The Book of Why - The New Science of Cause and Effect

1. 1State the counterfactual as a contrast between two worlds: the real one and a contradictory one. Write it as 'Given that Y was observed, what would Y have been had X been x?' In the passages' notation this is Y_x(u): the value Y would have taken for individual u had X been assigned x.
2. 2Draw the causal diagram before reasoning. Include every variable you can name, with arrows meaning 'listens to' or 'is caused by'. In the firing squad: the court order to the captain C, C to Soldiers A and B, A and B to the prisoner's death D. A diagram of arrows alone is incomplete. Behind each arrow sits a rule (possibly probabilistic) for how the effect would change if the cause changed. You may leave that rule unspecified when the structure alone answers the question.
3. 3Step 1, use rung one on the evidence. Condition on what you observed to infer the hidden state of the real world. Prisoner dead implies A shot, B shot, the captain gave the signal, and the court gave the order.
4. 4Step 2, perform surgery on the contradicted variable. Erase every arrow pointing into it (the arrow from C to A) to free it from the influences that set it in the real world. Set it by hand to the hypothetical value (A = false).
5. 5Step 3, keep the past fixed. The fictitious world retains the real world's history for every variable not downstream of the surgery. Leave the inferred facts about C, the court, and B as they were. The abducted background is carried over.
6. 6Step 4, propagate by ordinary logic through the modified diagram. Evaluate only the variables downstream of the changed node. Local changes need no re-evaluation of everything else, as when the hunters swap 'Hunters = 8' for 'Hunters = 9' and reassess success probability.
7. 7Step 5, compare the fictitious outcome with the real one. The answer is the difference between them. Because B also fired in the real world, the prisoner is still dead had A not shot.
8. 8Check the arrow structure against the question: ask whether the model could answer 'who listens to whom' at all. Data alone, however large, cannot predict what happens when you persuade marksman A not to shoot.

The prisoner lies dead after a firing squad execution. The diagram runs from the court order to captain C, from C to Soldiers A and B, and from A and B to the prisoner's death D. The question is: had A decided not to shoot, would the prisoner be alive? Step one uses observation. The death implies A shot, B shot, the captain signaled, and the court ordered. Step two builds the fictitious world: erase the arrow from C into A, because A is no longer obeying the captain, and set A to false. Step three leaves everything else in its real-world state, so C's signal and the court's order stay as they were. B is downstream of C, not of A, so B still fires. Step four propagates: with B's bullet, D is still true. The counterfactual answer is that the prisoner would still be dead. The passage says this question requires comparing the real world with a fictitious, contradictory one. A pure rung-two intervention (A decides to fire, arrow from C to A deleted, A set true) differs because it starts from a world not yet observed.

• If the question starts from something that already happened ('would he have survived had...'), then use the three-step counterfactual procedure: infer from evidence, do surgery, propagate. Do not run a plain intervention that ignores the evidence.
• If you are asking what will happen when someone acts, with no observed outcome to reconcile, then a rung-two intervention suffices: delete the arrows into the variable, set its value, and recompute.
• If a variable is downstream of the changed node, then recompute it; if it is not, then keep its real-world value. Nothing about A's decision should alter variables not downstream of A.
• If your only input is a large body of data without knowledge of who listens to whom, then stop. No amount of rung-one data lets you predict the result of persuading A not to shoot.
• If the arrows' exact functions are unknown, then check whether the diagram's structure alone answers the query before investing in estimating them.

• Treating a counterfactual as an observation: you condition on 'A did not shoot' instead of erasing the arrow into A. The tell: the captain's signal and the court's order get revised in your answer, as if A's silence were evidence about C.
• Rewriting the past: you change B or C along with A. The tell: your fictitious world no longer shares its history with the real one, so you cannot compare outcomes.
• Skipping the observation step: you fix A = false without first inferring from the dead prisoner that B also shot. The tell: you answer 'alive' without ever asking who else fired.
• Drawing arrows without meaning: a diagram that records association but not who responds to whom. The tell: you cannot say what rule determines the effect if the cause changed.
• Over-generalizing a counterfactual lesson, as the vacuum-cleaner robot would if told 'You shouldn't have' meant never vacuum upstairs again. The tell: the conclusion targets the action instead of the mechanism (noise wakes people up).

• ▸Counterfactual question: write 'Given Y observed, what would Y have been had X been x?'
• ▸Observed outcome(s): list what actually happened (e.g., prisoner dead).
• ▸Causal diagram: list each variable and its parent arrows ('who listens to whom').
• ▸Inferred real-world state: from the observations, list the hidden variables you can conclude are true or false.
• ▸Variable to change and its new value: name X and the contradicting value (A = false).
• ▸Arrows erased: list every arrow into X that the surgery deletes.
• ▸Facts held fixed: list the variables not downstream of X, with their real-world values.
• ▸Downstream recomputation: for each descendant of X, state its value in the fictitious world.
• ▸Comparison and answer: real outcome versus fictitious outcome, and what the difference says.

