Causal Machine Learning and Decision Intelligence

Introduction and Context

Machine learning is commonly evaluated by how accurately it predicts an outcome. Causal machine learning asks a different question: what would happen if an action, intervention, or policy were changed? This distinction is strategically important. A model may accurately predict customer churn without identifying which intervention will prevent it. Similarly, a model may predict hospital readmission while providing no reliable evidence that a particular treatment reduces readmission.

Causal machine learning combines causal inference with statistical and machine-learning methods. It supports decisions such as:

• Which customers should receive a retention offer?
• What is the effect of a pricing change on demand?
• Which patients benefit from a treatment?
• How would a public policy affect employment or inequality?
• Which intervention produces the greatest benefit per unit of cost?

The central challenge is that causal effects are fundamentally counterfactual. For an individual, we observe the outcome under the action that actually occurred, but not the outcome that would have occurred under an alternative action. Causal methods therefore require explicit assumptions, study designs, and careful interpretation.

Learning Objectives

After completing this chapter, students should be able to:

1. Distinguish prediction, association, and causation.

2. Represent causal assumptions using directed acyclic graphs.

3. Explain the potential-outcomes framework and the fundamental problem of causal inference.

4. Identify confounding, mediation, moderation, selection bias, and collider bias.

5. Compare randomized experiments with observational causal designs.

6. Estimate and interpret average, conditional, and heterogeneous treatment effects.

7. Explain uplift modeling and policy learning.

8. Conduct counterfactual analysis while recognizing identification and modeling assumptions.

9. Evaluate causal models for managerial, technical, ethical, and regulatory decisions.

1. Prediction, Association, and Causation

Prediction concerns the conditional distribution of an outcome given observed information. For example, a predictive model may estimate the probability that a customer will churn:

P(Y = 1 | X = x),

where Y is churn and X is a vector of customer characteristics. High predictive accuracy means that the model distinguishes customers with different observed risks. It does not establish that changing any component of X will change the outcome.

Association describes statistical dependence. If customers who contact support churn more frequently, support contact and churn are associated. The relationship may reflect dissatisfaction, an unobserved service failure, or reverse causality: customers may contact support because they are already considering cancellation.

Causation concerns the outcome change generated by an intervention. The causal effect of an action A is defined by comparing outcomes under alternative values of A, while holding the causal system and relevant conditions conceptually fixed. A predictive model can be useful for targeting even when it is not causal; however, using it to decide who should receive an intervention requires additional evidence.

A useful managerial distinction is:

• Predictive question: Who is likely to churn?
• Causal question: Who will be persuaded not to churn by an offer?
• Decision question: Which offer should be given to which customer, considering benefit, cost, and risk?

These questions can have different answers. A customer with high churn risk may be impossible to retain, while a customer with moderate risk may be highly responsive to an intervention.

2. Potential Outcomes and Counterfactuals

The potential-outcomes framework, associated with Neyman and Rubin, formalizes causal effects. Let A be a binary treatment, with A = 1 denoting treatment and A = 0 denoting control. Let Y(1) denote the outcome that would occur for a unit under treatment and Y(0) the outcome under control.

The individual treatment effect is:

τᵢ = Yᵢ(1) − Yᵢ(0).

Only one of these two potential outcomes is observed for each unit. If unit i receives treatment, Yᵢ(1) is observed and Yᵢ(0) is counterfactual. This is the fundamental problem of causal inference.

The average treatment effect (ATE) is:

ATE = E[Y(1) − Y(0)] = E[Y(1)] − E[Y(0)].

The expectation is taken over the target population. For a binary outcome, the ATE is a difference in probabilities and is measured in percentage points. For a continuous outcome, it is a difference in outcome units.

The average treatment effect on the treated (ATT) is:

ATT = E[Y(1) − Y(0) | A = 1].

ATT describes the average effect among those who actually received treatment. It can differ from the ATE when treatment effects vary across people and treatment assignment is selective.

For observed covariates X, the conditional average treatment effect (CATE) is:

τ(x) = E[Y(1) − Y(0) | X = x].

CATEs are central to personalization, uplift modeling, and policy targeting. They should not be interpreted as individual treatment effects without strong additional assumptions: a CATE is an average effect among units sharing covariate information, not a directly observed individual-level truth.

Identification Assumptions

Causal estimands become identifiable from data only under assumptions. Common assumptions include:

1. Consistency: If A = a, the observed outcome equals Y(a). This requires a well-defined treatment; vague or variable interventions make consistency questionable.

