Introduction

In previous lectures, we built multiple linear regression models that included several predictors. We interpreted each coefficient as the expected change in \(Y\) for a one-unit increase in \(X_j\), “holding all other predictors constant.” But we did not examine two critical methodological questions:

  1. Confounding: Does the estimated effect of our exposure change when we add or remove covariates? If so, those covariates are confounders, and we need them in the model to get an unbiased estimate.

  2. Interaction (Effect Modification): Is the effect of our exposure the same for everyone, or does it differ across subgroups? If the effect of sleep on physical health is different for men and women, we say that sex modifies the effect of sleep.

These are the two most important methodological concepts in associative (etiologic) modeling, the type of modeling most common in epidemiology.

Research question for today:

Is the association between sleep duration and physically unhealthy days modified by sex or education? And which variables confound this association?

Predictive vs. Associative Modeling

Before we dive in, it is important to revisit the distinction between the two primary goals of regression modeling, because confounding and interaction are only relevant in one of them.

Goal Question What matters most
Predictive Which set of variables best predicts \(Y\)? Overall model accuracy (\(R^2\), RMSE, out-of-sample prediction)
Associative What is the effect of a specific exposure on \(Y\), after adjusting for confounders? Accuracy and interpretability of a specific \(\hat{\beta}\)

In predictive modeling, we want the combination of variables that minimizes prediction error. We do not care whether individual coefficients are “correct” or interpretable, only whether the model predicts well.

In associative modeling, we care deeply about one (or a few) specific coefficients. We want to ensure that \(\hat{\beta}_{\text{exposure}}\) reflects the true relationship, free from confounding bias. This is the setting where confounding and interaction become critical.

In epidemiology, we are almost always doing associative modeling. We have a specific exposure of interest (e.g., sleep) and want to estimate its effect on a health outcome (e.g., physical health days) while controlling for confounders.

A Warning About Extrapolation

We should never extrapolate predictions from a statistical model beyond the range of the observed data. Extrapolation assumes that the relationship continues unchanged into regions where we have no data, which is often false.

A famous example: NASA engineers extrapolated O-ring performance data to predict behavior at temperatures colder than any previously tested. The resulting decision to launch the Space Shuttle Challenger in cold weather led to the 1986 disaster.

Rule of thumb: Your model is only valid within the range of the data used to build it.

Setup and Data

library(tidyverse)
library(haven)
library(janitor)
library(knitr)
library(kableExtra)
library(broom)
library(gtsummary)
library(GGally)
library(car)
library(ggeffects)
library(plotly)
library(lmtest)

options(gtsummary.use_ftExtra = TRUE)
set_gtsummary_theme(theme_gtsummary_compact(set_theme = TRUE))

Preparing the Dataset

We continue with the BRFSS 2020 dataset. For this lecture we shift our outcome to physically unhealthy days and examine sleep as the primary exposure, with sex, education, age, exercise, and general health as potential modifiers and confounders.

brfss_full <- read_xpt(
  "/Users/mm992584/Library/CloudStorage/OneDrive-UniversityatAlbany-SUNY/Spring 2024/Papers/BRFSS Project/Final Analysis/BRFSS Data 2013 - 2023/LLCP2020.XPT"
) |>
  clean_names()
brfss_ci <- brfss_full |>
  mutate(
    # Outcome: physically unhealthy days in past 30
    physhlth_days = case_when(
      physhlth == 88                    ~ 0,
      physhlth >= 1 & physhlth <= 30   ~ as.numeric(physhlth),
      TRUE                             ~ NA_real_
    ),
    # Primary exposure: sleep hours
    sleep_hrs = case_when(
      sleptim1 >= 1 & sleptim1 <= 14   ~ as.numeric(sleptim1),
      TRUE                             ~ NA_real_
    ),
    # Mentally unhealthy days (covariate)
    menthlth_days = case_when(
      menthlth == 88                    ~ 0,
      menthlth >= 1 & menthlth <= 30   ~ as.numeric(menthlth),
      TRUE                             ~ NA_real_
    ),
    # Age
    age = age80,
    # Sex
    sex = factor(sexvar, levels = c(1, 2), labels = c("Male", "Female")),
    # Education (4-level)
    education = factor(case_when(
      educa %in% c(1, 2, 3) ~ "Less than HS",
      educa == 4             ~ "HS graduate",
      educa == 5             ~ "Some college",
      educa == 6             ~ "College graduate",
      TRUE                   ~ NA_character_
    ), levels = c("Less than HS", "HS graduate", "Some college", "College graduate")),
    # Exercise in past 30 days
    exercise = factor(case_when(
      exerany2 == 1 ~ "Yes",
      exerany2 == 2 ~ "No",
      TRUE          ~ NA_character_
    ), levels = c("No", "Yes")),
    # General health status
    gen_health = factor(case_when(
      genhlth == 1 ~ "Excellent",
      genhlth == 2 ~ "Very good",
      genhlth == 3 ~ "Good",
      genhlth == 4 ~ "Fair",
      genhlth == 5 ~ "Poor",
      TRUE         ~ NA_character_
    ), levels = c("Excellent", "Very good", "Good", "Fair", "Poor")),
    # Income category (ordinal 1-8)
    income_cat = case_when(
      income2 %in% 1:8 ~ as.numeric(income2),
      TRUE             ~ NA_real_
    )
  ) |>
  filter(
    !is.na(physhlth_days),
    !is.na(sleep_hrs),
    !is.na(menthlth_days),
    !is.na(age), age >= 18,
    !is.na(sex),
    !is.na(education),
    !is.na(exercise),
    !is.na(gen_health),
    !is.na(income_cat)
  )

