Marginal semiparametric multivariate accelerated failure time model with generalized estimating equations

Abstract

The semiparametric accelerated failure time (AFT) model is not as widely used as the Cox relative risk model due to computational difficulties. Recent developments in least squares estimation and induced smoothing estimating equations for censored data provide promising tools to make the AFT models more attractive in practice. For multivariate AFT models, we propose a generalized estimating equations (GEE) approach, extending the GEE to censored data. The consistency of the regression coefficient estimator is robust to misspecification of working covariance, and the efficiency is higher when the working covariance structure is closer to the truth. The marginal error distributions and regression coefficients are allowed to be unique for each margin or partially shared across margins as needed. The initial estimator is a rank-based estimator with Gehan’s weight, but obtained from an induced smoothing approach with computational ease. The resulting estimator is consistent and asymptotically normal, with variance estimated through a multiplier resampling method. In a large scale simulation study, our estimator was up to three times as efficient as the estimateor that ignores the within-cluster dependence, especially when the within-cluster dependence was strong. The methods were applied to the bivariate failure times data from a diabetic retinopathy study.

Appendix

1.1 Sketches of the Proofs

We impose the following regularity conditions:

• A1: $\Vert X_i\Vert \le B$ for all $i = 1, \ldots , n$ and some nonrandom constant $B$, where $\Vert \cdot \Vert$ is matrix norm.

• A2: The density function of $F_{k, \beta }$ exists such that $\int _{-\infty }^\infty t^2{\mathrm {d}}F_{k, \beta }(t) < \infty$, for $k=1, \ldots , K$.

• A3: The distribution function $F_{k, \beta }$ is twice differentiable with density $f_{k, \beta }$ such that

  $$\begin{aligned} \int \limits _{-\infty }^\infty \left( \frac{f_{k, \beta }^\prime (t)}{f_{k, \beta }(t)}\right) ^2 {\mathrm {d}}F_{k, \beta }(t) < \infty \end{aligned}$$

  where $1 \le k \le K$, and both $f_{k, \beta }(t)$ and $f^\prime _{k, \beta }(t)$ are bounded functions.

• A4: $E[\exp (\theta \epsilon _{ik}^-)]+ \sup _{k\in \{1, \ldots , K\}} E[\exp (\theta C_{ik}^- )] < \infty$ for some $\theta > 0$, where $a^-=|a|I_{\{a\le 0\}}$.

• A5: $\sup _{| b | < \infty ; -\infty < t < \infty }\sum _{i=1}^n\sum _{k=1}^K \Pr (t \le C_{ik} - X_{ik}^\top b \le t +h) = O(nh)$ as $h \rightarrow 0$ and $nh \rightarrow \infty$.

• A6: As $n\rightarrow \infty$, $\hat{\alpha }_n$ is bounded and is $n^{1/2}$ consistent to $\alpha _0$ given $\beta$.

• A7: As $n\rightarrow \infty$, initial estimator $b_n$ is $n^{1/2}$ consistent to $\beta _0$ and $\sqrt{n}( b_n - \beta _0)$ is asymptoticly normal with zero mean.

• A8: The slope matrices $n^{-1} \partial U_n / \partial \beta$ and $n^{-1} \partial U_n / \partial b$ evaluated at $(\beta _0, \beta _0, \alpha _0)$ converge to nondegenerate, finite limit $A$ and $B$, respectively.

• A9: The derivative $\partial \Omega _i^{-1}(\alpha ) / \partial \alpha$ is finite for all $i = 1, 2, \ldots n$.

Conditions A1–A5 are standard and ensure the existence of the solution of Eq. (2) (Lai and Ying 1991). It is natural to assume that the working covariance matrix $\Omega$ in Eq. (4) is a symmetric positive definite matrix. Then there exist a $K\times K$ nonsingular matrix, $\Gamma$, such that $\Omega (\alpha _0) = \Gamma ^{1/2} \Gamma ^{1/2}$. Let $\mathbb X_i = \Gamma ^{-1/2} X_i$, $\mathbb T_i = \Gamma ^{-1/2} Y_i$, $\mathbb C_i = \Gamma ^{-1/2} C_i$, and $\omega _i = \Gamma ^{-1/2} \epsilon _i$. Then Eq. (4) evaluated at $\alpha = \alpha _0$ can be viewed as Eq. (2) with the transformed data $\mathbb X_i$ and $\mathbb Y_i = \min (\mathbb Y_i, \mathbb C_i)$, with error $\omega _i$, $i = 1, \ldots , n$. The existence of the solution to Eq. (4) can be verified by the same arguments as in Lai and Ying (1991), with assumptions similar to A1 to A5 on the transformed data. The consistency and asymptotic normality of the estimator given $\alpha = \alpha _0$ follow from the same arguments as in Jin et al. (2006a).