1. 1State the query in the language of the rung it belongs to. 'What is the effect of Drug D on Lifespan L?' is an interventional query. 'Would the prisoner be alive had A not shot?' is a counterfactual query. Rung one data (association) cannot answer either one by itself.
2. 2Draw the causal model before touching data. Put a dot for each relevant variable and an arrow for each believed direct influence. The passages treat this model as the 'Knowledge' input to the inference engine, and the engine cannot produce an estimand without it.
3. 3Translate the query into a graph operation. For an intervention, erase every arrow into the intervened variable and set it to its prescribed value (the firing squad's 'surgery' on A). For a counterfactual, keep the observed history, erase the arrows into the variable, and set it to the contradictory value.
4. 4Check whether the answer can be computed from the data you hold. Ask whether a recipe exists that turns the data and the diagram into the target quantity. In the passages' framing, the engine either produces an estimand (a recipe) or reports that none exists. Look for confounders such as age (Z) in the ad example, and for mediators such as Social Status or Acidity, because they change what must be measured.
5. 5List the variables the recipe requires and compare them to what was actually collected. If a required variable is unmeasured, look for a substitute: a set of measured variables, a small pilot observational study, or an instrument. Mendelian randomization in the cholesterol studies is the passages' example of a substitute.
6. 6Test transportability when the data come from a different population than the target (the Arkansas question). Draw the differences between populations as graph features and ask whether the effect can be computed without new data in the target.
7. 7Only after the recipe exists, estimate it from the data. Report the statistical uncertainty that comes from sample size, measurement error, and missing data. Keep that separate from identifiability, which is a property of the model and not of the sample.
8. 8Record the assumptions the answer depends on, namely which arrows you deleted or never drew. The answer is only as credible as those absences.

Take the firing squad. The court gives an order to Captain C, C signals soldiers A and B, and both shoot, so the prisoner D dies. Interventional query: what happens if A fires on his own? Delete the arrow from C to A, set A = true, and leave the arrow from A to D intact. The prisoner dies, and B (in all likelihood) did not shoot, because nothing about A's decision should affect variables that are not its descendants. This is answerable because the diagram tells you who listens to whom. No amount of rung-one data can predict what happens if you persuade A not to shoot, because that data lacks the structure. Counterfactual query: the prisoner is dead, so A shot, B shot, C signaled, and the court ordered. Would he be alive had A not shot? Keep the observed history, erase the arrow into A, set A = false, and compare with the real world. The answer is identifiable here because the model fixes the function from A to D and the other causes are inferred from the observation. B still shoots in the fictitious world, which is why the diagram, not the correlation, decides the answer.

• If the query asks 'what if we do X' and you hold only observational data, then you need the causal diagram to show an adjustment or recipe exists before you compute anything.
• If the query is counterfactual (a specific unit, a contradicted fact), then identify what the observed evidence fixes about that unit's history, and treat the question as harder than the population-level intervention.
• If a confounder such as age lies on a path between treatment and outcome and is unmeasured, then the effect is not estimable from that data unless you find an instrument, a substitute set, or new data.
• If a mediator is unknown or misidentified, as acidity was for citrus and scurvy, then direct and indirect effect queries are not answerable until the mediating path is specified.
• If the data come from a different population than the target, then express the differences as graph features and decide whether the effect transports before you collect new data.

• Reading a rung-one correlation as an interventional answer. The tell is a regression or machine-learning prediction offered as 'what happens if we change X' with no diagram behind it.
• Pooling groups and trusting the result. Pearson's combined skull data gave a correlation of 0.197 although males alone and females alone showed negligible correlation. The tell is an association that appears or vanishes when you stratify.
• Treating a wide confidence interval as the only problem. Uncertainty from sample size is not the same as non-identifiability, and more data will not fix a missing variable or missing arrow. The tell is that more data leaves the answer undetermined.
• Drawing arrows without thinking about what they imply. An arrow asserts a probability rule for how Y would change if X changed, and a missing arrow asserts none exists. The tell is a diagram built from convenience, such as double-headed arrows kept out of uncertainty, with no stated direction assumptions.
• Assuming the explanation is the one tradition favors without checking the mediator. Doctors assumed acidity carried citrus's effect and never proved it. The tell is a mechanism that is asserted and never tested.

• ▸Query (verbatim): write the question and mark its rung: association, intervention, or counterfactual.
• ▸Target quantity: name the treatment variable, the outcome variable, and the population or individual.
• ▸Causal diagram: list every variable and arrow, and mark double-headed arrows with the direction you assume.
• ▸Graph surgery: state which arrows are erased and the value each intervened variable is set to.
• ▸Confounders and mediators: list each with measured or unmeasured status.
• ▸Recipe found (yes/no): write the expression that converts the data into the target, or the reason none exists.
• ▸Data inventory: list the variables actually collected and the population they come from.
• ▸Gaps and substitutes: for each missing variable, name a substitute, instrument, or pilot study, or mark the question not answerable.
• ▸Estimate and uncertainty: give the estimate, with the sources of uncertainty (sample size, measurement error, missing data).

1. 1State the query first and place it on the Ladder of Causation. 'What is the effect of Drug D on Lifespan L?' is an interventional (rung two) question. 'Would the prisoner be alive had A not shot?' is counterfactual (rung three). 'Is there discrimination?' is a direct-effect question. The rung and the exact quantity decide everything downstream, and no amount of rung-one data substitutes for that choice.
2. 2Draw the causal diagram from your Knowledge (box 1 of the inference engine) before touching data. Include every variable you believe is a cause of the treatment, the outcome, or both. Arrows carry probability rules, so ask for each one whether you can say how Y would change if X changed. The diagram is the dashed box the engine needs but does not supply.
3. 3Pick the estimand that matches the query, then block or keep the right paths. For a total effect, do not condition on mediators. For a direct effect (the legal 'discrimination' standard of 'everything else the same'), disable every indirect path by stratifying on the mediator. In the Berkeley case that mediator is Department.
4. 4After each adjustment, check what it opens. Stratifying on a collider or on a descendant of one opens a spurious path. In Kruskal's version, conditioning on Department closes Gender → Department → Outcome but opens Gender → Department ← State of Residence → Outcome. Add State of Residence to the adjustment set to close it again.
5. 5For interventions, do the surgery: delete all arrows into the intervened variable, set it to its prescribed value, and recompute by ordinary logic. For the firing squad, delete the arrow C → A and set A = true. The arrow A → D stays, and variables like B are untouched.
6. 6For counterfactuals, work in three moves. First use the observed evidence to fix the facts (the prisoner is dead, so A shot, B shot, the captain signaled, the court ordered). Then erase the arrow into the variable you are changing and set it to the contrary value (A = false). Keep its past history as observed, and recompute the outcome in that fictitious world.
7. 7Let the engine's recipe and your Data produce the Estimate together with its statistical uncertainty. The uncertainty reflects sample size, measurement error, and missing data. Report both, and state the diagram assumptions the estimate depends on.
8. 8Cross-check by comparing estimates under different adjustment sets, as Kruskal did. Unadjusted, females show a lower admission rate. Adjusted for Department, they show a higher rate. Adjusted for Department and State of Residence, the remaining association is the direct effect. A reversal across sets tells you which paths each set opens or closes.

