The dynamics of events occurring in continuous time can be modelled using marked point processes, or multi-state processes. Here, we review and extend the work of Røysland et al. (2015) on causal reasoning with local independence graphs for marked point processes in the context of survival analysis. We relate the results to the decision-theoretic approach of Dawid & Didelez (2010) using influence diagrams, and present additional identifying conditions.
Dynamic dependence structures among the occurrence
of different types of events in continuous time can
be represented by local independence graphs as
developed by Didelez (2006, 2007, 2008). In
As we discuss here, the approach of
The notion of dynamic dependence on which we focus here can be stated as follows. For stochastic processes X(t), Y (t), Z(t) we say informally that X(t) is locally independent of Y (t) given Z(t) if the present of X(t) is independent of the past of Y (t) given the past of both X(t), Z(t). Slightly more formally we can write this as
X(t)⊥⊥ FtY− | FtX−,Z
where Ftk are filtrations generated by Xk(t), i.e. the
sets of information becoming available over time. Note
that this is an asymmetric type of independence as
discussed in detail in
More formally we consider a marked point process
(MPP) to describe the occurrence of different types
of events E ; this can be represented by a set of
counting processes {Nj (t)} for each type of event j ∈ E . It
may often be too detailed to model the dependence
structure between all possible types of events; e.g. the
event ‘stop treatment’ can necessarily only happen
after the event ‘start treatment’ and the two events
are therefore trivially dependent. Instead of a MPP
one can therefore group certain events together to
obtain a multi-state process with several components
YV (t) = Y(t) = (Y1(t), . . . , YK (t)), V = 1, . . . , K,
where e.g. Yk(t) describes the treatment process with
states ‘on / off treatment’. Note that the components
Yk(t) need to be such that none of them systematically
change state at the same time, i.e. Y(t) is composable
Under mild regularity conditions, the Doob–Meyer Theorem tells us that each counting process can be decomposed:
Yk(t) =
Λk(t) pre|d{iczta}ble +
Mk(t) , ma|rt{izng}ale Z t
0 where Λk(t) is predictable based on the history F V− t of whole YV and Mk(t) is an FtV –martingale. We will assume that the FtV –intensity processes λk(t) exist and have the following interpretation: Λk(t) = λk(s)ds,
λk(t)dt = E(Nk(dt) | FtV− ).
From the above we see that λk(t) fully describes the dependence of a process’ infinitesimal short-term expectation on the past. Any independencies must therefore be reflected in the structure of the intensity; if we find, for instance, that λk(t) remains unchanged regardless of whether an event of type j 6= k has occurred in the past, then we say there is a local independence. Indeed, the formal definition is that Yk is locally independent of Yj given YV \{j,k} if λk(t) is FtV \{j}– measurable, i.e. the intensity process remains the same when information on the past of Yj is omitted. We write this as Yj →/ Yk | YV \{j,k}. Note that FtV \{j} always contains the past of the component Yk itself. Meek’s (2014) approach allows for cases where λk(t) is FtV \{k}–measurable.
The local independence graph G = (V, E) of a multistate process YV (t) = (Y1(t), . . . , YK (t)) (or an MPP) is given such that the absence of a directed edge indicates a local independence, i.e.
(j, k) ∈/ E ⇒ Yj →/ Yk|YV \{j,k}.
The resulting graphs are directed, can have two directed edges between any two vertices, and can have cycles. Note that pa(k)∩ch(k) 6= ∅ is possible, and similar for ancestors and decendants etc.
Under regularity conditions, the definition implies that
the intensity process λk for Yk is F cl(k)–measurable
As for conditional independence graphs, certain separations on a local independence graph imply further local independencies. However, a different notion of separation is required, δ–separation: define GB as the graph obtained after deleting all edges emanating from Y4
Y3
nodes in set B; then we say that C δ–separates A from
B in the local independence graph G if it separates A
and B in the undirected graph (GBAn(A∪B∪C))m
obtained by moralising the subgraph of GB on the
ancestral set An(A ∪ B ∪ C). Note that δ–separation
is asymmetric, i.e. δ–separating A from B is not the
same as B from A.
A key result of
t Example I: The graph in Figure 1 encodes for instance that Y1 →/ Y4 | (Y2, Y3). Using δ–separation we can verify that this is not preserved without Y3, i.e. it is not the case that Y1 →/ Y4 | (Y2). This is because of the ‘selection effect’: knowing something about the past of Y2(t) makes the past of Y1(t) informative for past of Y3(t) and therefore predictive of Y4(t). 3
So far we described a notion, and graphical
representation, of dynamic (in)dependence based on how the
present of a subprocess depends or not on the past of
other processes; in other words, a notion of time-lagged
(in)dependence. As it is based on the intensity process
it can be considered as characterised by infinitesimal
short-term predictions, which is very much parallel to
so-called ‘Granger–causality’
Y p(xi|xpa(i)). i∈V σ1 Y4 Y3
Røysland et al. (2015) extend this notion of intervention to local independence graphs by assuming that the intervention replaces the intensity process λk of Yk by a different one λ˜k, which will typically be measurable with respect to a smaller subset of processes, e.g. those relevant to and observable by the decision maker.
Remember that for a given local independence graph
G, each intensity process λk is F cl(k)–measurable.
