Estimation for an accelerated failure time model with intermediate states as auxiliary information

Abstract

The accelerated failure time (AFT) model is a common method for estimating the effect of a covariate directly on a patient’s survival time. In some cases, death is the final (absorbing) state of a progressive multi-state process, however when the survival time for a subject is censored, traditional AFT models ignore the intermediate information from the subject’s most recent disease state despite its relevance to the mortality process. We propose a method to estimate an AFT model for survival time to the absorbing state that uses the additional data on intermediate state transition times as auxiliary information when a patient is right censored. The method extends the Gehan AFT estimating equation by conditioning on each patient’s censoring time and their disease state at their censoring time. With simulation studies, we demonstrate that the estimator is empirically unbiased, and can improve efficiency over commonly used estimators that ignore the intermediate states.

A Appendix

1.1 A.1 Proof that $\beta$ parameters for each transition are equal when individual transitions and overall survival time follow an AFT model

Consider a progressive model of $D+1$ states as in Fig. 1a. Let $T_{gh}$ indicate a continuous random variable for time to transition from state g to state h. Let $T = T_{01} + T_{12} + \cdots + T_{D-1,D}$. Further, suppose $T = \exp (\beta _0 X + \varepsilon )$, where $\varepsilon$ is independent of X. Let $T_{g-1,g} = \exp (\beta _{g}X + \varepsilon _g)$ for $g=1,\ldots ,D$, where $\varepsilon _g$ is also independent of X. Then we have:

$$\begin{aligned} T = \exp (\beta _0X + \varepsilon )&= \exp (\beta _{1}X +\varepsilon _1) + \cdots + \exp (\beta _{D}X +\varepsilon _D) \\&= \exp (\beta _{0}X)\frac{\exp (\beta _{1}X)}{\exp (\beta _{0}X)}\exp (\varepsilon _1) + \cdots + \exp (\beta _{0}X)\frac{\exp (\beta _{D}X)}{\exp (\beta _{0}X)}\exp (\varepsilon _D) \\&= \exp (\beta _{0}X) \lbrace \exp [(\beta _{1} - \beta _{0})X +\varepsilon _1)] + \cdots + \exp [(\beta _{D} - \beta _{0})X +\varepsilon _D)]\rbrace \end{aligned}$$

Since $T = \exp (\beta _0X)\exp (\varepsilon )$, It follows that $\varepsilon = \log \lbrace \exp [(\beta _{1} - \beta _{0})X +\varepsilon _1] + \cdots + \exp [(\beta _{D} - \beta _{0})X +\varepsilon _D] \rbrace$. Without loss of generality, suppose that $\beta _1 \ne \beta _0$. Then we have that $\varepsilon = \log \lbrace \exp [cX +\varepsilon _1] + \cdots + \exp [(\beta _{D} - \beta _{0})X +\varepsilon _D] \rbrace$ where c is a non-zero constant. In this case, $\varepsilon$ is not independent of X and cannot be independent of X unless $\varepsilon _1$ is not independent of X. But by our model assumptions, $\varepsilon$ and $\varepsilon _g$ are independent of X for $g=1,\ldots ,D$, so it must be the case that $\beta _1 = 0.$ Similarly, we will have that $\beta _2 = \cdots = \beta _D = \beta _0$.

A similar argument can be used for progressive models that have the form in Fig. 1b. We will show the case of the progressive illness-death model (Fig. 1c), but the proof is analogous for models with a larger state space. Suppose we have a 3-state model where subjects can transition from state $0 \rightarrow 1$, $1 \rightarrow 2$, and $0 \rightarrow 2$, with state 2 as the absorbing state. As before, let $T_{gh}$ denote the random variable for the direct transition from state g to state h, let T denote the absorbing failure time from origin, and assume $T = \exp (\beta _0 X + \varepsilon )$, where $\varepsilon$ is independent of X. Let $T_{gh} = \exp (\beta _{gh}X + \varepsilon _{gh})$ for $g=0,1$, $h=1,2$, and $h>g$, where $\varepsilon _{gh}$ is also independent of X. We have that

$$\begin{aligned} T&= I[\exp (\beta _{01}X +\varepsilon _{01}) \le \exp (\beta _{02}X +\varepsilon _{02})][\exp (\beta _{01}X +\varepsilon _{01}) + \exp (\beta _{12}X +\varepsilon _{12})] \nonumber \\&\quad + I[\exp (\beta _{01}X +\varepsilon _{01})> \exp (\beta _{02}X+\varepsilon _{02})][\exp (\beta _{02}X +\varepsilon _{02})] \nonumber \\&= \exp (\beta _{0}X)I[\exp (\beta _{01}X +\varepsilon _{01}) \le \exp (\beta _{02}X +\varepsilon _{02})]\lbrace \exp [(\beta _{01}-\beta _0)X +\varepsilon _{01}] + \exp [(\beta _{12}\nonumber \\&\quad -\beta _0)X +\varepsilon _{12}]\rbrace + \exp (\beta _{0}X)I[\exp (\beta _{01}X +\varepsilon _{01}) > \exp (\beta _{02}X+\varepsilon _{02})]\lbrace \exp [(\beta _{02}-\beta _0)X +\varepsilon _{02}]\rbrace \nonumber \\ \end{aligned}$$

