Pith. sign in

REVIEW 2 major objections 4 minor 79 references

Causality from the Point of View of Statistics

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Standard probability, not a separate causal calculus, is sufficient for the core results of statistical causality.

desk verdict Good teaching survey, but the advertised first proof of Pearl's rules is contradicted by the paper's own appendix disclaimer. read the letter →

arxiv 1908.07301 v8 pith:32UXI3FU submitted 2019-08-20 math.ST stat.TH

José A. Ferreira This is my paper · ORCID
classification math.STstat.TH MSC 62A9962F9962E9960J9960K9962P1062P2062P25
keywords causalityMarkovianmodelsidentifiabilityconfoundingpotentialoutcomesunconfoundednessinterventioncalculusrandomvariables
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that questions of cause and effect in statistics can be posed and answered within standard probability, using nothing more than random variables, conditional expectations, and carefully chosen functional equations. Its proposal is a single representational scheme for causal modelling in which each variable is defined as a function of earlier variables and of unobserved uniform "noise" variables, evaluated at one random draw. Within that scheme, a conditional-independence condition called unconfoundedness makes the law of the potential response to a fixed treatment equal to the observed conditional law of the response given treatment and situation, and that identity is shown to drive the main results of the intervention calculus and of potential-outcome analysis. The paper also gives elementary probability proofs of the first two rules of the calculus of intervention, and works out classical examples as routine calculations. If this is right, statistical causality is not a separate formalism but a chapter of applied probability, and the main schools of causal inference differ mostly in notation and emphasis rather than in substance.

What carries the argument

The load-bearing object is the random-variable model (4.1): $R_n=\rho(V_n,X_n,T_n)$ and $T_n=\tau(U_n,X_n)$, with $U_n,V_n$ standard uniform conditionally on $X_n$ and all variables defined as functions on one probability space, so observed data are one draw $\omega$. This model turns causal questions into questions about laws of random variables formed by freezing the treatment argument at a fixed value $t$, i.e. the potential outcomes; the key identity is (4.6), which equates $\mathcal{L}(R_n\,|\,X_n=x,T_n=t)$ with $\mathcal{L}(\rho(V_n,x,t))$ under unconfoundedness. The intervention model (4.2), obtained by replacing $T_n$ by a fixed sequence, represents what would happen if treatments were forced, and the paper shows that under unconfoundedness observational data estimate exactly that intervention model. The temporal precedence and acyclicity built into the functions $\rho$ and $\tau$ are what make the probability space a causal model rather than merely a joint distribution.

What would settle it

Simulate or observe a system with feedback, for example one where the current treatment and current response influence each other, apply the paper's unconfounded-adjustment estimator $\sum_x E(R\,|\,T=t,X=x)P(X=x)$ to the observational data, and compare it with the mean response under a randomized intervention that sets $T=t$; if the two disagree while the data satisfy the paper's conditional-independence checks, the representation (4.1) is not the true mechanism.

Watch

Extended reading notes

Core claim

The central discovery is that a causal model is a set of random variables defined recursively as functions of primitive random variables, with the random draw acting as the single source of randomness; distributional assumptions alone do not encode causal order. Concretely, the model $R_n = \rho(V_n,X_n,T_n)$, $T_n = \tau(U_n,X_n)$, with $U_n$ and $V_n$ standard uniform conditionally on $X_n$, says that each observation consists of one realized value of a system whose situation precedes treatment and whose treatment precedes response. Under the extra condition that $U_n$ and $V_n$ are conditionally independent given $X_n$ -- unconfoundedness -- the causal effect of treatment is fully captured by the laws $\mathcal{L}(\rho(V_n,x,t)\,|\,X_n=x)$, and these laws coincide with the observed conditional laws $\mathcal{L}(R_n\,|\,X_n=x,T_n=t)$. From this single identity follow the adjustment formula, the propensity-score reduction, the reduction of a richer causal graph to the basic model whenever a suitable set of variables is conditioned on, and a formula for the joint effect of two treatments applied over time. The same framework gives elementary proofs of the first two rules of the calculus of intervention, stated directly in probability language.

Load-bearing premise

The whole approach assumes that a real causal system can be captured by equations in which the situation is settled first, the treatment is then drawn as a function of the situation plus noise, and the response is then drawn as a function of the situation, treatment, and further noise, all observed at a single random draw; if the true system contains feedback, simultaneous determination, or interventions that change these functions, the identification formulas no longer apply.

