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Hein Stigum http://folk.uio.no/heins/ courses
DAGs intro, Statistics 8h (reordered ) DAG=Directed Acyclic Graph MF 9570: Causal Inference: Monday : Introduction to causal graphs (DAGs). Speaker: Stigum : Lunch break : Causal graphs (DAGs) continued. Speaker: Stigum : PC-lab Exercises with the program DAGitty. Instructor: Stigum. Tuesday : Analyzing DAGs with examples and exercises. Speaker: Stigum. Tider: 9-12: fram til Drawing DAGs PC lab: god tid til oppgaver, kan ta innledningen der. IPTW: begrunne metoden bedre. Metode som fjerner C-D? Outcome vs exposure modelling: begrunne bedre Avsluttning: ca min, bytt ut bounding factor med size of bias Outcome dependent selection: example? (immortal time bias?) Norsk beskrivelse: Introduksjon til kausale grafer (DAGs) Kausale grafer (Directed Acyclic Graphs) er nyttige verktøy for å forstå grunnleggende begreper som konfundering, mediering og seleksjonsfeil. Grafene kan finne variable som må justeres for, og variable som ikke bør justeres for. Og grafene er en presis beskrivelse av antagelsene i analysen. Kurset vil gi en introduksjon til kausale grafer med mye eksempler og lite formalisme. Velkommen under mottoet «Draw your assumptions before your conclusions» Engelsk beskrivelse: The causal graphs are useful tools to understand key concepts like confounding, mediation and colliding (selection bias). They help in the analysis by finding a group of variables that must be adjusted for (and variables that should not be adjusted for). And they give a clear statement of prior assumptions for the analysis. Hein Stigum courses Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. H.S. H.S. 1 1 1 1 1
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Agenda Background DAG concepts Analyzing DAGs
Causal thinking, Paths Analyzing DAGs Examples DAGs and stat/epi phenomena Mediation, Matching, Mendelian randomization, Selection More on DAGs Limitations, problems Exercises DAG concepts Define a few main concepts Paths: Surprisingly few rules needed Analyzing DAGs Examples: conf, intermediate, collider Selection bias, Information bias Rand, Mend Rand The two former: manual for use More on DAGs Deeper thoughts and problems With exercises, difficult to guess time Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. H.S. H.S. 2 2 2 2 2
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Background Potential outcomes: Neyman, 1923
Causal path diagrams: Wright, 1920 Causal DAGs: Pearl, 2000 Potential outcomes or counterfactual outcomes Jerzy Neyman (April 16, 1894 – August 5, 1981), born Jerzy Spława-Neyman, was a Polish mathematician and statistician who spent the first part of his professional career in various institutions in Warsaw, Poland, and the second part at the University of California, Berkeley. Neyman first introduced the modern concept of a confidence interval into statistical hypothesis testing[2] and co-devised null hypothesis testing (in collaboration with Egon Pearson). Sewall Green Wright (December 16, 1889 – March 3, 1988) was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. Judea Pearl (born 1936) is an Israeli-born American computer scientist and philosopher, best known for championing the probabilistic approach to artificial intelligence and the development of Bayesian networks (see the article on belief propagation). He is also credited for developing a theory of causal and counterfactual inference based on structural models (see article on causality) Dec-18 H.S.
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Regression purpose Prediction models Estimation models
Predict the outcome from the covariates Ex: Air pollution from distance to roads Estimation models Estimate effect of exposure on outcome Ex: Smokers have RR=20 for lung cancer DAGs are of no interrest DAGs are important Dec-18 H.S.
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Why causal graphs? Estimate effect of exposure on disease Problem
Association measures are biased Causal graphs help: Understanding Confounding, mediation, selection bias Analysis Adjust or not Discussion Precise statement of prior assumptions Dec-18 H.S.
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CONCEPTS Causal versus casual
(Rothman et al. 2008; Veieroed et al. 2012 Dec-18 H.S.
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DAG=Directed Acyclic Graph
god-DAG Causal Graph: Node = variable Arrow = cause E=exposure, D=disease DAG=Directed Acyclic Graph Read of the DAG: Causality = arrow Association = path Independency = no path Estimations: E-D association has two parts: ED causal effect keep open ECUD bias try to close Arrows=lead to or causes Time E- exposure D- disease C, V - cofactor, variable U- unmeasured Directed= arrows Acyclic = nothing can cause itself Conditioning (Adjusting): E[C]UD Time Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 7 7 7
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Association and Cause Association 3 possible causal structure E D
(reverse cause) E D Assume E precedes D in time Association: observe Cause: infer (extra knowledge) Causal structure force on the data Basic structures, may generalize with many more variables: use paths + more complicated structures Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 8 8 8
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Confounder idea + A common cause Adjust for smoking Smoking
Yellow fingers Smoking Lung cancer + + + Yellow fingers Lung cancer + A confounder induces an association between its effects Conditioning on a confounder removes the association Condition = (restrict, stratify, adjust) Paths Simplest form Causal confounding, (exception: see outcome dependent selection) “+” (assume monotonic effects) Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. H.S. 9 9 9 9 9
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Two causes for selection to study
Collider idea Two causes for selection to study Selected subjects Selected Yellow fingers Selected Lung cancer + + + Yellow fingers Lung cancer or + and Conditioning on a collider induces an association between its causes “And” and “or” selection leads to different bias Paths Simplest form “+” (assume monotonic effects) Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. H.S. 10 10 10 10 10
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Mediator Have found a cause (E) How does it work? Mediator (M) M E D
indirect effect How does it work? Mediator (M) Paths E direct effect D 𝑇𝑜𝑡𝑎𝑙 𝑒𝑓𝑓𝑒𝑐𝑡=𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡+𝑑𝑖𝑟𝑒𝑐𝑡 𝑀𝑒𝑑𝑖𝑎𝑡𝑒𝑑 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛= 𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡 𝑡𝑜𝑡𝑎𝑙 Use ordinary regression methods if: no E-M interaction and collapsible effect measures Otherwise, need new methods Dec-18 H.S.
