A historical & pedagogical exploration
An association tells us who gets sick. A causal explanation asks what would change if we acted. The tobacco debate makes that distinction concrete.
Begin the lessonKeep two levels separate. Smoking causes lung cancer; that conclusion rests on a large body of evidence. The probabilities and diagrams below are deliberately simplified teaching models. They are not estimates of real cancer risk. Surgeon General’s evidence summary.
01 / The historical problem
The smoking–cancer debate grew out of a changing society and a changing pattern of disease. To understand its importance, begin with the world in which cigarettes became ordinary and a causal conclusion became consequential. This account focuses on Britain and the United States within a wider international history of research.
Smoking tobacco long predated the modern cigarette industry. What changed in the late nineteenth and early twentieth centuries was the scale of cigarette production and promotion. Mechanized manufacture helped transform cigarettes into a mass-market product. By the 1950s, smoking was widespread in Britain and the United States, embedded in daily routines and supported by a powerful commercial industry. National Cancer Institute, historical overview.
At the same time, physicians were confronting a substantial rise in lung cancer. Wynder and Graham opened their 1950 paper by discussing increases recorded in hospitals, autopsies, and mortality statistics. The problem was therefore larger than explaining a few unusual patients: something appeared to be changing at the population level. Wynder & Graham, 1950.
The connection was difficult to read directly from everyday experience. Exposure and disease could be separated by decades. A habit might feel normal for years before its consequences became visible, and population mortality could lag behind changes in smoking. Comparing what was happening in the same calendar year could therefore be misleading. Royal College of Physicians, historical review.
A correlation between two rising trends did not settle the explanation. Researchers had to consider whether smokers and nonsmokers differed in other relevant ways. Air pollution and inherited susceptibility were among the alternatives discussed at the time. The 1962 Royal College of Physicians report explicitly distinguished agreement about the observed association from disagreement over its causal interpretation. RCP report, introduction.
There were also questions about the evidence itself. Did the comparison group represent the population that produced the cancer cases? Were smoking histories recorded reliably? Had exposure occurred before disease developed? These questions point to different problems—selection, measurement, and timing—and call for different ways of strengthening a study. They cannot all be resolved by increasing the number of observations.
Compare specific alternatives, seek evidence that distinguishes them, and ask whether findings survive changes in study design.
A causal explanation must speak to a change in the world. Predicting who becomes ill and identifying what prevents illness are different tasks.
The influential studies of 1950 were not the first warnings about smoking. They made the problem harder to dismiss by comparing substantial groups of people systematically. In the United States, Ernst Wynder and Evarts Graham studied 684 confirmed lung-cancer cases; in Britain, Doll and Hill compared smoking histories among hospital patients. Evidence was emerging from different investigators and settings. CDC account of Wynder’s work; Doll & Hill, 1950.
Prospective studies then changed the direction of inquiry: record smoking first, and follow subsequent mortality. Laboratory findings contributed another kind of evidence; experiments reported in 1953 produced tumors with cigarette-smoke condensate in mice. Such results did not reproduce human smoking exactly, but they informed the question of biological plausibility. The historical argument grew through the combination of evidence, with each approach answering some questions and leaving others open. Doll & Hill, 1954; Surgeon General’s historical review.
In January 1954, major tobacco companies issued A Frank Statement to Cigarette Smokers. The advertisement questioned the evidence linking cigarettes to cancer, assured readers of the manufacturers’ concern for health, and announced an industry research committee. It is a revealing primary source: uncertainty about causation was also being communicated by organizations with a commercial stake in the answer. Read the original statement in the UCSF archive.
For this lesson, the useful distinction is between an objection that can be investigated and a demand for certainty that leaves no practical route to a conclusion. Asking whether a particular common cause explains the association invites further evidence. Treating every remaining uncertainty as a reason to disregard all accumulated evidence does something different. This is a way to analyze the debate, not a claim that every scientific critic had the same motives.
