Chapter 7: Confounding

In Chapter 3, we introduced exchangeability as a key identifiability condition. In Chapter 6, we learned to represent causal relationships using DAGs and introduced the backdoor criterion for identifying confounding. This chapter provides a detailed examination of confounding—the most common threat to validity in observational studies.

This chapter is based on Hernán and Robins (2020, chap. 7, pp. 77-92).

1 7.1 The Structure of Confounding (pp. 77-80)

Confounding occurs when a common cause of treatment and outcome creates a non-causal association between them.

Definition 1 (Confounding Structure) A variable \(L\) is a confounder of the effect of \(A\) on \(Y\) if:

1. \(L\) causes \(A\) (or shares a common cause with \(A\))
2. \(L\) causes \(Y\) (or shares a common cause with \(Y\))
3. \(L\) is not affected by \(A\) (not a consequence of treatment)

Causal diagram representation:

L → A → Y
L → Y

The path \(A \leftarrow L \rightarrow Y\) is a backdoor path that creates non-causal association.

Why confounding creates bias:

Without confounding: \[E[Y | A = 1] - E[Y | A = 0] = E[Y^{a=1}] - E[Y^{a=0}]\] (association equals causation)

With confounding: \[E[Y | A = 1] - E[Y | A = 0] \neq E[Y^{a=1}] - E[Y^{a=0}]\] (association does not equal causation)

The observed association includes both:

• The causal effect of \(A\) on \(Y\)
• The confounding bias due to the common cause \(L\)

Common Confounding Scenarios

Example 1: Healthy worker bias

• Healthier individuals are more likely to be employed (employed → more likely to be exposed at work)
• Healthier individuals have better outcomes
• Comparing employed vs. unemployed introduces confounding by health status

Example 2: Confounding by indication

• Sicker patients receive more aggressive treatment
• Sicker patients have worse outcomes
• Treatment appears harmful when in fact it may be beneficial

Confounding can go in either direction:

• Positive confounding: Makes treatment appear more beneficial (or more harmful) than it truly is
• Negative confounding: Makes treatment appear less beneficial (or less harmful) than it truly is
• Zero confounding: The magnitude depends on the strength and direction of the \(L \rightarrow A\) and \(L \rightarrow Y\) relationships

2 7.2 Confounding and Exchangeability (pp. 80-82)

Confounding is equivalent to lack of (conditional) exchangeability.

No Confounding = Exchangeability

No confounding means: \[Y^a \perp\!\!\!\perp A \quad \text{for all } a\]

This is marginal exchangeability: the counterfactual outcomes are independent of treatment.

Confounding means exchangeability does not hold: \[Y^a \not\perp\!\!\!\perp A\]

The treated and untreated differ with respect to their potential outcomes.

Example 1 (Confounding and Exchangeability) Suppose exercise (\(A\)) affects heart disease (\(Y\)), and both are affected by age (\(L\)):

Without confounding:

• Young and old people equally likely to exercise
• \(E[Y^{a=1} | A = 1] = E[Y^{a=1} | A = 0]\) (exchangeable)

With confounding:

• Younger people more likely to exercise
  
• Younger people have lower baseline risk
• \(E[Y^{a=1} | A = 1] \neq E[Y^{a=1} | A = 0]\) (not exchangeable)
• Those who exercise would have had better outcomes even without exercising

Conditional Exchangeability

Even when marginal exchangeability fails, we may achieve conditional exchangeability by adjusting for confounders:

\[Y^a \perp\!\!\!\perp A \mid L \quad \text{for all } a\]

Within levels of \(L\), the treated and untreated are exchangeable.

Key insight: Confounding can be eliminated by conditioning on (adjusting for) the confounders.

Requirements: 1. We must identify all confounders based on subject-matter knowledge 2. We must measure them accurately 3. We must adjust for them appropriately in analysis

If these requirements are met, we can estimate causal effects from observational data.

3 7.3 Confounding and the Backdoor Criterion (pp. 82-85)

The backdoor criterion (Chapter 6) provides a graphical method for identifying confounding.

