PHTH 6202: Homework 1 - Solutions

Confounding, Study Design, and Causal Inference

Author

Solution Key

Published

January 1, 2026

Setup

library(dagitty)
library(ggdag)
library(tidyverse)
library(knitr)
library(kableExtra)

# Set theme for DAG plots
theme_set(theme_dag())

Question 1: Intent-to-Treat Analysis (10 points)

Question Prompt

A randomized controlled trial tested whether nicotine patches help smokers avoid relapse after quitting. 400 ex-smokers who recently quit were randomly assigned to receive either nicotine patches or placebo patches. The primary outcome was smoking relapse at 6 months.

Study results:

Table 1: Nicotine Patch Trial Results at 6 Months
Group Relapsed Abstinent Still Using Patch Total
Randomized to Nicotine Patch 70 130 140 200
Randomized to Placebo Patch 110 90 50 200
Total 180 220 190 400
Using Patch at 6m (nicotine or placebo) 45 145 - 190
Not Using Patch at 6m 135 75 - 210
Total 180 220 - 400

Note: “Smoking relapse” means the participant returned to smoking. “Using patch” means still using the assigned patch at 6 months.

Part a: Intent-to-Treat 2×2 Table (2 points)

Complete the 2×2 table using intent-to-treat analysis - classify participants by their original randomization group regardless of whether they were still using the patch.

Solution:

Intent-to-treat analysis uses the ORIGINAL RANDOMIZATION groups. From the data above:

itt_table <- tibble(
  Group = c("Nicotine Patch", "Placebo", "Total"),
  `Relapsed to Smoking` = c(70, 110, 180),
  `Remained Abstinent` = c(130, 90, 220),
  Total = c(200, 200, 400)
)

kable(itt_table, align = c("l", "c", "c", "c")) %>%
  kable_styling(bootstrap_options = c("striped", "bordered"), 
                full_width = FALSE) %>%
  row_spec(3, bold = TRUE)
Table 2: Intent-to-Treat Analysis
Group Relapsed to Smoking Remained Abstinent Total
Nicotine Patch 70 130 200
Placebo 110 90 200
Total 180 220 400

Grading: 2 points for correct table based on randomization groups.

Part b: Risk Ratio from ITT (1 point)

Calculate the risk ratio for smoking relapse using intent-to-treat analysis.

Solution:

# Calculate risks
risk_nicotine <- 70/200
risk_placebo <- 110/200

# Risk ratio
rr_itt <- risk_nicotine / risk_placebo

# Display
tibble(
  Group = c("Nicotine Patch", "Placebo"),
  `N Relapsed` = c(70, 110),
  `N Total` = c(200, 200),
  Risk = c(risk_nicotine, risk_placebo)
) %>%
  kable(digits = 3) %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Group N Relapsed N Total Risk
Nicotine Patch 70 200 0.35
Placebo 110 200 0.55

Risk Ratio = 0.636

Interpretation: Patients randomized to nicotine patches had a 36.4% lower risk of smoking relapse compared to those randomized to placebo (RR = 0.64).

Grading: 1 point for correct RR (0.64) with clear interpretation mentioning randomization groups.

Part c: As-Treated 2×2 Table (2 points)

Complete the 2×2 table using as-treated analysis - classify participants by whether they were actually using the patch at 6 months.

Solution:

As-treated analysis uses ACTUAL PATCH USE at 6 months. From the data table:

astreated_table <- tibble(
  Group = c("Using Patch at 6 months", "Not Using Patch at 6 months", "Total"),
  `Relapsed to Smoking` = c(45, 135, 180),
  `Remained Abstinent` = c(145, 75, 220),
  Total = c(190, 210, 400)
)

kable(astreated_table, align = c("l", "c", "c", "c")) %>%
  kable_styling(bootstrap_options = c("striped", "bordered"), 
                full_width = FALSE) %>%
  row_spec(3, bold = TRUE)
Table 3: As-Treated Analysis
Group Relapsed to Smoking Remained Abstinent Total
Using Patch at 6 months 45 145 190
Not Using Patch at 6 months 135 75 210
Total 180 220 400

Note: These groups are re-sorted based on 6-month patch use, NOT original randomization. The 190 “using patch” includes people from both the nicotine group (140 people) and placebo group (50 people).

Grading: 2 points for correct table based on actual patch use at 6 months.

Part d: Risk Ratio from As-Treated (1 point)

Calculate the risk ratio for smoking relapse using as-treated analysis.