2. Conditional exchangeability: Y(1), Y(0) are independent of A given X. In words, after adjusting for sufficient covariates, treatment assignment is as-if random.

3. Positivity: For every covariate pattern relevant to the target population, there is a positive probability of receiving each treatment: 0 < P(A = a | X = x) < 1.

4. No interference: One unit’s treatment does not alter another unit’s potential outcome, unless the design explicitly models spillovers.

Exchangeability cannot generally be verified from observational data. It is supported by domain knowledge, study design, measurement quality, and sensitivity analysis.

3. Causal Graphs and Structural Reasoning

A directed acyclic graph (DAG) represents variables as nodes and direct causal relationships as arrows. DAGs help analysts determine which variables should be adjusted for and which should not.

Consider treatment A, outcome Y, and a common cause X:

X → A

X → Y

A → Y

Here X is a confounder. Because X influences both treatment and outcome, the observed association between A and Y mixes the treatment effect with differences in X. Adjustment for X can block the noncausal backdoor path A ← X → Y.

The backdoor criterion states, informally, that a set of variables can identify the causal effect of A on Y if it blocks every path from A to Y that begins with an arrow entering A and does not include descendants of A. This criterion is a graphical guide, not a substitute for substantive knowledge.

Variables Requiring Careful Treatment

Mediators lie on the causal pathway. For example:

Training → Skill → Productivity.

Adjusting for Skill estimates a direct effect that excludes the pathway through skill, rather than the total effect of training. If the objective is the total effect, mediators should generally not be adjusted for.

Colliders are variables caused by two other variables. If A → C ← U, conditioning on C can create a spurious association between A and U. For example, conditioning on admission to a selective program may induce association between prior preparation and an unmeasured factor influencing selection.

Moderators or effect modifiers alter the size or direction of an effect. If treatment works differently by age, region, or baseline risk, these variables are important for estimating heterogeneous effects, even when they are not confounders.

Post-treatment variables are affected by treatment. Adjusting for them can remove part of the treatment effect or introduce bias. A reliable analysis establishes the temporal order of variables before selecting adjustment covariates.

Structural Causal Models

A structural causal model represents variables as functions of their causes and exogenous disturbances. For example:

Y = f(A, X, U),

where U represents unmeasured causes. An intervention do(A = a) replaces the mechanism determining A with an external assignment. The interventional distribution P(Y | do(A = a)) is generally different from the observational conditional distribution P(Y | A = a). This distinction is summarized by the warning that conditioning on treatment is not automatically equivalent to intervening on treatment.

4. Randomized Experiments and Observational Studies

Randomized Controlled Trials

Randomization assigns treatment independently of potential outcomes, in expectation. Under proper implementation, the difference in mean outcomes between treatment and control estimates the ATE:

ATE estimate = Ȳ₁ − Ȳ₀,

where Ȳ₁ is the sample mean outcome in the treatment group and Ȳ₀ is the sample mean in the control group. For a binary outcome, these are sample proportions.

Randomized experiments provide strong internal validity, but they may have limited generalizability, high cost, ethical constraints, noncompliance, attrition, or interference. Cluster randomization may be needed when individuals affect one another, but it requires analysis that accounts for dependence within clusters.

Important experimental practices include preregistering outcomes, defining treatment and control precisely, checking randomization, measuring attrition, avoiding outcome switching, and reporting uncertainty through confidence intervals.

Observational Causal Designs

When randomization is infeasible, analysts may use:

• Regression adjustment
• Matching and propensity scores
• Inverse probability weighting
• Doubly robust estimators
• Difference-in-differences
• Instrumental variables
• Regression discontinuity
• Synthetic controls

A propensity score is e(X) = P(A = 1 | X), the probability of treatment conditional on measured covariates. It can support weighting or matching, but it does not solve unmeasured confounding. A propensity model should be assessed for overlap and covariate balance, not merely predictive accuracy.

In difference-in-differences, treatment and comparison groups are observed before and after an intervention. The key identifying assumption is parallel trends: absent treatment, the average change in the two groups would have been the same. This assumption should be investigated using pre-intervention trends, though it cannot be proven from a single post-treatment comparison.

An instrumental variable affects treatment assignment but influences the outcome only through treatment and is independent of unmeasured causes of the outcome. These assumptions are strong and often difficult to defend. Under additional conditions, instrumental-variable estimates may represent a local average treatment effect for compliers rather than the effect for the entire population.

