Lecture 1: Introduction to Causal Inference

Causal Inference and Statistical Designs · Michaelmas 2026 · Qingyuan Zhao

Four words to think about

  • What is mathematics?
  • What is statistics?
  • What is causality?
  • What is design?

Up to three words or phrases.

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Our answers

Mathematics · Statistics · Causality · Design

We will ask again later in the course.

If you believe your answers improve, I shall try not to take full causal credit.

For pleasure: xkcd, “Correlation”.

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One topic per question: Lecture N - Keywords (e.g. Lecture 1 - Potential outcomes)

Further information including scribed lecture notes can be found at https://www.statslab.cam.ac.uk/~qz280/teaching/causal-2026/.

A little etymology

Marble bust of Aristotle, a Roman copy after a Greek original; Palazzo Altemps. Photograph by Jastrow (2006).
Mathematics Greek mathēma that which is learned
Statistics German Statistik describing the state
Causality Latin causa cause, reason
Design Latin designare mark out, designate

Aristotle (384–322 BCE)

A bronze statue:

  • Material: the bronze.
  • Formal: the figure represented.
  • Efficient: the sculptor’s craft.
  • Final: its purpose, e.g. commemoration.

Aristotle, Physics II.3. Word histories: mathematics, statistics, cause, design. Bust: Roman marble copy after Lysippos; photograph Jastrow (2006), public domain.

David Hume (1711–1776)

David Hume, painted by Allan Ramsay in 1766; Scottish National Portrait Gallery.

Empiricism: knowledge of the world rests on experience.

  • “The sun always rises in the east.”
  • “When someone dies, it’s never me.”

The Problem of Induction “It worked before” uses induction to justify induction.

“We are determined by custom alone to suppose the future conformable to the past.”

Hume (1748), An Enquiry concerning Human Understanding, §IV, ¶¶2, 6, 18–21. Closing quotation: Abstract (1740), ¶15, SBN 652. Examples are teaching illustrations. Portrait: Allan Ramsay (1766), public domain.

Charles Sanders Peirce (1839–1914)

Charles Sanders Peirce, photographed by Napoleon Sarony in 1891.

Connected inquiry stages above three labelled syllogisms. Rule: all the beans from this bag are white. Case: these beans are from this bag. Result: these beans are white. Abduction infers the case from rule and result; deduction infers the result from rule and case; induction infers the rule from case and result. Feedback returns from induction to deduction for further predictions, or to abduction for a new hypothesis.

Peirce (1878), “Deduction, Induction, and Hypothesis”, p. 472. Arrows reconstruct his later account of inquiry: SEP §3. Portrait: Napoleon Sarony (1891), public domain.

Law of Errors

An elliptical orbit with imperfectly measured positions around it; an illustration, not historical data.
  • Legendre (1805) · Gauss (1809) - Astronomical measurements, least squares.

An open nineteenth-century Swedish population register, with handwritten household entries and ruled columns.
  • Laplace (1814) — regularities in births, marriages, misaddressed letters.
  • Quetelet (1835) — government records; “social physics”.
  • Darwin (1859) → Galton (1886) — heredity, natural selection, variation.

Legendre, Nouvelles méthodes… (1805); Gauss, Theoria motus… (1809). Orbit: teaching illustration. Laplace, Essai (1814); Quetelet, Sur l’homme (1835); Galton, Natural Inheritance (1889). Swedish population register, nineteenth century, public domain. For pleasure: Menand (2001), The Metaphysical Club, ch. 8, “The Law of Errors”.

Laplace’s demon

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past could be present before its eyes.

Pierre Simon Laplace, A Philosophical Essay on Probabilities (1814). English quotation.

George Udny Yule (1871-1951)

Did relief outside the workhouse increase pauperism?

Metropolitan unions Constant Out-relief ratio Aged proportion Population
1871–1881 13.19 0.755 −0.022 −0.322
1881–1891 1.36 0.324 1.370 −0.369

Coefficients in linear models predicting changes in pauperism.

On interpreting these coefficients as causes:

“Strictly, for ‘due to’ read ‘associated with.’”

Yule (1899), “An Investigation into the Causes of Changes in Pauperism in England…”, JRSS 62, 249–286. Table C excerpt (p. 259), re-typeset; quotation: p. 270, n. 25. See Freedman’s discussion.

Randomized experiments in psychophysics

Coover (1917): telepathy. Can a sender mentally transmit a card’s identity?

One trial in order. First shuffle and cut a 40-card pack, keeping the target hidden from the receiver, and roll a die. Second, the sender follows one condition: 2, 4 or 6 leaves the card unknown to the sender; 1 means looking at it; 3 means glancing, concealing it, then imagining saying its name; 5 combines looking with imagined speech and sound. After the interval the receiver records a guess, then the sender records the card. Guesses and cards are compared after the set. Assignment means the die face; exposure means the sender’s condition; outcome means whether the guess is correct.

Coover reported no evidence of thought-transference.

Coover, Experiments in Psychical Research (1917), pp. 51–54, conclusion p. 124. Diagram is a teaching reconstruction.

For pleasure: Hacking (1988), “Telepathy: Origins of Randomization in Experimental Design”, Isis 79, 427–451.

Smoking and lung cancer

Deaths per 1,000/year · age-standardized · male doctors, age 35+

Cause Non-smokers 1–14 g/day 15–24 g/day 25+ g/day Ratio† Difference†
Lung cancer 0.07 0.47 0.86 1.66 23.71 1.59
Other cancer 2.04 2.01 1.56 2.63 1.29 0.59
Other respiratory 0.81 1.00 1.11 1.41 1.74 0.60
Coronary thrombosis 4.22 4.64 4.60 5.99 1.42 1.77
Other causes 6.11 6.82 6.38 7.19 1.18 1.08

† 25+ g/day versus non-smokers; ratios/differences calculated from rounded rates. Tobacco g/day; former smokers classified at quitting.

Doll & Hill (1956), “Lung Cancer and Other Causes of Death in Relation to Smoking”, Table V, p. 1073. Berkson (1958), “Smoking and Lung Cancer: Some Observations on Two Recent Reports”, JASA 53, 28–38.

Potential outcomes and randomization

  • Potential outcomes: assignments, exposure and consistency.
  • Neyman–Rubin model: no interference.
  • The fundamental problem of causal inference — Holland (1986).
  • Average treatment effect and the difference-in-means estimator.
  • Complete randomization: unbiasedness and Neyman’s variance formula (1923).

Neyman (1923), “On the Application of Probability Theory to Agricultural Experiments”; Rubin (1974), “Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies”, Journal of Educational Psychology 66, 688–701.

Holland (1986), “Statistics and Causal Inference”, JASA 81, 945–960, names the fundamental problem.

For pleasure: Salsburg (2001), The Lady Tasting Tea, opening chapter.