Matched survival data in a co-twin control design

Abstract

When using the co-twin control design for analysis of event times, one needs a model to address the possible within-pair association. One such model is the shared frailty model in which the random frailty variable creates the desired within-pair association. Standard inference for this model requires independence between the random effect and the covariates. We study how violations of this assumption affect inference for the regression coefficients and conclude that substantial bias may occur. We propose an alternative way of making inference for the regression parameters by using a fixed-effects models for survival in matched pairs. Fitting this model to data generated from the frailty model provides consistent and asymptotically normal estimates of regression coefficients, no matter whether the independence assumption is met.

Appendix

The proofs follow closely the lines of Gross and Huber (1987) and build on the following conditions.

• (A1)
  $$\begin{aligned} \int \limits _0^{\tau }\alpha _0(t)dt<\infty . \end{aligned}$$

  (8)

• (A2) There exists a neighborhood $\mathcal{B }$ of $\beta _0$ such that for each of the in (10)–(16) defined scalar, vector or matrix random functions, $S(t,\beta )$, there is a deterministic function, $s(t,\beta )$ satisfying that

  $$\begin{aligned}&\sup _{\beta \in \mathcal{B },t\in [0,\tau ]}\Vert S(t,\beta )-s(t,\beta )\Vert \stackrel{P}{\rightarrow }0, \end{aligned}$$

  (9)
  $$\begin{aligned}&S_{L0}(s,\beta )=\frac{1}{n}\sum _{i=1}^n E[Z_i\mid \mathcal{F }_{s-}]\log \left( \frac{S_{0i}(s,\beta )}{S_{0i}(s,\beta _0)}\right) S_{0i}(s,\beta _0), \end{aligned}$$

  (10)
  $$\begin{aligned}&S_{L1}(s,\beta )=\frac{1}{n}\sum _{i=1}^n E[Z_i\mid \mathcal{F }_{s-}]\log \left( \frac{S_{0i}(s,\beta )}{S_{0i}(s,\beta _0)}\right) S_{1i}(s,\beta _0), \end{aligned}$$

  (11)
  $$\begin{aligned}&S_{L2}(s,\beta )=\frac{1}{n}\sum _{i=1}^n E[Z_i\mid \mathcal{F }_{s-}]\left\{ \log \left( \frac{S_{0i}(s,\beta )}{S_{0i}(s,\beta _0)}\right) \right\} ^2S_{0i}(s,\beta _0), \end{aligned}$$

  (12)
  $$\begin{aligned}&S_1(s,\beta )=\frac{1}{n}\sum _{i=1}^n E[Z_i\mid \mathcal{F }_{s-}]\frac{S_{0i}(s,\beta )}{S_{0i}(s,\beta _0)}S_{1i}(s,\beta ), \end{aligned}$$

  (13)
  $$\begin{aligned}&S_2(s,\beta )=\frac{1}{n}\sum _{i=1}^n E[Z_i\mid \mathcal{F }_{s-}]\frac{S_{0i}(s,\beta )}{S_{0i}(s,\beta _0)}S_{2i}(s,\beta ), \end{aligned}$$

  (14)
  $$\begin{aligned}&S_{\Delta 2}(s,\beta )=\frac{1}{n}\sum _{i=1}^n E[Z_i\mid \mathcal{F }_{s-}]S_{0i}(s,\beta _0)V_i(s,\beta ), \end{aligned}$$

  (15)
  $$\begin{aligned}&S_{\Delta 4}(s,\beta )=\frac{1}{n}\sum _{i=1}^nE[Z_i\mid \mathcal{F }_{s-}]S_{0i}(s,\beta _0)V_i(s,\beta )^{\otimes 2}. \end{aligned}$$

  (16)

• (A3) The deterministic $s$-functions, i.e. the limits (10)–(16) are bounded for $t\in [0,\tau ]$ and $\beta \in \mathcal{B }$, and the matrix

  $$\begin{aligned} \Sigma _\tau =\int \limits _0^\tau s_{\Delta 2}(s,\beta _0)\alpha _0(s)ds \end{aligned}$$

  (17)

  is positive definite.

• (A4) Derivatives of the functions $s(\cdot ,\beta )$ are limits in probability of the derivatives of the corresponding random functions, $S(\cdot ,\beta )$.

Proof

(Theorem 1) Define

$$\begin{aligned} \Delta (\beta ,t)=\frac{1}{n}\left( C_t(\beta )-C_t(\beta _0)\right) . \end{aligned}$$

(18)

The compensator of this process is

$$\begin{aligned} \tilde{\Delta }(\beta ,t)=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^2\int \limits _0^t\left[ (\beta -\beta _0)^T \mathbf{X}_{ij}(s)-\log \left( \frac{S_{0i}(s,\beta )}{S_{0i}(s,\beta _0)}\right) \right] d\Lambda _{ij}^\mathcal{O }(s)) \end{aligned}$$

(19)

where $\Lambda _{ij}^\mathcal{O }(t)=\int _0^t\lambda _{ij}^\mathcal{O }(s)ds$, cf. (4). The corresponding martingale, i.e. the difference between (18) and (19) is