Regression discontinuity exploits a treatment rule based on a threshold. Units just above and below the cutoff may be comparable, but the resulting effect is local to that threshold and depends on no manipulation of the assignment variable.

5. Machine Learning for Causal Effect Estimation

Machine learning is useful when the relationship between covariates, treatment, and outcome is complex or high-dimensional. Its role is often to estimate nuisance functions such as:

• The outcome regression m(a, x) = E[Y | A = a, X = x]
• The propensity score e(x) = P(A = 1 | X = x)

Flexible prediction alone does not guarantee causal validity. Causal estimators still depend on design, identification assumptions, appropriate estimands, and uncertainty quantification.

Doubly Robust Estimation

For a binary treatment, a common augmented inverse-probability expression for the ATE is based on the influence function:

φᵢ = m(1, Xᵢ) − m(0, Xᵢ) + Aᵢ[Yᵢ − m(1, Xᵢ)]/e(Xᵢ) − (1 − Aᵢ)[Yᵢ − m(0, Xᵢ)]/[1 − e(Xᵢ)].

The ATE can be estimated by averaging φᵢ. Under suitable regularity conditions and exchangeability, consistency, and positivity, this estimator is called doubly robust because it can remain consistent if either the outcome models or the propensity model is correctly specified, subject to technical conditions. “Doubly robust” does not mean robust to arbitrary confounding, positivity violations, poor measurement, or treatment misspecification.

Cross-fitting, in which nuisance models are trained on one data fold and evaluated on another, can reduce overfitting bias and support valid inference under modern machine-learning conditions. The exact inferential guarantees depend on sample size, complexity, rates of convergence, and estimator construction.

Heterogeneous Treatment Effects

An effect is heterogeneous when it varies across units or groups. Analysts may estimate CATE using causal forests, Bayesian additive regression trees, meta-learners, neural models, or other methods. Common meta-learners include:

• S-learner: one model includes treatment as a feature.
• T-learner: separate outcome models are fitted for treatment and control.
• X-learner: treatment effects are imputed and then modeled, often useful under treatment imbalance.
• R-learner: treatment effects are estimated after residualizing outcomes and treatment.

No learner automatically creates causal information. The data must contain credible treatment variation, and effect estimates require validation. Splitting data for discovery and confirmation, prespecifying subgroup hypotheses, calibrating intervals, and evaluating policy value are preferable to reporting many uncorrected subgroup findings.

6. Uplift Modeling and Policy Learning

Uplift modeling estimates the incremental effect of an intervention rather than the probability of an outcome. For a binary positive outcome, uplift may be represented as:

u(x) = P(Y = 1 | do(A = 1), X = x) − P(Y = 1 | do(A = 0), X = x).

In marketing, an uplift model can distinguish four conceptual groups:

• Persuadables: treatment changes the outcome positively.
• Sure things: positive outcome occurs with or without treatment.
• Lost causes: negative outcome occurs with or without treatment.
• Sleeping dogs: treatment causes harm or reduces the desired outcome.

Targeting only high-risk individuals can waste resources on lost causes. Uplift targeting seeks people for whom the intervention creates incremental benefit.

A policy π(x) maps observed features to an action. A simple binary-treatment policy assigns treatment when estimated benefit exceeds cost. If the incremental benefit is Δ(x), treatment cost is c(x), and the benefit is valued in compatible monetary units, a treatment rule is:

π(x) = 1 if Δ(x) − c(x) > 0; otherwise π(x) = 0.

When treatment has risks, constraints, fairness requirements, capacity limits, or multiple actions, policy optimization becomes a constrained decision problem rather than simple ranking.

Policy value is the expected outcome under a policy:

V(π) = E[Y(π(X))].

In practice, value may combine outcomes, costs, harms, and operational constraints. Evaluation should use a randomized holdout, an unbiased policy evaluator, or an appropriate off-policy estimator. Ranking metrics alone are insufficient because the best ranking model may not produce the best operational policy.

7. Worked Example: Retention Intervention

A subscription company tests a retention message. Among 10,000 randomly assigned customers, 5,000 receive the message and 5,000 receive standard service. After one month, 4,250 treated customers remain active, while 4,000 control customers remain active.

The treated retention rate is:

4,250 / 5,000 = 0.85.