The Royal College of Physicians’ 1962 report brought the evidence to a broad public and called for measures to reduce smoking. Its institutional history records disagreement even over whether a medical college should advise the public on what to do. Scientific assessment, professional responsibility, commercial interests, and government policy were becoming intertwined. The U.S. Surgeon General’s report followed in 1964. RCP, 1962 report; RCP’s account of the institutional debate.
What this history teaches: causal knowledge developed by confronting competing explanations with different kinds of evidence. The diagrams later in this lesson give that reasoning a precise language. They simplify the scientific problem; they do not replace the historical work that made a causal conclusion credible.
Read each milestone as a contribution to an argument. The later mathematical formulation should not be projected backward onto how the early studies were conducted.
Richard Doll and Austin Bradford Hill compared smoking histories of hospital patients with lung cancer and patients with other conditions. The case–control study found a strong association. Its design also invited questions about how patients were selected and how past exposure was recorded.
Method lesson: an association is evidence to explain.The British doctors study recorded smoking habits and then followed mortality; initial results appeared in 1954. Recording exposure before the outcome strengthened the temporal argument and provided a different route to evidence. The study was still observational: doctors were not randomly assigned to smoke.
Method lesson: change the design, then compare the evidence.R. A. Fisher challenged the causal interpretation, including the possibility that inherited characteristics could influence both smoking and cancer. Such a possibility is a confounding hypothesis. Drawing it is useful for reasoning; drawing it does not establish that it explains the evidence.
Method lesson: make the alternative explanation explicit.Fisher’s 1957 letter · Fisher, Nature, 1958 (archived transcription)
The Royal College of Physicians published Smoking and Health, argued for the harm caused by smoking, and urged public-health measures and advice to patients. It helped move the discussion from research findings to institutional responsibility and action.
Method lesson: deciding what the evidence means also raises the question of what to do.The first U.S. Surgeon General’s report concluded that cigarette smoking caused lung cancer in men; its assessment for women was then “probable.” This landmark synthesized accumulated evidence. It was not a deduction from one correlation table or the later front-door formula.
Method lesson: a historical verdict concerns a body of evidence.Hill discussed viewpoints such as strength, consistency, timing, dose–response, and biological plausibility. He did not present them as a mechanical checklist that guarantees causation. They guide judgment about the whole explanation.
Method lesson: statistical significance does not settle causation.Judea Pearl formalized rules for identifying intervention effects from observational distributions and causal diagrams. Front-door identification provides a striking example: a measured mediator can sometimes allow identification even when a common cause is hidden.
Method lesson: distinguish evidence for assumptions from consequences of assumptions.A question to carry forward: What did each new study or argument add that a bigger version of the same correlation table could not?
02 / Two different questions
P(C = 1 | S = 1)
Among people observed to smoke, what proportion have cancer? We select people whose smoking status is 1. They may also differ in other ways.
Conditioning filters the population. It does not change how smoking was determined.
P(C = 1 | do(S = 1))
In the model, what proportion would have cancer if smoking were externally set to 1? We replace the process that normally determines smoking.
In a causal graph, remove arrows entering S. Keep arrows leaving S.
The intervention notation and graph operation follow Pearl (1995). Here do describes a hypothetical model operation, not a proposal for a tobacco experiment.
The comparison we want: P(C = 1 | do(S = 1)) − P(C = 1 | do(S = 0)). This is a difference in probabilities, reported below in percentage points. It is a population comparison, not a prediction of one person’s outcome.
03 / Three causal stories
Consider a hypothetical population with the probabilities shown below. Smoking and cancer are associated in these synthetic data. For this exercise, treat the probabilities as known exactly; sampling uncertainty is a separate issue.
| Observed group | Share of population | P(C = 1 | group) |
|---|---|---|
| S = 0 | 60% | 10% |
| S = 1 | 40% | 30% |
Overall P(C = 1) = 0.60 × 0.10 + 0.40 × 0.30 = 18%. The observed difference is 20 percentage points.