Query: does gender directly affect admission at Berkeley? Under the court's standard, discrimination means the employer would have acted differently had the person been of a different gender and everything else been the same. That is the direct effect of Gender on Admission, not the total effect. In the simple diagram (Figure 9.4), Gender affects Admission directly and through Department. Stratifying by department closes the indirect path Gender → Department → Outcome, so Bickel and Hammel's partition by department gives a valid direct-effect estimate. Kruskal's diagram (Figure 9.5) adds State of Residence as a cause of both Department and Outcome. Now Department is a collider on Gender → Department ← State of Residence → Outcome. Stratifying by Department closes the mediated path but opens this spurious one, so you must also adjust for State of Residence. The numbers show the same sequence. Unadjusted, females have a lower admission rate. Adjusted for Department, they appear to have a higher rate. Adjusted for both, whatever association remains is attributed to the direct path Gender → Outcome. The valid estimate comes from the diagram, the estimand, and the adjustment set aligning, not from the data alone.

• If the query is about the total effect, then do not condition on mediators. If it is about the direct effect, then condition on them.
• If you stratify on a variable that is a collider on some path, then check for the path it opens and adjust for the other cause on that path, as with State of Residence.
• If the question is 'what if I make X happen?', then delete the arrows into X and set it. If it is 'what if X had been different given what I saw?', then fix the observed facts first, then do the surgery.
• If different adjustment sets give opposite signs (lower rate, higher rate), then do not choose by preference. Return to the diagram and decide which paths your estimand needs closed.
• If the arrows' functional forms are unknown, then rely on the diagram's structure where it suffices to identify the effect. Estimate the probability rules from data only where the structure leaves them undetermined.

• Reading an association as the answer to a causal query. The tell: the estimate comes from rung-one data (observing) but is reported as what would happen if you intervened. Data alone cannot predict the result of persuading marksman A not to shoot.
• Conditioning on a mediator while claiming a total effect. The tell: the effect shrinks or reverses once you stratify by a variable lying on the path from treatment to outcome.
• Adjusting for a collider and creating bias. The tell: a new association appears after stratifying, as with Department opening the path through State of Residence.
• Reporting a point estimate without uncertainty or without the diagram. The tell: no statement of sample-size, measurement-error, or missing-data uncertainty, and no stated assumptions.
• Treating the counterfactual world as a fresh one. The tell: you reset the variable's past instead of keeping its history as observed and erasing only the arrows into it.

• ▸Query (verbatim): write the why/what-if question in one sentence.
• ▸Rung and estimand: interventional, counterfactual, total effect, or direct effect.
• ▸Diagram: list the variables and arrows, and what knowledge supports each arrow.
• ▸Paths to keep / paths to block: name the direct path, indirect paths, and spurious paths.
• ▸Adjustment set: list the variables to stratify on and why each is included.
• ▸Collider check: for each adjusted variable, state any path it opens and the variable that closes it.
• ▸Surgery or counterfactual steps: arrows deleted, values set, and facts held fixed from evidence.
• ▸Estimate and uncertainty: the number, plus sample-size, measurement-error, and missing-data caveats.
• ▸Robustness comparison: results under alternative adjustment sets and an explanation of any differences.

1. 1State the query in causal terms before touching data. Write it as a question about an intervention or a counterfactual, for example 'What is the effect of Drug D on Lifespan L?' A query that stays at the level of association ('who takes D and who lives longer?') cannot be answered causally, however much data you have.
2. 2Draw the causal model as a diagram of dots and arrows. Each dot is a variable. Each arrow says some probability rule determines how the effect would change if the cause changed. Include every cause you believe matters, including non-deterministic ones (hunters, rain, and so on) that change the probability of success without guaranteeing it.
3. 3Encode who listens to whom. List what each variable responds to (the court order C drives the captain's signal, which drives A and B). Data alone will not tell you this, so it comes from your knowledge: past observation, past action, education, cultural mores.
4. 4Identify confounders before estimating anything. Look for the fork X ← Z → Y. Z mixes the true effect of X on Y with a spurious correlation. If Z is unmeasured, you cannot disentangle the two. If Z is measured, deconfounding is easy.
5. 5Answer intervention questions by surgery. To model 'making A happen,' delete every arrow pointing into A, set A to its prescribed value, and run ordinary logic on the rest. Leave the arrows going out of A intact. Variables A does not influence (B, in the firing squad) stay as they were.
6. 6Answer counterfactual questions in three moves. First, use the observed facts to infer the real world (the prisoner is dead, so A shot, B shot, the captain signaled, the court ordered). Second, build the fictitious world by erasing the arrow into A and setting A to false, keeping the past history the same. Third, compare outcomes in the two worlds.
7. 7Test the explanation against mechanism. Ask which mediator carries the effect. Citrus fruits were assumed to work through acidity (Citrus → Acidity → Scurvy), and that wrong mediator contributed to the remedy being dismissed. Name the mediator you believe in, then check whether it would predict the result.
8. 8Feed the model and the data to the estimand. Let the diagram tell you whether the query is answerable and which recipe to use. Then estimate the answer with its uncertainty, which reflects sample size, measurement error, and missing data. The mathematical form of each arrow often stays unspecified.