Solution:

# Calculate risks
risk_using <- 45/190
risk_notusing <- 135/210

# Risk ratio
rr_astreated <- risk_using / risk_notusing

# Display
tibble(
  Group = c("Using Patch", "Not Using Patch"),
  `N Relapsed` = c(45, 135),
  `N Total` = c(190, 210),
  Risk = c(risk_using, risk_notusing)
) %>%
  kable(digits = 3) %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Group N Relapsed N Total Risk
Using Patch 45 190 0.237
Not Using Patch 135 210 0.643

Risk Ratio = 0.368

Interpretation: People who were using a patch at 6 months (regardless of original assignment) had a 63.2% lower risk of relapse compared to those not using a patch (RR = 0.37).

Grading: 1 point for correct RR (0.38) with interpretation noting this is about actual use.

Part e: Which Analysis is Better? (4 points)

Which analysis provides a better estimate of the causal effect? Explain addressing:

  • How each analysis relates to randomization and exchangeability
  • Why the two risk ratios differ (0.64 vs 0.38)
  • What question each analysis answers

Solution:

Answer: Intent-to-Treat analysis provides the better estimate of the causal effect.

Comparison of Risk Ratios:

comparison <- tibble(
  Analysis = c("Intent-to-Treat", "As-Treated"),
  `Risk Ratio` = c(rr_itt, rr_astreated),
  `Risk Reduction` = paste0(round((1 - c(rr_itt, rr_astreated))*100, 1), "%"),
  `Preserves Exchangeability?` = c("Yes ✓", "No ✗"),
  `Valid Causal Estimate?` = c("Yes ✓", "No ✗")
)

kable(comparison, align = c("l", "c", "c", "c", "c")) %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Table 4: Comparison of ITT and As-Treated Results
Analysis Risk Ratio Risk Reduction Preserves Exchangeability? Valid Causal Estimate?
Intent-to-Treat 0.6363636 36.4% Yes ✓ Yes ✓
As-Treated 0.3684211 63.2% No ✗ No ✗

Detailed Explanation:

1. How each relates to randomization and exchangeability:

Intent-to-Treat:

  • Analyzes participants according to their original randomization group
  • Preserves exchangeability created by randomization
  • At baseline, randomization ensured the nicotine and placebo groups would have the same risk of relapse if given the same treatment
  • The placebo group serves as the valid counterfactual for the nicotine group
  • Can estimate the causal effect because groups remain exchangeable

As-Treated:

  • Re-groups participants based on patch use at 6 months (after randomization)
  • Breaks exchangeability created by randomization
  • Patch use at 6 months is NOT random - it’s influenced by:
    • Treatment response (people who relapsed may have stopped patch)
    • Side effects (nausea, skin irritation → stop using)
    • Motivation levels (more motivated continue)
    • Early success (doing well → continue)
  • People still using patch at 6 months are systematically different from those not using it
  • Groups are no longer exchangeable
  • Cannot validly estimate causal effect - non-patch-users do not represent what would have happened to patch-users without the patch

2. Why the risk ratios differ:

The as-treated RR (0.38) shows a larger protective effect than ITT RR (0.64) because:

As-treated compares selected/different groups:

  • “Using patch” group at 6 months: More motivated, tolerating treatment well, many haven’t relapsed (that’s why still using it), selected good-prognosis group
  • “Not using patch” group: Includes many who already relapsed and stopped patch, experienced side effects, less motivated, selected poor-prognosis group
  • Comparing these groups mixes the effect of the patch with the effect of being in a good-prognosis vs. poor-prognosis group

ITT includes all participants as randomized:

  • Nicotine group includes those who stopped using the patch
  • “Dilutes” the effect (includes non-adherers)
  • More conservative estimate
  • But groups remain comparable/exchangeable

The difference (0.64 vs 0.38) reflects selection bias from re-sorting groups after randomization.

3. What question each analysis answers:

Intent-to-Treat answers: “What is the effect of a policy of offering nicotine patches to ex-smokers trying to quit?”

  • Realistic scenario including non-adherence
  • Reflects what would happen in practice
  • Policy-relevant for healthcare decisions
  • Accounts for the fact that some people will stop using the patch

As-Treated answers: “What is the effect among people who continue using the patch?”

  • Applies only to a selected subgroup (adherers)
  • Not generalizable to all people offered the treatment
  • Groups are not comparable (non-exchangeable)
  • Cannot estimate causal effect validly, though may give some insight into efficacy among compliers

Grading Rubric (4 points):

Exchangeability explanation (2 points):

  • 2 pts: Clearly explains ITT preserves exchangeability (placebo is valid counterfactual); as-treated breaks it (groups differ beyond patch use)
  • 1.5 pts: Mentions exchangeability and that as-treated breaks randomization but less clear
  • 1 pt: Basic statement that ITT maintains randomization
  • 0.5 pts: Vague statement about ITT being better
  • 0 pts: Incorrect or says as-treated is better

Why RRs differ (1 point):