Backdoor Paths and Confounding

A backdoor path from \(A\) to \(Y\):

• Starts with an arrow into \(A\) (i.e., \(\cdot \rightarrow A\))
• Connects \(A\) to \(Y\) through any sequence of arrows

Confounding exists if backdoor paths are open (unblocked).

Example 2 (Identifying Confounders with the Backdoor Criterion) Diagram 1:

L → A → Y
L → Y

Backdoor path: \(A \leftarrow L \rightarrow Y\) Confounders: \(L\) Solution: Adjust for \(L\)

Diagram 2:

U → L → A → Y
      L → Y

Backdoor paths: \(A \leftarrow L \rightarrow Y\), \(A \leftarrow L \leftarrow U \rightarrow Y\) (if U causes Y) Confounders: \(L\) (and \(U\) if it affects \(Y\)) Solution: Adjust for \(L\) (and \(U\) if measured)

Diagram 3:

A → M → Y
L → A
L → Y

Backdoor path: \(A \leftarrow L \rightarrow Y\) Confounders: \(L\) Do NOT adjust for \(M\): \(M\) is a mediator (on the causal path), not a confounder

Common mistakes:

1. Adjusting for mediators: Variables on the causal path from \(A\) to \(Y\) should NOT be adjusted for, as this blocks the causal effect we’re trying to estimate

2. Adjusting for colliders: Variables caused by both \(A\) and \(Y\) should NOT be adjusted for, as this induces bias

3. Failing to adjust for all confounders: If even one confounder is unmeasured or unadjusted, bias remains

4. Overadjustment: Including unnecessary variables (especially descendants of treatment) can introduce bias

4 7.4 Confounding and Confounders (pp. 85-87)

The traditional definition of “confounder” in epidemiology differs slightly from the causal DAG perspective.

Traditional Confounder Definition

Traditionally, a variable \(L\) is considered a confounder if: 1. \(L\) is associated with treatment \(A\) 2. \(L\) is associated with outcome \(Y\) (among the untreated) 3. \(L\) is not affected by treatment \(A\)

DAG-Based Definition

From the DAG perspective, \(L\) is a confounder if:

• \(L\) opens a backdoor path from \(A\) to \(Y\)

Differences:

The traditional definition is based on associations (statistical relationships). The DAG definition is based on causal structure (graphical relationships).

Why this matters:

1. A variable can be associated with both \(A\) and \(Y\) without being a confounder
   • Example: A collider caused by both \(A\) and \(Y\)
   • Adjusting for it would induce bias, not remove it
2. A variable can be a confounder without being associated with both \(A\) and \(Y\) in the data
   • Example: A confounder whose effects cancel out, leaving no association
   • Failing to adjust for it would leave bias

Best practice: Use DAGs to identify confounders based on causal structure, not associations alone.

5 7.5 Single-World Intervention Graphs (pp. 87-89)

Single-World Intervention Graphs (SWIGs) are an extension of DAGs that explicitly represent interventions and counterfactual outcomes.

SWIGs vs. DAGs

• Standard DAGs: Represent relationships among observed variables
• SWIGs: Represent relationships among counterfactual variables under specified interventions

SWIG notation:

• \(Y_a\): Counterfactual outcome under intervention \(do(A = a)\)
• \(A_a\): Treatment value set to \(a\) by intervention
• Edges represent causal effects in the counterfactual world where \(A = a\)

Example SWIG: For the causal effect of \(A\) on \(Y\) with confounder \(L\):

L → A_a → Y_a
L → Y_a

SWIGs make explicit:

• Which variables are set by intervention
• Which variables remain as observed
• Which counterfactual outcome we’re interested in

Advantages:

• Clearer representation of counterfactuals
• Explicit about the intervention
• Useful for complex scenarios (time-varying treatments, mediation)

Disadvantages:

• More complex notation
• Require more assumptions to be specified
• Not yet as widely used as standard DAGs

For most purposes in this book, standard DAGs suffice. SWIGs are mentioned for completeness and for readers interested in advanced topics.