The control retention rate is:

4,000 / 5,000 = 0.80.

Because assignment was randomized and the treatment was well defined, the estimated ATE for retention is:

0.85 − 0.80 = 0.05,

or 5 percentage points. The estimated incremental retained customers among the 5,000 treated customers is 5,000 × 0.05 = 250.

Suppose the message costs £2 per treated customer and the expected contribution margin from an additionally retained customer is £30. The estimated incremental contribution is 250 × £30 = £7,500. The treatment cost is 5,000 × £2 = £10,000. The estimated net incremental contribution is therefore £7,500 − £10,000 = −£2,500.

The intervention increases retention but is not profitable under these assumptions. A heterogeneous-effect analysis might show that it is profitable for a subgroup with lower contact cost or higher customer margin. Such subgroup targeting must be evaluated on new randomized data to avoid optimistic estimates caused by model selection.

8. Counterfactual Analysis for Decision Intelligence

Decision intelligence connects causal estimates to organizational choices. A sound decision process includes:

1. Define the decision, action alternatives, population, time horizon, and outcome.

2. Specify the estimand: ATE, ATT, CATE, incremental profit, risk reduction, or another quantity.

3. Draw a causal diagram and document assumptions.

4. Select a design capable of identifying the estimand.

5. Estimate effects with uncertainty intervals and sensitivity analyses.

6. Translate effects into value, constraints, and stakeholder consequences.

7. Pilot the policy and monitor realized outcomes.

8. Update the model as behavior, markets, and treatment mechanisms change.

Counterfactual analysis can also compare strategic scenarios, such as demand under alternative prices or employment under alternative training access. However, extrapolating beyond observed treatment levels is dangerous. A model trained on small price changes should not automatically be used to predict the effect of doubling prices.

Decision makers should distinguish:

• Internal validity: whether the estimated effect is credible for the study population and design.
• External validity: whether it applies to another population, time, setting, or treatment implementation.
• Operational validity: whether the organization can deliver the intervention as assumed.
• Economic validity: whether the outcome improvement exceeds full costs and risks.

9. Illustrative Case: Predictive Risk Versus Treatment Response

Background

A fictional bank wants to reduce loan delinquency. It has historical records containing income, credit history, account activity, prior late payments, and whether customers previously received financial counseling.

Problem

A predictive model identifies customers with the highest probability of delinquency. Managers propose sending counseling to the top 10 percent of high-risk customers. However, counseling is costly, and some high-risk customers may be unlikely to respond.

Analysis

Historical counseling was not randomly assigned. Customers in financial difficulty were more likely to receive it, creating confounding by indication. A naive comparison may conclude that counseling increases delinquency because recipients were already at higher risk.

The bank first defines the estimand as the incremental probability of avoiding delinquency within six months under counseling versus standard communication. It draws a DAG containing prior risk, counseling assignment, delinquency, income, and account activity. It then conducts a randomized pilot, stratified by baseline risk and region. Treatment delivery, noncompliance, and missing outcomes are recorded.

A causal model estimates CATE by baseline risk and financial circumstances. The bank evaluates both effect estimates and expected net value, incorporating counselor cost, avoided loss, customer burden, and regulatory constraints. It discovers that the highest-risk group has a small response because many cases involve severe financial shocks. A moderate-risk group has a larger positive treatment effect and produces the greatest net value.

Approach and Outcome

The bank deploys counseling to the moderate-risk segment, retains a randomized holdout, and monitors delinquency, complaints, approval disparities, and effects across protected groups. It does not use protected attributes to deny service. Instead, it audits whether the policy creates unequal access or unequal benefit and investigates legitimate, non-discriminatory reasons for differences.

Learning Points

The case demonstrates that risk prediction is not treatment-effect estimation. A causal pilot can reveal where an intervention works, while policy evaluation must incorporate cost, implementation, fairness, and uncertainty. It also illustrates why historical treatment data may be unsuitable for direct causal interpretation.

10. Challenges, Ethics, and Governance

Unmeasured Confounding

Observational estimates may be biased by variables not measured or not included. Sensitivity analysis can assess how strong an unmeasured confounder would need to be to eliminate the observed effect. Sensitivity analysis does not repair the data; it communicates fragility.

Measurement and Missingness

Poorly measured confounders may leave residual confounding. Missing outcomes can bias results when missingness depends on treatment or unobserved outcomes. Analysts should describe missingness mechanisms, use appropriate methods, and report robustness across plausible assumptions.