  • 1 pt: Explains as-treated compares different/selected groups; patch users at 6m differ from non-users in prognosis
  • 0.5 pts: Notes selection into groups but incomplete
  • 0 pts: No explanation or incorrect

What questions answered (1 point):

  • 1 pt: Distinguishes policy/real-world question (ITT) from efficacy in subset (as-treated)
  • 0.5 pts: Some distinction but unclear
  • 0 pts: Missing or wrong

Question 2: Confounding in Observational Studies (10 points)

Question Prompt

Researchers conducted an observational study to examine whether sugar-sweetened beverage (SSB) consumption is causally related to BMI. They collected cross-sectional data on SSB consumption (high vs. low/none), BMI, and potential confounders.

Study results are provided in Tables 1 and 2 below.

table1 <- tribble(
  ~Variable, ~`Low/No SSB (n=363)`, ~`High SSB (n=60)`,
  "Age in years, mean (SD)", "61.8 (12.8)", "55.0 (10.4)",
  "Male, n (%)", "148 (40.9%)", "24 (40.3%)",
  "Race, n (%)", "", "",
  "   White", "141 (38.8%)", "38 (63.3%)",
  "   Black", "18 (5.0%)", "11 (18.3%)",
  "   Asian", "171 (47.1%)", "1 (1.7%)",
  "   Other", "33 (9.1%)", "10 (16.7%)",
  "Current smoker, n (%)", "58 (16.5%)", "29 (48.3%)",
  "No physical activity, n (%)", "56 (15.4%)", "19 (31.7%)",
  "Diabetes, n (%)", "26 (7.2%)", "9 (15.0%)",
  "Hypertension, n (%)", "72 (19.8%)", "21 (35.0%)"
)

kable(table1, align = c("l", "c", "c")) %>%
  kable_styling(bootstrap_options = c("striped", "condensed"), 
                full_width = FALSE) %>%
  row_spec(3, italic = TRUE)
Table 5: Table 1. Baseline Characteristics by SSB Consumption Level
Variable Low/No SSB (n=363) High SSB (n=60)
Age in years, mean (SD) 61.8 (12.8) 55.0 (10.4)
Male, n (%) 148 (40.9%) 24 (40.3%)
Race, n (%)
White 141 (38.8%) 38 (63.3%)
Black 18 (5.0%) 11 (18.3%)
Asian 171 (47.1%) 1 (1.7%)
Other 33 (9.1%) 10 (16.7%)
Current smoker, n (%) 58 (16.5%) 29 (48.3%)
No physical activity, n (%) 56 (15.4%) 19 (31.7%)
Diabetes, n (%) 26 (7.2%) 9 (15.0%)
Hypertension, n (%) 72 (19.8%) 21 (35.0%) 
table2 <- tribble(
  ~Variable, ~`Beta (SE)`, ~`P-value`,
  "High SSB Consumption", "4.6 (0.9)", "<0.001",
  "Age (per year)", "-0.04 (0.03)", "0.14",
  "Male sex", "-0.8 (0.7)", "0.27",
  "Asian race (vs White)", "-5.7 (0.7)", "<0.001",
  "Current smoking (vs never)", "1.7 (0.9)", "0.06",
  "Moderate physical activity (vs none)", "-3.7 (0.9)", "<0.001",
  "Diabetes", "9.9 (1.1)", "<0.001",
  "Hypertension", "6.5 (0.7)", "<0.001"
)

kable(table2, align = c("l", "c", "c")) %>%
  kable_styling(bootstrap_options = c("striped", "condensed"), 
                full_width = FALSE)
Table 6: Table 2. Univariable Associations with BMI
Variable Beta (SE) P-value
High SSB Consumption 4.6 (0.9) <0.001
Age (per year) -0.04 (0.03) 0.14
Male sex -0.8 (0.7) 0.27
Asian race (vs White) -5.7 (0.7) <0.001
Current smoking (vs never) 1.7 (0.9) 0.06
Moderate physical activity (vs none) -3.7 (0.9) <0.001
Diabetes 9.9 (1.1) <0.001
Hypertension 6.5 (0.7) <0.001

Note: These are results from simple linear regression models with BMI as outcome.

Part a: Exchangeability and Counterfactual (2 points)

Explain why exchangeability is central to causal inference. Then describe the conditional counterfactual for the high SSB consumption group.

Solution:

Why exchangeability is central:

Exchangeability is central to causal inference because we cannot directly observe the counterfactual outcome. We must use a comparison group as a proxy, and exchangeability ensures this proxy is valid.