Editorial extensions

If this is right

  • If model (4.1) holds with unconfoundedness, causal effects can be estimated by stratifying or matching on the situation variables, recovering the classical adjustment formula and the propensity-score method.
  • The first two rules of the intervention calculus follow from elementary conditional-probability manipulations, so those rules can be taught and verified without specialized graph-algebra machinery.
  • A detailed second-level causal graph can be reduced to the basic model whenever a suitable conditioning set is chosen; the paper gives a criterion for admissible sets and shows that conditioning on the wrong variables can create confounding.
  • For a two-stage treatment plan, the joint effect of the two treatments on the final response can be written as a sum of observed conditional probabilities, a special case of the general longitudinal formula.
  • Classical paradoxes such as Simpson's and Lord's become transparent illustrations of confounding and of the distinction between what happened and what may happen, once the underlying random-variable model is fixed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not say this, but if its reconciliation is right, the disputes among the main schools of statistical causality are largely notational: their identification formulas are the same identities expressed in different languages, so a common exposition is possible.
  • One testable extension is to carry the same elementary-probability derivations through the third rule of the intervention calculus and through continuous-time or dynamic settings, which the paper leaves largely unworked.
  • Because the paper insists that observed data alone cannot distinguish a causal order from a reversed order with the same joint distribution, it implies that no amount of purely observational data can identify causal direction without subject-matter assumptions.
  • A practical check suggested by the construction is to simulate from a known structural system, estimate the paper's adjustment formula from the simulated observational data, and compare it with the true randomized-intervention distribution; disagreement flags violations of unconfoundedness or of the assumed functional order.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes a unified, conventional-probability foundation for statistical causality. Its core model (4.1) represents each observation as functions of standard uniform random variables, with treatment generated from situation and external factors and the response generated from situation, treatment and further external factors; unconfoundedness is identified with conditional independence of those external factors. In this framework, Section 4 derives the standard identification of potential-outcome laws from observed conditional laws (Eq. 4.6) and the adjustment/integration formulas (4.7)–(4.8), with worked examples including Simpson's paradox and Lord's paradox. Section 5 develops second-level causal models and a reduction to first-level causality, including a version of Pearl's back-door criterion, and derives a Robins-type formula for joint treatment effects in a two-stage model (Eq. 5.19). Section 6 solves several benchmark examples from the causal-inference literature using elementary probability, including smoking and genotype, eelworms and fumigants, direct effects in a two-stage plan, sex discrimination, and case-control studies. An appendix is said to treat Pearl's calculus of intervention, but the text repeatedly describes it as an unfinished attempt that is not used in the paper.

Significance. If the central derivations are accepted, the paper makes a valuable contribution by showing that many results from the Pearl, Robins and Rubin schools can be derived in a single probability-space framework using elementary random-variable constructions, without invoking a separate causal calculus. The worked examples in Sections 4 and 6 are mostly self-contained and genuinely illuminating, and the treatment of unconfoundedness as conditional independence of the driving uniforms is conceptually clear. The paper is also honest about the observational assumptions that underlie causal conclusions. However, the strongest advertised novelty — a first elementary proof of Pearl's Rules 1 and 2 — is not delivered by the manuscript as it stands, since the relevant appendix is described by the author themselves as unfinished. In addition, the proof of Pearl's criterion in Section 5.1 is heuristic and the author explicitly states dissatisfaction with existing proofs. These issues affect the abstract's central claim and the completeness of Section 5.1, though they do not, in my view, undermine the value of the remaining derivations.