The firing squad shows all three rungs. The chain is Court order C → Captain signal → Soldiers A and B → Prisoner's death D. Rung one (association): the prisoner is dead, so you infer that A shot, B shot, the captain signaled, and the court ordered. Rung two (intervention): Soldier A decides to fire on his own. You delete the arrow from C into A and set A = true. The arrow from A to D is untouched, so the prisoner dies. B, however, is not affected by A's decision, so you judge that B (in all likelihood) did not shoot. No amount of observational data, absent knowledge of who listens to whom, lets you predict the effect of persuading marksman A not to shoot. Rung three (counterfactual): the prisoner is dead, and you ask whether he would be alive had A decided not to shoot. You keep the real-world history, erase the arrow into A, set A = false, and compare the two worlds. The same structure serves all three questions, which is what makes it a causal model and not a table of correlations.

• If the question contains 'what if I do X' or 'what would happen if', then treat it as an intervention: delete the arrows into X and set X. Do not condition on X as observed.
• If the question starts from an observed outcome and asks 'would it have happened had...', then use the three-step counterfactual: infer the real world, edit the fictitious world, compare.
• If a third variable Z points into both X and Y, then it is a confounder. Measure it and adjust for it. If it is unmeasured, state that the true and spurious effects cannot be separated.
• If the data show a trend that reverses between the whole population and its subgroups (Simpson's paradox), then use the causal diagram to decide whether to stratify. With Age as a confounder of Exercise and Cholesterol, stratify by age and conclude exercise is beneficial.
• If an explanation names a mechanism, then identify the mediator and check it. An untested mediator can bury a true effect, as with citrus and scurvy.

• Reading regression or correlation as a causal direction. The tell: you claim fathers cause sons' regression, yet the same loss of height appears going backward in time. Regression to the mean is symmetric between cause and effect, and Galton mistook it for a physical process.
• Conditioning when you should intervene. The tell: you treat 'A fired' as observed evidence and revise your beliefs about the captain and B, instead of cutting the arrow into A and leaving B alone.
• Stratifying or pooling by habit. The tell: two statisticians, using the same data on Lord's paradox, reach opposite conclusions because neither used a diagram to decide which comparison is legitimate.
• Drawing arrows with no function behind them. The tell: you cannot say how Y would change if X changed. An arrow implicitly asserts a probability rule, even if you leave its form unspecified.
• Reporting a point answer without separating uncertainty sources. The tell: no distinction between limited sample size, measurement error, and missing data in the estimate's uncertainty.

• ▸Query: write it as an intervention ('effect of X on Y') or a counterfactual ('had X been different, would Y...').
• ▸Rung: mark association, intervention, or counterfactual, whichever the query actually requires.
• ▸Variables and arrows: list each dot and each arrow with the direction of influence.
• ▸Knowledge sources: note the past observations, actions, education, or cultural mores that justify each arrow.
• ▸Confounders: list every Z with arrows into both X and Y, and mark each as measured or unmeasured.
• ▸Surgery or counterfactual edit: state which incoming arrows you delete, what value you assign, and which arrows stay intact.
• ▸Mediator check: name the variable carrying the effect and state how you would verify it.
• ▸Estimate and uncertainty: record the answer plus its sources of uncertainty (sample size, measurement error, missing data).
• ▸Transport note: state which parts of the model would stay valid in a different setting.

1. 1List every variable the query touches: the treatment, the outcome, and the factors you believe influence either one. For 'What is the effect of Drug D on Lifespan L?', start with D and L, then add whatever you know acts on them. Include the Knowledge behind each inclusion (past observation, past action, education), since the model is the repository of that knowledge.
2. 2Draw one arrow for each direct causal influence, pointing from cause to effect, and ask the 'who listens to whom' question for each variable: which other variables does it respond to? In the firing squad, the court's order reaches the captain C, C reaches soldiers A and B, and A and B reach the prisoner's death D.
3. 3Treat every missing arrow as an assumption. Omitting an arrow says there is no direct causal relationship, and that is the strongest claim in the diagram. Write each omission down so it can be challenged later.
4. 4Treat each arrow as shorthand for a probability rule or function: 'some rule specifies how Y would change if X were to change.' Do not specify the rule at this stage. The structure alone often supports causal and counterfactual estimates, whether the relationships are linear or nonlinear, deterministic or probabilistic.
5. 5Keep the model modular. You should be able to change one thing locally (e.g., 'Hunters = 8' to 'Hunters = 9') and re-evaluate without recomputing every other factor from scratch.
6. 6Test the model with the three kinds of query. For observation, read off associations. For intervention, delete all arrows pointing into the intervened variable and set it to its value (A = true, arrow from C to A erased). For counterfactual, keep the observed history, erase the arrows into the variable, and set it to the contrary value (A = false).
7. 7Use the diagram to choose adjustments, not habit. Trace each path: block indirect paths by conditioning on mediators if you want the direct effect, and check whether conditioning on a collider opens a spurious path. Then add a variable only if it closes that path.
8. 8Challenge the assumptions you cannot defend. Following Cornfield's sensitivity analysis, ask how strong an omitted relationship would have to be to explain the observed data, then judge whether that strength is plausible.