Specifically:

  1. The counterfactual is unobservable: We cannot observe what would have happened to the high SSB consumers if they had not consumed SSB
  2. We need a comparison group: We use the low/no SSB group’s observed outcome to estimate what the high SSB group’s outcome would have been without SSB
  3. Exchangeability makes this valid: If the groups are exchangeable, they would have the same expected outcome under the same exposure level. The low SSB group’s outcome then validly estimates the high SSB group’s counterfactual outcome
  4. Confounding = lack of exchangeability: When groups differ in ways that affect the outcome (like age, smoking, activity), they are not exchangeable. The comparison group’s outcome does not represent the counterfactual, creating confounding

Mathematical expression: With exchangeability: \(E[Y^{a=1}|A=0] = E[Y^{a=1}|A=1]\)

Where \(Y^{a=1}\) is the potential outcome under exposure, A is actual exposure status.

This allows us to estimate causal effects: \(E[Y|A=1] - E[Y|A=0]\) estimates \(E[Y^{a=1}] - E[Y^{a=0}]\)

Conditional counterfactual for high SSB group:

The conditional counterfactual is the BMI that the high SSB consumers would have had if they had instead consumed low/no SSB, holding all other characteristics constant up to the point of SSB exposure.

This includes keeping identical:

  • Age, sex, race, genetic factors
  • Socioeconomic status
  • Baseline health status
  • Dietary preferences and patterns (except SSB)
  • Physical activity levels
  • Smoking status
  • Family environment
  • All other factors

The only difference in the counterfactual scenario is the level of SSB consumption. This is “conditional” on maintaining all pre-exposure characteristics - essentially a time machine scenario where we replay their lives with only SSB consumption changed.

Grading: 2 points for clear exchangeability explanation + accurate counterfactual mentioning holding other factors constant.

Part b: Draw DAG (3 points)

Draw a directed acyclic graph including: SSB consumption, BMI, diabetes, physical activity, age, and smoking.

Solution:

# Define DAG structure
dag_ssb <- dagify(
  BMI ~ SSB + Age + Smoking + PhysicalActivity,
  SSB ~ Age + Smoking + PhysicalActivity,
  Diabetes ~ BMI,
  Hypertension ~ BMI,
  
  # Set coordinates for better layout
  coords = list(
    x = c(SSB = 1, BMI = 3, Age = 0, Smoking = 0.5, 
          PhysicalActivity = 1.5, Diabetes = 4, Hypertension = 4),
    y = c(SSB = 2, BMI = 2, Age = 4, Smoking = 3, 
          PhysicalActivity = 1, Diabetes = 3, Hypertension = 1)
  ),
  
  exposure = "SSB",
  outcome = "BMI"
)

# Plot
ggdag(dag_ssb, text_size = 3.5, node_size = 18) +
  theme_dag() +
  labs(title = "Causal Structure: SSB Consumption and BMI",
       subtitle = "Confounders (Age, Smoking, Physical Activity) point to both SSB and BMI\nDiabetes and Hypertension are downstream of BMI") +
  theme(plot.title = element_text(hjust = 0.5, size = 14, face = "bold"),
        plot.subtitle = element_text(hjust = 0.5, size = 10))
Figure 1: DAG for SSB Consumption and BMI

Explanation of DAG Structure:

Confounders (create backdoor paths):

  • Age → SSB: Older people tend to consume less SSB
  • Age → BMI: Age affects metabolism and weight
  • Smoking → SSB: Smokers tend to consume more SSB
  • Smoking → BMI: Smoking affects weight and metabolism
  • Physical Activity → SSB: Less active people consume more SSB
  • Physical Activity → BMI: Activity strongly affects weight

These create backdoor paths from SSB to BMI that must be blocked.

Main causal pathway:

  • SSB → BMI: The causal effect of interest (dotted arrow in some DAG conventions)

Downstream consequences:

  • BMI → Diabetes: High BMI causes/increases diabetes risk
  • BMI → Hypertension: High BMI causes hypertension

These are consequences of BMI, not confounders.

Alternative: Show paths

# Show all paths
ggdag_paths(dag_ssb, text_size = 3, node_size = 16) +
  theme_dag() +
  labs(title = "All Paths from SSB to BMI",
       subtitle = "Red paths are backdoor paths (confounding)") +
  theme(plot.title = element_text(hjust = 0.5, size = 14, face = "bold"),
        plot.subtitle = element_text(hjust = 0.5, size = 10))
Figure 2: Backdoor Paths from SSB to BMI

Backdoor paths identified:

  1. SSB ← Age → BMI
  2. SSB ← Smoking → BMI
  3. SSB ← Physical Activity → BMI

Grading: 3 points for correct DAG structure with age/smoking/PA as confounders and diabetes/HTN downstream.

Part c: Identify Adjustment Variables (2 points)

Based on your DAG, identify which variables to adjust for. For each variable, state whether to include or exclude it and explain why.