major comments (2)
  1. [Abstract; §1, footnote 8; §6, opening paragraph (p. 52)] The abstract states that Pearl's first two rules of intervention are 'formulated and proved by means of elementary probability for the first time,' but the manuscript itself says in Section 1, footnote 8 that Appendix A is 'an unfinished attempt at understanding Pearl's calculus of intervention,' and Section 6 says 'Appendix A presents an attempt at understanding Pearl's calculus, but we shall make no use of it here.' Since Appendix A is the only place where Rules 1 and 2 could be proved, the advertised novelty is not supported by the manuscript as it stands. This is a load-bearing discrepancy: either the appendix must be completed to contain a rigorous derivation of the two rules, or the abstract and Section 1 must be revised to state that the calculus is only examined or attempted, not proved.
  2. [§5.1, Eqs. (5.10)–(5.14) and footnote 50] The proof of Pearl's criterion is presented as a sequence of informal 'constraint' arguments and concludes that 'eventually' the conditioning equations reduce to forms such as (5.13)–(5.14). This does not constitute a proof of the graph-theoretic criterion for arbitrary second-level models: the argument does not rigorously treat collider configurations, descendants of colliders, or non-atomic distributions, and footnote 50 admits 'We have not been satisfied with any proof of this result.' As written, Section 5.1 provides a heuristic justification rather than a theorem. Since the criterion is used repeatedly in the paper, the section should either be replaced by a complete proof or be explicitly presented as a motivation with the criterion attributed to Pearl and with the central claims of the paper not relying on it as a proved result.
minor comments (4)
  1. [§5.2, derivation of Eq. (5.19)] The derivation multiplies and divides by assignment probabilities such as P(τ(Ũ, x) = t) and P(τ'(Ũ', x', t, r) = t') without stating positivity conditions; the paper should state the support assumptions needed for these displayed identities, especially because the paper elsewhere works with discrete variables and zero-probability conditioning events.
  2. [§4, Eqs. (4.3)–(4.6)] The identification formulas are written for discrete situations and treatments, with continuous cases addressed only in a footnote; since these formulas are central, a brief formal statement with the appropriate density or kernel notation for the continuous case would improve clarity.
  3. [Throughout] There are several typographical errors, including 'it it is undeniable' in Section 1 and 'footone 66' in Section 6.4; these should be corrected in a final pass.
  4. [§6.1–6.5] The worked examples are valuable but often rely on support/positivity assumptions implicitly (e.g., dividing by P(Y = y, Z = z) in Section 6.1); adding a sentence about such assumptions in each example would make the derivations fully rigorous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the advertised Pearl-rule proof is internally disclaimed as an unfinished, unused attempt, but that is a novelty/overclaim issue, not a circular derivation.

full rationale

I find no step in which a claimed derivation or prediction reduces by construction to its inputs. The paper contains no data fitting: its quantities are identified via elementary probability identities (e.g., (4.6), (5.3), (6.14)) from the explicitly stated structural model (4.1), R_n = rho(V_n, X_n, T_n), T_n = tau(U_n, X_n), together with unconfoundedness; unconfoundedness is an assumption used as an input, not a conclusion smuggled into the outputs. Pearl's criterion in Section 5.1 is attributed to Pearl, but the paper states the criterion, illustrates it, and attempts an independent proof, so the derivation is not circular merely because the result is not new. The genuine defect is that the abstract's claim that Pearl's first two rules are 'formulated and proved by means of elementary probability for the first time' is contradicted by the paper's own statements: near the end of Section 1 it calls Appendix A 'an unfinished attempt at understanding Pearl's calculus of intervention', and Section 6 says 'Appendix A presents an attempt at understanding Pearl's calculus, but we shall make no use of it here'. This is an internal overclaim risk, not a circularity: an unfinished appendix cannot support the claimed novelty, but failing to deliver a proof is different from assuming the conclusion. There are no load-bearing self-citations (the author does not cite his own prior work), and no fitted parameter is renamed as a prediction. Thus the derivation chain is self-contained relative to its stated modeling assumptions, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free parameters and no new physical or mathematical entities. Its central claim rests on four structural assumptions: standard probability, sequential acyclic representation, unconfoundedness, and the graphical-summary interpretation of second-level models. These are all modeling assumptions about how real causal systems relate to probability spaces, not fitted constants.

assumptions (4)
  • standard math Kolmogorov probability theory and the construction of random variables on a common probability space, as in Billingsley and other standard texts (Section 2).
    The paper relies on standard results about random variables, independence, conditional laws, and the existence of sequences of independent uniforms.
  • domain assumption Causal systems can be represented by acyclic equations of the form R = rho(V, X, T) and T = tau(U, X), with U and V standard uniform conditionally on X (model (4.1)).
    This representation is the foundation of the paper's treatment of causality. It presumes a temporal ordering and excludes feedback and simultaneous causation.
  • domain assumption Unconfoundedness: U and V are independent conditionally on X, stated in Section 4 as necessary for identifying treatment effects.
    All estimation formulas in the paper, including L(R|T=t, X=x) = L(rho(V,x,t)|X=x), depend on this conditional independence.
  • domain assumption For second-level models, the graph structure and Pearl's back-door criterion are assumed to capture conditional independence relations.
    The proof of the criterion in Section 5.1 relies on a graphical representation where each node is a function of earlier nodes and independent uniform disturbances.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Causality from the Point of View of Statistics." pith.science (2026). https://pith.science/paper/32UXI3FU