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Since $T = \exp (\beta _0 X + \varepsilon )$, it follows that $\varepsilon = \log \big \lbrace I[\exp (\beta _{01}X +\varepsilon _{01}) < \exp (\beta _{02}X +\varepsilon _{02})]\lbrace \exp [(\beta _{01}-\beta _0)X +\varepsilon _{01}] + \exp [(\beta _{12}-\beta _0)X +\varepsilon _{12}]\rbrace + I[\exp (\beta _{01}X +\varepsilon _{01}) > \exp (\beta _{02}X+\varepsilon _{02})]\lbrace \exp [(\beta _{02}-\beta _0)X +\varepsilon _{02}]\rbrace \big \rbrace$. In order for the indicator functions to be independent of X, we would need $\beta _{01}=\beta _{02}$, and in order for the non-indicator terms to be independent of X, we need $\beta _{01} = \beta _{12}=\beta _{02} = \beta _0$. If at least one of $\beta _{01}, \beta _{12}, \beta _{02}$ are not equal to $\beta _0$, then $\varepsilon$ is not independent of X, which contradicts our model assumption.

1.2 A.2 Justification for estimating equation

Consider the formulation of the estimating equation given in (5):

$$\begin{aligned} U_P(\beta ) = \frac{1}{n^2}\sum _i\sum _{j< i} (X_i - X_j)\left[ P\left( \tilde{e_i}^{\beta } < \tilde{e_j}^{\beta } | S_i(e_i^{\beta }), S_j(e_j^{\beta })\right) - P\left( \tilde{e_i}^{\beta } > \tilde{e_j}^{\beta } | S_i(e_i^{\beta }), S_j(e_j^{\beta })\right) \right] \end{aligned}$$

We can think of the probabilities as expectations of an indicator function conditional on what we observe:

$$\begin{aligned} \frac{1}{n^2}\sum _i\sum _{j< i} (X_i - X_j) E\left[ I\left( \tilde{e_i}^{\beta } < \tilde{e_j}^{\beta } | S_i(e_i^{\beta }), S_j(e_j^{\beta })\right) - I\left( \tilde{e_i}^{\beta } > \tilde{e_j}^{\beta } | S_i(e_i^{\beta }), S_j(e_j^{\beta })\right) \right] \end{aligned}$$

where the expectation is taken with respect to the distribution of the residual failure times conditional on the disease states at the residual follow-up times. This function can be seen to be centered at 0 when $\beta = \beta _0$, as its expectation is:

$$\begin{aligned} (X_i - X_j) E\left\{ E\left[ I\left( \tilde{e_i}^{\beta } < \tilde{e_j}^{\beta } | S_i(e_i^{\beta }), S_j(e_j^{\beta })\right) - I\left( \tilde{e_i}^{\beta } > \tilde{e_j}^{\beta } | S_i(e_i^{\beta }), S_j(e_j^{\beta })\right) \right] \right\} \end{aligned}$$

where the outside expectation is taken with respect to the distribution of the observed states at the residual follow-up times. By the law of iterated expectations, this is simply equal to:

$$\begin{aligned} (X_i - X_j) \left[ P\left( \tilde{e_i}^{\beta } < \tilde{e_j}^{\beta }\right) - P\left( \tilde{e_i}^{\beta } > \tilde{e_j}^{\beta }\right) \right] \end{aligned}$$

Since $\tilde{e_i}^{\beta }$ and $\tilde{e_j}^{\beta }$ are i.i.d. and independent of $X_i$ and $X_j$ when $\beta = \beta _0$, it follows that the expectation is 0 under boundedness of the residual failure time and log censoring time densities, and the covariates.

1.3 A.3 Variance estimation for $\hat{\beta }$: Gaussian quadrature method

First, we give the assumptions in Jin et al. for validity of their Monte Carlo Method and Gaussian Quadrature Method of variance estimation (Jin et al. 2014). Suppose we denote the estimating equation as $U(\beta )$, and $\beta _0$ is the true parameter vector:

Assumption 1$\sqrt{n}U(\beta _0)$ is asymptotically normal with mean 0 and covariancematrix D.

Assumption 2 The estimator $\hat{\beta }$ is root-n consistent, and $\sqrt{n}(\hat{\beta } - \beta _0)$ is asymptotically normal with mean 0 and covariance matrix V.

