right) . \end{aligned}$$

Because of $\Phi (-x)\le (\sqrt{2\pi }x)^{-1}\exp \{-x^2/2\}$, we can obtain $\lim \nolimits _{x\rightarrow +\infty }x\Phi (-x)=0$. Due to the fact $r_{ij}=\sqrt{\frac{(X_j-X_i)^{'}(X_j-X_i)}{m}}$, the term $|\frac{e_j(\theta _0) -e_i(\theta _0)}{r_{ij}}|\Phi (-|\frac{e_j(\theta _0) -e_i(\theta _0)}{r_{ij}}|)=|\frac{\sqrt{m}[e_j(\theta _0) -e_i(\theta _0)]}{\sqrt{(X_j-X_i)^{'}(X_j-X_i)}}| \Phi (-|\frac{\sqrt{m}[e_j(\theta _0) -e_i(\theta _0)]}{\sqrt{(X_j-X_i)^{'}(X_j-X_i)}}|)$ goes to zero as m goes to infinity. Therefore, Lemma 2 holds by applying the strong law of large number (Pollard 1990). $\square$

Proof of Theorem 1:

Due to the fact that ${\tilde{U}}_{m,G}(\theta )$ is the gradient of the convex objective function

$$\begin{aligned} L_m(\theta )=m^{-1}\sum \limits _{i=1}^m \sum \limits _{j=1}^m\delta _iW_iW_j(e_j(\theta )-e_i(\theta ))I(e_j(\theta )-e_i(\theta )\ge 0), \end{aligned}$$

(7.7)

a parameter estimator could be obtained by minimizing $L_m(\theta )$ with respect to $\theta$ and the resulting set of solutions is also convex. However, the lack of smoothness also presents computational challenges. We can use standard results for normal random variables and integration by parts to obtain

$$\begin{aligned} {\bar{L}}_m(\theta )= & {} m^{-1}\sum \limits _{i=1}^m \sum \limits _{j=1}^m\delta _iW_iW_j\left[ (e_j(\theta )-e_i(\theta ))\Phi \left( \frac{e_j (\theta )-e_i(\theta )}{r_{ij}}\right) \right. \nonumber \\&+r_{ij}\phi \left( \frac{e_j (\theta )-e_i(\theta )}{r_{ij}}\right) \left. \right] , \end{aligned}$$

(7.8)

where the function $\phi (\cdot )$ is a standard normal density function. A straightforward calculation can show that ${\bar{U}}_{m,G}(\theta )=\partial {\bar{L}}_m(\theta )/\partial \theta$. The smoothed objective function ${\bar{L}}_m(\theta )$ is convex and continuously differentiable. Hence, the standard numerical methods can be used to obtain ${\widehat{\theta }}_m=\arg \min \nolimits _{\theta \in {\mathcal {B}}} {\bar{L}}_m(\theta )$. By Lemmas 1 and 2 of Johnson and Strawderman (2009), the respective minimizers ${\tilde{\theta }}_m$ and ${\widehat{\theta }}_m$ of $L_m(\theta )$ and ${\bar{L}}_m(\theta )$ thus converge almost surely to $\theta _0$ (Andersen and Gill 1982, Corollary II.2). By Taylor expansion of ${\bar{U}}_{m,G}(\theta )$ around $\theta _0$, we have

$$\begin{aligned} {\bar{U}}_{m,G}(\theta )- {\bar{U}}_{m,G}(\theta _0)=\frac{\partial {{\bar{U}}_{m,G}}(\theta )}{\partial \theta }|_{\theta ^*} ( \theta -\theta _0), \end{aligned}$$

where $\theta ^*$ is between $\theta$ and $\theta _0$. Inserting ${\widehat{\theta }}_{m}$ in the above equation, we can obtain

$$\begin{aligned} m^{-1/2}{{\bar{U}}_{m,G}}(\theta _0)= \left\{ -m^{-1}\frac{\partial {\bar{U}}_{m,G}(\theta ^*)}{\partial \theta } \right\} \sqrt{m} ( {\widehat{\theta }}_{m}-\theta _0) \end{aligned}$$

with $\theta ^*$ being between ${\widehat{\theta }}_{m}$ and $\theta _0$. The asymptotic normality of ${\widehat{\theta }}_{m}$ can be established based on Lemmas 1 and 2, Condition (C5), and the consistency of ${\widehat{\theta }}_{m}$. Hence, Theorem 1 holds. $\square$