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Concepts: Summing up E D M E D C E D K E D
Associations visible in data. Causal structure from outside the data. DAG: no arrow means independence E D Cause M Cause with Mediator E D C Cause with Confounder E D K Cause with Collider E D Dec-18 H.S.
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Causal thinking in analyses
Dec-18 H.S.
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Aims in papers Standard aim (in introduction) Problems Solution
“We what to estimate the association between E and D” Problems Imprecise many E-D association Why adjust gives no rationale for adjusting Solution Be bold: “We what to estimate the effect of E on D” Or more realistic: “We what to estimate the association with smallest bias” Dec-18 H.S.
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Regression before DAGs
Risk factors for D: Use statistical criteria for variable selection Variable OR Comments E 2.0 C 1.2 Surprisingly low association Report all variables in the model as equals Association Both can be misleading! C E D Dec-18 H.S.
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Statistical criteria for variable selection
- Want the effect of E on D (E precedes D) - Observe the two associations C-E and C-D - Assume statistical criteria dictates adjusting for C (likelihood ratio, Akaike (赤池 弘次) or 10% change in estimate) C E D The undirected graph above is compatible with three DAGs: C C C E D E D E D Confounder 1. Adjust Mediator 2. Direct: adjust 3. Total: not adjust Collider 4. Not adjust Hirotugu Akaike 赤池 弘次 Conclusion: The data driven method is correct in 2 out of 4 situations Need information from outside the data to do a proper analysis DAGs variable selection: close all non-causal paths Dec-18 H.S.
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Reporting variable as equals: Association versus causation
Risk factors for D: Use statistical criteria for variable selection Variable OR Comments E 2.0 C 1.2 Surprisingly low association Report all variables in the model as equals Association Causation Base adjustments on a DAG C C Report only the E-effect or use different models for each exposure variable E D E D Symmetrical Directional C is a confounder for E-D C is a confounder for ED E is a confounder for C-D E is a mediator for CD Westreich & Greenland 2013 Dec-18 H.S.
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Exercise: report variables as equals?
Risk factors for Fractures Interpret as effect of: Variable OR Comments (surprises) Diabetes 2 2.0 Physical activity 1.2 Protective in other studies? Obesity 1.0 No effect? Bone density 0.8 Diabetes adjusted for all other vars. Phy. act. adjusted for all other vars. Obesity adjusted for all other vars. Bone d. adjusted for all other vars. physical activity P P is a confounder for E→D, but is E a confounder for P→D? Which effects are reported correctly in the table? diabetes 2 E fractures D obesity O bone density B 5 min Dec-18 H.S.
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Exercise: Stratify or not
Want the effect of action(A, exposure/treatment) on disease (D). Have stratified on C. Make a guess at the population effect of A on D Calculate the population effect of A on D What is the correct analysis (and RR)? OBS several answers possible! P=0.16 Re=2.0 Rd=5.0 Red=0.6 Low Bp=30% C could be low/high blood pressure Population = crude Stratified = adjusted for C C A D 10 min Hernan et al. 2011 Dec-18 H.S.
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Causal thinking: Summing up
Make a clear aim Data driven analyses do not work. Need causal information from outside the data. (Data driven prediction models OK though). Reporting table of adjusted associations is misleading. Simpson’s paradox: causal information resolves the paradox. Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 20 20
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Paths The Path of the Righteous Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 21
Ezekiel 25:17. "The Path of the Righteous Man Is Beset on All Sides by The inequities of the Selfish and the Tyranny of Evil Men." (Pulp Fiction version) Paths Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 21 21
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Path definitions Path: any trail from E to D (without repeating itself) Type: causal, non-causal State: open, closed Path 1 E®D 2 E®M®D 3 E¬C®D 4 E®K¬D Four paths: Notice: path with or against the arrows Paths show potential association Goal: Keep causal paths of interest open Close all non-causal paths Dec-18 Dec-18 H.S. H.S. 22
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Four rules 1. Causal path: ED 2. Closed path: K
(all arrows in the same direction) otherwise non-causal Before conditioning: 2. Closed path: K (closed at a collider, otherwise open) Conditioning on: 3. a non-collider closes: [M] or [C] 4. a collider opens: [K] (or a descendant of a collider) Dec-18 H.S.
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ANALYZING DAGs Dec-18 H.S.
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Confounding examples Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 H.S.
Informal, no strict notation/def Casual about the causal! Confounding examples Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. H.S. H.S. 25 25 25 25 25 25
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Vitamin and birth defects
Is there a bias in the crude E-D effect? Should we adjust for C? What happens if age also has a direct effect on D? Unconditional Path Type Status 1 E®D Causal Open 2 E¬C®U®D Non-causal Bias This is an example of confounding Noncausal open=biasing path Both C and U are confounders Problem that we have ”forgotten” arrow C->D? Conditioning on C Path Type Status 1 E®D Causal Open 2 E¬[C]®U®D Non-causal Closed Question: Is U a confounder? No bias 3 E¬[C] ®D Non-causal Closed Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 26 26 26
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Exercise: Physical activity and Coronary Heart Disease (CHD)
We want the total effect of Physical Activity on CHD. Write down the paths. Are they causal/non-causal, open/closed? What should we adjust for? Noncausal open=biasing path 5 minutes Dec-18 Dec-18 Dec-18 H.S. H.S. 27 27
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Intermediate variables
Direct and indirect effects Intermediate variables Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. H.S. H.S. 28 28 28 28 28 28
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Exercise: Tea and depression
Write down the paths. You want the total effect of tea on depression. What would you adjust for? You want the direct effect of tea on depression. What would you adjust for? Is caffeine an intermediate variable or a variable on a confounder path? Tea and depression: Finnish study Caffeine reduces depression: Nurses Health Study 10 minutes Hintikka et al. 2005 Dec-18 H.S.