Berkeley admissions, from the passages. The simple diagram (Figure 9.4) runs Gender to Department to Outcome, with Gender also having a direct arrow to Outcome. The legal question concerns discrimination, which the court frames as the direct effect of gender: would the employer have acted the same had the person been of a different gender and everything else the same. To estimate it you must disable the indirect path Gender → Department → Outcome, so you stratify by Department. Under the Figure 9.4 assumptions, Bickel and Hammel were right to partition by department. Kruskal's version (Figure 9.5) adds State of Residence, which influences Outcome and is influenced by Department. Now stratifying by Department closes the indirect path but opens the spurious path Gender → Department ← State of Residence → Outcome, because Department is a collider on it. Controlling for State of Residence as well closes that path, so any remaining correlation is due to the direct path. The numbers match the diagram: unadjusted, females have a lower admission rate; adjusted for Department, females appear to have a higher rate. Kruskal had to argue with numbers because he lacked diagrams; the model makes the adjustment set derivable.

• If a query is about making something happen (an intervention), then delete all arrows into the intervened variable and set it to its value. Do not just condition on it, because observing and doing differ.
• If you want the total effect of X on Y, then do not condition on a mediator. If the question is about the direct effect (as in discrimination law), then condition on the mediators to freeze the indirect paths.
• If conditioning on a variable that is a collider on a path, then expect to open that path. Add a further variable that closes it, as State of Residence does in Kruskal's version.
• If you cannot defend an omitted arrow, then run a sensitivity analysis: ask how strong the alternative relationship would have to be to explain the data, and judge that strength for plausibility.
• If the question is counterfactual (what if A had not shot, given the prisoner is dead), then keep the observed history, erase the arrows into A, and set A = false. Do not rebuild the world from scratch.

• Treating the model as just arrows. The tell: you cannot say what rule or function sits behind an arrow, or you assume no one needs to ask. Each arrow implies a probability rule for how Y would change if X changed.
• Substituting a data table for a model. The tell: you treat potential outcomes as missing values to be imputed, as with Table 8.1, which has no information on whether education affects salary or the reverse and cannot represent it.
• Hiding assumptions in omitted arrows. The tell: the diagram looks simple and no one can list what was assumed absent, so the strongest claims go unchallenged.
• Conditioning on the wrong variables. The tell: adjusting for Department opens a spurious path through a collider, or you adjust for a mediator while claiming a total effect. Unadjusted and adjusted results then disagree, as in the admissions data, without explanation.
• Believing data alone answers the query. The tell: you or a learning algorithm predict the result of persuading marksman A not to shoot from data alone, with no model of who listens to whom.

• ▸Query: state the exact causal question (e.g., effect of Drug D on Lifespan L) and whether it is observational, interventional, or counterfactual.
• ▸Variables: list every variable, marking treatment, outcome, and mediators or colliders.
• ▸Arrows (who listens to whom): for each variable, name the variables it responds to.
• ▸Knowledge source: for each arrow, record the past observation, action, or education that supports it.
• ▸Omitted arrows: list each pair deliberately left unconnected, with the reason you believe there is no direct influence.
• ▸Intervention surgery: name the variable to set, the arrows to be deleted, and the value assigned.
• ▸Paths and adjustment: list the direct and indirect paths, the colliders on them, and the variables to condition on or avoid.
• ▸Sensitivity check: for each weakest omitted arrow, state how strong it would have to be to explain the data and whether that is plausible.

1. 1Write the question as P(Y | do(X)), not P(Y | X). Say which world you want to know about: everyone receiving X, as in the farmer's 'uniform application of Fertilizer 1 to the entire field'. Seeing and doing are different questions, and only the second is causal.
2. 2Draw the causal diagram before touching data. List every arrow into X and every path between X and Y, including the back-door paths that start with an arrow into X. Pearl's point is that the diagram, not the data, tells you which procedure applies: identical-looking data can require opposite analyses (Lord's paradox).
3. 3Apply the surgery mentally. Erase every arrow pointing into X, set X to its value, and keep all other arrows intact. In the firing squad, forcing A to fire deletes the arrow from C to A but leaves A→D, so the prisoner dies and B is unaffected. Anything left that still links X and Y is the confounding to remove.
4. 4Prefer randomization when you can control assignment. It simulates the do-world by making X listen only to the random device, so all old confounders are disabled and no new ones are added. Check the diagram for any arrow from the random device to Y that bypasses X (plants cannot read cards; human subjects can). Generate a fresh random assignment for each experiment.
5. 5If you cannot randomize and the confounders are measured, use back-door adjustment. Find a set of measured variables that blocks every back-door path between X and Y without opening a collider, then adjust for that set. Do not adjust for a collider or for a descendant of one (the birth-weight paradox made smoking look beneficial this way).
6. 6If a confounder is unmeasured, test whether a front-door path exists. It needs a mediator M such that X→M→Y carries the whole effect, X→M has no unblocked back door, and the back door from M to Y can be blocked by adjusting for X. Estimate X→M and M→Y separately and combine them.
7. 7If there is no mediator, look for an instrument Z. Z must affect X, must affect Y only through X, and must share no cause with Y. In a linear model, divide the Z→Y regression coefficient by the Z→X coefficient (b = r_ZY / r_ZX). With binary variables you need further assumptions such as monotonicity. In Mendelian randomization, a gene for HDL level plays the role of Z.
8. 8State the untestable assumptions behind your chosen strategy (no Card→Yield arrow, no Z→Y bypass, no Z=0 & X=1 individuals) and report them with the estimate. If no strategy identifies the effect, say so rather than adjusting by habit.

Smoking and cancer, with a possible unmeasured Smoking Gene. Back-door adjustment fails because the path Smoking ← Smoking Gene → Cancer cannot be blocked without data on the gene. The diagram offers a front door: Smoking → Tar → Cancer, with no direct Smoking→Cancer arrow and no other indirect pathway. Step one is the effect of Smoking on Tar. The only back-door route, Smoking ← Gene → Cancer ← Tar, is already blocked by the collider at Cancer, so you simply observe P(tar | smoking) against P(tar | no smoking), and the difference is the causal effect. Step two is the effect of Tar on Cancer. Here the back door Tar ← Smoking ← Gene → Cancer is open, so you block it by adjusting for Smoking. Combining the two estimates gives the effect of Smoking on Cancer without ever measuring the gene.