The extra complexity here comes from the fact that Eq. (4) is solved at $\alpha = \hat{\alpha }_n$, an estimator of $\alpha _0$. Under condition A9, the $i$th term in the summation of $\partial U_n / \partial \alpha$ evaluated at $(\beta _0, \beta _0, \alpha _0)$ is a linear function of $\hat{Y}_i(\beta _0)-X_i^\top \beta _0$, $i = 1, \ldots , n$, with expectation zero. By the law of large number, $n^{-1}\partial U_n/\partial \alpha$ evaluated at $(\beta _0, \beta _0, \alpha _0)$ converges to zero in probability.

1.1.1 Proof of Theorem  1

At the solution $\hat{\beta }_n^{(1)}$ given $b_n$ and $\hat{\alpha }_n$, we have $n^{-1} U_n(\hat{\beta }_n^{(1)}, b_n, \hat{\alpha }_n) = 0$. Taylor expansion at $(\beta _0, \beta _0, \alpha _0)$ gives

$$\begin{aligned} 0&= \frac{1}{n}U_{n}(\beta _0, \beta _0, \alpha _0) + \frac{1}{n} \frac{\partial }{\partial \beta }\left[ U_n(\beta _0, \beta _0, \alpha _0) \right] (\hat{\beta }_n^{(1)}-\beta _0)\nonumber \\&+\, \frac{1}{n}\frac{\partial }{\partial b}\left[ U_n(\beta _0, \beta _0, \alpha _0) \right] (b_n-\beta _0) + \frac{1}{n} \frac{\partial }{\partial \alpha }\left[ U_n(\beta _0, \beta _0, \alpha _0) \right] (\hat{\alpha }_n-\alpha _0)\nonumber \\&+\,\, o_p(n^{-1/2}) \nonumber \\&= \frac{1}{n}U_n(\beta _0, \beta _0, \alpha _0) + A_n (\hat{\beta }_n^{(1)}-\beta _0) +B_n (b_n-\beta _0)+C_n(\hat{\alpha }_n-\alpha _0) + o_p(n^{-1/2}).\nonumber \\ \end{aligned}$$

(10)

With regularity conditions A1–A5, the first term converges in probability to zero by the law of large number. The convergence of $b_n$ and $\alpha _n$ in A6 and A7, combined with the limit condition in A8 and A9, then gives consistency of $\hat{\beta }_n^{(1)}$ to $\beta _0$. By induction, $\hat{\beta }^{(m)}_n$ is consistent for $\beta _0$ at every $m$.

1.1.2 Proof of Theorem  2

Under regularity conditions $\sqrt{n}(\hat{\beta }_n^{(1)}-\beta _0)$ can be expressed as

$$\begin{aligned}&\sqrt{n}(\hat{\beta }_n^{(1)}-\beta _0)\nonumber \\&\quad =\left[ A_n\right] ^{-1}\left[ \frac{1}{\sqrt{n}} U_n(\beta _0, \beta _0, \alpha _0)+B_n\sqrt{n}(b_n-\beta _0)+ C_n\sqrt{n}(\hat{\alpha }_n - \alpha _0)\right] + o_p(1).\nonumber \\ \end{aligned}$$

(11)

With condition A9, $C_n$ converges to zero in probability, and, hence, with $\sqrt{n}$ consistency of $\hat{\alpha }_n$, $C_n \sqrt{n} (\hat{\alpha }_n - \alpha _0) = o_p(1)$. Equation (11) is then asymptotically equivalent to

$$\begin{aligned} \left[ A_n\right] ^{-1}\left[ \frac{1}{\sqrt{n}} U_n(\beta _0, \beta _0, \alpha _0)+B_n\sqrt{n}(b_n-\beta _0)\right] . \end{aligned}$$