Solution:

We can use dagitty to identify the minimal sufficient adjustment set:

# Identify adjustment sets
adj_sets <- adjustmentSets(dag_ssb, exposure = "SSB", outcome = "BMI")

cat("Minimal sufficient adjustment set(s):\n")
Minimal sufficient adjustment set(s):
print(adj_sets)
{ Age, PhysicalActivity, Smoking }
# Visualize what needs to be adjusted
ggdag_adjustment_set(dag_ssb, text_size = 3, node_size = 16) +
  theme_dag() +
  labs(title = "What to Adjust For",
       subtitle = "Green = Adjust, Red = Do Not Adjust, Gray = Not in model") +
  theme(plot.title = element_text(hjust = 0.5, size = 14, face = "bold"))

Detailed Explanation for Each Variable:

Age: INCLUDE ✓

  • Reasoning: Age is a confounder - it creates a backdoor path (SSB ← Age → BMI)
  • From DAG: Age has arrows pointing to BOTH SSB and BMI (common cause)
  • Not downstream of SSB (temporal order: age comes first)
  • Must adjust to block the backdoor path through age
  • Meets all 3 properties: Determinant of outcome, correlated with exposure, not downstream

Smoking: INCLUDE ✓

  • Reasoning: Smoking is a confounder - creates backdoor path (SSB ← Smoking → BMI)
  • From DAG: Arrows point to both SSB and BMI
  • Not downstream of SSB
  • Must adjust to block this backdoor path
  • Meets all 3 properties

Physical Activity: INCLUDE ✓

  • Reasoning: Physical activity is a confounder - creates backdoor path (SSB ← Physical Activity → BMI)
  • From DAG: Arrows point to both SSB and BMI
  • Not downstream of SSB
  • Must adjust to block this backdoor path
  • Meets all 3 properties

Diabetes: EXCLUDE ✗

  • Reasoning: Diabetes is DOWNSTREAM of BMI (BMI → Diabetes)
  • From DAG: Diabetes is a consequence/effect of BMI, not a cause
  • This is a mediator: If SSB → BMI → Diabetes, adjusting for diabetes would block part of the causal pathway
  • Violates property 3: Occurs downstream of the outcome
  • Do NOT adjust: Would create over-adjustment bias and block the causal pathway we’re trying to estimate
  • Not a confounder: Doesn’t create a backdoor path (arrow goes FROM BMI, not TO it)

Hypertension: EXCLUDE ✗

  • Reasoning: Same as diabetes - hypertension is DOWNSTREAM of BMI
  • From DAG: BMI → Hypertension (consequence, not cause)
  • Mediator: Part of the causal pathway
  • Violates property 3
  • Do NOT adjust: Would create over-adjustment bias
  • Not a confounder

Summary:

Minimal sufficient adjustment set: {Age, Smoking, Physical Activity}

Adjusting for these three variables blocks all backdoor paths from SSB to BMI while avoiding adjustment for downstream consequences (diabetes, hypertension).

Grading: 2 points for correctly identifying all variables with clear causal reasoning using DAG/temporal concepts.

Part d: Interpret Table 1 (2 points)

Describe what you observe in Table 1. Which characteristics show largest differences? What does this tell you about exchangeability?

Solution:

Observed Imbalances:

Let’s quantify the key differences:

imbalances <- tribble(
  ~Characteristic, ~`Low SSB`, ~`High SSB`, ~`Difference/Ratio`,
  "Age (mean)", "61.8 years", "55.0 years", "6.8 years younger",
  "Current smoking", "16.5%", "48.3%", "2.9-fold higher",
  "No physical activity", "15.4%", "31.7%", "2.1-fold higher",
  "Asian race", "47.1%", "1.7%", "27.6-fold difference",
  "White race", "38.8%", "63.3%", "1.6-fold higher",
  "Diabetes", "7.2%", "15.0%", "2.1-fold higher",
  "Hypertension", "19.8%", "35.0%", "1.8-fold higher",
  "Male sex", "40.9%", "40.3%", "Nearly identical"
)

kable(imbalances, align = c("l", "c", "c", "l")) %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE) %>%
  row_spec(c(1,2,3,4), background = "#FFF3CD") %>%  # Highlight major imbalances
  row_spec(8, background = "#D4EDDA")  # Highlight balanced variable
Table 7: Key Imbalances Between SSB Groups
Characteristic Low SSB High SSB Difference/Ratio
Age (mean) 61.8 years 55.0 years 6.8 years younger
Current smoking 16.5% 48.3% 2.9-fold higher
No physical activity 15.4% 31.7% 2.1-fold higher
Asian race 47.1% 1.7% 27.6-fold difference
White race 38.8% 63.3% 1.6-fold higher
Diabetes 7.2% 15.0% 2.1-fold higher
Hypertension 19.8% 35.0% 1.8-fold higher
Male sex 40.9% 40.3% Nearly identical

