These impediments have changed dramatically in the past three decades; for example, a mathematical language has been developed for managing causes and effects, accompanied by a set of tools that turn causal analysis into a mathematical game, like solving algebraic equations or finding proofs in high-school geometry. These tools permit scientists to express causal questions formally, codify their existing knowledge in both diagrammatic and algebraic forms, and then leverage data to estimate the answers. Moreover, the theory warns them when the state of existing knowledge or the available data is insufficient to answer their questions and then suggests additional sources of knowledge or data to make the questions answerable.

The development of the tools has had a transformative impact on all da-ta-intensive sciences, especially social science and epidemiology, in which causal diagrams have become a second language. 14, 34 In these disciplines, causal diagrams have helped scientists extract causal relations from associations and deconstruct paradoxes that have baffled researchers for decades. 23, 25

I call the mathematical framework that led to this transformation “ structural causal models” (SCM), which consists of three parts: graphical models, structural equations, and counterfactual and interventional logic. Graphical models serve as a language for representing what agents know about the world. Counterfactuals help them articulate what they wish to know. And structural equations serve to tie the two together in a solid semantics.

Figure 2 illustrates the operation of SCM in the form of an inference engine. The engine accepts three inputs—Assumptions, Queries, and Data—and produces three outputs— Estimand, Estimate, and Fit indices.

Questions Answered
with a Causal Model

Consider the following five questions:

• How effective is a given treatment in preventing a disease?;

• Was it the new tax break that caused our sales to go up?;

• What annual health-care costs are attributed to obesity?;

• Can hiring records prove an employer guilty of sex discrimination?; and

• I am about to quit my job, but should I?

The common feature of these ques-
tions concerns cause-and-effect rela-
tionships. We recognize them through
such words as “preventing,” “cause,”
“attributed to,” “discrimination,” and
“should I.” Such words are common
in everyday language, and modern so-
ciety constantly demands answers to
such questions. Yet, until very recently,
science gave us no means even to ar-
ticulate them, let alone answer them.

Unlike the rules of geometry, mechanics, optics, or probabilities, the rules of cause and effect have been denied the benefits of mathematical analysis.

To appreciate the extent of this denial readers would likely be stunned to learn that only a few decades ago scientists were unable to write down a mathematical equation for the obvious fact that “Mud does not cause rain.” Even today, only the top echelon of the scientific community can write such an equation and formally distinguish “mud causes rain” from “rain causes mud.”

Figure 1. The causal hierarchy. Questions at level 1 can be answered only if information from level i or higher is available.

Level (Symbol) Typical Activity Typical Questions Examples

1. Association

P(y|x)

Seeing What is? How would seeing X change my belief in Y?

What does a symptom tell me about a disease? What does a survey tell us about the election results?

2. Intervention

P(y|do(x), z)

Doing,
Intervening
What if? What if I do X What if I take aspirin,

will my headache be cured? What if we ban cigarettes?

3. Counterfactuals

P(yx|x′, y′)

Imagining,
Retrospection
Why? Was it X that
caused Y What if I had
acted differently?

Was it the aspirin that stopped my headache? Would Kennedy be alive had Oswald not shot him? What if I had not been smoking the past two years?

Figure 2. How the SCM “inference engine” combines data with a causal model (or assumptions) to produce answers to queries of interest.

ES F

E;S Assumptions (Graphical model)

Data Fit Indices
Estimate
(Answer to query)
Query
answering the query)
(Recipe for
Estimand
Inputs Outputs

Figure 3. Graphical model depicting causal assumptions about three variables; the task is to estimate the causal effect of X on Y from non-experimental data on {X, Y, Z}.

 

XY Z