On collapsibility and confounding bias in Cox and Aalen regression models

Abstract

We study the situation where it is of interest to estimate the effect of an exposure variable $X$ on a survival time response $T$ in the presence of confounding by measured variables $Z$. Quantifying the amount of confounding is complicated by the non-collapsibility or non-linearity of typical effect measures in survival analysis: survival analyses with or without adjustment for $Z$ typically infer different effect estimands of a different magnitude, even when $Z$ is not associated with the exposure, and henceforth not a confounder of the association between exposure and survival time. We show that, interestingly, the exposure coefficient indexing the Aalen additive hazards model is not subject to such non-collapsibility, unlike the corresponding coefficient indexing the Cox model, so that simple measures of the amount of confounding bias are obtainable for the Aalen hazards model, but not for the Cox model. We argue that various other desirable properties can be ascribed to the Aalen model as a result of this collapsibility. This work generalizes recent work by Janes et al. (Biostatistics 11:572–582, 2010).

Appendix: Large sample properties

1.1 Aalen model

Let us first look at the Aalen model. Let $Y(t)$ be the $n\times 3$-matrix with $i$th row equal to $Y_i(t)(1,X_i,Z_i)$ where $Y_i(t)$ is the at-risk indicator, and let $Y^m(t)$ be the $n\times 2$-matrix with $i$th row equal to $Y_i(t)(1,X_i)$. We let $Y^{-}(t)=\{Y^T(t)Y(t)\}^{-1}Y^T(t)$ and similarly with $\{Y^m\}^{-}(t)$. Also let $A=(0,1,0)$ and $A^m=(0,1)$, and let $N(t)$ denote the vector-counting process $(N_1(t),\ldots ,N_n(t))^T$ with $N_i(t)=I(U_i\le t,\delta _i)$. Since, with $B^m(t)=\{B_0^m(t),B_X^m(t)\}$,

$$\begin{aligned} {\hat{B}}_X(t)=\int _0^tAY^{-}(s)dN(s),\quad {\hat{B}}_X^m(t)=\int _0^tA^m\{Y^m\}^{-}(s)dN(s), \end{aligned}$$

using the ordinary least squares inverses, it follows that

$$\begin{aligned} \epsilon _i^B(t)=\int _0^tAR^{-1}(s)Y_i(s)dM_i(s)- \int _0^tA^m\{R^m\}^{-1}(s)Y^m_i(s)dM^m_i(s), \end{aligned}$$

where $M_i$ and $M_i^m$ denote the counting process martingales with respect to the marginal (conditioning on $X$) and the conditional (conditioning on $X$ and $Z$) filtrations. In the latter display, $R(t)$ is the limit in probability of $n^{-1}Y^T(t)Y(t)$ and similarly with $R^m(t)$.

1.2 Cox model

We here sketch the proofs of the asymptotic results for the Cox model. The process $W_n(t)$ consists of differences like

$$\begin{aligned} n^{1/2}[\log {[{\hat{S}}_j \{x,{\hat{\theta }}(t)\}]}- \log {[S_j \{x,\theta (t)\}]}]\!&= \!\frac{n^{1/2}[{\hat{S}}_j \{x,{\hat{\theta }}(t)\}-S_j \{x,\theta (t)\}]}{S_j\{x,\theta (t)\}}\!+\!o_p(1)\nonumber \\ n^{1/2}[\log {[{\hat{S}}_j^x\{x,{\hat{\theta }}(t)\}]}{-}\log {[S_j^x\{x,\theta (t)\}]}]\!&= \!\frac{n^{1/2}[\hat{S}_j^x\{x,{\hat{\theta }}(t)\}{-}S_j^x\{x,\theta (t)\}]}{S_j^x\{x,\theta (t)\}}\!{+}\!o_p(1),\nonumber \\ \end{aligned}$$

(10)

where

$$\begin{aligned} S_j\{x,\theta (t)\}=E[S_j\{x,\theta (t);Z\}],\; {\hat{S}}_j\{x,\theta (t)\}=n^{-1}\sum _{i=1}^nS_j\{x,\theta (t);Z_i\}, \end{aligned}$$

and

$$\begin{aligned} S_j^x\{x,\theta (t)\}&= E[S_j\{x,\theta (t);Z\}|X=x],\; {\hat{S}}_j^x\{x,\theta (t)\}\\&= n_x^{-1}\sum _{i=1}^nS_j\{x,\theta (t);Z_i\}I(X_i=x) \end{aligned}$$

with $n_x=\sum _{i=1}^nI(X_i=x)$ and $x$ being fixed. We now first show that

$$\begin{aligned} n^{1/2}[{\hat{S}}_j\{x,{\hat{\theta }}(t)\}-S_j\{x,\theta (t)\}]=n^{-1/2}\sum _{i=1}^n{\tilde{\epsilon }}_i^{S_j}(t,x)+o_p(1) \end{aligned}$$

(11)

where $\left\{ {\tilde{\epsilon }}_i^{S_j}(t)\right\} _i$ are zero-mean iid terms. This is similar to the proof in (Chen et al. (2010), p. 724–725). Let $\mathcal{P }_n$ and $P$ denote the empirical measure and the distribution under the true model, respectively. We write $\int fdQ$ as $Qf$, where $Q$ here denotes a measure. With this notation, the left hand side of (11) can then be written as

$$\begin{aligned}&n^{1/2}(\mathcal{P }_n-P)[S_j\{x,\theta (t);Z\}-S_j\{x,\theta (t)\} ]+Pn^{1/2}[S_j\{x,{\hat{\theta }}(t);Z\}\nonumber \\&\quad -S_j\{x,\theta (t);Z\}]+n^{1/2}(\mathcal{P }_n-P)[S_j\{x,{\hat{\theta }}(t);Z\}-S_j\{x,\theta (t);Z\} ], \end{aligned}$$