Positivity and Distribution Shift

If a subgroup almost never receives one treatment, its causal effect cannot be reliably learned from that data. Artificially extrapolating into unsupported regions can produce confident but invalid recommendations. Changes in markets, policies, user behavior, or treatment delivery can also invalidate historical effects.

Fairness

Causal targeting raises questions beyond predictive fairness. A policy may have equal average effects but unequal access, unequal error costs, or unequal exposure to intervention harms. Fairness constraints can reduce aggregate value or create trade-offs among groups. These trade-offs should be made transparently with affected stakeholders.

Privacy and Consent

Causal systems often combine sensitive behavioral, financial, health, and demographic data. Data minimization, lawful processing, secure access, purpose limitation, and appropriate consent are essential. In high-impact domains, explanations should address not only why a person was predicted to be at risk, but why an intervention was recommended and what evidence supports its expected benefit.

Reproducibility and Accountability

Organizations should maintain data and code provenance, treatment definitions, analysis plans, model versions, subgroup results, uncertainty intervals, and monitoring records. A causal claim should identify its population, estimand, assumptions, study period, and limitations. Governance should assign responsibility for approving, deploying, auditing, and stopping a policy.

11. Contemporary Developments and Best Practices

Modern causal machine learning increasingly combines representation learning, causal discovery, transportability, dynamic treatment regimes, reinforcement learning, and synthetic or privacy-preserving data. Causal discovery algorithms can suggest graphs from data, but temporal information, interventions, domain expertise, and testable assumptions remain essential. Large language models may assist with extracting variables or generating hypotheses, but they do not establish causal identification merely by describing plausible mechanisms.

Best practices include:

• Start with a precise decision and estimand rather than a preferred algorithm.
• Separate predictive features from justified adjustment variables.
• Establish temporal ordering and treatment definitions.
• Prefer randomized experimentation when feasible.
• Diagnose overlap and covariate balance.
• Use cross-fitting and honest validation for flexible models.
• Report uncertainty, sensitivity analyses, and subgroup performance.
• Evaluate policy value, not only CATE prediction error.
• Test deployment through controlled pilots.
• Monitor drift, treatment fidelity, harms, and fairness.
• Document assumptions and permit independent audit.

Chapter Summary

Causal machine learning extends machine learning from predicting what is likely to happen to estimating what would happen under alternative actions. Prediction, association, and causation answer different questions. The potential-outcomes framework defines treatment effects through unobserved counterfactual outcomes, while causal graphs make assumptions about confounding, mediation, colliders, and adjustment explicit.

Randomized experiments provide strong evidence when correctly designed, whereas observational methods depend on assumptions such as exchangeability, positivity, consistency, and no interference. Machine learning can flexibly estimate outcome and treatment-assignment mechanisms, but it cannot compensate for unidentified causal effects. CATE, uplift modeling, and policy learning support targeted decisions by estimating who is likely to benefit, not merely who is at risk.

Decision intelligence translates causal effects into policies that account for value, cost, uncertainty, fairness, implementation, and regulation. Responsible systems define an estimand, validate assumptions, evaluate policy value, monitor outcomes, and preserve human and organizational accountability.

Key Terms

Causal effect: Change in an outcome caused by changing an intervention.

Counterfactual: The outcome that would have occurred under an alternative action.

Potential outcome: Outcome defined under a particular treatment level.

ATE: Average treatment effect in a target population.

ATT: Average treatment effect among treated units.

CATE: Average treatment effect conditional on observed covariates.

Confounding: Distortion caused by a variable influencing both treatment and outcome.

DAG: Directed acyclic graph representing causal assumptions.

Backdoor path: A noncausal path from treatment to outcome beginning with an arrow into treatment.

Mediator: Variable on the causal pathway from treatment to outcome.

Collider: Variable caused by two other variables, where conditioning can induce bias.

Exchangeability: Conditional independence of treatment assignment and potential outcomes.

Positivity: Positive probability of each treatment for relevant covariate patterns.

Consistency: Alignment between observed treatment and the corresponding potential outcome.

Uplift modeling: Estimation of an intervention’s incremental effect.

Policy value: Expected outcome under a specified decision policy.

Heterogeneous treatment effect: Treatment effect that varies across units or groups.

Propensity score: Probability of treatment conditional on measured covariates.