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Exercise: Statin and CHD
Write down the paths. You want the total effect of statin on CHD. What would you adjust for? If lifestyle is unmeasured, can we estimate the direct effect of statin on CHD (not mediated through cholesterol)? Is cholesterol an intermediate variable or a collider? C cholesterol U lifestyle E statin D CHD Statin: lipid (cholesterol) lowering drug 10 minutes Dec-18 H.S. H.S. 30 30
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Direct and indirect effects
So far: Controlled (in)direct effect Causal interpretation: no E-M interaction and collapsible measures New concept: Natural (in)direct effect Causal interpretation also for: E-M interaction or non-collapsible measures Controlled effect = Natural effect if no E-M interaction and collapsible measures 3 h Hafeman and Schwartz 2009; Lange and Hansen 2011; Pearl 2012; Robins and Greenland 1992; VanderWeele 2009, 2014 Dec-18 H.S.
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Mixed Confounder, collider and mediator Dec-18 Dec-18 Dec-18 Dec-18
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Diabetes and Fractures
We want the total effect of Diabetes (type 2) on fractures Conditional Path Type Status 1 E→D Causal Open 2 E→F→D 3 E→B→D 4 E←[V]→B→D Non-causal Closed 5 E←[P]→B→D Unconditional Path Type Status 1 E→D Causal Open 2 E→F→D 3 E→B→D 4 E←V→B→D Non-causal 5 E←P→B→D Questions: Paths ←→? More paths? B a collider? V and P ind? Diabetes->eye disease->fall, could have ->eye disease->physical activity-> Diabetes II reduces bone density, BMI increases bone density Questions: more paths (E-B-P-E-D)? Two (or three) arrows are colliding in B, is B a collider? Mediators Confounders Dec-18 H.S.
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Drawing DAGs Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 34 34 34
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Technical note on drawing DAGs
Drawing tools in Word (Add>Figure) Use Dia Use DAGitty Hand-drawn figure. Dec-18 H.S.
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Direction of arrow C Does physical activity reduce smoking, or
does smoking reduce physical activity? ? E Phys. Act. D Diabetes 2 H Health con. C Smoking Maybe another variable (health consciousness) is causing both? E Phys. Act. D Diabetes 2 Dec-18 H.S.
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Drawing a causal DAG Start: E and D 1 exposure, 1 disease
add: [S] variables conditioned by design add: C-s all common causes of 2 or more variables in the DAG C C must be included common cause V may be excluded exogenous M may be excluded mediator K may be excluded unless we condition V E D M K Dec-18 H.S.
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DAGitty Free program to draw and analyze DAGS Dec-18 Dec-18 Dec-18
1h presentation+exercises or Just exersises (add more) DAGitty Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 38 38 38
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DAGitty background DAGitty Web page Draw DAGs Analyze DAGs Test DAGs
Run or download Johannes Textor, Theoretical Biology & Bioinformatics group, University of Utrecht Dec-18 HS
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Interface Dec-18 HS
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Draw model Draw new model New variables, connect, rename
Model>New model, Exposure, Outcome New variables, connect, rename n new variable (or double click) c connect (hit c over V1 and over V2 to connect) r rename d delete Status (toggle on/off) e exposure o outcome u unobserved a adjusted Draw Viatmin->Birth defects example Draw E and D New Age and Obesity Connect a-adjust, u-unobserved Dec-18 HS
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Export DAG Export to Word or PowerPoint
“Zoom” the DAGitty drawing first (Ctrl-roll) Use “Snipping tool” or use Model>Export as PDF Without zooming With zooming Dec-18 HS
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Model code Variable x y Age 1 @ 0.151, 0.840
Birth%20defects O @ 0.468, Obesity 1 @ 0.470, Vitamin E @ 0.145, x Arrow list Age Obesity Vitamin Obesity Birth%20defects Vitamin Birth%20defects y May change the x and y values to align the variables Dec-18 HS
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Changed model code Aligning x and y coordinates (no space after ,)
Age 1 @0.1,0.8 Birth%20defects O @0.5,1.0 Obesity 1 @0.5,0.8 Vitamin E @0.1,1.0 Age Obesity Vitamin Obesity Birth%20defects Vitamin Birth%20defects 0.1 0.5 x 0.8 1.0 Copy, paste and Update DAG y Dec-18 HS
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Exercises Dec-18 HS
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Excercise Draw the Vitamin-Birth defects DAG
Use Obesity as an observed variable. Interpret the “Causal effect identification” Interpret the “Testable implications” Add arrow from Age to Birth defects Make obesity an unobserved variable Dec-18 HS
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Excercise Draw the Statin-CHD DAG Use Lifestyle as an unobserved variable. Interpret the “Causal effect ident.” for total effects Interpret the “Causal effect ident.” for direct effects Interpret the “Testable implications” Dec-18 HS
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Exercise: Drawing survivor bias
We what to study the effect of exposure early in life (E) on disease (D) later in life. Exposure (E) decreases survival (S) in the period before D (deaths from other causes than D). A risk factor (R) reduces survival (S) in the period before D. The risk factor (R) increases disease (D). Only survivors are available for analysis (look at Collider idea). Draw and analyze the DAG 10 minutes Dec-18 Dec-18 H.S. H.S. 48
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Real world examples Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 49 49
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Endurance training and Atrial fibrillation
Tobacco Cardiovascular factors * Alcohol consumption Socioeconomic Status ** BMI Diabetes Endurance training Atrial fibrillation Genetic disposition Health *** consciousness Hyperthyreosis Height Gender Missing arrows: for example from Age and Gender to BMI, Tobacco, Cardio, … Want direct effect of ET->AF, means close many paths, all missing paths are also closed. Infinite chain of causality! Close factor distant factor Age Long-distance racing Several arrows missing! *Hypertension, heart disease, high cholesterol ** Socioeconomic status: Education, marital status *** Unmeasured factors (Blue: Mediators, red: confounders, violet: colliders) Myrstad et al. 2014b
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Methods to remove confounding
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Methods to remove confounding
Action DAG effect C Condition: Restrict, Stratify, Adjust Close path E D C Cohort matching, Propensity Score Inverse Probability Treatment Weighting Remove CE arrow E D Stratification: non-parametric adjustment Regression: parametric adjustment Matching in cohort (C=age): for every exposed person, find an unexposed of the same age. Matching in CaseControl: for every case, find a control of the same age C Case-Control matching? Other methods? Remove CD arrow E D Dec-18 H.S.