Interactive model explorer
U represents an unobserved factor that could influence both smoking and cancer. Its role is a hypothesis to examine in these teaching models.
Observed probability of cancer
The observed distribution and this graph do not determine the intervention effect. More than one causal system can fit them and give different effects.
Removing U → S defines the intervention, but does not reveal the unknown causal mechanisms. P(S, C) alone cannot supply their unique numerical answer.
The two observed smoking groups contain different mixtures of U. That can make their cancer probabilities differ even without an arrow S → C. Setting S externally leaves U—and therefore C—distributed as before. Both intervention groups have the original population cancer probability, 18%.
The zero effect follows from this graph’s assumption that smoking has no causal route to cancer. It is not a conclusion about real tobacco.
“Not identifiable” does not mean “no effect.” Model 3 permits an effect but does not determine its size from these data. Even an infinitely large sample of S and C would not by itself resolve this ambiguity. Additional design, measurements, or justified assumptions are needed.
| Model | Observe S = 0 / 1 | do(S = 0) / do(S = 1) | Causal difference |
|---|---|---|---|
| Direct effect | 10% / 30% | 10% / 30% | +20 pp |
| Common cause only | 10% / 30% | 18% / 18% | 0 pp |
| Both pathways | 10% / 30% | Not identifiable | Not identifiable |
Dataset A is a synthetic probability distribution for exploring the three hypotheses. The historical timeline should not be read as a literal sequence of these diagrams.
04 / The front-door idea
Now introduce a measured mediator, T = Tar, between smoking and cancer. We now use dataset B, a different synthetic distribution. The new variable comes with new assumptions—not an automatic solution to hidden confounding.
S → T → C
U → S and U → C
U still confounds the relation between S and C. T lies on the causal route from S to C. “Tar present” is a binary abstraction used for this exercise.
Also require the relevant observed combinations to have positive probability, so the conditionals can be estimated.
Front-door conditions: Pearl (1995), §3.2. A back-door path starts with an arrow pointing into the exposure under discussion.
A mathematical illustration, not a biological claim. Complete mediation through this one binary “Tar” variable is a strong teaching assumption. This tutorial does not establish that it holds for real smoking and cancer, or that this was how the historical causal conclusion was reached.
P(S = 0) = 0.60 and P(S = 1) = 0.40.
| Smoking | P(T = 1 | S) | P(C = 1 | S, T = 0) | P(C = 1 | S, T = 1) |
|---|---|---|---|
| S = 0 | 0.10 | 0.10 | 0.20 |
| S = 1 | 0.30 | 0.30 | 0.40 |
Here the observed cancer probabilities are 0.90 × 0.10 + 0.10 × 0.20 = 11% for S = 0, and 0.70 × 0.30 + 0.30 × 0.40 = 33% for S = 1. Do not mix these with dataset A’s 10% and 30%.
With no confounding between S and T, P(T | do(S = s)) equals the observed P(T | S = s). This supplies the weights for the final average.
For the effect of T on C, S blocks the path T ← S ← U → C. Average P(C | T, S) over the population smoking distribution P(S), not P(S | T). Call the result Q(t).
Because all of smoking’s effect travels through T, average Q(t) using the tar distribution induced by setting S. Every ingredient is an observed probability.
s is the smoking value we set. s′ ranges over the observed population when computing Q(t).
Interactive front-door calculator
With all three structural conditions, the intervention probability is identified from dataset B.
Inner average: use the same population weights, 0.60 and 0.40, for each tar level.
For S = 1, 70% have T = 0 and 30% have T = 1 in dataset B.
The observational probabilities stay fixed. Changing the assumed graph can change whether they identify a causal answer.
The second comparison is justified only for the valid front-door graph. These differences describe the synthetic example; neither is an estimate of the effect of real smoking.
Valid graph: do(S = 0) gives 0.90 × 0.18 + 0.10 × 0.28 = 19%; do(S = 1) gives 0.70 × 0.18 + 0.30 × 0.28 = 21%.
