A causal proportional hazards estimator under homogeneous or heterogeneous selection in an IV setting

Abstract

In this paper we present a framework to do estimation in a structural Cox model when there may be unobserved confounding. The model is phrased in terms of a selection bias function and a baseline model that describes how covariates affect the survival time in a scenario without exposure. In this way model congeniality is ensured. The method uses an instrumental variable. Interestingly, the formulated model turns out to have similarities to the so-called Cox–Aalen survival model for the observed data. We exploit this to enhance estimation of the unknown parameters. This also allows us to derive large sample properties of the proposed estimator.

Appendix: Large sample properties

Large sample properties of the estimator of A

The consistency of ${{\hat{A}}}(t, \phi _0)$ may be shown similar to what is done in Martinussen et al. (2017b). We will here focus on the asymptotic distribution of ${{\hat{A}}}(t, \phi _0)$. To this end, define the $p \times n$ matrix H as

$$\begin{aligned} H (s, \phi , b)&= \{Y(s, \phi , b)^TW(s)Y(s, \phi , b)\}^{-1}Y(s, \phi , b)^TW(s) \end{aligned}$$

for $p=k+2$ and k the dimension of C. Then we can write

$$\begin{aligned} {{\hat{A}}}(t, \phi _0)&= \int _0^t H\{s, \phi _0, {{\hat{B}}}_X(s-)\}dN(s). \end{aligned}$$

Notice that we can write the true nuisance parameter $A_0$ as

$$\begin{aligned} A_0(t)&= \int _0^tH\{s, \phi _0, B^0_X(s)\}Y\{s,\phi _0, B^0_X(s)\}dA_0(s)\\&=\int _0^tH\{s, \phi _0, B^0_X(s)\} dN(s) - \int _0^tH \{s, \phi _0, B^0_X(s)\}dM(s) . \end{aligned}$$

Now $n^{1/2}\{{{\hat{A}}} (t, \phi _0)-A_0(t)\}$ can be expressed as

$$\begin{aligned}&n^{1/2}\int _0^tH \{s, \phi _0, B^0_X(s)\}dM(s)\\&\quad +\, n^{1/2}\int _0^t[H \{s, \phi _0,{{\hat{B}}}_X(s-)\}-H \{s, \phi _0,B^0_X(s-)\}]dN(s). \end{aligned}$$

First we take a closer look at the last integral in this expression. By a Taylor approximation we have

$$\begin{aligned}&n^{1/2}\int _0^t[H \{s, \phi _0,{{\hat{B}}}_X(s-)\}-H \{s, \phi _0,B^0_X(s-)\}]dN(s)\\&\quad =n^{1/2}\int _0^tV(ds) \{{{\hat{A}}} (s-, \phi _0)-A_0(s-)\} +o_P(1) \end{aligned}$$

where V is the $p\times p$ matrix

$$\begin{aligned} V(s)=\frac{\partial H \{s, \phi _0, B_X^0(s-)\}N(s)}{\partial A_0(s)^T}. \end{aligned}$$

As H only depends on the first element of $A_0$ namely $B_X^0$ then the first column of V is non-zero and the rest consist of zeros. Define $Z(t, \phi _0) = n^{1/2}\{{{\hat{A}}} (t, \phi _0)-A_0(t)\}^T$ and from the above calculations we see that it satisfies the following Volterra equation (Andersen et al. 1993, p. 91)

$$\begin{aligned} Z(t, \phi _0)= n^{1/2}\int _0^t M(ds)^T H \{s, \phi _0,B^0_X(s-)\}^T + \int _0^t Z(s-, \phi _0) V(ds)^T , \end{aligned}$$

and the solution is

$$\begin{aligned} n^{1/2} \int _0^t M(ds)^TH\{s, \phi _0, B^0_X(s-)\}^T{\mathcal {Q}}(s,t) \end{aligned}$$

where ${\mathcal {Q}}$ is the product integral

as defined in Andersen et al. (1993). Finally, we have that

$$\begin{aligned} n^{1/2}\{{{\hat{A}}}(t, \phi _0)-A_0(t)\}&=n^{1/2} \int _0^t \mathcal Q(s,t)^T H\{s, \phi _0,B^0_X(s-)\} dM(s). \end{aligned}$$