• If you control who gets X and the assignment cannot affect Y except through X, randomize. Otherwise move to the diagram.
• If all confounders on back-door paths are measured and none is a collider or collider descendant, use back-door adjustment. If one is unmeasured, you cannot.
• If a confounder is unmeasured and a mediator exists that satisfies the front-door conditions, use front-door adjustment.
• If there is a variable that moves X but touches Y only through X, use it as an instrument. In the linear case, divide the Z→Y coefficient by the Z→X coefficient.
• If the data look the same but the diagram differs, trust the diagram. The correct analysis can flip between cases with identical data.

• Treating P(Y | X) as P(Y | do(X)). The tell: you are 'controlling' for variables without a diagram or a stated back-door path.
• Adjusting for a collider or its descendant, as in the birth-weight paradox. The tell: adding a covariate makes the effect reverse sign or the treatment look beneficial, and the story contradicts known mechanisms.
• Naive 'experiments' that bring in new confounders: Fertilizer 1 on the high end and Fertilizer 2 on the low end of the field (Drainage), or one fertilizer per year (Weather). The tell: the treatment assignment correlates with some feature of place or time.
• Assuming randomization is clean when subjects can respond to the assignment itself (an arrow from Card to Yield) or do not comply. The tell: assignment differs from actual treatment received, and the outcome moves with assignment through a route other than treatment.
• Using an invalid instrument. The tell: Z plausibly has its own arrow to Y or shares a cause with Y, such as a gene whose effect on the outcome could run through something other than the exposure.

• ▸Causal question: write it as P(Y | do(X = x)) and name the population and the intervention.
• ▸Diagram: list all nodes and arrows, marking which variables are measured and which are unmeasured.
• ▸Arrows into X: list them, since these are what the do-operator or randomization removes.
• ▸Randomization feasible?: yes or no. If yes, list any possible arrow from the assignment device to Y that bypasses X.
• ▸Back-door paths and blocking set: list each path from X to Y that starts with an arrow into X, and the measured set that blocks them. Flag any collider.
• ▸Front-door mediator: name M, then check that X→M is unconfounded (or blockable) and that M→Y is blockable by adjusting for X.
• ▸Instrument candidate: name Z, then check that it moves X, has no path to Y except through X, and shares no cause with Y.
• ▸Chosen strategy and formula: state which method applies and the estimator.
• ▸Assumptions stated and untestable: list each assumption (monotonicity, no bypass arrow, no unmeasured collider path) and who would need to accept it.

1. 1Draw the causal diagram before touching the data. Put X (treatment), Y (outcome), and every plausible common cause as nodes, with arrows only where you believe a causal influence exists. A confounder is the variable at the center of a fork X ← Z → Y.
2. 2List every back-door path from X to Y: any path that starts with an arrow pointing into X. Mark each path as open or blocked. A path through a collider (→ B ←) is already blocked.
3. 3Choose a deconfounding set that blocks all open back-door paths and contains no descendant of X. A variable that is only associated with X and Y is not enough. If the diagram shows no arrows into X, you need no adjustment.
4. 4Check each candidate for a collider before you control for it. Controlling for a collider opens a path that was closed and creates a new spurious association. The fix is to add a variable on that path (such as A or C in Game 4) that recloses it.
5. 5Check measurability. If the confounder is unobserved (the seat-belt case, where B is not likely to be observable), look for another set that blocks the same path, such as a variable upstream or downstream of it. If none exists, you cannot disentangle the effect from the data alone.
6. 6Deconfound by stratifying. Compare treatment and control within each level of Z (for example, each age group), then average the stratum effects, weighting each stratum by its share of the target population.
7. 7Cross-check with an alternative valid set. Different strategies (no control, or controlling for C and D) should give the same answer if the model is right and the sample is large. A disagreement points to a flaw in the diagram.
8. 8If you can intervene, randomize. Randomization erases all arrows into X (do-operator logic) without adding new ones. Confirm there is no arrow from the randomizing device to Y, since plants cannot read cards but human subjects can.

The book's drug example: patients given a drug are younger on average than controls, so age sits in a fork Age → Drug and Age → Outcome. The raw comparison mixes the true drug effect with the spurious age-driven correlation. With no age data, you cannot separate the two. With age recorded, you compare treated and control patients within each age group. You then average the group-specific effects, weighting each age group by its percentage in the target population. The same logic appears in the walking-and-mortality diagram, where Age is a confounder of walking and mortality, and physical condition is another candidate, since casual walkers may have walked less because they could not walk more. The farmer's fertilizer case is a design version of the problem. Choosing Fertilizer 1 for the wetter plots makes Drainage a confounder, and switching fertilizer by year makes Weather a confounder. Randomizing with Fisher's deck of cards erases the arrows into Fertilizer and gives an unconfounded comparison.

• If no arrows point into X, then there are no back-door paths. Control for nothing, even if a variable like B is associated with X and Y.
• If a back-door path contains a collider and you have not conditioned on it, then leave the path alone. It is already blocked.
• If a candidate covariate is a collider (or a descendant of one) on a path between X and Y, then do not control for it alone. Add a variable that recloses the opened path, or skip it.
• If the confounder is unobserved, then search for another variable set that blocks the same back-door path. If none exists, state that the effect is not identifiable from the data.
• If you can assign treatment, then randomize. It disables old confounders without creating new ones. Then check whether the assignment mechanism could affect Y directly (a serious concern for human subjects).

