Overview

Learning Goal: Understand regression adjustment as an alternative to IPTW for controlling confounding.

Key Questions:

How does regression “adjust” for confounding?
When does regression give us causal effects?
How does regression compare to IPTW?

Dataset: NHEFS smoking cessation study (males 25-35 years old)

Treatment: Quit smoking between 1971-1982
Outcome: Change in BMI
Confounders:
- Comorbidities: Baseline BMI, cholesterol, diabetes, high blood pressure
- Lifestyle Variables: Exercise frequency, alcohol consumption
- Demographics: Age, education (school years), race, marital status

Section A: IPTW Bridge & Review

Recap: What is IPTW?

Inverse Probability of Treatment Weighting (IPTW):

Goal: Create balance on observed confounders by reweighting
Method: Weight by inverse of propensity score

Propensity Score: \(e(X) = P(A=1 | X)\)

Probability of treatment given confounders
Calculated from logistic regression model

IPTW Weights:

Treated: \(w = \frac{1}{e(X)}\)
Control: \(w = \frac{1}{1-e(X)}\)

Result: Creates pseudo-population where treatment is independent of confounders

DAG: Confounders → Treatment

Initial Imbalance: Love Plot

Key Observation: Several variables show substantial imbalance (SMD > 0.1) before weighting, indicating confounding.

Propensity Score Distributions

Key Insight: IPTW reweights observations so that propensity score distributions become similar between treatment groups.

Balance After IPTW

Key Result: After IPTW, all variables have SMD < 0.1, indicating good balance.

Key Insight: PS Model Matters

What IPTW models: Treatment assignment \(P(A|X)\)

Critical point: You can only balance variables included in PS model!

Let’s see this with NHEFS data…

Balance Check: Love Plots

Observation:

Model 1 balances comorbidities ✓, but NOT lifestyle or demographics
Model 2 balances comorbidities + lifestyle ✓, but NOT demographics
Key point: You can only balance variables included in the PS model!

IPTW Estimates

Effect of Quitting Smoking on BMI Change: IPTW Estimates
	Naive	IPTW: Comorbidities	IPTW: +Lifestyle	IPTW: +Demographics
+ p < 0.1, * p < 0.05, p < 0.01, * p < 0.001
Quit Smoking	1.505*	1.826**	1.915**	2.213**
	(0.584)	(0.681)	(0.703)	(0.757)
Num.Obs.	218	218	218	218

Key point: Different PS models → different estimates

Now let’s see a different approach: Regression adjustment

Section B: Regression Mechanics

Simple Regression: Concept

Linear regression model: \(Y = \beta_0 + \beta_1 X + \varepsilon\)

\(\beta_1\) = expected change in Y for 1-unit change in X
Fitted line minimizes squared residuals
This is descriptive association, not causal

Let’s see two examples with NHEFS data…

Example 1: BMI Over Time

Question: How does baseline BMI predict follow-up BMI?

Result: \(\hat{\beta}_1 \approx\) 0.95 (p < .001)

Interpretation: A 1-unit increase in 1971 BMI is associated with 0.95-unit increase in 1982 BMI on average.

Example 2: BMI Change

Question: How does baseline BMI predict BMI change?

Result: \(\hat{\beta}_1 \approx\) -0.053 (p = 0.119)

Interpretation: Weak, non-significant association between baseline BMI and BMI change.

Key point: Different outcome → different relationship

Multiple Regression: Concept

Multiple regression: \(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_p X_p + \varepsilon\)

Critical interpretation: Each \(\beta\) represents effect “holding other variables constant”

This “holding constant” language is KEY to adjustment strategy!

Connection to Week 5: Like stratification, but allows continuous variables

Adding Treatment Variable

Now add quit smoking as a predictor:

Model: BMI Change = \(\beta_0 + \beta_1 \times\) Quit Smoking \(+ \beta_2 \times\) Baseline BMI

Results:

Quit smoking: \(\hat{\beta}_1 \approx\) 1.53 (p = 0.009)
Baseline BMI: \(\hat{\beta}_2 \approx\) -0.056 (p = 0.1)

Interpretation:

After adjusting for baseline BMI, quitting smoking is associated with 1.53-unit increase in BMI
Coefficient for quit smoking is now “adjusted” for baseline BMI

Visualizing Multiple Regression

Key insights:

Two parallel lines (quitters vs non-quitters)
Vertical distance = treatment effect (1.53 units)
Parallel slopes = assumes constant effect across all BMI levels

Progressive Adjustment

What if we add MORE confounders?