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Matching: Cohort vs Case-Control
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Matching in cohort, binary E
For every exposed person with a value of C, find an unexposed person with the same value of C S C E D selected based on E and C E independent of C after matching All open paths between C and E must C C→E C→[S]←E sum to “null” E D Cohort matching removes confounding Unfaithful DAG Stratification: non-parametric adjustment Regression: parametric adjustment Matching in cohort (C=age): for every exposed person, find an unexposed of the same age. Matching in CaseControl: for every case, find a control of the same age Cohort matching is not common, except in propensity score matching Mansournia et al. 2013; Shahar and Shahar 2012 Dec-18 H.S.
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Matching in Case Control, binary D
For every case with a value of C, find a control with the same value of C C S E D selected based on D and C D independent of C after matching All open paths between C and D must C S C→D C→[S]←D sum to “null” E D C→E→D Stratification: non-parametric adjustment Regression: parametric adjustment Matching in cohort (C=age): for every exposed person, find an unexposed of the same age. Matching in CaseControl: for every case, find a control of the same age Matched CC analysis: 1-1 matching: conditional logistic (condition on matched pair) frequency matching: adjust for C Case-Control matching does not removes confounding, unless E→D=0 (or C→E=0) must adjust for C in all analyses Case-Control matching common, may improve precision (Mansournia et al. 2013; Shahar and Shahar 2012 Dec-18 H.S.
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Inverse probability weighting
Stabilized weights. Marginal Structural Models Explain background, why IPTW Inverse probability weighting Dec-18 H.S.
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Marcov decomposition DAG implies:
Joint probability can be factorized into the product of conditional distributions of each variable given its parents C 𝑃 𝐶,𝐸,𝐷 = 𝑃(𝐶) ∙𝑃(𝐸|𝐶) ∙𝑃(𝐷|𝐸,𝐶) E D Pearl 2000 Dec-18 H.S.
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Inverse Probability Weighting
C Have observed data distribution: Factorization: 𝑃(𝐶)∙𝑃(𝐸|𝐶)∙𝑃(𝐷|𝐸,𝐶) E D C Want the RCT distribution: Factorization: 𝑃(𝐶)∙𝑃(𝐸)∙𝑃(𝐷|𝐸,𝐶) E D Can reweight the observed data with weights: 𝑃(𝐸)/𝑃(𝐸|𝐶) to obtain the RCT distribution IPW knocks out arrows in the DAG Dec-18 H.S.
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Marginal Structural Model
DAG for the reweighted pseudo data E D MSM: The expected value of a counterfactual outcome D under a hypothetical exposure e: 𝐸 𝐷 𝑒 = 𝛼 0 + 𝛼 1 𝑒 effect =𝐸 𝐷 1 −𝐸 𝐷 0 Veieroed, Lydersen et al Daniel, Cousens et al Rothman, Greenland et al. 2008 Dec-18 HS
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Outcome versus exposure modeling
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Remove confounding, variables and methods
How can we remove confounding? A C B What variables should be involved? C- A- B- yes E D no maybe Close ECD by conditioning (ordinary) regression model outcome modeling E(D| E,C) Remove the EC arrow, binary E Propensity score matching Inverse probability weighting exposure modeling P(E|C) Combine exposure and outcome modeling: doubly robust models A) Pearl 2010; B) Robinson and Jewell 1991; Xing and Xing 2010
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Outcome vs exposure modeling
Outcome model: E(D|E,C) (and possibly B) Ex: Nano-particlesCardioVascularDisease Know little about risk for nano-particles Know a lot about risk factors for CVD A C B E D Do outcome model Exposure model: E(E|C) Ex: SmokingBladder cancer Know a lot about risk for smoking Know little about risk factors for bladder cancer A C B E D Do exposure model Same confounder set, but functional forms may differ Dec-18 H.S.
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Doubly robust methods Combine the outcome and the exposure models:
Do the regression E(D|E,C) with inverse probability weighting (IPTW) Will be unbiased if the outcome- or the exposure-model is correct Doubly robust methods: Twice as right! Dec-18 H.S.
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DAGs and other causal models
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Greenland and Brumback, Causal modeling methods, Int J Epid 2002
DAGs and causal pies SCC Sufficient Component Causes background causes assumed! 4. … DAGs are less specific than causal pies DAGs are scale free, interaction is scale dependent Greenland and Brumback, Causal modeling methods, Int J Epid 2002 Dec-18 H.S.