Adding S → C violates complete mediation. Adding U → T opens confounding of S → T and an unblocked T ← U → C path. In either expanded graph, the front-door calculation is no longer an identified total effect; this tutorial therefore withholds that causal number.
All inputs and calculations for dataset B are shown above. You can reproduce each result with a calculator or by hand.
05 / From models to evidence
A calculation is the last part of an argument. Before using it, we need a clear question, a defensible model, and data that match the question. The historical studies and the interactive examples illuminate different parts of that work.
Specify the population, the exposure change, and when the outcome will be measured. In this lesson, S = 0 and S = 1 are simple binary states. In a real study, never smoking, quitting after years of smoking, and reducing consumption would describe different changes.
Use timing, substantive knowledge, and evidence from different study designs to support the proposed mechanisms. Ask what U could represent. An absent arrow also makes a claim: removing S → C from the front-door graph requires all of the effect to pass through T.
Could two causal systems satisfy the same assumptions and fit the same observations, yet predict different intervention effects? If so, the effect is not identified. Model 3 illustrated this problem; the front-door assumptions supplied one way to overcome it in a different model.
With an identified formula, estimate its ingredients from observations. Consider sampling uncertainty, measurement error, and how people entered the study. Then ask how the conclusion would change if a key causal assumption were wrong.
A question about the data and causal assumptions, even if the observed probabilities were known exactly.
A question about learning the identified quantity from a finite sample. Our worked examples set this uncertainty aside.
A question about the credibility of the mechanisms, measurements, and study design. A formula cannot answer it on its own.
Return to the historical problem. A prospective study could strengthen the temporal argument. Repeated findings across different designs could challenge particular explanations of bias. These contributions help justify a causal interpretation; the diagrams make explicit what that interpretation permits us to calculate.
On weighing evidence: Hill (1965). On identification from causal diagrams: Pearl (1995).
These additional thought experiments test whether we distinguish observing a variable from setting it. In the valid front-door graph, intervening on C cuts its incoming arrows and cannot change its ancestors S or T.
| Query (dataset B) | Result | Reason |
|---|---|---|
| P(S = 1 | do(C = 1), T = 1) | 66.7% | Observing tar is informative about smoking: 0.4 × 0.3 / (0.6 × 0.1 + 0.4 × 0.3). |
| P(S = 1 | do(C = 1)) | 40% | Setting a downstream variable leaves the smoking distribution unchanged. |
| P(S = 1 | do(C = 1), do(T = 1)) | 40% | Setting tar severs S → T; it does not select people whose natural tar value was 1. |
| P(T = 1 | do(C = 1), do(S = 1)) | 30% | The unconfounded S → T mechanism supplies P(T = 1 | S = 1). |
These are formal queries within the toy model. “Set cancer to 1” is a mathematical operation, not a clinical procedure.
06 / Check your reasoning
Each question targets a different confusion. Select an answer for immediate feedback; you can revise it.
Think about which arrows remain after intervening.
Separate uncertainty about a probability from uncertainty about a causal explanation.
Which structural requirement links the two stages of the formula?
Distinguish the chronology of evidence from a later mathematical illustration.
Answers: 1: no change in C; 2: multiple compatible causal answers; 3: no, complete mediation fails; 4: historical evidence and mathematical illustration have different roles.
For a seminar: Choose one arrow you would question in the front-door graph. What substantive evidence would support or challenge it? Would a new variable, a different study design, or a sensitivity analysis be more informative than a larger sample of the same variables?
For an instructor: Ask learners to commit to a prediction before switching the model. Then have them explain the changed result without using a formula.
The historical task is to justify a causal explanation with evidence. The mathematical task is to work out what that explanation, together with the data, permits us to calculate. Good causal reasoning needs both.
07 / Sources & further study
These readings document the historical debate and the formal ideas used in this lesson. The probability tables are synthetic teaching examples; they are not observations from the historical studies.