In analogy to the influence diagrams of
t
Similar to the case of longitudinal data, it turns out that inference about the dynamics between events under the intervened system can be obtained by re-weighting. Specifically the weights are given as W (t) :=
Y s≤t λ˜k(s) !ΔNk(s) λk(s) exp
For these to be well-defined, in particular for P˜ << P , we need W (t) to be uniformly integrable which can be interpreted as λk(t), λ˜k(t) not being ‘too different’, e.g. W (t) could be uniformly bounded. In fact, if Λk(t) is assumed absolutely continuous such that λk(t) exists, then it is e.g. not possible to re-weight with an intervention that has discrete jumps of Nk(t) at fixed time points. Note that this can be regarded as correspondent of the ‘positivity’ condition typically made in many causal inference contexts.
In the context of survival or duration data it is almost inevitable to have censoring (e.g. due to the end of the study). Censoring in itself can be regarded as an event and modelled with a counting process that jumps when the observation is censored. This then allows us to express assumptions about the censoring in terms of its intensity process. A common assumption is independent censoring which can be stated as the relevant process (e.g. survival) being locally independent of the censoring process, possibly conditional on other observed processes. The most obvious violation of this assumption occurs when there are unobserved common causes for censoring and survival.
Moreover, censoring can be linked to the above ideas
of intervention and re-weighting in the following sense.
The target of inference is typically a population where
no censoring occurs (e.g. future patients) or where
censoring is entirely random and stochastically
independent of other processes. Hence we can say that the
target is to replace the censoring intensity by a
different intensity that does not depend on the past. When
this is possible given the observed processes
therefore depends among others on whether the local
independence graph on all events including censoring is
causally valid wrt. the censoring process.
In the following we assume that the index set of processes is V = V0 ∪ X ∪ L ∪ U where V0 are observable processes of interest (‘outcome’ processes), X (or counting process NX ) is the process in which we want to intervene changing its intensity, L is a set of observable processes in which we are not interested, and U is a set of unobservable processes.
Let G be the local independence graph for processes
V = V0 ∪ X ∪ L ∪ U ; assume causal validity wrt. X.
Consider an intervention in X that changes its
observational F V –intensity λX to a F V0 –intensity λ˜X . We
say that the effect of such an intervention on V0 is
identified by L if the F V0 –intensities for every
counting process N ∈ V0 under the intervention exist and
are given by re-weighting with the above weights W (t).
In the situation of Definition 1, if U →/ X | (V0 ∪ L), then the effect on V0 of intervening in X is identified by L.
Example I (ctd.): In Figure 2, assume we are interested in the effect of an intervention in X = Y1 on V0 = Y4 and let L = Y2 and U = Y3. Then we see that Proposition 2 is satisfied, meaning that re-weighting will allow us to compute aspects of the possibly modified behaviour of Y4 under an intervention that changes the intensity process of Y1, where the weights require no observation of Y3.
The condition of Proposition 2 is the point
process analogue of sequential randomisation in
With the preconditions of Definition 1, and the augmented local independence graph Gσ with intervention node σX , we define that simple stability holds if σX →/ (L ∪ V0) | X.
We conjecture that identification can in fact be obtained under the wider assumption of simple stability.
Assume the preconditions of Definition 1, and the augmented local independence graph Gσ (i.e. causal validity wrt. X).
If simple stability holds, then the effect on V0 of intervening in X is identified by L.
The condition of Proposition 2 implies simple stability. We can now formulate a result corresponding to Dawid L V0 & Didelez’ (2010) notion of ‘sequential irrelevance’; this condition allows unobserved processes in U to affect the treatment process X as long as they are ‘irrelevant’ to the other processes of interest.
Assume the preconditions of Definition 1, and the augmented local independence graph Gσ (i.e. causal validity wrt. X). Then U →/ (V0 ∪ L) | X implies simple stability.
Both, Corollary 5 and 6 are sufficient but not necessary for simple stability as the following example demonstrates.
Example II: The graph in Figure 3 shows a situation where U = (U1, U2) satisfies neither Proposition 2 nor Corollary 6. However, simple stability is satisfied. Note that U1 alone fulfills Corollary 6 and U2 alone Proposition 2. All these would be destroyed by an edge between U1 and U2. 5
More generality? In the time-discrete case, more
general conditions for causal effect identification can and
have been given than those analogous to simple
stability. Specific to sequential decisions in longitudinal
data these are for example addressed in
Why an intervention indicator? The decision theoretic
approach to causality makes it formally and
graphically explicit that an intervention in a particular node
is being considered and what assumptions are involved
Causal Search? We assumed that the local
independence graph is given and that subject matter
knowledge justifies causal validity wrt. certain events or
processes.
Proof of Conjecture 4: see
Remember that in the augmented local independence graph Gσ, assuming causal validity wrt. X, there is only a single edge involving σX pointing into X. Further, the condition of Proposition 2 is satisfied iff U ∩pa(X) = ∅ in G. The graphical check of δ–separation for simple stability involves removing all outgoing edges from V0 ∪ L; in the resulting graph before moralisation, there are no edges into X except the one from σX . Hence, in the moral graph, σX only has an edge with X and Definition 3 is satisfied.
As above, in the augmented local independence graph Gσ there is only a single edge involving σX pointing into X. The graphical check of δ–separation for simple stability, furthermore, involves removing all outgoing edges out of V0 ∪ L and with the condition of Corollary 6 this means that there are no edges between U and V0 ∪ L at all. Hence, even if there are moral edges between σX and U these do not lead to paths between V0 ∪ L and σX in the relevant moral graph and Definition 3 is satisfied.
Financial support f