Assumption 3$U(\beta )$ is locally asymptotically linear in a neighborhood of $\beta _0$.

Let B be the limiting slope matrix of $U(\beta _0)$. B is difficult to estimate because the estimating function U is not smooth in $\beta$. First, we define $\varGamma = n^{-1/2}V^{1/2}$, where $V = B^{-1}DB^{-1}$, i.e. the variance of $\sqrt{n}(\hat{\beta } - \beta _0)$. We are ultimately interested in estimating $\varGamma$, which depends on B. Jin et al. show that the derivative B of a smoothed version of the estimating equation satisfies the following expression:

$$\begin{aligned} B(\varGamma ; \beta ) = E_Z\left[ U(\beta +\varGamma Z)Z^T\varGamma ^{-1}\right] . \end{aligned}$$

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We can use Gaussian quadrature or Monte Carlo methods to approximate $B(\varGamma ; \beta )$ and evaluate $\varGamma$, but notice that $B(\varGamma ; \beta )$ also depends on $\varGamma$, resulting in an iterative algorithm. We describe our implementation of the algorithm for the Gaussian Quadrature Method below:

• 1. Calculate an estimate $\hat{D}$ for D, the covariance matrix of $\sqrt{n}U(\beta )$. This can be done using the formula in (6), or a bootstrap procedure. Set $\varGamma _0 = n^{-1/2}I$.

• 2. Suppose the dimension of $\beta$ is p. Choose m nodes $x_j$, $j = 1,\ldots ,m$, based on one-dimensional Gauss–Hermite quadrature, and let $z_1,z_2,\ldots ,z_{m^p}$ each be a $p \times 1$ vector for a unique single combination of the m nodes among p points. For example, if we choose 5 1-D Gauss–Hermite quadrature nodes, and we had 2 $\beta 's$ to estimate, we would have $5^2$ unique vectors $z_j$ of 2-dimensional nodes for estimating the (double) integral of interest; these two dimensional nodes would be $(x_1, x_1), (x_1, x_2),\ldots ,(x_2, x_1),\ldots ,(x_5, x_5)$ (see Fig. 2). Let $w_j$ be the $p \times 1$ vector of Gaussian quadrature weights corresponding to the nodes in $z_j$. Thus, we will have a grid of points over which we approximate the p-dimensional integral $B(\varGamma ;\beta )$. We are interested in computing the integral, $\int _{-\infty }^{\infty }(\frac{1}{\sqrt{2\pi }})^p e^{-x_1^2/2}\cdots e^{-x_p^2/2}[U(\beta + \varGamma x)x^T\varGamma ^{-1}]dx_1\cdots dx_p$. Since Gauss–Hermite quadrature computes integrals of the form $\int _{-\infty }^{\infty }e^{-x^2}f(x)dx$, we use a change of variable on x so that we can write the integral in this form. Set $x^{*} = \sqrt{2}x$, then the integral becomes $\int _{-\infty }^{\infty }(\frac{1}{\sqrt{\pi }})^pe^{-x_1^{*2}}\cdots e^{-x_p^{*2}}[U(\beta + \varGamma x^{*})x^{*T}\varGamma ^{-1}]dx_1^{*}\cdots dx_p^{*}$. Thus, let $z_j^{*} = \sqrt{2}z_j$ for all j, and proceed.

• 3. Compute at the $k\mathrm{th}$ step:

  $$\begin{aligned} B_k = B\left( \varGamma _{k-1}; \hat{\beta }\right) = \dfrac{1}{\sqrt{\pi }^p}\sum _{j=1}^m U\left( \hat{\beta } + \varGamma _{k-1}z_j^{*}\right) z_j^{*T}\varGamma _{k-1}^{-1}\prod _{l=1}^p w_{jl} \end{aligned}$$

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  where $w_{jl}$ is the $l\mathrm{th}$ element of the weight vector $w_j$.

• 4. Calculate $G_k = B_k^{-1}\hat{D}B_k^{-1}$ and let $\varGamma _k = G_k^{1/2}n^{-1/2}$.

• 5. Repeat steps 3 and 4 until $\varGamma _k$ converges within a specified tolerance level.

The diagonal of the matrix $\varGamma _k$ at the last iteration yields the standard error estimates for the vector $\hat{\beta }$. The MCM is the same as the above method, except that in step 2 the $z_j$ vectors are randomly generated from a standard multivariate normal distribution, and in step 3 $B_k$ is estimated as $B(\varGamma _{k-1}; \hat{\beta }) = \frac{1}{m}\sum _{j=1}^m U(\hat{\beta } + \varGamma _{k-1}z_j)z_j^{T}\varGamma _{k-1}^{-1}$. In simulations, we found that as few as 8–10 Gauss–Hermite nodes worked reasonably well for the variance estimation when there is a single covariate.