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Exercise: causal pies H E D hospital diabetes fractures 10 minutes
Write down the causal pies for getting into hospital based on the DAG. Show that the DAG is compatible with at least 3 different combinations of sufficient causes. Selection bias: Discuss how the different combinations of sufficient causes for getting into hospital might affect the estimate of E on D among hospital patients (perhaps difficult). H hospital E diabetes D fractures 10 minutes Dec-18 H.S.
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Structural Equation Models, SEM
Causal assumptions + statistical model + data SEM: parametric DAG X is unmeasured (latent), x-I and y are measured variables Legg in bilde fra AMOS som eksempel December 18December 18 H.S.
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Causal models compared
DAGs qualitative population assumptions sources of bias (not easily seen with other approaches) Causal Pies (SCC) specific hypotheses about mechanisms of action SEM quantitative analysis of effects Dec-18 H.S.
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Selection bias Three concepts Dec-18 Dec-18 Dec-18 Dec-18 Dec-18
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Selection bias: concept 1 Simple version
“Selected different from unselected” Prevalence (D) Old have lower prevalence than young Old respond less to survey Selection bias: prevalence overestimated Effect (E→D) Old have lower effect of E than young Selection bias: effect of E overestimated Selection bias often based on idea of difference: the selected are different from unselected. Must be different in what we are measuring. Different in prevalence Different in E-D effect Weight by stratum size or inverse stratum variance Dec-18 H.S.
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Selection bias: concept 1 “Selected different from unselected”
Paths Type Status smoke®CHD Causal Open S age smoke CHD Normally, selection variables unknown Selection bias often based on idea of difference: the selected are different from unselected. Must be different in what we are measuring. Different in prevalence Different in E-D effect Weight by stratum size or inverse stratum variance Properties: - Need smoke-age interaction - Cannot be adjusted for, but stratum effects OK True RR=weighted average of stratum effects RR in “natural” range ( ) Scale dependent Name: interaction based? Dec-18 H.S.
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Selection bias: concept 2 Simple version
“Distorted E-D distributions” DAG Collider bias Words Selection by sex and/or age Distorted sex-age distribution Old have more disease Men are more exposed Distorted E - D distribution Selection bias often based on idea of difference: the selected are different from unselected. Must be different in what we are measuring. Different in prevalence Different in E-D effect Weight by stratum size or inverse stratum variance Dec-18 H.S.
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Selection bias: concept 2 “Distorted E-D distributions”
Paths Type Status smoke®CHD Causal Open smoke¬sex®[S]¬age®CHD Non-causal sex age smoke CHD Properties: Open non-causal path (collider) Does not need interaction Can be adjusted for (sex or age) Not in “natural” range (“surprising bias”) Name: Collider stratification bias Common table of properties? Both types of selection may operate in the named examples. Ref to Pearl Selection bias types: Berkson’s, loss to follow up, nonresponse, self-selection, healthy worker Hernan et al, A structural approach to selection bias, Epidemiology 2004 Dec-18 H.S.
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1) “Exclusive or” selection
Low EMF-Low IQ: not interested High EMF-Low IQ: interested Low EMF-high IQ: positive to research High EMF-high IQ: know there is no effect Show “exclusive or” because of easely seen striking effects Dec-18 H.S.
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Exercise: Dust and COPD Chronic Obstructive Pulmonary Disease
Calculate the RR of dust (high/low) on COPD in good and poor health groups. Write down the paths for the effect of E on D. E0 and D0 are unknown (past) measures. What would you adjust for? Suppose the crude effect of dust on COPD is RR=0.7 and the true RR=2. What do you call this bias? Could the concept 1 (interaction based) selection bias work here? S cur. worker D0 diseases H health E0 prior dust E cur. dust D COPD COPD risks: COPD: Chronic obstructive pulmonary disease Risk factors: smoking, air pollution, genetics, workplace dust 10 minutes Dec-18 H.S.
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Convenience sample, homogenous sample
hospital Convenience: Conduct the study among hospital patients? E diabetes D fractures 2. Homogeneous sample: Population data, exclude hospital patients? Unconditional Path Type Status 1 E→D Causal Open 2 E→H←D Non-causal Closed Conditional Path Type Status 1 E→D Causal Open 2 E→[H]←D Non-Causal Collider, selection bias Collider stratification bias: at least on stratum is biased Dec-18 H.S.
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Selection bias: concept 3 Outcome dependent selection
Selection into the study based on D. Get bias among selected. E D U Explanation: Always have exogenous U. D is a collider on E→D←U, S is a descendant of collider D. Conditioning on (a descendant of) a collider opens the E→D←U path, and U becomes associated with E. U now acts a confounder for E→D. Selection depends on: Strength of E→D. Strength of U→D Example of non-causal confounding Unmatched Case-Control Dec-18 H.S.
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Selection bias summing up
Concept 1 Concept 2 Concept 3 Selected differ from unselected in E-D effects Selected differ from unselected in E-D distributions Outcome dependent selection Interaction Collider bias Non-causal confounder “natural” effects “surprising” effects Bias depends on: E→D, U→D strength Report stratum effects Adjust Adjust if possible smoke CHD age S smoke CHD age S sex smoke CHD S U Quite different concepts Dec-18 H.S.
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MORE ON DAGs Dec-18 H.S.
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Back door, front door, D-separated
Paths from E to D, all are “leaving” E Paths open before conditioning: back door . non-causal open need to close front door . causal open E Plus paths closed at a collider If all paths from A to B are closed d-separated Pearl 2000 Dec-18 H.S.
