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Summary of Causality Hein Stigum Presentation, data and programs at: November 18November 18 H.S.
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Contents Concepts Causal models Causal inference Methods of adjustment
Statistics and Causality Counterfactuals, Actions Causal models DAGs, Pies, SEM, (MSM) Causal inference Exchangeability, Positivity, Consistency Methods of adjustment Causal inference: much theory, will focus on practical advice November 18November 18 H.S.
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Concepts Statistics and Causality (J. Pearl) November 18November 18
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Traditional Statistics
P Joint distribution Q(P) (Aspects of P) Data Inference Infer if customers who by product A will also by product B Q=P(B|A) November 18November 18 H.S.
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From Statistical to Causal analysis
P Joint distribution P’ Joint distribution Q(P’) (Aspects of P’) Data change Intervention: P changes to P’ Infer if customers who by product A will also by product B when we double the price Statistics deals with static relations, P does not tell us how it ought to change: P’(v)P(v|price=2) Need assumptions about aspects of P that stay invariant under intervention (change) The DAG specifies such aspects The structural equations specify such aspects The causal pies specify such aspects November 18November 18 H.S.
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Statistical and Causal concepts
Statistical and causal concepts do not mix No causes in – no causes out Causal assumptions + Statistical assumptions + Data Standard mathematics Causal assumptions cannot be expressed Non-standard mathematics Structural Equation Models (Wright 1920, Simon 1969) Counterfactuals (Neyman-Rubin) Do-operator (Pearl) Statistical Causal Association Randomization / intervention Controlling for / Conditioning Confounding Independence Instrumental variable Collapsibility Endogeneity = causal conclusions November 18November 18 H.S.
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Concepts Potential outcome, Counterfactual November 18November 18 H.S.
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Individual causal effect
Outcome if exposed Y1 Outcome if unexposed Y0 Causal effect if Y1 Y0 Important Clear definition Notation mathematical proofs Notation new methods Estimate individual effect? No, (but Crossover design) Potential outcomes Counterfactual Potential outcomes: both are possible outcomes Counterfacual: counter to what happened The definitions are keys to understand designs November 18November 18 H.S.
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Population causal effect
Average causal effect Expected outcome if all exposed E(Y1) Expected outcome if all unexposed E(Y0) Causal effect if E(Y1) E(Y0) Causal effect measures RDcausal= E(Y1) - E(Y0) RRcausal= E(Y1) / E(Y0) Estimate average effect? Yes, Randomized Controlled Trial Proof that rand trial gives the average effect Treated versus not treated must be clearly defined November 18November 18 H.S.
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Actions Modifiable risk factors Not modifiable risk factors Examples
Smoking, Radon Actions Reduce prevalence of smoking from 15% to 10% Examples Sex, Age Actions ? Causal effects are strictly speaking only defined for actions November 18November 18 H.S.
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DAGs, Pies and SEM Causal models November 18November 18 H.S. 11
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Causal models Four models Causal graphs (DAGs)
Causal Pies, Sufficient Component Cause (SCC) Structural Equation Models (SEM) Potential outcome models Marginal Structural Models (MSM) November 18November 18 H.S.
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Causal graphs, DAGs C Causal assumptions
Units: individuals (also other units) E->D reads E causes D, any definition of cause Qualitative (non-parametric): the E->D may be linear, threshold, U-shaped, … Simple, only 4 rules needed for analysis No estimation, only qualitative results: confounding yes/no Non-action (immutable) variables as exogenous, all rules apply New understanding: collider E D November 18November 18 H.S.
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Causal Pies (SCC) Causal assumptions Units: causal mechanisms
Any definition of cause No estimation, only qualitative results: interaction yes/no Logically finer than DAGs U A B A D B U A U B U A B 1 DAG 5 SCCs Additive scale (interaction) New understanding: sufficient-,necessary cause, interaction November 18November 18 H.S.
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Structural Equation Models, SEM
Causal assumptions + statistical model + data Units: individuals (also other units) Any definition of cause Quantitative (parametric): linear Estimation: direct and indirect effects Ordinary regression: association of actual covariates with actual outcomes SEM: effect of actions on potential outcomes SEM: parametric DAG Legg in bilde fra AMOS som eksempel November 18November 18 H.S.
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Causal Inference November 18November 18 H.S. 16
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Causal inference question
Counterfactual definition of cause Cannot estimate individual causal effects Can estimate average causal effects from RCTs Can we estimate average causal effects from observational data? Find conditions needed for causal inference Examine RCTs for conditions Apply to observational studies November 18November 18 H.S.
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Randomized Controlled Trial, RCT
U Observational study Suffers from unmeasured confounders Randomized trial If full compliance: R=E No arrow from U to E Three (trivial) conditions in RCTs : Exchangeability: exposed and unexposed may be switched Positivity: have both exposed and unexposed Consistency: well defined treatment U R E D Nov-18 Nov-18 H.S. H.S. 18
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RCTs versus Observational studies
RCT get Observational need strength test Exchangeability Conditional exchangeability weaker untestable Positivity Conditional positivity stronger testable Consistency - November 18November 18 H.S.