@misc{pith2026190807301,
  author       = {Pith},
  title        = {Pith review of: Causality from the Point of View of Statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/32UXI3FU}},
  note         = {Machine review of arXiv:1908.07301}
}
read the original abstract

We present a basis for studying questions of cause and effect in statistics which subsumes and reconciles the models proposed by Pearl, Robins, Rubin and others, and which, as far as mathematical notions and notation are concerned, is entirely conventional. In particular, we show that, contrary to what several authors had thought, standard probability can be used to treat problems that involve notions of causality, and in a way not essentially different from the way it has been used in the area generally known (since the 1960s, at least) as 'applied probability'. Conventional, elementary proofs are given of some of the most important results obtained by the various schools of 'statistical causality', and a variety of examples considered by those schools are worked out in detail. Pearl's 'calculus of intervention' is examined anew, and its first two rules are formulated and proved by means of elementary probability for the first time since they were stated 25 years or so ago. Note: Corrected and extended parts of this paper will soon be published as a book of the same title.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

79 extracted references · 78 canonical work pages

  1. [1]

    and Rubin, D.B

    Angrist, J.D., Imbens, G.W. and Rubin, D.B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statis- tical Association, Vol. 91, No. 434, pp. 444–455

    work page 1996

  2. [2]

    and Parner, J

    Arjas, E. and Parner, J. (2004). Causal reasoning from longitudinal data. Scandinavian Journal of Statistics 31 171–187

    work page 2004

  3. [3]

    Arjas, E. (2012). Causal inference from observational data: a Bayesian p re- dictive approach. Causality: Statistical Perspectives and Applications, Ed. C. Berzuini, A.P. Dawid and L. Bernardinelli, pp. 71–84. Chichester, UK

    work page 2012

  4. [4]

    and Prakasa Rao, B.L.S

    Basawa, I.V. and Prakasa Rao, B.L.S. (1980). Statistical Inference for Stochastic Processes. Academic Press, New York

    work page 1980

  5. [5]

    (1961).Statistical Inference for Markov Processes

    Billingsley, P. (1961).Statistical Inference for Markov Processes. The Uni- versity of Chicago Press

    work page 1961

  6. [6]

    Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York

    work page 1995

  7. [7]

    and Hunter, W.G

    Box, G.E.P., Hunter, S. and Hunter, W.G. (2005). Statistics for Exper- imenters: Design, Innovation and Discovery , 2nd ed. Wiley, New Jersey

    work page 2005

  8. [8]

    (1998).Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues

    Br´ emaud, P. (1998).Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues . Springer-Verlag

    work page 1998

Show all 79 references
  1. [9]

    and Davis, R.A

    Brockwell, P.J. and Davis, R.A. (1991). Time series: Theory and Meth- ods, 2nd ed. Springer-Verlag

    1991

  2. [10]

    Coles, S.A. (2001). An Introduction to Statistical Modelling of Extreme Values. Springer-Verlag, London

    2001

  3. [11]

    Cornfield, J. (1951). A method for estimating comparative rates from clinical data. Applications to cancer of the lung, breast, and cervix . Journal of the National Cancer Institute , 11, pp. 1269–1275

    1951

  4. [12]

    and Snell, E.J

    Cox, D.R. and Snell, E.J. (1980). Applied Statistics: Principles and Examples. Chapman & Hall, London

    1980

  5. [13]

    and Wermuth, N

    Cox, D.R. and Wermuth, N. (2004). Causality: a statistical view. Inter- national Statistical Review , 72, 3, pp. 285–305

    2004

  6. [14]

    and Wood, F.S

    Daniel, C. and Wood, F.S. (1971). Fitting Equations to Data: Computer Analysis of Multifactor Data for Scientists and Engineers (written with the assistance of J.W. Gorman). Wiley, New York

    1971

  7. [15]

    Dawid, A. (2000). Causal inference without counterfactuals. Journal of the American Statistical Association , 95, pp. 407–448

    2000

  8. [16]

    Dawid, A. (2015). Statistical causality from a decision-theoretic perspec- tive. Annual Review of Statistics and Its Application , 2, pp. 273–303