Interpretation:

Largest differences observed:

  1. Age: High SSB consumers are 6.8 years younger on average - a substantial and meaningful difference
  2. Race/Ethnicity: Dramatic differences - Asians represent 47% of low consumers but only 2% of high consumers (27-fold difference!)
  3. Smoking: Current smoking rate is nearly 3 times higher in high SSB group (48% vs 17%)
  4. Physical Activity: Sedentary lifestyle is twice as common in high SSB group (32% vs 15%)

What this tells us about exchangeability:

The groups are NOT exchangeable. High and low SSB consumers differ substantially on multiple characteristics that independently affect BMI:

  • Age affects BMI: Younger people may have different metabolism and weight patterns
  • Smoking affects BMI: Smoking influences weight and metabolism
  • Physical activity affects BMI: Strong independent effect on weight
  • Race/ethnicity may reflect BMI: Different genetic backgrounds and cultural dietary patterns

Implication for confounding:

Because these groups differ on factors that affect BMI, they would likely have different average BMI even in the absence of any SSB consumption. The low SSB group’s observed BMI (lower) does not represent what the high SSB group’s BMI would be if they didn’t consume SSB - it represents a different group of people with different risk profiles.

This is the essence of confounding: Lack of exchangeability between exposure groups. Any crude comparison of BMI between these groups will mix:

  1. True causal effect of SSB (if any)
  2. Effects of age differences
  3. Effects of smoking differences
  4. Effects of activity differences
  5. Effects of racial/ethnic differences

Note on diabetes and hypertension differences:

The higher rates of diabetes (15% vs 7%) and hypertension (35% vs 20%) in the high SSB group likely reflect consequences of their higher BMI rather than confounders. These are downstream outcomes.

Sex is balanced:

The only characteristic showing no imbalance is sex (41% vs 40% male), suggesting sex is unlikely to confound this relationship.

Visualization:

# Create data for visualization
imbalance_data <- tribble(
  ~Variable, ~Ratio,
  "Asian race", 27.6,
  "Current smoking", 2.9,
  "No physical activity", 2.1,
  "Diabetes", 2.1,
  "Hypertension", 1.8,
  "Male sex", 1.0
) %>%
  mutate(Variable = fct_reorder(Variable, Ratio))

# Plot
ggplot(imbalance_data, aes(x = Ratio, y = Variable)) +
  geom_col(aes(fill = Ratio > 1.5)) +
  geom_vline(xintercept = 1, linetype = "dashed", color = "gray50") +
  scale_fill_manual(values = c("#D4EDDA", "#FFF3CD"), guide = "none") +
  scale_x_continuous(trans = "log2", breaks = c(1, 1.5, 2, 3, 5, 10, 20, 30)) +
  labs(title = "Imbalances Between High and Low SSB Groups",
       subtitle = "Fold-difference in prevalence (log scale)",
       x = "Fold Difference (High SSB / Low SSB)",
       y = NULL) +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5, face = "bold"),
        plot.subtitle = element_text(hjust = 0.5, size = 9))
Figure 3: Magnitude of Imbalances Between SSB Groups

Grading: 2 points for identifying major imbalances AND explicitly connecting to lack of exchangeability/confounding.

Part c: Identify Confounders (2 points)

Based on your DAG, identify which variables to adjust for.

Solution Using dagitty:

# Check each variable using dagitty
cat("Is Age a confounder?\n")
Is Age a confounder?
cat("Creates backdoor path:", 
    "SSB <- Age -> BMI" %in% paths(dag_ssb, "SSB", "BMI")$paths, "\n")
Creates backdoor path: TRUE 
cat("Conclusion: INCLUDE (confounder)\n\n")
Conclusion: INCLUDE (confounder)
cat("Is Smoking a confounder?\n")
Is Smoking a confounder?
cat("Creates backdoor path:", 
    "SSB <- Smoking -> BMI" %in% paths(dag_ssb, "SSB", "BMI")$paths, "\n")
Creates backdoor path: TRUE 
cat("Conclusion: INCLUDE (confounder)\n\n")
Conclusion: INCLUDE (confounder)
cat("Is Physical Activity a confounder?\n")
Is Physical Activity a confounder?
cat("Creates backdoor path:", 
    "SSB <- PhysicalActivity -> BMI" %in% paths(dag_ssb, "SSB", "BMI")$paths, "\n")
Creates backdoor path: TRUE 
cat("Conclusion: INCLUDE (confounder)\n\n")
Conclusion: INCLUDE (confounder)
cat("Is Diabetes a confounder?\n")
Is Diabetes a confounder?
cat("Position in DAG: Downstream of BMI (BMI -> Diabetes)\n")
Position in DAG: Downstream of BMI (BMI -> Diabetes)
cat("Conclusion: EXCLUDE (mediator/downstream)\n\n")
Conclusion: EXCLUDE (mediator/downstream)
cat("Is Hypertension a confounder?\n")
Is Hypertension a confounder?
cat("Position in DAG: Downstream of BMI (BMI -> Hypertension)\n")
Position in DAG: Downstream of BMI (BMI -> Hypertension)
cat("Conclusion: EXCLUDE (mediator/downstream)\n")
Conclusion: EXCLUDE (mediator/downstream)