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Exchangeability, Positivity and Consistency
Conditions for estimating causal effect: 1. Cond. Exchangeability Exch Also include open selection paths Pos radioactivity and cancer, C=living close to Tjernobyl or Fukushima, (C, U1) makes the arrow stochastic, U1 not normally drawn Con Overweight and CHD, As action: reduce weight by diet, exercise or smoke. Different effect on CHD. As exposure: overweight for many reasons with different effect on CHD No open non-causal paths 2. Cond. Positivity Arrows into E not deterministic 3. Consistency Causal paths well defined Nov-18 H.S.
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Conditional positivity example
Prior knowledge Dose response is linear Positivity problem Estimate dose response for each sex? Depends on the strength of the biological knowledge of a linear dose response, and on similar biology for the sexes.
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Conditional positivity, Common support
= exposed and unexposed for all values of C C E D strong effect of C on D may affect positivity C influence continuous E, then dichotomize Positivity between dotted lines, only this can be used for non-parametric causality With parametric assumption we can use all data Parametric assumption: linear “dose response” November 18November 18 H.S.
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Well defined intervention and contrast
Consistency Consistency = Well defined intervention and contrast November 18November 18 H.S. 23
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Air pollution no consistency Excess mortality from air pollution?
Standard method: estimate attributable fraction Implicit contrast: current levels versus zero Implicit intervention: not existent Attributable fraction * mortality type=number of deaths Calc problematic for 2 reasons: unknown intervention (the int may have side effects), and unrealistic contrast no consistency November 18November 18 H.S.
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Body Mass Index no consistency Excess mortality from obesity?
Standard method: estimate attributable fraction Implicit contrast: 30 versus <25 Exercise Implicit intervention: Diet Mortality Smoking Deaths=AF*CHD mort Contrast: unrealistic large shift in BMI distribution Intervention: many different int with highly different effect on mortatlity May also affect exch: next no consistency November 18November 18 H.S. 25
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Methods of adjustment G-methods versus Stratification based methods
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Adjusting for confounding
G-methods Stratification-methods Simulated population with exchangeability Sub population with C constant Causal effect valid for entire population Causal effect valid for sub population MSM marginal structural models NMS nested structural models IPW, standarization ←Non-parametric→ Stratification, matching MSM, NSM ←parametric→ regression Nov-18 H.S.
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Stratification based adjustment
H chocolate We want the direct effect of tea on depression O coffee C caffeine U low carb Try stratification based adjustment E tea D depression Fails: one non-causal path is left open Nov-18 H.S.
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Inverse probability weighting
chocolate H We want the direct effect of tea on depression O coffee C caffeine U low carb Try adjustment by IPW: Choose a variable V and weight by the inverse of P(V| direct causes) Try C E tea D depression Works: all non-causal paths are closed, only direct effect left Nov-18 H.S.
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Summing up Concepts Models Causal inference Adjustment
Causal definition: counterfactual (potential outcome) Causal conclusion requires causal assumptions Models DAGs, Pies causal assumptions SEM, MSM statistical + causal assumptions Causal inference Exchangeability: comparable E+ and E- Positivity: E+ and E- in all strata Consistency: well defined intervention and contrast Adjustment Stratification based stratification, matching, regression G-methods IPW, MSM November 18November 18 H.S.
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Recommended reading Books Papers
Hernan, M. A. and J. Robins. Causal Inference. Web: Rothman, K. J., S. Greenland, and T. L. Lash. Modern Epidemiology Papers Greenland, S., J. Pearl, and J. M. Robins. "Causal diagrams for epidemiologic research." Epidemiology 1999 Hernandez-Diaz, S., E. F. Schisterman, and M. A. Hernan. "The birth weight "paradox" uncovered?" Am J Epidemiol 2006 Hernan, M. A., S. Hernandez-Diaz, and J. M. Robins. "A structural approach to selection bias." Epidemiology 2004 Greenland, S. and B. Brumback. "An overview of relations among causal modeling methods." Int J Epidemiol 2002 Weinberg, C. R. "Can DAGs clarify effect modification?" Epidemiology 2007 Hernan and Robins Causal inference (web) Hernan a struct approach Hernandez- From causal Shahar Rothman Nov-18 Nov-18 Nov-18 Nov-18 H.S. H.S. H.S. 31 31 31
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References Chen, L., et al. "Alcohol intake and blood pressure: a systematic review implementing a Mendelian randomization approach." PLoS Med 2008 Greenland, S. and B. Brumback. "An overview of relations among causal modelling methods." Int J Epidemiol 2002 Hernan, M. A., S. Hernandez-Diaz, and J. M. Robins. "A structural approach to selection bias." Epidemiology 2004 Hernan, M. A. and S. R. Cole. "Causal diagrams and measurement bias." Am J Epidemiol 2009 Sheehan, N. A., et al. "Mendelian randomisation and causal inference in observational epidemiology." PLoS Med 2008 VanderWeele, T. J. and J. M. Robins. "Directed acyclic graphs, sufficient causes, and the properties of conditioning on a common effect." Am J Epidemiol 2007 VanderWeele, T. J., M. A. Hernan, and J. M. Robins. "Causal directed acyclic graphs and the direction of unmeasured confounding bias." Epidemiology 2008 VanderWeele, T. J. "The sign of the bias of unmeasured confounding." Biometrics 2008 Nov-18 H.S.
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