Rubin Causal Model Holland 1986 Holland PW 1986

Rubin Causal Model Holland (1986)

Holland PW (1986). Statistics and causal inference. JASA 81: 945 -960. • Potential Outcomes: – Yt(u): response to exposure S=t for unit u – Yc(u): response to exposure S=c for unit u – Ys(u): response to exposure S=s for unit u. (More general. ) • Causal effect of t (relative to c) on u – Yt(u) - Yc(u) • Fundamental Problem of Causal Inference: – It is impossible to observe the value of Yt(u) and Yc(u) on the same unit, and therefore, it is impossible to observe the effect of t on u. • All causal inference is based on assumptions that cannot be derived from observations alone.

Holland (1986) • Average Causal Effect: T = E(Yt – Yc) = E(Yt) – E(Yc) • Prima Facie (at first appearance) Causal Effect: TPF = E(Ys|S=t) – E(Ys|S=c) • T ≠TPF

Assumptions to Identify Causal Effects • Temporal Stability and Causal Transcience (a) Value of Yc(u) does not depend on when the sequence ‘apply c to u then measure Y on u’ occurs (b) The value of Yt(u) is not affected by the prior exposure of u to the sequence in (a) • Example: Turning the switch (t=on, c=off) causes the light to go on (Ys(u)=1) because we assume Yc(u)=dark does not depend on when S=c, and that Yt(u)=light is not affected by the prior exposure S=c.

Assumptions to Identify Causal Effects • Unit Homogeneity Yt(u 1)=Yt(u 2) and Yc(u 1)=Yc(u 2) • Causal effect is therefore Yt(u 1) – Yc(u 2) • Example: Laboratory scientists convince themselves that units are homogeneous in all relevant aspects except exposure.

Assumptions to Identify Causal Effects • In general: – E(Yt|S=t) ≠ E(Yt) – E(Yc|S=c)≠ E(Yc) – Example: Units in population assigned treatment are different from units in population assigned control, therefore looking at average outcome only among those treated is not representative of the average outcome had everyone been treated. • If S is independent of Yt and Yc: – E(Yt|S=t) = E(Yt) – E(Yc|S=c) = E(Yc) – Example: Treatment vs. control is randomly assigned.

Constant effect • T=E(Yt(u) – Yc(u)) = Yt(u) – Yc(u) for all u. • Individual causal effect = Average causal effect for all units. • Strong assumption.

Holland (1986), Sections 5 -6 • Section 5: – Holland compares and contrasts the Rubin Causal Model with ideas from philosophers. • Section 6: – Holland compares and contrasts the Rubin Causal Model with ideas from statisticians. – Rubin Causal Model comes from Rubin (1974, 1977, 1978, 1980) – Some of the potential outcome ideas are particularly apparent in Neyman (1935); sometimes called the Neyman -Rubin Causal Model. – Also hints of it in the work of Kempthorne (1952), Cox (1958), and Fisher (1926).

Holland (1986). Section 7 • What can be a cause? • Example questions: (A) She did well on the exam because she is a woman. (B) She did well on the exam because she studied for it. (C) She did well on the exam because she was coached by her teacher. • Holland argues (A) Refers to an attribute, not a cause. (Cannot assign someone to be female) (B) Is not well defined. (Studying compared to not studying? How much studying? ) (C) Well defined. (Exposure is coached by teacher, control is not coached by teacher. )

Holland (1986). Section 7 • No causation without manipulation • I’m not sure I completely agree: – My y chromosome causes me to be male. – I cannot be assigned an X chromosome, so manipulation is not possible. (Or is manipulation possible? ) – So does that mean that my Y chromosome is not a cause? • Poorly defined exposures: – E. g. , race • Is race a person’s skin color, genetics, culture, life experience, or all of the above?

Holland (1986). Section 8. • Comments on causal inference in various disciplines • Medicine – Koch’s postulates for deciding when organism causes disease – Hill’s criteria for interpreting association as causation • Economics – Granger Causation • Social Science – Causal models / Path diagrams / Structural equation models