With the assumption that $b_n-\beta _0$ is asymptoticly normal, there exist some nonrandom functions $\eta _i$ with zero mean such that,

$$\begin{aligned} \sqrt{n}(b_n - \beta _0) = n^{-1/2}\sum _{i=1}^n\eta _i + o_p\left( \Vert b_n-\beta _0\Vert \right) . \end{aligned}$$

On the other hand, $U_n(\beta _0, \beta _0, \alpha _0)$ is a sum of independent and identically distributed quantities with zero mean, denoted by $\phi _i$’s, $i = 1, \ldots , n$. Equation (11) reduces to

$$\begin{aligned} \sqrt{n}(\hat{\beta }_n^{(1)}-\beta _0) = \left[ A_n\right] ^{-1}\left[ n^{-1/2}\sum _{i=1}^n \left( \phi _i+B_n\eta _i\right) \right] + o_p\left( \Vert b_n-\beta _0\Vert \right) . \end{aligned}$$

By multivariate central limit theorem for sums of independent random vectors, the asymptotic distribution for $\hat{\beta }_n^{(1)}$ is zero mean multivariate normal as $n\rightarrow \infty$. The limit covariance matrix $\Sigma$ have the form $A^{-1}\Phi A^{-1}$, where $\Phi = \lim _{n\rightarrow \infty }n^{-1}\sum _{i=1}^n \imath _i \imath _i^{\top }$ with $\imath _i = \phi _i + B\eta _i$. Induction then implies that $\hat{\beta }_n^{(m)}$ is multivariate normal for every $m$.

1.2 Analytic details of $W(t, x)$ in model checking

Using arguments similar to those in Jin et al. (2006a) and Novák (2013), $W(t, x)$ can be shown to have the same asymptotic distribution as $\hat{W}(t, x)$, where

$$\begin{aligned} \hat{W}(t, x)&= n^{-1/2}\sum _{i=1}^n\sum _{k=1}^K\int \limits _0^t \\&\quad \left( \omega _{ik}(x) - \frac{\sum _{j=1}^n\sum _{l=1}^K \omega _{jl}(x) I[e_{jl}(\hat{\beta }_n^{(m)})\ge u]}{\sum _{j=1}^n\sum _{l{=}1}^KI[e_{jl}(\hat{\beta }_n^{(m)}\ge u])}\right) {\mathrm {d}}\hat{M}_{ik}(u, \hat{\beta }_n^{(m)}) (Z_i {-}1)\\&\quad -n^{1/2}\left( \hat{f}_n(t, x) + \int \limits _0^t\hat{f}_Y(u, x){\mathrm {d}}\hat{\Lambda }(u, \hat{\beta }_n^{(m)})\right) ^\top (\hat{\beta }_n^{(m)}-\hat{\beta }_n^{(m)*})\\&\quad -n^{{-}1/2}\int \limits _0^t\sum _{i{=}1}^n\sum _{k{=}1}^K\omega _{ik}(x)I[e_{ik}(\hat{\beta }_n^{(m)}){\ge } u]{\mathrm {d}}\left( \hat{\Lambda }(u, \hat{\beta }_n^{(m)}) {-} \hat{\Lambda }(u, \hat{\beta }_n^{(m)*})\right) \!, \end{aligned}$$

where $\hat{f}_N(t, x) = n^{-1}\sum _{i=1}^n\sum _{k=1}^K\Delta _{ik}\omega _{ij}(x) \hat{f}_0(t) X_{ik}$, $\hat{f}_Y(t, x) = n^{-1}\sum _{i=1}^n$ $\sum _{k=1}^K \omega _{ik}(x) \hat{g}_0(t) X_{ik}$, and $f_0(t)$ and $g_0(t)$ are the baseline densities of $\epsilon _{ik}$ and $e_{ik}(\beta _0)$, respectively, with kernel estimate $\hat{f}_0(t)$ and $\hat{g}_0(t)$ (e.g., Novák 2013), obtained with $\beta _0$ replaced with $\hat{\beta }_{n}^{(m)}$. Note that the multipliers $Z_i$’s used to obtain the bootstrap samples $\hat{\beta }_{n}^{(m)*}$ are used again here.