Feedback and Mediation in Causal Inference Illustrated by Stochastic Process Models

The concept of causality is naturally related to processes developing over time. Central ideas of causal inference like time-dependent confounding (feedback) and mediation should be viewed as dynamic concepts. We shall study these concepts in the context of simple dynamic systems. Time-dependent confounding and its implications are illustrated in a Markov model. We emphasize the distinction between average treatment effect, ATE, and treatment effect of the treated, ATT. These effects could be quite different, and we discuss the relationship between them. Mediation is studied in a stochastic differential equation model. A type of natural direct and indirect effects is considered for this model. Mediation analysis of discrete measurements from such processes may give misleading results, and one needs to consider the underlying continuous process. The dynamic and time-continuous view of causality and mediation is an essential feature, and more attention should be payed to the time aspect in causal inference.

Propagation Models for Biochemical Reaction Networks

Maria-Emanuela-Canini Mateescu

In this thesis we investigate different ways of approximating the solution of the chemical master equation (CME). The CME is a system of differential equations that models the stochastic transient behaviour of biochemical reaction networks. It does so by describing the time evolution of probability distribution over the states of a Markov chain that represents a biological network, and thus its stochasticity is only implicit. The transient solution of a CME is the vector of probabilities over the states of the corresponding Markov chain at a certain time point t, and it has traditionally been obtained by applying methods that are general to continuous-time Markov chains: uniformization, Krylov subspace methods, and general ordinary differential equation (ODE) solvers such as the fourth order Runge-Kutta method. Even though biochemical reaction networks are the main application of our work, some of our results are presented in the more general framework of propagation models (PM), a computational formalism that we introduce in the first part of this thesis. Each propagation model N has two associated propagation processes, one in discrete-time and a second one in continuous-time. These propagation processes propagate a generic mass through a discrete state space. For example, in order to model a CME, N propagates probability mass. In the discrete-time case the propagation is done step-wise, while in the continuous-time case it is done in a continuous flow defined by a differential equation. Again, in the case of the chemical master equation, this differential equation is the equivalent of the chemical master equation itself where probability mass is propagated through a discrete state space. Discrete-time propagation processes can encode methods such as the uniformization method and the fourth order Runge-Kutta integration method that we have mentioned above, and thus by optimizing propagation algorithms we optimize both of these methods simultaneously. In the second part of our thesis, we define stochastic hybrid models that approximate the stochastic behaviour of biochemical reaction networks by treating some variables of the system deterministically. This deterministic approximation is done for species with large populations, for which stochasticity does not play an important role. We propose three such hybrid models, which we introduce from the coarsest to the most refined one: (i) the first one replaces some variables of the system with their overall expectations, (ii) the second one replaces some variables of the system with their expectations conditioned on the values of the stochastic variables, (iii) and finally, the third one, splits each variable into a stochastic part (for low valuations) and a deterministic part (for high valuations), while tracking the conditional expectation of the deterministic part. For each of these algorithms we give the corresponding propagation models that propagate not only probabilities but also the respective continuous approximations for the deterministic variables.

EPFL2011

Causal inference in continuous time: an example on prostate cancer therapy

Pal Christie Ryalen, Mats Julius Stensrud

In marginal structural models (MSMs), time is traditionally treated as a discrete parameter. In survival analysis on the other hand, we study processes that develop in continuous time. Therefore, Røysland (2011. A martingale approach to continuous-time marginal structural models. Bernoulli 17, 895–915) developed the continuous-time MSMs, along with continuous-time weights. The continuous-time weights are conceptually similar to the inverse probability weights that are used in discrete time MSMs. Here, we demonstrate that continuous-time MSMs may be used in practice. First, we briefly describe the causal model assumptions using counting process notation, and we suggest how causal effect estimates can be derived by calculating continuous-time weights. Then, we describe how additive hazard models can be used to find such effect estimates. Finally, we apply this strategy to compare medium to long-term differences between the two prostate cancer treatments radical prostatectomy and radiation therapy, using data from the Norwegian Cancer Registry. In contrast to the results of a naive analysis, we find that the marginal cumulative incidence of treatment failure is similar between the strategies, accounting for the competing risk of other death.

2020

Propagation Models for Biochemical Reaction Networks

Maria-Emanuela-Canini Mateescu

EPFL2011