Watch the coefficient for 'Quit Smoking' change as we add confounders
	Baseline	+Comorbidities	+Lifestyle	+Demographics
+ p < 0.1, * p < 0.05, p < 0.01, * p < 0.001
(Intercept)	4.494**	6.212***	5.615**	4.630
	(1.511)	(1.809)	(1.858)	(3.112)
qsmkTRUE	1.530**	1.800**	1.887**	2.071***
	(0.582)	(0.598)	(0.596)	(0.609)
bmi71	-0.056+	-0.045	-0.040	-0.032
	(0.034)	(0.035)	(0.035)	(0.035)
cholesterol		-0.009	-0.008	-0.008
		(0.007)	(0.007)	(0.007)
diabetes		2.108	2.274	1.989
		(1.536)	(1.527)	(1.526)
hbp		-2.465	-2.618+	-2.359
		(1.564)	(1.556)	(1.554)
exercise1			-0.320	-0.252
			(0.564)	(0.563)
exercise2			1.158+	1.108+
			(0.630)	(0.632)
alcoholfreq			0.055	0.039
			(0.213)	(0.219)
marital				0.231
				(0.230)
race1				1.287
				(0.799)
school				-0.106
				(0.091)
age				0.040
				(0.078)
Num.Obs.	218	218	218	218
R2	0.042	0.066	0.092	0.117

Observation: Quit smoking coefficient changes from 1.53 → 2.07

This is omitted variable bias in action!

Section C: Omitted Variable Bias

The Problem

Omitted Variable Bias (OVB): What happens when we leave out a confounder?

Setup:

True model: \(Y = \beta_0 + \beta_1 A + \beta_2 Z + \varepsilon\)
Fitted model: \(Y = \beta_0 + \beta_1 A + \varepsilon\) (omit Z)
Result: \(\hat{\beta}_1\) is BIASED

When does bias occur? Need BOTH:

Z affects Y (\(\beta_2 \neq 0\))
Z correlated with A (Z is a confounder!)

OVB Direction Formula

Direction of bias:

\[\text{Sign(Bias)} = \text{Sign}(\beta_2) \times \text{Sign}(\text{Corr}(A, Z))\]

\(\beta_2\) (Z → Y)	Corr(A,Z)	Bias Direction
Positive (+)	Positive (+)	Positive (overestimate)
Positive (+)	Negative (−)	Negative (underestimate)
Negative (−)	Positive (+)	Negative (underestimate)
Negative (−)	Negative (−)	Positive (overestimate)

Let’s test this with NHEFS data…

NHEFS Example: Omitting Exercise

Scenario: What if we omit “exercise” from the model?

Results:

\(\beta_2\) (exercise → outcome): 0.476 (POSITIVE)
Corr(quit smoking, exercise): -0.013 (NEGATIVE)
Predicted bias direction: NEGATIVE (underestimate)
Observed bias: -0.079 (NEGATIVE)
Match? YES ✓

Verification Table

Effect of Including vs Omitting Exercise
	Without Exercise	With Exercise
+ p < 0.1, * p < 0.05, p < 0.01, * p < 0.001
Quit Smoking	1.800**	1.880**
	(0.598)	(0.594)
Num.Obs.	218	218
R2	0.066	0.091

Key takeaways:

OVB formula correctly predicts bias direction
This is the regression version of unmeasured confounding
If we don’t measure/include confounder → biased estimate
Same problem IPTW faces!

NHEFS Example 2: Omitting Age

New question: What if we omit age from our model?

Analysis:

\(\beta_2\) (age → outcome): 0.08 (POSITIVE)
Corr(quit smoking, age): 0.081 (POSITIVE)
Predicted bias direction: POSITIVE
Observed bias: 0.04 (POSITIVE)
Match? YES ✓

Interpretation:

Age has a positive effect on BMI change (older people gain more weight)
Age is positively correlated with quitting smoking
Omitting age creates POSITIVE bias in our treatment effect estimate

NHEFS Example 3: Omitting Baseline BMI

New question: What if we omit baseline BMI from our model?

Note: This is interesting because baseline BMI is a baseline measure of our outcome variable!

Analysis:

\(\beta_2\) (baseline BMI → outcome): -0.053 (NEGATIVE)
Corr(quit smoking, baseline BMI): 0.026 (POSITIVE)
Predicted bias direction: NEGATIVE
Observed bias: -0.025 (NEGATIVE)
Match? YES ✓

Interpretation:

Baseline BMI has a negative effect on BMI change (regression to the mean)
Higher baseline BMI is positively correlated with quitting
Omitting baseline BMI creates NEGATIVE bias

Why this matters: Baseline measures of the outcome are always important confounders to include!

Comparison Table: Multiple OVB Examples

	Omitted Variable	β₂ (Z → Y)	Corr(A, Z)	Predicted Bias	Observed Bias	Formula Works?
as.numeric(exercise)	Exercise	0.476 (+)	-0.013 (-)	NEGATIVE (underestimate)	-0.079 (-)	✓ YES
age	Age	0.08 (+)	0.081 (+)	POSITIVE	0.04 (+)	✓ YES
bmi71	Baseline BMI	-0.053 (-)	0.026 (+)	NEGATIVE	-0.025 (-)	✓ YES

Key Takeaways:

OVB formula works consistently across different types of confounders
Different confounders can create bias in different directions
Baseline measures of the outcome are critical confounders
All measured confounders should be included in the model!

Section D: IPTW Makes Regression Robust

The Key Insight

Powerful discovery: When we use IPTW weighting, the treatment effect estimate stays stable regardless of which confounders we include in the regression!

