Quality & Quantity 35: 49–60, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Graphics for the Minimal Sufﬁcient Cause Model
Center for Health Research, Kaiser Permanente Northwest Division, 3800 N. Interstate Avenue,
Portland, OR 97227, U.S.A., e-mail: MikelAickin@kp.org
Abstract. Graphical representations of causation have been used for at least seventy years, and
the modern development of directed acyclic graphs to portray causal systems continues the trend. It
is sometimes difﬁcult to understand, however, what it is about these diagrams that is ‘causal’. The
approach to causal graphics that is taken here is to base the development of graphics on the concepts
of an underlying causal theory, the minimal sufﬁcient cause model. This leads to deﬁning a Boolean
‘and’ for arrows that represent causal pathways, and a Boolean ‘or’ for collections of pathways.
Complementation is a more complex operation in the minimal sufﬁcient cause theory than simply
inverting the sense of a causal relationship, and this also is represented in the graphics. By using
diagrams that are more faithful to underlying causal systems, and using a coherent causal theory,
both the perception of causation and its analysis might be enhanced.
Key words: causal diagrams, disease etiology, binary variables
There are many examples in science and mathematics where one can choose to take
either an analytic approach or a visual approach. In the calculus, for example, one
can think of the derivative in terms of limiting ratios, or as a drawing showing the
tangent line to a curve at a point. In simple systems, the two approaches usually
complement each other, but in complex systems the reverse is the case. Thus, if
one were to write a computer program to simulate a complex system, then the
list of equations and rules that deﬁned the system would be essential, no matter
how long the ensuing program became. In order to understand how the system
works, however, one would almost surely rely on some kind of diagrammatic
In the case of simple causal systems, Sewall Wright developed a blend of
the analytic and the visual, in the special case of genetic inheritance (Wright,
1934). The nodes of Wright’s graphs were characteristics of people, and the
arrows denoted transmission of genetic information. The underlying analytic for-
mulas were all based on the standard formula for decomposition of a covariance
(cov(x, y) = E[cov(x, y|z)]+cov(E[x|z],E[y|z])) repeated over the multiple
regression equations relating the nodes. Wright made the connection between the
two approaches by assigning a coefﬁcient to each arrow, and then giving rules by