Using instrumental variables to estimate a Cox’s proportional hazards regression subject to additive confounding

Abstract

The estimation of treatment effects is one of the primary goals of statistics in medicine. Estimation based on observational studies is subject to confounding. Statistical methods for controlling bias due to confounding include regression adjustment, propensity scores and inverse probability weighted estimators. These methods require that all confounders are recorded in the data. The method of instrumental variables (IVs) can eliminate bias in observational studies even in the absence of information on confounders. We propose a method for integrating IVs within the framework of Cox’s proportional hazards model and demonstrate the conditions under which it recovers the causal effect of treatment. The methodology is based on the approximate orthogonality of an instrument with unobserved confounders among those at risk. We derive an estimator as the solution to an estimating equation that resembles the score equation of the partial likelihood in much the same way as the traditional IV estimator resembles the normal equations. To justify this IV estimator for a Cox model we perform simulations to evaluate its operating characteristics. Finally, we apply the estimator to an observational study of the effect of coronary catheterization on survival.

Appendices

Appendix 1: The marginal is Cox proportional hazards model

Here we show that if the treatment $X$ is applied independently of $U$ in model (1) that the marginal distribution of the time-to-event with respect to $X$ is proportional hazards. This is relevant because it shows that parameter $\beta$ of that model has an intepretation as the logarithm of the hazard ratio comparing the potential outcome if a subject recieves $X=1$ to the potential outcome if a subject recieves $X=0$ and $U$ is unknown (i.e. not conditioned on).

The survival curve corresponding to model (1), $S(t;x|U)={\mathrm{Pr}}[\widetilde{T}(x)\ge t|U]$ is

$$\begin{aligned} \exp \left[ -\int \limits _0^t \bigl \{ \lambda _{01}(t)\exp (\beta ^{\prime }x)+ h(u,t) \}dt \right] = \exp \left[ -\Lambda _{01}(t)\exp (\beta ^{\prime }x)\right] \exp \left[ -H(u,t) \right] \nonumber \end{aligned}$$

where $H(u,t)=\int _0^t h(u,t^{\prime })dt^{\prime }$. If $X$ is applied independently of $U$ then the marginal causal model for the survival curve, obtained by integrating over $U$ equals

$$\begin{aligned} S(t;x)&= \int S(t;x|u) dF_U = \int \exp \biggl [-\Lambda _{01}(t)\exp (\beta ^{\prime }x)\biggr ] \exp \biggl [-H(u,t) \biggr ] dF_U(u) \nonumber \\&= \exp \biggl [-\Lambda _{01}(t)\exp (\beta ^{\prime }x)\biggr ] \int \exp \biggl [-H(u,t) \biggr ] dF_U(u). \end{aligned}$$

(7)

Suppose $H(u,t)=A(t)u + B(t)$. Then (7) becomes

$$\begin{aligned} S(t;x)&= \exp \biggl [-\Lambda _{01}(t)\exp (\beta ^{\prime }x)\biggr ] \exp \biggl [-B(t) \biggr ] \int \exp \biggl [-A(t)u \biggr ] dF_U(u) \end{aligned}$$

(8)

$$\begin{aligned}&= \exp \biggl [-\Lambda _{01}(t)\exp (\beta ^{\prime }x)\biggr ] \exp \biggl [-B(t) \biggr ] {\mathrm{mgf}}_U[-A(t)]. \end{aligned}$$

(9)

Therefore if we take $B(t)=\ln {\mathrm{mgf}}_U[-A(t)]$ the marginal causal survival curve is $S(t;x)=\exp \biggl [-\Lambda _{01}(t)\exp (\beta ^{\prime }x)\biggr ]$ which is a Cox proportional hazards model.

This particular causal conditional model has the property that E$[h(u,t)|\widetilde{T}(x) \ge t]$ equals zero as demonstrated below:

$$\begin{aligned} {\mathrm{E}}[h(U,t)|\widetilde{T}(x) \ge t]&= \int h(u,t) d{\mathrm{Pr}}[U \le u| \widetilde{T}(x) \ge t] \nonumber \\&= \int h(u,t) {\mathrm{Pr}}[\widetilde{T}(x) \ge t|U=u]\,dF_U(u) / S(t;x) \nonumber \\&= \int h(u,t) \exp [-\exp (\beta ^{\prime }x)\Lambda _0(t) -H(u,t)]\,dF_U(u) / S(t;x) \nonumber \\&= \exp [-\exp (\beta ^{\prime }x)\Lambda _0(t)] \int h(u,t) \exp [-H(u,t)]\,dF_U(u) / S(t;x) \nonumber \\&= S(t;x) \int h(u,t) \exp [-H(u,t)]\,dF_U(u) / S(t;x) \nonumber \\&= \frac{d}{dt}\biggl \{ -\int \exp [-A(t)u-B(t)]\,dF_U(u) \biggr \} \nonumber \\&= -\frac{d}{dt}\biggl \{\exp [-B(t)] {\mathrm{mgf}}_U(-A(t)) \biggr \} \nonumber \\&= 0, \end{aligned}$$

(10)

when $B(t)=\ln {\mathrm{mgf}}_U[-A(t)]$.

Appendix 2: R code for implementing the estimating equation