What this means:

With IPTW weights: qsmkTRUE coefficient ≈ 3.5 whether we control for 0, 1, or 10 confounders
Without IPTW weights: coefficient changes dramatically as we add/remove confounders

Why this matters:

IPTW has already balanced the confounders through weighting
Regression doesn’t need to “work hard” to adjust for them
This provides double protection against confounding

Let’s demonstrate this with our smoking data…

Four Weighted Regression Models

We’ll fit 4 models with progressively more confounders, all using IPTW weights:

Weighted Regression: Stable Estimates Across Model Specifications
	Model Specification	qsmkTRUE Estimate	Std Error	95% CI Lower	95% CI Upper
qsmkTRUE	1. No confounders	2.213	0.757	0.729	3.697
qsmkTRUE1	2. + Baseline BMI	2.222	0.758	0.735	3.709
qsmkTRUE2	3. + Comorbidities	2.192	0.734	0.753	3.630
qsmkTRUE3	4. + All confounders	2.305	0.670	0.993	3.617

KEY OBSERVATION:

All four estimates cluster around 2.23
Range across models: Only 0.113
Confidence intervals all overlap substantially
Conclusion: With IPTW weighting, which confounders we include doesn’t matter much!

Why This Works: Double Robustness

The key insight: When we achieve balance through IPTW, the weighted data mimics randomization.

Two complementary approaches to remove confounding:

IPTW: Creates balance by reweighting so confounders are uncorrelated with treatment
Regression adjustment: Directly controls for confounders in the outcome model

Together = “Double Robustness”: Using IPTW + regression provides extra protection. We saw this! Weighted Model 1 (no confounders) ≈ Weighted Model 4 (all confounders) because IPTW already removed the confounding. This approach is called doubly robust estimation and is popular in practice because you need to misspecify BOTH models to get biased estimates.

Comparison: RCT vs Observational Study with IPTW

Aspect	RCT (Randomized Trial)	Observational Study + IPTW
Balance	Created by randomization	Created by reweighting
Exchangeability	Holds by design	Assumed (conditional on X)
Confounding	None (by construction)	Removed if all confounders measured
Regression sensitivity	Low (already balanced)	Low (after IPTW balancing)
Model dependence	Minimal	Minimal (after good balance)
Key assumption	Random assignment worked	Unconfoundedness + correct PS model

Main insight: IPTW attempts to recreate the balance that randomization provides naturally in RCTs.

Section E: Synthesis & Comparison

When Does Regression Give Causal Effects?

Case 1: Randomized Control Trial

If quit smoking was randomly assigned:

Randomization → quit smoking ⊥ (all confounders)
Exchangeability holds by design

What happens with regression:

Simple model (Y ~ A): Already unbiased
Multiple model (Y ~ A + X): Increases precision, still unbiased
Functional form doesn’t matter much (we just demonstrated this!)

Bottom line: In RCT, regression is “nice to have” not “must have”

Case 2: Observational Study

Requirements for unbiased estimate:

Requirement 1: Unconfoundedness \((Y_1, Y_0) \perp A | X\)

Must measure and include ALL confounders
No unmeasured confounding
Same as IPTW!

Requirement 2: Correct functional form

Linear relationships (or include polynomials/interactions)
No missing interactions
Correct link function

Critical point: Regression requires unconfoundedness (like IPTW) PLUS correct outcome model

IPTW vs Regression: Side-by-Side

Aspect	IPTW	Regression
Strategy	Reweight to create balance	Model outcome to adjust
What you model	Treatment assignment P(A\|X)	Outcome E[Y\|A,X]
Unconfoundedness	✓ Required	✓ Required
Additional assumption	Correct PS model	Correct outcome model
Can check	Covariate balance	Residual diagnostics
Vulnerable when	PS model misspecified	Outcome model misspecified
What to get right	Treatment selection	Outcome prediction
Robust to functional form?	Yes, if balanced	No, unless balanced

Key Takeaways

1. Both methods adjust for confounding

IPTW: Adjusts via reweighting
Regression: Adjusts via modeling (“holding constant”)
Both aim to remove confounding

2. Both require same fundamentals

Unconfoundedness: (Y₁, Y₀) ⊥ A | X
Only adjust for OBSERVED confounders
Neither fixes unmeasured confounding
Results still observational, not experimental

3. Different modeling requirements

IPTW: Must correctly model P(A|X)
Regression: Must correctly model E[Y|A,X]
Both vulnerable to misspecification

4. Balance helps both methods

IPTW creates balance by design (when PS model correct)
Regression benefits from balance (less sensitive to functional form)
We demonstrated this empirically!

5. Critical reading skills

When reading papers, look for:

“Adjusted for…” → Regression
“Propensity score weighted” → IPTW
“Doubly robust” → Combines both

Always ask:

Is unconfoundedness plausible?
Were balance/diagnostics checked?
Could unmeasured confounding explain results?

Summary

Today we learned:

Regression mechanics: Simple → multiple → omitted variable bias
OVB formula: Predicts direction of bias from omitted confounders
Balance matters: Functional form misspecification less critical when balanced
IPTW vs Regression: Different strategies, same fundamental requirements

Next class: Practice applying these concepts to real studies