    2015

  9. [17]

    Diaconis, P. (1998). A place for philosophy? The rise of modeling in statis- tical science.Quarterly of Applied Mathematics , Vol. L VI, No. 4, pp. 797–805

    1998

  10. [18]

    (2002).Daniel Bernoulli’s epidemi- ological model revisited

    Dietz, K., and Heesterbeek, J.A.P. (2002).Daniel Bernoulli’s epidemi- ological model revisited. Mathematical Biosciences, 180, pp. 1–21

    2002

  11. [19]

    Doob, J.L. (1997). The Development of Rigor in Mathematical Probability (1900-1950). The American Mathematical Monthly, Vol. 103, No. 7, pp. 586–95

    1997

  12. [20]

    and Haggstrom, O

    Eriksson, K. and Haggstrom, O. (2014). Lord’s paradox in a continuous setting and a regression artifact in numerical cognition research. PLOS ONE, Volume 9, Issue 4. Author/Causality: Viewpoint of Statistics 84

    2014

  13. [21]

    Feller, W. (1968). An Introduction to Probability Theory and its Appli- cations, Vol. I, 3rd ed. Wiley, New York

    1968

  14. [22]

    Ferreira, J.A. (2015). Some models and methods for the analysis of observational data. Statistics Surveys 9, pp. 106–208

    2015

  15. [23]

    Freedman, D.A. (1997). From association to causation via regression. Ad- vances in Applied Mathematics 18, pp. 59–110

    1997

  16. [24]

    Freedman, D.A. (2009). Statistical Models: Theory and Practice , revised ed. New York: Cambridge University Press

    2009

  17. [25]

    Freedman, D.A. (2010). Statistical Models and Causal Inference: A Dialogue with the Social Sciences . Edited by D. Collier, J.S. Sekhon and P.B. Stark New York: Cambridge University Press

    2010

  18. [26]

    and Kotz, S

    Galambos, J. and Kotz, S. (1978). Characterization of Probability Dis- tributions. Lecture Notes in Mathematics, Vol. 675, Springer-Verlag, Berlin

    1978

  19. [27]

    and , Safko, J

    Goldstein, H., Poole, C. and , Safko, J. (2001). Classical Mechanics, 3rd ed. Addison-Wesley

    2001

  20. [28]

    Guttorp, P. (1995). Stochastic Modeling of Scientific Data . Chapman & Hall, Boca Raton

    1995

  21. [29]

    Heyde, C.C. (1997). Quasi-Likelihood and its Application: a General Ap- proach to Optimal Parameter Estimation . Springer-Verlag, New York

    1997

  22. [30]

    Causal diagrams for empirical research

    Imbens, G.W . & Rubin, D.B. (1995). Discussion of “Causal diagrams for empirical research” by J. Pearl. Biometrika, 82, 4, pp. 694–695

    1995

  23. [31]

    & Rubin, D.B

    Imbens, G.W . & Rubin, D.B. (2015). Causal Inference in Statistics, So- cial, and Biomedical Sciences: An Introduction . New York: Cambridge Uni- versity Press

    2015

  24. [32]

    Ioannidis, J.P. (2005). Why most published research findings are false. PLOS Medicine, Vol. 2, Issue 8, pp. 696–701

    2005

  25. [33]

    Isham, V. (2005). Stochastic models for epidemics. Celebrating Statistics: Papers in honour of Sir David Cox on his 80th birthday , Ed. A.C. Davison, Y. Dodge and N. Wermuth. Oxford University Press, New York

    2005

  26. [34]

    and Nisio, M

    Ito, K. and Nisio, M. (1968). On the convergence of sums of independent Banach space valued random variables. Osaka J. Math. , 5, pp. 35–48

    1968

  27. [35]

    Jørgensen, B. (1993). The Theory of Linear Models . Chapman & Hall, New York

    1993

  28. [36]

    and Buhlmann, P

    Kalisch, M. and Buhlmann, P. (2014). Causal structure learning and inference: a selective review. Quality Technology & Qualitative Management , 11, 1, pp. 3–21

    2014

  29. [37]

    and Taylor, H.M

    Karlin, S. and Taylor, H.M. (1975). A First Course in Stochastic Pro- cesses, 2nd ed. Academic Press, New York

    1975

  30. [38]

    and Rootzen, H., (1983)