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n\sum _{j=1}^2\int \limits _0^t\left[ (\beta -\beta _0)^T \mathbf{X}_{ij}(s)-\log \left( \frac{S_{0i}(s,\beta )}{S_{0i}(s,\beta _0)}\right) \right] dM_{ij}^\mathcal{O }(s) \end{aligned}$$

(20)

which has predictable variation process

$$\begin{aligned}&\frac{1}{n^2}\sum _{i=1}^n\sum _{j=1}^2\int \limits _0^t \left[ (\beta -\beta _0)^T X_{ij}(s)-\log \left( \frac{S_{0i}(s,\beta )}{S_{0i}(s,\beta _0)}\right) \right] ^2\\&\quad \times Y_{ij}(s)\exp \left( \beta _0^TX_{ij}(s)\right) E[Z_i\mid \mathcal{F }_{s-}]\alpha _0(s)ds. \end{aligned}$$

Due to the conditions stated in (8) and (9) (use (11), (12) and (14)), this converges in probability to $0$ as $n\rightarrow \infty$. Now, using Lenglart’s inequality (Andersen et al. 1993, Sect. II.5.2), (20) converges to $0$ in probability as $n\rightarrow \infty$.

It further follows from (8) and (9) (use (10) and (13)) that for $\beta \in \mathcal{B }$ and $n\rightarrow \infty , \tilde{\Delta }(\beta ,\tau )$ and thereby also $\Delta (\beta ,\tau )$ converges pointwise in probability towards

$$\begin{aligned} f(\beta )=\int \limits _0^\tau \left[ (\beta -\beta _0)^T s_1(s,\beta _0)- s_{L0}(s,\beta )\right] \alpha _0(s)ds. \end{aligned}$$

(21)

Now,

$$\begin{aligned} \frac{\partial }{\partial \beta }f(\beta )=\int \limits _0^\tau \left[ s_1(s,\beta _0)-\frac{\partial }{\partial \beta }\left( s_{L0}(s,\beta )\right) \right] \alpha _0(s)ds \end{aligned}$$

with

$$\begin{aligned} \frac{\partial s_{L0}(s,\beta )}{\partial \beta }= \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{i=1}^n E[Z_i\mid \mathcal{F }_{s-}]\frac{S_{1i}(s,\beta )}{S_{0i}(s,\beta )}S_{0i}(s,\beta _0) \end{aligned}$$

which for $\beta =\beta _0$ equals $s_1(s,\beta _0)$ and hence

$$\begin{aligned} \frac{\partial f(\beta )}{\partial \beta }\bigg |_{\beta =\beta _0}=0. \end{aligned}$$

Furthermore,

$$\begin{aligned} \frac{\partial ^2 f(\beta )}{\partial \beta ^2} =\int \limits _0^\tau s_{\Delta 2}(s,\beta )\alpha _0(s)ds \end{aligned}$$

which is positive definite in $\beta =\beta _0$, cf. (17). This ensures that $f(\beta )$ is concave with a unique maximum in $\beta _0$. Then the proof can be completed following the same arguments as (Andersen et al. (1993), p. 498) (using Andersen and Gill 1982, Theorem II.1). $\square$

Proof

(Theorem 2) First, we use the martingale CLT (Andersen et al. 1993, Theorem II.5.1) to show that

$$\begin{aligned} n^{-1/2}U_\tau ({\beta _{0}})\mathop {\rightarrow }\limits ^\mathcal{D }\mathcal{N }(0,\Sigma _\tau ). \end{aligned}$$

(22)

This requires a Lindeberg condition similar to the one in (Gross and Huber (1987), p. 33) and follows the lines of (Andersen et al. (1993), pp. 499, 523) as well as Gross and Huber (1987) using (9) on (16). Next, a Taylor expansion of $U_\tau (\beta )$ around $\beta _0$ gives that

$$\begin{aligned} U_\tau (\beta )-U_\tau (\beta _0)=-\mathcal{I }(\beta ^*)\cdot (\beta -\beta _0) \end{aligned}$$

where $\beta ^*$ is a $p$-vector for which the $j$th coordinate is on the line segment between the $j$th coordinates of $\beta$ and $\beta _0$. In particular, since $U(\widehat{\beta })=0$ we get that

$$\begin{aligned} n^{-1/2}U_\tau (\beta _0)=\frac{1}{n}\mathcal{I }(\beta ^*) n^{-1/2}\left( \widehat{\beta }-\beta _0\right) . \end{aligned}$$

In order to show (22), it must be shown that

$$\begin{aligned} \langle n^{-1/2}U_\tau ({\beta _0})\rangle \mathop {\rightarrow }\limits ^\mathcal{P }\Sigma _\tau . \end{aligned}$$

(23)

This follows from the fact that

$$\begin{aligned} \left\langle n^{-1/2} U(\beta _0)\right\rangle (\tau ) = \frac{1}{n}\sum _{i=1}^n\int \limits _0^\tau \alpha _0(s)E[Z_i\mid \mathcal{F }_{t-}] S_{0i}(s,\beta _0)V_i(s,\beta _0)ds, \end{aligned}$$

cf. (15) and (17).$\square$