Difference-in-differences: Design comparing changes over time between treatment and comparison groups.

Decision intelligence: Use of evidence, causal analysis, and constraints to support consequential decisions.

Review Questions

Short Answer Questions

1. How does a causal question differ from a predictive question?

2. Define the average treatment effect and the average treatment effect on the treated.

3. What is the purpose of a causal DAG?

4. Why can adjusting for a mediator distort estimation of a total treatment effect?

5. State two identification assumptions required for observational causal inference.

Long Answer Questions

1. Explain the potential-outcomes framework, including the fundamental problem of causal inference and the interpretation of ATE, ATT, and CATE.

2. Compare randomized controlled trials with difference-in-differences, instrumental variables, and regression discontinuity designs.

3. Explain confounding, mediation, collider bias, and effect modification using suitable causal structures.

4. Discuss how causal machine learning methods estimate heterogeneous treatment effects and why predictive accuracy alone is insufficient.

5. Explain how an organization can translate causal estimates into a governed, economically viable, and fair intervention policy.

Analytical / Application Questions

1. A firm observes that customers receiving discounts have higher churn than customers receiving no discount. Identify at least two causal explanations for this association and propose a design capable of estimating the discount’s causal effect.

2. A health program is offered only to patients whose risk score exceeds a threshold. Explain how regression discontinuity could be used, state its key assumptions, and identify the population to which the estimate applies.

3. A retention intervention costs $3 per customer and increases retention by 4 percentage points among 2,000 treated customers. If each additionally retained customer contributes $25, calculate the estimated incremental contribution, treatment cost, and net contribution. Discuss why subgroup validation is still required before deployment.

Multiple Choice Questions with Answers

Question 1

Which statement best distinguishes prediction from causation?

A. Prediction always requires randomized data.

B. Causation concerns outcomes under alternative interventions.

C. Causation is simply a stronger correlation.

D. Prediction and causation are identical when accuracy is high.

Correct Answer: B. Causation concerns outcomes under alternative interventions.

Question 2

Which variable is typically a confounder?

A. A common cause of treatment and outcome

B. A variable caused only by the treatment

C. A variable occurring after the outcome

D. A random measurement error unrelated to all variables

Correct Answer: A. A common cause of treatment and outcome

Question 3

What does positivity require?

A. Every unit must receive both treatments.

B. Treatment effects must be identical across groups.

C. Each relevant covariate pattern has a positive probability of each treatment.

D. The outcome must be positive for every observation.

Correct Answer: C. Each relevant covariate pattern has a positive probability of each treatment.

Question 4

In the retention example, the estimated treatment effect was 0.85 − 0.80. What does this equal?

A. 1 percentage point

B. 5 percentage points

C. 15 percentage points

D. 80 percentage points

Correct Answer: B. 5 percentage points

Question 5

Why should a mediator generally not be adjusted for when estimating a total treatment effect?

A. It is always a collider.

B. It is unrelated to the outcome.

C. It may block part of the treatment pathway.

D. It guarantees positivity.

Correct Answer: C. It may block part of the treatment pathway.

Question 6

What does uplift modeling estimate?

A. Only the baseline risk of an outcome

B. The incremental effect of an intervention

C. The probability that treatment was assigned randomly

D. The measurement error in the outcome

Correct Answer: B. The incremental effect of an intervention

Question 7

Which assumption is central to a basic difference-in-differences analysis?

A. Equal treatment effects for every individual

B. Parallel trends absent the intervention

C. No treatment group is needed

D. The outcome must be binary

Correct Answer: B. Parallel trends absent the intervention

Question 8

A policy value is best described as:

A. The accuracy of a treatment-assignment classifier

B. The expected outcome under a specified policy

C. The number of variables in a causal graph

D. The probability of observing a collider

Correct Answer: B. The expected outcome under a specified policy

Question 9

In the worked retention example, the estimated incremental contribution was £7,500 and treatment cost was £10,000. What was the net incremental contribution?

A. £17,500

B. £2,500

C. −£2,500

D. −£17,500

Correct Answer: C. −£2,500

Question 10

Which statement about doubly robust estimation is correct?

A. It is immune to unmeasured confounding.

B. It can remain consistent if either the outcome or propensity model is correctly specified, under required conditions.

C. It requires no positivity assumption.

D. It proves that a treatment is effective.

Correct Answer: B. It can remain consistent if either the outcome or propensity model is correctly specified, under required conditions.