In this expression $[n^{-1}Y\{s, \phi _0, B^0_X(s-)\}^TW(s)Y\{s, \phi _0, B^0_X(s-)\}]^{-1}$ converges in probability to some $p\times p$ matrix that we denote $l_1(s)$. Also ${\mathcal {Q}}(s, t)^T$ converges to some limit in probability that is denoted l(s, t). Further, one can show the convergence in distribution of $n^{-1/2}Y\{s, \phi _0, B^0_X(s-)\}^TW(s) d M(s)$ to a mean zero process. Then we have the i.i.d. representation of ${{\hat{A}}}(t, \phi _0)$

$$\begin{aligned} n^{1/2}\{{{\hat{A}}} (t, \phi _0)-A_0(t)\}&= n^{-1/2}\sum _{i=1}^n \epsilon _i^A (t) + o_P(1) \end{aligned}$$

with

$$\begin{aligned} \epsilon _i^A(t)&= \int _0^t l (s,t) l_1(s)Y_i\{s, \phi _0,B^0_X(s-)\}^TW_i(s) d M_i(s) , \end{aligned}$$

where the elements of $\epsilon _i^A$ are denoted $(\epsilon _i^{B_X}, \epsilon _i^{\varOmega _C}, \epsilon _i^{\varOmega _0})^T$. This representation ensures convergence of the finite dimensional distribution. Convergence in distribution as a process can be shown similarly to what is done in Martinussen et al. (2017b).

Large sample properties of ${{\hat{\psi }}}$

We first note that

$$\begin{aligned} n^{-1/2}U({{\hat{\psi }}})&=n^{-1/2}U({{\hat{\psi }}},{{\hat{\theta }}}) =n^{-1/2}U(\phi _0)+\{n^{-1}D_{\psi }U\}n^{-1/2}({{\hat{\psi }}}\\&\quad -\psi _0)+\{n^{-1}D_{ \theta }U\}n^{-1/2}({{\hat{\theta }}}- \theta _0)+o_p(1) \end{aligned}$$

so

$$\begin{aligned} n^{-1/2}({{\hat{\psi }}}-\psi _0)=-\{n^{-1}D_{\psi }U\}^{-1}[n^{-1/2}U(\phi _0)+\{n^{-1}D_{ \theta }U\}n^{-1/2}({{\hat{\theta }}}- \theta _0)]+o_p(1) \end{aligned}$$

and since we have assumed ${{\hat{\theta }}}$ to be RAL we just need to find the asymptotic distribution of $n^{-1/2}U(\phi _0)$. We can write this function as

$$\begin{aligned} n^{-1/2}U(\phi _0)&= n^{-1/2}\int _0^\tau X^T[dN(t) - Y\{t, \phi _0,B^0_X(t)\}dA_0(t)]\nonumber \\&\quad - n^{-1/2}\int _0^\tau X^TY\{t, \phi _0, {{\hat{B}}}_X(t-)\}\{d{{\hat{A}}}(t, \phi _0)- d A_0(t)\}\end{aligned}$$

(14)

$$\begin{aligned}&\quad -\, n^{-1/2}\int _0^\tau X^T[Y \{t, \phi _0, {{\hat{B}}}_X(t-)\}-Y \{t, \phi _0,B^0_X(t)\}] d A_0(t) . \end{aligned}$$

(15)

The first term on the right hand side of the latter display is the martingale

$$\begin{aligned} n^{-1/2}\int _0^\tau X^TdM(t)&= n^{-1/2}\sum _{i=1}^n\int _0^\tau X_idM_i (t). \end{aligned}$$

Note that $Y \{t, \phi _0, {{\hat{B}}}_X(t-)\}$ and $Y \{t, \phi _0,B^0_X(t)\}$ share all entries except for the first column. If we let $Y_{*1}$ denote the first column of Y then (15) can be written as

$$\begin{aligned} -n^{-1/2}\int _0^\tau X^T[Y_{*1} \{t, \phi _0, {{\hat{B}}}_X(t-)\}-Y_{*1} \{t, \phi _0,B^0_X(t)\}] dB^0_X(t) . \end{aligned}$$

Using a Taylor expansion this is asymptotically equivalent to

$$\begin{aligned}&-n^{-1/2}\int _0^\tau \frac{\partial X^T Y_{*1}\{t, \phi _0,B^0_X(t)\}}{\partial B_X^0(t)}\{{{\hat{B}}}_X(t-)-B_{X0}(t)\} dB^0_X(t). \end{aligned}$$