    Leadbetter, M.R., Lindgren, G. and Rootzen, H., (1983). Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York

    1983

  31. [39]

    Lehmann, E.L. (1983). Theory of Point Estimation . Wiley, New York

    1983

  32. [40]

    Lehmann, E.L. (1986). Testing Statistical Hypotheses, 2nd ed. Wiley, New York

    1986

  33. [41]

    Lehmann, E.L. (1990). Model specification: the views of Fisher and Ney- man, and later developments. Statistical Science, Vol. 5, no. 2, pp. 166–168. Author/Causality: Viewpoint of Statistics 85

    1990

  34. [42]

    Lindley, D.V. (2002). Seeing and doing: the concept of causation. Inter- national Statistical Review , 70, 2, pp. 191–197

    2002

  35. [43]

    Lord, F.M. (1967). A paradox in the interpretation of group comparisons. Psychological Bulletin , 68, 5, pp. 304–305

    1967

  36. [44]

    Nuzzo, R. (1967). Statistical errors. Nature, Vol. 506, pp. 150–152

    1967

  37. [45]

    Pearl, J. (1995a). Causal diagrams for empirical research. Biometrika, 82, 4, pp. 669–710

    1995

  38. [46]

    Causal diagrams for em- pirical research

    Pearl, J. (1995b). Rejoinder to discussions of “Causal diagrams for em- pirical research”. Biometrika, 82, 4, pp. 702–710

    1995

  39. [47]

    Pearl, J. (2009a). Causality: Models, Reasoning and Inference , 2nd ed. Cambridge University Press

    2009

  40. [48]

    Pearl, J. (2009b). Causal inference in statistics: an overview. Statistics Surveys, 3, pp. 96–146

    2009

  41. [49]

    Pearl, J. (2009c). Myth, confusion, and science in causal analysis. Techni- cal Report R-348, Department of Computer Science, University of California, Los Angeles

    2009

  42. [50]

    Separated at Birth

    Pearl, J. (2011). Statistics and causality: separated to reunite— Commentary on Bryan Dowd’s “Separated at Birth” . Health Serv. Res. 46 (2), pp. 421–429

    2011

  43. [51]

    Pearl, J. (2012). The Do-Calculus Revisited. Proceedings of the Twenty- Eighth Conference on Uncertainty in Artificial Intelligenc e, Corvallis, Ore- gon, edited by N. de Freitas and K. Murphy: AUAI Press, pp. 4–11, 2012

    2012

  44. [52]

    Pearl, J. (2015). Indirect confounding and causal calculus. (On three pa- pers by Cox and Wermuth.) Technical Report R-457, Department of Com- puter Science, University of California, Los Angeles

    2015

  45. [53]

    Pearl, J. (2016). Lord’s Paradox Revisited—(Oh Lord! Kumbaya!). Tech- nical Report R-436, Department of Computer Science, University of Califor- nia, Los Angeles

    2016

  46. [54]

    and Jewel, N.P

    Pearl, J., Glymour, M. and Jewel, N.P. (2016). Causal Inference in Statistics: a Primer , Wiley

    2016

  47. [55]

    and Maathuis, M.H

    Perkovi´c, E., Textor, J., Kalisch, M. and Maathuis, M.H. (2018). Complete graphical characterization and construction of adjust ment sets in Markov equivalence classes of ancestral graphs. Journal of Machine Learning Research, 18, pp. 1–62

    2018

  48. [56]

    (1986).A new approach to causal inference in mortality stud- ies with a sustained exposure period—applications to control of the healthy workers survivor effect

    Robins, J.M. (1986).A new approach to causal inference in mortality stud- ies with a sustained exposure period—applications to control of the healthy workers survivor effect. Mathematical Modeling, 7, pp. 1393–512

    1986

  49. [57]

    Causal diagrams for empirical re- search

    Robins, J.M. (1995). Discussion of “Causal diagrams for empirical re- search” by J. Pearl. Biometrika, 82, 4, pp. 695–698

    1995

  50. [58]

    and W asserman, L

    Robins, J.M. and W asserman, L. (1997). Estimation of effects of se- quential treatments by reparameterizing directed acyclic graphs . Proceed- ings of the Thirteenth Conference on Uncertainty in Artifici al Intelligence (UAI1997), pp. 409–420

    1997

  51. [59]