6 7.6 Confounding Adjustment (pp. 89-92)

Once confounders are identified, several methods can adjust for them.

Methods for Confounding Adjustment

1. Stratification: Estimate effects within strata of \(L\), then combine (standardization)

2. Regression adjustment: Include \(L\) as covariates in a regression model

3. Inverse probability weighting: Weight by \(1/Pr[A | L]\) to create a pseudo-population where \(A\) and \(L\) are independent (Chapter 12)

4. Matching: Match treated and untreated individuals on \(L\)

Example 3 (Comparing Adjustment Methods) Data: Effect of smoking (\(A\)) on lung cancer (\(Y\)), adjusting for age (\(L\))

Stratification:

• Estimate effect separately for age = 40, 50, 60, 70
• Combine using weighted average

Regression:

glm(Y ~ A + L, family = binomial())

IP weighting (Chapter 12):

weight <- 1 / predict(glm(A ~ L, family = binomial()), type = "response")
glm(Y ~ A, weights = weight, family = binomial())

Matching:

• For each smoker, find non-smoker of same age
• Compare outcomes

Choosing an adjustment method:

Stratification:

• ✓ Transparent, easy to understand
• ✓ Allows checking for effect modification
• ✗ Limited to discrete confounders
• ✗ Requires large sample sizes for fine strata

Regression:

• ✓ Handles continuous confounders
• ✓ Efficient (uses all data)
• ✗ Relies on model assumptions
• ✗ Can obscure effect modification

IP weighting:

• ✓ Flexible, can handle complex confounding
• ✓ Estimates marginal (population-average) effects
• ✗ Can be unstable with extreme weights
• ✗ More complex to implement

Matching:

• ✓ Intuitive, creates comparable groups
• ✗ Discards unmatched data
• ✗ Largely superseded by other methods

Modern recommendation: Use IP weighting or doubly robust methods (combine regression and weighting) for flexibility and robustness.

7 Summary

This chapter provided a detailed examination of confounding.

Key concepts:

1. Confounding structure: Common causes of treatment and outcome create backdoor paths

2. Exchangeability: Confounding = lack of exchangeability; conditional exchangeability can be achieved by adjusting for confounders

3. Backdoor criterion: Provides a graphical method to identify which variables to adjust for

4. DAG vs. traditional definitions: DAG-based confounding identification is preferred over association-based criteria

5. Adjustment methods: Stratification, regression, IP weighting, and matching can all adjust for confounding

6. Critical assumptions:

   • All confounders must be identified (no unmeasured confounding)
   • All confounders must be measured accurately
   • Adjustment must be done correctly

Practical guidelines for dealing with confounding:

1. Draw a DAG based on subject-matter knowledge before analyzing data

2. Identify confounders using the backdoor criterion

3. Measure confounders as accurately as possible

4. Choose an adjustment method appropriate for your data and confounders

5. Check assumptions:

   • Positivity: Do all \((A, L)\) combinations occur?
   • Model fit: Are regression model assumptions met?
   • Balance: After adjustment, are confounders balanced?

6. Conduct sensitivity analyses: How robust are findings to unmeasured confounding?

7. Be transparent: Report the DAG, adjustment set, and method clearly

Limitations:

Even with perfect adjustment for measured confounders, bias can remain if:

• Important confounders are unmeasured (Chapter 19 covers sensitivity analysis)
• Confounders are measured with error (Chapter 9)
• Adjustment methods are applied incorrectly
• Positivity is violated

Confounding control is necessary but not sufficient for valid causal inference.

Looking ahead:

• Chapter 8: Selection bias
• Chapter 9: Measurement bias
• Chapters 12-15: Advanced methods for confounding adjustment

8 References

Hernán, Miguel A, and James M Robins. 2020. Causal Inference: What If. Boca Raton: Chapman & Hall/CRC. https://miguelhernan.org/whatifbook.