The stochastic system approach for estimating dynamic treatments effect

Abstract

The problem of assessing the effect of a treatment on a marker in observational studies raises the difficulty that attribution of the treatment may depend on the observed marker values. As an example, we focus on the analysis of the effect of a HAART on CD4 counts, where attribution of the treatment may depend on the observed marker values. This problem has been treated using marginal structural models relying on the counterfactual/potential response formalism. Another approach to causality is based on dynamical models, and causal influence has been formalized in the framework of the Doob–Meyer decomposition of stochastic processes. Causal inference however needs assumptions that we detail in this paper and we call this approach to causality the “stochastic system” approach. First we treat this problem in discrete time, then in continuous time. This approach allows incorporating biological knowledge naturally. When working in continuous time, the mechanistic approach involves distinguishing the model for the system and the model for the observations. Indeed, biological systems live in continuous time, and mechanisms can be expressed in the form of a system of differential equations, while observations are taken at discrete times. Inference in mechanistic models is challenging, particularly from a numerical point of view, but these models can yield much richer and reliable results.

Appendix

1.1 Proof of Lemma 1

The proof can be made by induction. The result holds for $t=1$ since $f^1_{\bar{V}_{0}|\bar{A}_{0},\bar{Y}_{0}}=f^2_{\bar{V}_{0}|\bar{A}_{0},\bar{Y}_{0}}=f_{V_0}$. Assuming $f^1_{\bar{V}_{t-1}|\bar{A}_{t-1},\bar{Y}_{t-1}}=f^2_{\bar{V}_{t-1}|\bar{A}_{t-1},\bar{Y}_{t-1}}$, we prove that we have $f^1_{\bar{V}_{t}|\bar{A}_{t},\bar{Y}_{t}}=f^2_{V_{t}|\bar{A}_{t},\bar{Y}_{t}}$.

We have first that $f^1_{\bar{V}_{t}|\bar{A}_{t},\bar{Y}_{t}}= f^1_{V_t|\bar{V}_{t-1}} f^1_{\bar{V}_{t-1}|\bar{A}_{t},\bar{Y}_{t}}$ (because and ). Using the decomposition $(\bar{A}_{t},\bar{Y}_{t})=(Y_t,A_t,\bar{A}_{t-1},\bar{Y}_{t-1})$ and letting all densities conditioned on $(\bar{A}_{t-1},\bar{Y}_{t-1})$, we have from Bayes theorem:

$$\begin{aligned} f^1_{\bar{V}_{t-1}|\bar{A}_{t},\bar{Y}_{t}}=\frac{f^1_{Y_t,A_t|\bar{A}_{t-1},\bar{V}_{t-1},\bar{Y}_{t-1}}f^1_{\bar{V}_{t-1}|\bar{A}_{t-1},\bar{Y}_{t-1}}}{\int f^1_{Y_t,A_t|\bar{A}_{t-1},\bar{V}_{t-1}=\bar{v}_{t-1},\bar{Y}_{t-1}}f^1_{\bar{V}_{t-1}|\bar{A}_{t-1},\bar{Y}_{t-1}}~{\mathrm{d}}\bar{v}_{t-1}}. \end{aligned}$$

Next, we have $f^1_{Y_t,A_t|\bar{A}_{t-1},\bar{V}_{t-1},\bar{Y}_{t-1}}=f^1_{A_t|\bar{A}_{t-1},\bar{V}_{t-1},\bar{Y}_{t}}f^1_{Y_t|\bar{A}_{t-1},\bar{V}_{t-1},\bar{Y}_{t-1}}$. Because we have that $A_t \perp \perp _{\bar{A}_{t-1},\bar{Y}_t} \bar{V}_{t-1}$. Thus $f^1_{A_t|\bar{A}_{t-1},\bar{V}_{t-1},\bar{Y}_{t}}=f^1_{A_t|\bar{A}_{t-1},\bar{Y}_{t-1}}$. Since this term does not depend on $\bar{v}_{t-1}$ it can be taken out of the integral in the denominator and deleted from numerator and denominator. It is easy to see that all the other terms are equal under ${\mathrm{P}}^1$ and ${\mathrm{P}}^2$ using the assumptions of the Lemma and our inductive assumption. Hence the Lemma.