(12)

where the expectations $\mathcal{P }_nS_j\{x,{\hat{\theta }}(t);Z\}$ and $PS_j\{x,{\hat{\theta }}(t);Z\}$ are taken with respect to $Z$. As in Chen et al. (2010), the third term in (12) can be shown to converge to zero in probability using Lemma 19.24 of van der Vaart (1998) and because of the asymptotic properties of the partial likelihood estimator of the Cox model. The first term in (12) is already of the desired form so we can concentrate on the second term. We have

$$\begin{aligned} n^{1/2}[S_j\{x,{\hat{\theta }}(t);z\}-S_j\{x,\theta (t);z\}] =\dot{S}_j\{x,\theta (t);z\}n^{-1/2} \sum _{i=1}^n\epsilon _i^{\theta }(t)+o_p(1) \end{aligned}$$

with $\dot{S}_j$ denoting the derivative of $S_j$ with respect to the second (vector)-argument, and where $\{{\epsilon }_i^{\theta }(t)\}_i$ is the iid decomposition corresponding to $n^{1/2}\{{\hat{\theta }}(t)-\theta (t)\}$; expressions for these can be found in Martinussen and Scheike (2006) Ch. 7, and their empirical counterparts can be retrieved from the cox.aalen-function in the R-package timereg. Hence, (11) is fulfilled and therefore also

$$\begin{aligned} n^{1/2}\left[\log {[{\hat{S}}_j\{x,{\hat{\theta }}(t)\}]}- \log {[S_j\{x,\theta (t)\}]}\right]=n^{-1/2}\sum _{i=1}^n \epsilon _i^{S_j}(t,x)+o_p(1), \end{aligned}$$

where

$$\begin{aligned} \epsilon _i^{S_j}(t,x)=\left[S_j\{x,\theta (t);Z_i\}-S_j\{x,\theta (t)\} +E[\dot{S}_j\{x,\theta (t);Z\}]\epsilon _i^{\theta }(t)\right]/S_j\{x,\theta (t)\} \end{aligned}$$

are zero-mean iid terms. To handle (10) we let $n_x^*=n^{-1}\sum _{i=1}^nI(X_i=x)$ and assume that its limit in probability, $p_x$, say, is larger than zero. Now, rewrite the numerator of the right hand side of (10) as

$$\begin{aligned} n^{1/2}[{\tilde{S}}_j^x\{x,{\hat{\theta }}(t)\}-S_j^x\{x,\theta (t)\}]- \bigg [n^{-1/2}\sum _{i=1}^n\epsilon _i^{p_x} \bigg ]S_j^x\{x,\theta (t)\}p_x^{-1}+o_p(1), \end{aligned}$$

(13)

where

$$\begin{aligned} {\tilde{S}}_j^x\{x,\theta (t)\}=n^{-1}\sum _{i=1}^nS_j\{x,\theta (t);Z_i\}I(X_i=x)p_x^{-1},\; \epsilon _i^{p_x}=I(X_i=x)-p_x. \end{aligned}$$

The first term in (13) can be dealt with as we did with the left hand side of (11), and the second term in (13) is already on the desired form. Thus

$$\begin{aligned} n^{1/2}\left[\log {[{\hat{S}}_j^x\{x,{\hat{\theta }}(t)\}]}- \log {[S_j^x\{x,\theta (t)\}]}\right]=\sum _{i=1}^n \epsilon _i^{S_j^x}(t,x)+o_p(1), \end{aligned}$$

where

$$\begin{aligned} \epsilon _i^{S_j^x}(t,x)&= \biggl [S_j^x\{x,\theta (t);X_i,Z_i\}-S_j^x\{x,\theta (t)\} +E[\dot{S}_j^x\{x,\theta (t);X,Z\}]\epsilon _i^{\theta }(t)\\&\quad -\epsilon _i^{p_x} S_j^x\{x,\theta (t)\}p_x^{-1}\biggr ]/S_j^x\{x,\theta (t)\} \end{aligned}$$

are zero-mean iid terms with $S_j^x\{x,\theta (t);X,Z\}=S_j\{x,\theta (t);Z\}I(X=x)p_x^{-1}$, and with $\dot{S}_j^x\{x,\theta (t);X,Z\}$ being its derivative with respect to the second (vector)-argument. Hence, $W_n(t)=n^{-1/2}\sum _{i=1}^n\epsilon _i^{\Delta }(t)+o_p(1),$ with

$$\begin{aligned} \epsilon _i^{\Delta }(t)=\epsilon _i^{\Delta _1}(t)-\epsilon _i^{\Delta _2}(t) \end{aligned}$$

(14)

being zero-mean iid terms. In (14),

$$\begin{aligned} \epsilon _i^{\Delta _1}(t)&= \epsilon _i^{S_1}(t,1)-\epsilon _i^{S_0}(t,1)+\epsilon _i^{S_0}(t,0)-\epsilon _i^{S_1}(t,0)\\ \epsilon _i^{\Delta _2}(t)&= \epsilon _i^{S_1^1}(t,1) -\epsilon _i^{S_0^1}(t,1)+\epsilon _i^{S_0^0}(t,0)-\epsilon _i^{S_1^0}(t,0). \end{aligned}$$

It readily follows that $\epsilon _i^{\mathcal{T }}(t)=\epsilon _i^{\Delta _1}(t)+\epsilon _i^{\beta _X},$ where $\epsilon _i^{\beta _X}$ is the first component of $\epsilon _i^{\theta }(t)$. Also $\epsilon _i^{\Delta _{nl}}(t)=\epsilon _i^{\Delta _1}(t).$