Detailed Explanations:

Age: INCLUDE ✓

From the DAG, age creates a backdoor path: SSB ← Age → BMI

  • Age causes/influences both SSB consumption and BMI
  • Common cause = confounder
  • Not on the causal pathway we’re studying
  • Must adjust to block this backdoor path

Smoking: INCLUDE ✓

Backdoor path: SSB ← Smoking → BMI

  • Smoking influences both SSB consumption and BMI
  • Common cause = confounder
  • Not downstream
  • Must adjust to block this path

Physical Activity: INCLUDE ✓

Backdoor path: SSB ← Physical Activity → BMI

  • Physical activity influences both SSB and BMI
  • Common cause = confounder
  • Not downstream
  • Must adjust to block this path

Diabetes: EXCLUDE ✗

From DAG: BMI → Diabetes

  • Diabetes is DOWNSTREAM of BMI (consequence, not cause)
  • High BMI causes diabetes
  • This is a mediator on the pathway: SSB → BMI → Diabetes
  • Adjusting would block part of the causal effect
  • Violates property 3 (occurs downstream)
  • Do NOT adjust

Hypertension: EXCLUDE ✗

From DAG: BMI → Hypertension

  • Same reasoning as diabetes
  • Downstream of BMI (consequence)
  • Mediator: SSB → BMI → Hypertension
  • Do NOT adjust

Visual Representation:

# Show adjustment set visually
ggdag_adjustment_set(dag_ssb, 
                     exposure = "SSB", 
                     outcome = "BMI",
                     text_size = 3.5,
                     node_size = 18) +
  theme_dag() +
  labs(title = "What to Adjust For: SSB → BMI",
       subtitle = "Adjust = Green | Unadjusted = Red | Outcome = Blue | Exposure = Red") +
  theme(plot.title = element_text(hjust = 0.5, size = 14, face = "bold"),
        plot.subtitle = element_text(hjust = 0.5, size = 10))
Figure 4: Minimal Sufficient Adjustment Set

Summary: Adjust for Age, Smoking, and Physical Activity. Do NOT adjust for Diabetes or Hypertension.

Grading: 2 points for all five correctly classified with clear causal reasoning.

Part d: Interpret Table 1 (2 points)

See solution in Part d above (already provided with Table 1).

Main points:

  • Major imbalances in age (6.8 years), race (27-fold for Asian), smoking (3-fold), physical activity (2-fold)
  • Groups are NOT exchangeable
  • Would have different BMI even without SSB differences
  • This indicates confounding is present

Grading: 2 points for identifying imbalances and connecting to lack of exchangeability.

Part e: Interpret Crude vs. Adjusted (1 point)

The crude association is β=4.6. After adjusting for age, race, smoking, and physical activity, β=2.5. Interpret this change.

Solution:

# Create data
estimates <- tibble(
  Analysis = factor(c("Crude", "Adjusted"), 
                   levels = c("Crude", "Adjusted")),
  Beta = c(4.6, 2.5),
  Label = c("4.6", "2.5")
)

# Plot
ggplot(estimates, aes(x = Analysis, y = Beta)) +
  geom_col(aes(fill = Analysis), width = 0.6, show.legend = FALSE) +
  geom_text(aes(label = Label), vjust = -0.5, size = 5, fontface = "bold") +
  scale_fill_manual(values = c("Crude" = "#E74C3C", "Adjusted" = "#3498DB")) +
  annotate("segment", x = 1, xend = 2, y = 4.6, yend = 2.5,
           arrow = arrow(length = unit(0.3, "cm")), 
           color = "gray30", size = 1) +
  annotate("text", x = 1.5, y = 3.8, 
           label = "2.1 units\n(46% reduction)\ndue to confounding",
           size = 3.5, color = "gray20") +
  ylim(0, 5.5) +
  labs(title = "Effect of Adjustment for Confounders",
       subtitle = "Change from Crude to Adjusted Association",
       y = "Beta Coefficient (BMI units)",
       x = NULL) +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5, face = "bold", size = 14),
        plot.subtitle = element_text(hjust = 0.5, size = 10),
        axis.text = element_text(size = 11))
Figure 5: Confounding in SSB-BMI Association

Interpretation:

The reduction from crude (β = 4.6) to adjusted (β = 2.5) indicates substantial confounding was present in the crude association.