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3 strategies for estimating causal effects
D C Back-door criterion Condition to close all no-causal paths (between E and D) E D U M1 M2 Front-door criterion Condition an all intermediate variables (between E and D) Instrumental Variables Use an IV to control the effect (of E on D) IV criteria: IV must affect E No direct IV-D effect IV and D no common causes U 3 IV 1 E D 2 Pearl 2009, Glymor and Greenland, 2008 Dec-18 Dec-18 H.S. H.S. 81
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Example: front–door criterion
Weight and Coronary Heart Disease U lifestyle Assume: adjusted for sex, age and smoke lifestyle is unmeasured no other mediators (between E and D) C cholesterol E weight D CHD B blood pressure Can estimate effect of E on D Path Type Status 1 E→C→D Causal Open 2 E→B→D 3 E←U→D Non-causal Difference = causal Crude Adjusted for B and C Weight is not a good “action” Dec-18 H.S.
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Confounding versus selection bias
Path: Any trail from E to D (without repeating itself) Open non-causal path = biasing path Confounding and selection bias not always distinct May use DAG to give distinct definitions: C E D B A Confounding: Non-causal path without colliders K E D B A Selection bias: Non-causal path open due to conditioning on a collider E D B A Causal Note: interaction based selection bias not included Hernan et al, A structural approach to selection bias, Epidemiology 2004 Dec-18 H.S.
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Testable implications
The DAG implies: C is independent of D given E and O O test Regress D on E, C and O, if the C coefficient is different from zero we reject the DAG or rather add the arrow. DAGitty gives a list of testable implications Textor et al. 2011 Dec-18 H.S.
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Definitions Dec-18 H.S.
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Causal graphs: definitions
Graph showing causal relations and conditional independencies between variables G={V,E} Vertices=random variables Edges=associations or cause Edges undirected or → directed Path Sequence of connected edges: [(L,A),(A,Y)] Parent → child Ancestors → → descendants Exogenous: variables with no parents U L A Y U Dec-18 H.S.
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Directed Acyclic Graphs
Ordinary DAG Arrows = associations Causal DAG Arrows = cause All common causes of any pair of variables in the DAG are included Two types of variables Immutable sex, age Mutable exposure (actions), smoking Mixing variables in a DAG is OK All dependence/independence conclusions valid L A Y U Dec-18 H.S.
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Variables and arrows Variable at least two values
cause, almost any causal definition will work E D usually on the individual level, “at least one subject with an effect of the exposure” ? age only possible on group level E D +/-, the dose response can be linear, threshold, U-shaped or any other (DAGs are non-parametric) DAGs are non-parametric Dec-18 H.S.
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DAG units DAG units individuals populations C C gene D gene D
(almost) No variable can influence a gene in an individual No confounding A variable can influence gene frequency in a population Dec-18 H.S.
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D-separation, moralization
Directed graph-separation two variables d-separated if no open path otherwise d-connected 2 DAG analyses Paths (Pearl) Moralization (Lauritzen) equivalent Dec-18 H.S.
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DAGs and probability theory
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DAGs rules and statistical independence
DAG correct? Two assumptions Compatibility: separated independent Faithfulness: separated independent = connected dependent Weak faithfulness: connected variables may be dependent B B A Y A Y Pearl 2009, Glymor and Greenland, 2008 Dec-18 H.S.
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Limitations, problems and extensions of DAGs
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Limitations and problems of DAGs
New tool relevance debated, focus on causality Focus on bias precision also important Bias or not direction and magnitude Interaction scale dependent Static may include time varying variables Simplified infinite causal chain Simplified do not capture reality Dec-18 H.S.
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DAG focus: bias, not precision
Should we adjust for C? DAG: no bias from C, need not adjust E D May include C to improve precision, depends on model! E->D=1 in linear regr C->D=2 in linear regr D2=10% in logistic, crude E->D=1.17, adjusted E->D=1.43, no E-C interaction Including C: better precision Including C: worse precision OR not collapsible Robinson and Jewell 1991; Xing and Xing 2010 Dec-18 H.S.
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Signed DAGs and direction of bias
U M Positive or negative bias from confounding by U? + + Neg True E→D Pos E D on average Average monotonic effect + - X Y → for all Y=y Distributional monotonic effect To find direction of bias, multiply signs: Need distributional monotonic effects except at each end Positive bias from this confounding VanderWeele, Hernan & Robins, 2008 Dec-18 H.S.
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Simplified bounding factor
If RREU and RRUD have the same magnitude, then to explain away and observed effect=RR: 𝑅𝑅 𝐸𝑈 = 𝑅𝑅 𝑈𝐷 ≥𝑅𝑅+ 𝑅𝑅(𝑅𝑅−1) Observed RR 3 2 1,5 1,3 Confounder 5.4 3.4 2.4 1.9 U >2.4 >2.4 E D RRobs=1.5 A confounder of strength 2.4 (on both sides) could completely explain away an observed RR=1.5 but a weaker confounder could not. (Ding and VanderWeele 2016) Dec-18 HS
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Interaction in DAGs + = DAG Causal pie Extended DAG C C C D E,C D E E
Mech- anisms C C C C E C + = D E,C D E E E E Red arrow = interaction Specify scale VanderWeele and Robins 2007 Dec-18 H.S.
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DAGs and time processes
DAGs often static, but may have time varying A1, A2,… Want total effect of A-s, Time Dependent Confounding DAG Process graph HDL HDL A1 A2 CHD Alcohol CHD The process graph is simpler but less specific The process graph allows feedback loops and has a clear time component Aalen et al. 2012 Dec-18 HS
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Most paths involving variables back in the chain (U)
Infinite causal chain U we adjust for variables E D in the analysis Most paths involving variables back in the chain (U) will be closed Dec-18 H.S.
