Quality & Quantity 38: 533–545, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Center for Health Research, Kaiser Permanente Northwest Region, 3800 North Interstate Avenue,
Portland, Oregon 97227-1098
Abstract. The Directed Acyclic Graph (DAG) theory of causation is based on the assumption that
randomly sampling the variables of a causal system will yield a joint probability distribution that
satisﬁes the Markovian condition. It is shown here that this condition can be split into two parts,
one of which is named the Millsian condition. It is further shown that the Millsian condition alone
implies that causally unrelated sets of variables are conditionally independent given their common
causes, very likely a key requirement stated by John Stuart Mill 150 years ago. In Millsian causation,
unlike Markovian causation, it is possible for an indirect cause to be associated with its effect even
when controlling for the intermediate direct causes. This phenomenon is explained by taking into
account the existence of potential causal modulation.
Key words: causal inference, conditional independence, directed acyclic graphs, hypertension,
interaction, Markov property, modulator
In 1856, John Stuart Mill wrote that
Whatever phenomenon varies in any manner whenever another phenomenon
varies in some particular manner, is either a cause or an effect of that phe-
nomenon, or is connected with it through some fact of causation (Mill,
In the same work he also wrote
If an instance in which the phenomenon under investigation occurs, and an
instance in which it does not occur, have every circumstance in common save
one, that one occurring only in the former; the circumstance in which alone
the two instances differ, is the effect, or cause, or a necessary part of the cause,
of the phenomenon.
At the risk of translating 19th Century words too easily into 21st Century concepts,
we might say that Mill was getting at the following two basic aspects of causation.
First, all covariation between variables is due to causal connections between them
(which might involve other variables). Secondly, if none of the common causes of
two variables change over time, but the two variables do, then one is (at least part