Magnitude of confounding:

  • Absolute reduction: 4.6 - 2.5 = 2.1 BMI units
  • Relative reduction: (2.1 / 4.6) × 100 = 46% of the crude association

What this means:

Nearly half (46%) of the crude association between high SSB consumption and higher BMI was due to confounding by age, race, smoking, and physical activity rather than a direct causal effect of SSB.

Interpretation:

High SSB consumers had BMI 4.6 units higher in the crude analysis, but this included:

  1. True causal effect of SSB (approximately 2.5 units)
  2. Confounding from other factors (approximately 2.1 units):
    • Being younger (age affects BMI)
    • Higher smoking rates (smoking affects BMI)
    • Less physical activity (activity affects BMI)
    • Different racial composition (may reflect genetic/cultural factors affecting BMI)

After accounting for these confounding factors, the estimated causal effect is 2.5 BMI units, which is still substantial and statistically significant (p=0.008) but notably smaller than the crude estimate.

Conclusion: Yes, there was substantial confounding. Adjustment was necessary to obtain a less biased estimate of the causal effect. The adjusted estimate (2.5) is a better approximation of the true causal effect, though residual confounding from unmeasured factors is still possible.

Grading: 1 point for clearly stating substantial confounding (~46%) with accurate interpretation of what this means.

Summary of Key Concepts

From Question 1:

  • Intent-to-treat preserves exchangeability created by randomization
  • Placebo group serves as valid counterfactual for treatment group
  • As-treated breaks exchangeability by re-sorting based on post-randomization behavior
  • ITT answers policy question; as-treated answers efficacy in selected subgroup
  • Causal effects require exchangeable comparison groups

From Question 2:

  • Exchangeability is fundamental - comparison group must be valid counterfactual
  • DAGs show causal structure - arrows represent causation
  • Confounders create backdoor paths (common causes)
  • Mediators are on the causal pathway (downstream)
  • Temporal order matters - distinguish upstream causes from downstream consequences
  • Confounding = lack of exchangeability - groups differ on outcome-relevant factors
  • Adjustment magnitude quantifies degree of confounding

Integration:

Both questions apply the counterfactual framework and concept of exchangeability:

  • RCTs create exchangeability through randomization
  • Observational studies must adjust for confounders to approximate exchangeability
  • Valid causal inference requires exchangeable comparison groups

R Code Summary

Packages Used:

library(dagitty)      # DAG creation and analysis
library(ggdag)        # DAG visualization
library(tidyverse)    # Data manipulation and plotting
library(knitr)        # Table formatting
library(kableExtra)   # Enhanced tables

Key DAG Functions:

# Create DAG
dag <- dagify(
  outcome ~ exposure + confounder,
  exposure ~ confounder,
  coords = list(x = c(...), y = c(...))
)

# Visualize
ggdag(dag)

# Find adjustment sets
adjustmentSets(dag, exposure = "X", outcome = "Y")

# Show paths
ggdag_paths(dag)

# Show what to adjust
ggdag_adjustment_set(dag, exposure = "X", outcome = "Y")

Creating Custom DAG Layouts:

For better visualization, specify coordinates manually:

# Horizontal left-to-right layout
coords = list(
  x = c(exposure = 1, outcome = 3, confounder1 = 0, confounder2 = 0),
  y = c(exposure = 2, outcome = 2, confounder1 = 3, confounder2 = 1)
)

# Vertical top-to-bottom  
coords = list(
  x = c(exposure = 2, mediator = 2, outcome = 2, confounder = 1),
  y = c(exposure = 3, mediator = 2, outcome = 1, confounder = 3)
)

This allows you to control the visual appearance for clearer communication!

Additional Resources

DAGitty Online Tool:

Students who prefer not to code can use: http://www.dagitty.net

  • Browser-based DAG drawing
  • Automatic identification of adjustment sets
  • Export as image
  • Free and easy to use

R Packages Documentation:

  • dagitty: https://cran.r-project.org/package=dagitty
  • ggdag: https://ggdag.malco.io/

Further Reading on DAGs:

  • Hernán MA, Robins JM. Causal Inference: What If. Chapter 6 (Graphical representation of causal effects)
  • Pearl J. Causality. 2nd ed. Cambridge University Press, 2009
  • Greenland S, Pearl J, Robins JM. Causal diagrams for epidemiologic research. Epidemiology 1999;10:37-48