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DAGs are simplified DAGs are models of reality
must be large enough to be realistic, small enough to be useful Dec-18 H.S.
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Better discussion based on DAGs before your conclusions
Summing up Data driven analyses do not work. Need causal information from outside the data. DAGs are intuitive and accurate tools to display that information. Paths show the flow of causality and of bias and guide the analysis. DAGs clarify concepts like confounding and selection bias, and show that we can adjust for both. Better discussion based on DAGs Draw your assumptions before your conclusions Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 102 102
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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, 2008. Morgan and Winship, Counterfactuals and Causal Inference, 2009 Pearl J, Causality – Models, Reasoning and Inference, 2009 Veierød, M.B., Lydersen, S. Laake,P. Medical Statistics. 2012 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 Berk, R.A. An introduction to selection bias in sociological data, Am Soc R 1983 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 Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. 103 103 103
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References 1 Aalen OO, Roysland K, Gran JM, Ledergerber B Causality, mediation and time: A dynamic viewpoint. Journal of the Royal Statistical Society Series A 175: Chen L, Davey SG, Harbord RM, Lewis SJ Alcohol intake and blood pressure: A systematic review implementing a mendelian randomization approach. PLoS Med 5:e52. Daniel RM, Cousens SN, De Stavola BL, Kenward MG, Sterne JAC Methods for dealing with time-dependent confounding. Statistics in Medicine 32: Greenland S, Schlesselman JJ, Criqui MH Re: "The fallacy of employing standardized regression coefficients and correlations as measures of effect". AJE 125: Greenland S, Robins JM, Pearl J Confounding and collapsibility in causal inference. Statistical Science 14:29-46. Greenland S, Brumback B An overview of relations among causal modelling methods. Int J Epidemiol 31: Greenland S, Mansournia MA Limitations of individual causal models, causal graphs, and ignorability assumptions, as illustrated by random confounding and design unfaithfulness. Eur J Epidemiol. Greenland SM, Malcolm; Schlesselman, James J.; Poole, Charles; Morgenstern, Hal Standardized regression coefficients: A further critique and review of some alternatives. Epidemiology 2:6. Hafeman DM, Schwartz S Opening the black box: A motivation for the assessment of mediation. International Journal of Epidemiology 38: Hernan MA, Hernandez-Diaz S, Werler MM, Mitchell AA Causal knowledge as a prerequisite for confounding evaluation: An application to birth defects epidemiology. AJE 155: Hernan MA, Hernandez-Diaz S, Robins JM A structural approach to selection bias. Epidemiology 15: Hernan MA, Cole SR Causal diagrams and measurement bias. AJE 170: Hernan MA, Clayton D, Keiding N The simpson's paradox unraveled. Int J Epidemiol. Hintikka J, Tolmunen T, Honkalampi K, Haatainen K, Koivumaa-Honkanen H, Tanskanen A, et al Daily tea drinking is associated with a low level of depressive symptoms in the finnish general population. European Journal of Epidemiology 20: Lange T, Hansen JV Direct and indirect effects in a survival context. Epidemiology 22: Mansournia MA, Hernan MA, Greenland S Matched designs and causal diagrams. International Journal of Epidemiology 42: McCaffrey DF, Ridgeway G, Morral AR Propensity score estimation with boosted regression for evaluating causal effects in observational studies. Psychological Methods 9: Myrstad M, Lochen ML, Graff-Iversen S, Gulsvik AK, Thelle DS, Stigum H, et al. 2014a. Increased risk of atrial fibrillation among elderly norwegian men with a history of long- term endurance sport practice. Scand J Med Sci Spor 24:E238-E244. Dec-18 H.S.
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References 2 Pearl J Causality: Models, reasoning, and inference. Cambridge:Cambridge Univeristy Press. Pearl J The causal mediation formula-a guide to the assessment of pathways and mechanisms. Prev Sci 13: Robins JM, Greenland S Identifiability and exchangeability for direct and indirect effects. Epidemiology 3: Robins JM Data, design, and background knowledge in etiologic inference. Epidemiology 12: Robinson LD, Jewell NP Some surprising results about covariate adjustment in logistic-regression models. Int Stat Rev 59: Rothman KJ, Greenland S, Lash TL Modern epidemiology. Philadelphia:Lippincott Williams & Wilkins. Shahar E Causal diagrams for encoding and evaluation of information bias. Journal of evaluation in clinical practice 15: Shahar E, Shahar DJ Causal diagrams and the logic of matched case-control studies. Clinical epidemiology 4: Sheehan NA, Didelez V, Burton PR, Tobin MD Mendelian randomisation and causal inference in observational epidemiology. PLoS Med 5:e177. Textor J, Hardt J, Knuppel S Dagitty a graphical tool for analyzing causal diagrams. Epidemiology 22: VanderWeele TJ, Robins JM Directed acyclic graphs, sufficient causes, and the properties of conditioning on a common effect. AJE 166: VanderWeele TJ, Hernan MA, Robins JM Causal directed acyclic graphs and the direction of unmeasured confounding bias. Epidemiology 19: VanderWeele TJ Mediation and mechanism. Eur J Epidemiol 24: VanderWeele TJ, Arah OA Bias formulas for sensitivity analysis of unmeasured confounding for general outcomes, treatments, and confounders. Epidemiology 22:42-52. VanderWeele TJ, Hernan MA Results on differential and dependent measurement error of the exposure and the outcome using signed directed acyclic graphs. AJE 175: VanderWeele TJ A unification of mediation and interaction: A 4-way decomposition. Epidemiology 25: Veieroed M, Lydersen S, Laake P Medical statistics in clinical and epidemiological research. Oslo:Gyldendal Akademisk. Westreich D, Greenland S The table 2 fallacy: Presenting and interpreting confounder and modifier coefficients. AJE 177: Xing C, Xing GA Adjusting for covariates in logistic regression models. Genet Epidemiol 34: Dec-18 H.S.