• Applying the classical three-step test (associated with X, associated with Y among the unexposed, not on the causal path). B in Game 1 passes this test yet is not a confounder, and adjusting for it increases bias. The tell: you labeled a variable a confounder without tracing the back-door paths.
• Controlling for every pre-treatment variable on the theory that there is no harm. Conditioning on B or C in Game 2 opens a noncausal path and confounds X and Y. The tell: adjusting changes the estimate in an unexpected direction.
• Conditioning on a collider such as birth weight in the smoking study. The path Smoking → Birth Weight ← Birth Defect → Mortality opens and makes smoking look beneficial. The tell: an implausible protective effect, which epidemiologists argued over for forty years.
• Mixing up total and direct effects when adjusting for a mediator. Stratifying by Department closes Gender → Department → Outcome and opens a path via State of Residence at the collider unless you also control for State. The tell: the sign flips as variables are added (females lower unadjusted, higher with Department adjusted).
• Assuming a randomization device has no side effects. The tell: an arrow from the card (assignment) to Y, such as subjects who can tell which arm they are in, which breaks the unconfounded Model 3.

• ▸Treatment X and outcome Y: Name both and state the causal question (total or direct effect).
• ▸Candidate common causes: List each variable you think causes both X and Y (e.g., age, physical condition, drainage, weather).
• ▸Arrows into X: List every variable pointing into X. If there are none, write 'no back-door paths' and stop.
• ▸Back-door paths: Write out each path from X to Y that begins with an arrow into X.
• ▸Colliders on each path: Mark every variable where two arrowheads meet. Note whether the path is already blocked.
• ▸Proposed adjustment set: List the variables you will control for. Confirm none is a descendant of X.
• ▸Collider check: For each variable in the set, state whether controlling for it opens a path. If it does, name the variable that recloses it.
• ▸Observability: Mark each adjustment variable measured or unmeasured. For unmeasured ones, name an alternative blocking set.
• ▸Alternative valid set and result comparison: Write a second valid set and record whether both give the same estimate.

1. 1Draw the causal diagram before touching the data. Put every variable you might adjust for on it, and mark each arrow's direction. Without the diagram you cannot see which variables are colliders.
2. 2Mark every collider: a variable with two arrowheads pointing into it (A → B ← C). A collider blocks the path through it by default, so the two causes stay independent until you condition on the collider.
3. 3List every back-door path from treatment X to outcome Y, meaning each path that starts with an arrow into X. Label each path open or blocked. A path is blocked if it contains a collider you have not conditioned on. Example: X ← A → B ← D ← E → Y is already blocked at B.
4. 4Do not call a variable a confounder just because it is associated with both X and Y. Apply the back-door criterion instead. In M-bias (Game 5), B passes the traditional three-step test but is not a confounder, because X and Y are unconfounded until you control for B.
5. 5Before adding any control, ask what it is a common effect of. If it is caused by two things on separate paths, adjusting for it opens a path: a pretreatment variable can still be a collider, so 'measured before treatment' is not a safe criterion. Skip it unless you can re-close the path.
6. 6If you must condition on a collider, or the design does it for you (hospital patients, a restricted birth-weight group), look for a variable that re-closes the opened path. In Game 4, controlling for B opens the pipe and controlling for A or C closes it. If those variables are unobserved, do not condition on B.
7. 7Check for hidden selection. Studying only hospitalized patients conditions on Hospitalization, and restricting to low-birth-weight babies conditions on Birth Weight. Name each such restriction as a conditioning step and put it on the diagram.
8. 8Check for equivalent strategies. In Game 2, a researcher who adjusts for nothing and one who adjusts for C and D should get the same answer, given a correct model and a large enough sample. If your adjustment set disagrees with the minimal one, find the collider you opened.

The birth-weight paradox. Maternal smoking appears to protect low-birth-weight babies: among them, mortality is lower for babies of smokers. The causal diagram is Smoking → Birth Weight ← Birth Defect → Mortality, plus the direct effect Smoking → Mortality. Birth Weight is a collider. Restricting the analysis to low-birth-weight babies conditions on it and opens the back-door path Smoking → Birth Weight ← Birth Defect → Mortality. That path is noncausal because one arrow points the wrong way. Yet it induces a spurious correlation between Smoking and Mortality, large enough that smoking appears beneficial. The fix is to stop conditioning on birth weight when estimating the direct effect of smoking on mortality. Epidemiologists argued about this paradox for forty years, and the diagram makes the source of the bias obvious. Tyler VanderWeele's 2014 explanation uses the same diagram.

• If a candidate control has two arrows pointing into it, treat it as a collider and do not adjust for it unless you also close the path it opens.
• If the back-door paths from X to Y are all blocked by colliders, adjust for nothing. In Games 1, 2 and 5, no control is needed.
• If the only back-door path runs through an unobserved confounder and no observed variable blocks it (Game 3), you cannot fix it by adjusting for a collider, so state that the effect is not identifiable from the data you have.
• If you must condition on a collider (Game 4: B), then also condition on a variable that re-closes the opened path (A or C), and only if it is observable.
• If your sample is restricted by hospitalization, admission, or a birth-weight cutoff, treat the restriction as conditioning on a collider and check the diagram for the spurious path it opens.

• Adjusting for everything measured before treatment on the assumption it does no harm. The tell: the control is a common effect of two ancestors, as with B and C in Game 2, and the estimate shifts when you add it.
• Using the classical three-step test (associated with X, associated with Y given X, not on the causal path) to define confounders. The tell: a variable like B in M-bias passes the test even though X and Y were unconfounded before you controlled for it.
• Reading an association found in a selected sample (hospital patients, low-birth-weight babies) as a real relationship. The tell: the association vanishes in the general population, as in Berkson's paradox, or the sign flips, as when smoking looks beneficial.
• Stratifying on a mediator-like variable to get the direct effect and ignoring the new path that stratifying opens. The tell: stratifying by Department opens Gender → Department ← State of Residence → Outcome (Kruskal's point), and you have not also controlled for State of Residence.
• Treating the Monty Hall result as a probability puzzle instead of a collider. The tell: you ignore that Door Opened depends on both Your Door and Location of Car. When Monty picks at random (Let's Fake a Deal) and the arrow from the car is removed, conditioning does nothing.