Let $d_{B_X}(t)$ denote the limit in probability of the derivative $n^{-1}\frac{\partial X^T Y_{*1}\{t, \phi _0,B^0_X(t)\}}{\partial B_X^0(t)}$. By the i.i.d. representation of $n^{1/2}\{{{\hat{B}}}_X(t-)-B_X^0(t)\}$ derived earlier in this Appendix, (15) is seen to be asymptotically equivalent to the following sum of zero-mean i.i.d. terms

$$\begin{aligned} -n^{-1/2}\sum _{i=1}^n\int _0^\tau d_{B_X}(t) \epsilon ^{B_X}_i(t-) dB^0_X(t). \end{aligned}$$

For the latter of these integrals, (14), we have convergence in distribution of its integrand $n^{1/2}\{{{\hat{A}}}(t, \phi _0)-A_0(t)\}$ and the integral can be written as (Kosorok 2008, Lemma 4.2)

$$\begin{aligned} -n^{-1/2}\int _0^\tau X^TY\{t, \phi _0,B^0_X(t)\}\{d{{\hat{A}}}(t, \phi _0)- d A_0(t)\} \end{aligned}$$

since $n^{-1}[X^TY\{t, \phi _0, {{\hat{B}}}_X(t-)\}-X^TY\{t, \phi _0,B^0_X(t)\}]$ converges in probability to 0. Denote the limit in probability of $n^{-1}X^TY\{t, \phi _0, B_X^0(t)\}$ as $l_{XY}(t)$. Thus it is asymptotically equivalent to

$$\begin{aligned} -n^{-1/2}\sum _{i=1}^n\int _0^\tau l_{XY}(t) d\epsilon _i^{A}(t) \end{aligned}$$

Finally, we have that $n^{-1/2} U(\psi _0) = n^{-1/2}\sum _{i=1}^n\epsilon ^U_i+o_p(1)$, where

$$\begin{aligned} \epsilon ^U_i = \int _0^\tau X_idM_i (t) -\int _0^\tau d_{B_X}(t) \epsilon ^{B_X}_i(t) dB^0_X(t)-\int _0^\tau l_{XY}(t) d\epsilon _i^{A}(t). \end{aligned}$$

Finally,

$$\begin{aligned} n^{1/2}({{\hat{\psi }}}-\psi _0) =n^{-1/2}\sum _{i=1}^n\epsilon ^{\psi }_i+o_p(1) \end{aligned}$$

where

$$\begin{aligned} \epsilon ^{\psi }_i=-\mathcal{J}_{\psi }^{-1}\{\epsilon ^U_i+{{{\mathcal {J}}}}_{\theta }\epsilon ^{\theta }_i\}, \end{aligned}$$

(16)

with ${{{\mathcal {J}}}}_{\psi }$ denotes the limit in probability of $n^{-1}D_{\psi }U$ and similarly with ${{{\mathcal {J}}}}_{\theta }$. Based on the above derivations, the i.i.d. decomposition of $n^{1/2} \{ {{\hat{A}}}(t, {{\hat{\phi }}})-A_0(t)\}$ can easily be obtained since

$$\begin{aligned} n^{1/2}\{{{\hat{A}}}(t,{{\hat{\phi }}})-A_0(t)\}&= n^{1/2}\{{{\hat{A}}}(t,\hat{\phi })-{{\hat{A}}}(t, \phi _0)\} + n^{1/2}\{{{\hat{A}}}(t,\phi _0)- A_0(t)\} \\&= n^{1/2} \{D_{\psi } {{\hat{A}}}(t, \phi _0)\}\{{{\hat{\psi }}}-\psi _0\} + n^{1/2} \{D_{\theta } {{\hat{A}}}(t, \phi _0)\}\{{{\hat{\theta }}}-\theta _0\}\\&\quad +\, n^{1/2}\{{{\hat{A}}}(t,\psi _0)- A_0(t)\} \end{aligned}$$

where $D_{\psi } {{\hat{A}}}(t, \psi _0)$ converges in probability to some limit as $n\rightarrow \infty$. Hence also,

$$\begin{aligned} n^{1/2}\{{{\hat{A}}}(t,{{\hat{\phi }}})-A_0(t)\}=n^{-1/2}\sum _{i=1}^n {{\tilde{\epsilon }}}_i^A (t) + o_p(1), \end{aligned}$$

where

$$\begin{aligned} {{\tilde{\epsilon }}}_i^A (t)= \epsilon _i^A (t)+ \mathcal{A}_{\psi }\epsilon ^{\psi }_i + {{{\mathcal {A}}}}_{\theta }\epsilon ^{\theta }_i, \end{aligned}$$

with ${{{\mathcal {A}}}}_{\psi }$ denoting the limit in probability of $D_{\psi } {{\hat{A}}}(t, \phi _0)$, and similarly with $\mathcal{A}_{\theta }$.