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Extra material Dec-18 H.S.
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Non-collapsibility of the odds ratio
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Non-collapsibility of the OR
D Dec-18 H.S.
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Non-collapsibility of the OR
Non-collapsibility depends on frequency of D E D Not collapsible Appr. collapsible C:\Users\hest\Documents\Courses\Common files\Collapsibility of RR and OR Collapsible Dec-18 H.S.
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Information bias Hernan and Cole 2009; Shahar 2009;
VanderWeele and Hernan 2012 Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 Dec-18 H.S. H.S. H.S. H.S. H.S. 111 111 111 111 111 111
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Depicting measurement error
Error in E E=true exposure E*=measured exposure UE=process giving error in E b) Error in E and D D=true disease D*=diagnosed disease UD=process giving error in D E=fat intake, E* fat intake from questionnaire D=infarctions, D*= diagnosed infarctions a) and b) Can test H0 b) shows independent non-differential errors Dec-18 H.S.
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Dependent errors, differential errors
a) E and D measures both temp dependent b) Alcohol in preg and malformations c) Air pollution and asthma No assumption of additive error or linear effect of E on D needed to explain concepts, but needed to estimate effect of errors Dependent errors: Temp. in lab Differential error: Recall bias in Case-Control study Differential error: Investigator bias in cohort study Hernan and Cole, Causal Diagrams and Measurement Bias, AJE 2009 Dec-18 H.S.
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Exercise: Hair dye and congenital malformations
We study the effect of hair dye (E) during pregnancy on malformations (D) in the baby in a traditional case–control study. Mothers are asked after birth how often they dyed their hair during pregnancy. Draw a DAG of the situation were mothers do not recall exactly how often they dyed their hair, and were the recall is different for mothers with malformed babies. Use E for the correct amount of hair dye, and E* for the reported. Malformations are assumed to be without misclassification. Show the paths for the effect of E on D. Will there be a bias? Show the paths for the effect of E* on D. Will there be a bias? Can E* be associated with D even if E→D is zero? 10 minutes Dec-18 H.S.
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Selection bias depicted
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Simplified example S E and D continuous, Z, normal
Selection S E and D continuous, Z, normal True effect of E on D: =0 E D EMF IQ Stratification Selection Selection in quadrants (common understanding for continuous and binary variables) Change scales? Focus: selection types and bias patterns (Clarity over realism) Dec-18 HS
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1) “Exclusive or” selection
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2) “Inclusive or” selection
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3) “And” selection Dec-18 HS All: all states and all bp-s
Only Hypertensive will participate (just 5% participation among low bp) Perform the study in a state with high fluoride in water (just 5% participation from low fluoride (private) water) Dec-18 HS
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4) “Gradient” selection
Assume published studies show EMF->IQ effects. Dec-18 H.S.
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Z-scores Dec-18 H.S.
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Birth weight paradox M U E D Results: birth weight
Maternal smoking increases neonatal mortality overall Maternal smoking decreases neonatal mortality among low birth weight Possible explanation: conditioning on M opens collider path via U Some advocate standardizing birth weight with respect to smoking, i.e. Z-scores birth weight M U E smoke D neonatal mort Dec-18 H.S. H.S. 122 122
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Z-scores M U Z E D 𝑍 𝑖𝑗 = 𝑏𝑖𝑟𝑡ℎ 𝑤𝑒𝑖𝑔𝑡ℎ 𝑖𝑗 − 𝑚 𝑗 𝑠𝑑 𝑗 𝑗=0,1 (𝑠𝑚𝑜𝑘𝑒)
𝑍 𝑖𝑗 = 𝑏𝑖𝑟𝑡ℎ 𝑤𝑒𝑖𝑔𝑡ℎ 𝑖𝑗 − 𝑚 𝑗 𝑠𝑑 𝑗 𝑗=0,1 (𝑠𝑚𝑜𝑘𝑒) E(smoke) independent of Z All open paths between E and Z must E→Z E→M→Z sum to “null” when we condition on Z birth weight M U Adjusting for Z estimates the total effect of E on D Z No gain, crude model also estimates the total effect of E on D E smoke D neonatal mort “unfaithful” Dec-18 H.S. H.S. 123 123
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Z-scores G U E D Should not adjust for gestational age: gest. age
removes part of the effect of exposure opens up a collider path involving U Some advocate standardizing birth weight with respect to gestational age, i.e. Z-scores but this also represents some type of adjustment for gestational age G gest. age U E toxicant D birth weight Dec-18 H.S. H.S. 124 124
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Odds of Treatment weighting (OT)
Previous weighting targeted the Average Causal Effect (ATE) Want now the effect among the exposed (treated) instead (ATT) 𝑃(𝐸=1|𝐶) is the probability of treatment Can reweight the observed data with weights: 𝑃(𝐸=1|𝐶)/𝑃(𝐸|𝐶) to obtain the RCT distribution among the exposed 𝑃(𝐸=1|𝐶)/𝑃 𝐸=1 𝐶 =1 if E=1 𝑃(𝐸=1|𝐶)/𝑃 𝐸 𝐶 = 𝑃(𝐸=1|𝐶)/𝑃 𝐸=0 𝐶 =OT if E=0 McCaffrey et al. 2004 Dec-18 H.S.
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