• ▸Treatment (X) and outcome (Y): name them exactly.
• ▸Candidate controls: list every variable you plan to adjust for or restrict on.
• ▸Arrows into each candidate: list the parents of each variable, to find any with two or more arrowheads pointing in.
• ▸Colliders found: mark each candidate with two incoming arrowheads on some path.
• ▸Hidden conditioning: list any sample restriction (hospitalized, admitted, birth-weight cutoff) and the variable it conditions on.
• ▸Back-door paths X to Y: write each path and mark it open or blocked, and say where the block is.
• ▸Path opened by adjustment: for each collider you would condition on, write the new path it opens.
• ▸Re-closing variable: name an observed variable that blocks each newly opened path, or write 'none'.
• ▸Final adjustment set and decision: list what you adjust for, what you leave alone, or state 'not identifiable'.

1. 1Name the pair first: write the cause X and effect Y in one line (e.g., Education → Salary). A variable has no role until it is assigned relative to a specific pair.
2. 2Draw the diagram before touching the data. Every arrow says "listens to": the child is a function of its parents, as in S = f(EX, ED). Include any cause of one variable that also directly affects another variable; the source says such a cause "must be represented explicitly."
3. 3For each candidate third variable Z, locate it on the paths between X and Y. If Z sits at the center of a fork (X ← Z → Y), it is a confounder: it mixes the true effect with spurious correlation.
4. 4If Z sits on the chain X → Z → Y, it is a mediator: it carries part of the effect you are trying to measure.
5. 5Check whether Z is a common effect (X → Z ← Y), the third role in the definition. The supplied passages do not work a collider example, so use the definition's wording. Adjusting for it introduces spurious association. The source's warning is that a variable can pass the epidemiologists' three-part statistical test and still increase bias if adjusted for.
6. 6Act on the role. Confounder: adjust by comparing X groups within each level of Z, then average the effects, weighting each level by its share of the target population (the age example). Mediator: do not adjust when you want the effect that flows through it. Collider: do not adjust.
7. 7Stress-test the arrow direction. Reverse the arrow between X and Z and see whether the role flips. If the role flips, your adjustment decision depends on an assumption you must justify from causal knowledge, not from the data.
8. 8If you can intervene, use the surgery logic: delete all arrows into the intervened variable. Randomization does this, disabling old confounders without introducing new ones, so no adjustment for Z is needed.

Source example: the causal diagram for the effect of Education (ED) and Experience (EX) on Salary (S). The diagram says Experience listens to Education, and Salary listens to both. The employee table lists Alice (EX 6, ED 0), Bert (9, 1), Caroline (9, 2), David (8, 1), Ernest (12, 1) and Frances (13, 0). Bert and Caroline share EX = 9, so matching on experience looks tempting. The tempting move is to match employees with similar experience, which is treating EX as a confounder. But the diagram has ED → EX, so Experience is a mediator, and matching on it blocks part of Education's effect on Salary. The source calls mistaking a mediator for a confounder "one of the deadliest sins in causal inference." Had the arrow run EX → ED, EX would sit in a fork over ED and S, be a confounder, and matching on it would be "completely appropriate." The data are identical in both cases; only the arrow direction changes the correct action. The role comes from the diagram, not from the table.

• If Z causes both X and Y (a fork), then adjust: compare within levels of Z and average, weighting by each level's share of the target population.
• If Z lies on the path from X to Y (X → Z → Y) and you want the total effect, then do not adjust; it forbids adjustment and would block part of the effect.
• If you cannot defend the arrow direction between X and Z from causal knowledge, then do not decide the adjustment by correlation or by passing a statistical test. State both diagrams and report which role each implies.
• If Z is a common effect of X and Y (the collider role), then leave it unadjusted, because conditioning on it creates spurious association.
• If X is randomized or set by intervention, then delete the arrows into X in your diagram and do not adjust for former confounders.

• Treating Experience-like variables as confounders because they correlate with both X and Y. The tell: your justification is "it is related to both," and you never drew an arrow direction.
• Relying on the epidemiologists' three-part statistical test to select covariates. The tell: a variable passes the test, yet adjusting for it increases bias, as Greenland and Robins showed.
• Omitting a variable from the diagram that directly affects two nodes. The tell: a cause of X that also affects another node is missing as an explicit node, so a hidden common cause goes unadjusted.
• Assuming no data on the confounder means the effect is estimable. The tell: no age data on walkers, yet a causal claim; without measurements, the true and spurious effects cannot be disentangled.
• Adjusting for everything available as a precaution. The tell: your covariate list contains mediators or colliders, because 'more adjustment' is treated as safer.

• ▸Cause X and effect Y: state the pair in one line.
• ▸Candidate variable Z: name the third variable under review.
• ▸Arrows drawn: list the arrows between Z, X and Y, with direction.
• ▸Justification for each arrow direction: write the causal reason, not a correlation.
• ▸Role: confounder (X ← Z → Y), mediator (X → Z → Y), or collider (X → Z ← Y).
• ▸Flip test: if the Z–X arrow reversed, would the role change? Yes or no, and what changes.
• ▸Estimand wanted: total effect of X on Y, or something else.
• ▸Action: adjust (stratify and weight by population share), do not adjust, or randomize.
• ▸Unmeasured common causes: list any cause of two nodes not yet in the diagram.