    Rosenbaum, P.R. (2002). Observational Studies , 2nd ed. Springer

    2002

  52. [60]

    and Rubin, D.B

    Rosenbaum, P.R. and Rubin, D.B. (1983).The central role of the propen- sity score in observational studies for causal effects. Biometrika 70 41–55. Author/Causality: Viewpoint of Statistics 86

    1983

  53. [61]

    Ross, S.M. (1970). Applied Probability Models with Optimization Applica- tions. Holden-Day, San Francisco

    1970

  54. [62]

    Ross, S.M. (2010). Introduction to Probability Models , 10th ed. Academic Press

    2010

  55. [63]

    Rubin, D. (1990). Formal modes of statistical inference for causal effect s. Journal of Statistical Planning and Inference , Vol. 25, pp. 279–292

    1990

  56. [64]

    Rubin, D. (2009). Author’s reply.Statistics in Medicine , Vol. 28, pp. 1420– 1422

    2009

  57. [65]

    Senn, S. (2006). Change from baseline and analysis of covariance revisited. Statistics in Medicine , Vol. 25, pp. 4334–4344

    2006

  58. [66]

    and Tobin, M.(2008)

    Shehan, N., Didelez, V ., Burton, P . and Tobin, M.(2008). Mendelian randomisation and causal inference in observational epidemiology. PLOS Medicine, Vol. 5, Issue 8, pp. 1205–1210

    2008

  59. [67]

    Shrier, I. (2008). Letter to the editor. Statistics in Medicine , Vol. 27, pp. 2740–2741

    2008

  60. [68]

    Simpson, E.H. (1951). The interpretation of interaction in contingency tables. Journal of the Royal Statistical Society , Series B, 13, 2, pp. 238–241

    1951

  61. [69]

    Seeing and doing: the concept of causation

    Singpurwalla, N.D. (2002). On causality and causal mechanisms. Com- ment on Dennis Lindley’s “Seeing and doing: the concept of causation ”. International Statistical Review , Vol. 70, No. 2, pp. 198–206

    2002

  62. [70]

    Snell, J.L. (1997). A conversation with Joe Doob. Statistical Science, Vol. 12, No. 4, pp. 301–311

    1997

  63. [71]

    Sturtevant, A.H. (1965). A History of Genetics. Harper & Row: New York. Republished in 2000 by Electronic Scholarly Publishing Project& Cold Spring Harbor Laboratory Press: http://www.esp.org/books/st urt/history

    1965

  64. [72]

    and Rising, G.R (1989)

    Tanur, J.M., Mosteller, F., Kruskal, W.H., Lehmann, E.L., L ink, R.F., Pieters, R.S. and Rising, G.R (1989). Statistics: A Guide to the Unknown, 3rd ed. Wadsworth & Brooks, California

    1989

  65. [73]

    an d Ellison G

    Textor, J., van der Zander B., Gilthorpe M., Liskiewicz M. an d Ellison G. (2016). Robust causal inference using directed acyclic graphs: the R package ‘dagitty’.International Journal of Epidemiology , 18, pp. 1–62

    2016

  66. [74]

    and Marks, M

    Trafimow, D. and Marks, M. (2015). Editorial. Basic and Applied So- cial Psychology, Vol. 37, pp. 1–2

    2015

  67. [75]

    Tucci, R.R. (2013). Introduction to Judea Pearl’s do-calculus. arXiv.org, paper arXiv:1305.5506 available at https://arxiv.org/pdf/1305.550 6.pdf

    2013 arXiv

  68. [76]

    and Shpitser, I

    V anderWeele, T.J. and Shpitser, I. (2013). On the definition of a confounder. Annals of Statistics , Vol. 41, no. 1, pp. 196–220

    2013

  69. [77]

    and Lazar, N.A

    W asserstein, R.L. and Lazar, N.A. (2016). The ASA statement on p-values: context, process, and purpose. The American Statistician , 70:2, pp. 129–133

    2016

  70. [78]

    and Brown, L.M

    W ayner, H. and Brown, L.M. (2007). Three statistical paradoxes in the interpretation of group differences: Illustrated with medical scho ol admission and licensing data. In Handbook of Statistics , Vol. 26: Psychometrics (edited by C.R. Rao and S. Sinharay), pp. 893–918

    2007

  71. [79]

    and Cox, D.R

    Wermuth, N. and Cox, D.R. (2008). Distortion of effects caused by in- direct confounding. Biometrika, 95, 1, pp. 17–33

    2008

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.