# Reproducible random sample
set.seed(1220)
brfss_ci <- brfss_ci |>
  select(physhlth_days, sleep_hrs, menthlth_days, age, sex,
         education, exercise, gen_health, income_cat) |>
  slice_sample(n = 5000)

# Save for lab activity
saveRDS(brfss_ci,
  "/Users/mm992584/Library/CloudStorage/OneDrive-UniversityatAlbany-SUNY/Spring 2026/Epi 553/Lectures Notes/Confounding Interaction/brfss_ci_2020.rds")

tibble(Metric = c("Observations", "Variables"),
       Value  = c(nrow(brfss_ci), ncol(brfss_ci))) |>
  kable(caption = "Analytic Dataset Dimensions") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)
Analytic Dataset Dimensions
Metric Value
Observations 5000
Variables 9

Descriptive Statistics

brfss_ci |>
  select(physhlth_days, sleep_hrs, age, sex, education, exercise, gen_health) |>
  tbl_summary(
    label = list(
      physhlth_days ~ "Physically unhealthy days (past 30)",
      sleep_hrs     ~ "Sleep (hours/night)",
      age           ~ "Age (years)",
      sex           ~ "Sex",
      education     ~ "Education level",
      exercise      ~ "Any physical activity (past 30 days)",
      gen_health    ~ "General health status"
    ),
    statistic = list(
      all_continuous()  ~ "{mean} ({sd})",
      all_categorical() ~ "{n} ({p}%)"
    ),
    digits = all_continuous() ~ 1,
    missing = "no"
  ) |>
  add_n() |>
  bold_labels() |>
  italicize_levels() |>
  modify_caption("**Table 1. Descriptive Statistics — BRFSS 2020 (n = 5,000)**") |>
  as_flex_table()
**Table 1. Descriptive Statistics — BRFSS 2020 (n = 5,000)**

Characteristic

N

N = 5,0001

Physically unhealthy days (past 30)

5,000

3.4 (7.9)

Sleep (hours/night)

5,000

7.1 (1.4)

Age (years)

5,000

54.1 (17.0)

Sex

5,000

Male

2,291 (46%)

Female

2,709 (54%)

Education level

5,000

Less than HS

276 (5.5%)

HS graduate

1,263 (25%)

Some college

1,378 (28%)

College graduate

2,083 (42%)

Any physical activity (past 30 days)

5,000

3,929 (79%)

General health status

5,000

Excellent

1,031 (21%)

Very good

1,762 (35%)

Good

1,507 (30%)

Fair

524 (10%)

Poor

176 (3.5%)

1Mean (SD); n (%)

Part 1: Guided Practice — Confounding and Interactions

1. Interaction (Effect Modification)

1.1 What Is Interaction?

Interaction (also called effect modification) is present when the relationship between an exposure and an outcome is different at different levels of a third variable. In regression terms, the slope of the exposure-outcome relationship changes depending on the value of the modifier.

For example, if the effect of sleep on physical health is stronger for women than for men, then sex modifies the effect of sleep. The two variables (sleep and sex) have a multiplicative, not merely additive, effect on the outcome.

This is fundamentally different from confounding:

Concept Question Implication
Confounding Is the crude estimate of the exposure effect biased by a third variable? Must adjust for the confounder to get a valid estimate
Interaction Does the effect of the exposure differ across subgroups? Must report stratum-specific effects, not a single overall estimate

Critical point: Always assess interaction before confounding. If interaction is present, a single “adjusted” coefficient for the exposure is misleading because the effect is not the same for everyone. You must stratify or include interaction terms.

1.2 Interaction Between a Continuous and a Dichotomous Variable

Consider a model with a continuous exposure \(X_1\) (sleep hours) and a dichotomous variable \(X_2\) (sex, where Male = 1 and Female = 0):

\[Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2 + \varepsilon\]

The term \(\beta_3 X_1 X_2\) is the interaction term. Let’s see what happens when we plug in the values for each group:

For males (\(X_2 = 1\)): \[E(Y | X_1, \text{Male}) = (\beta_0 + \beta_2) + (\beta_1 + \beta_3) X_1\]

For females (\(X_2 = 0\)): \[E(Y | X_1, \text{Female}) = \beta_0 + \beta_1 X_1\]

The key insight:

  • The intercept for males is \(\beta_0 + \beta_2\), while for females it is \(\beta_0\)
  • The slope for males is \(\beta_1 + \beta_3\), while for females it is \(\beta_1\)
  • \(\beta_3\) is the difference in slopes between the two groups

If \(\beta_3 = 0\), the slopes are equal and the lines are parallel (no interaction). If \(\beta_3 \neq 0\), the lines are not parallel (interaction is present).

# Create synthetic data to illustrate the concept
set.seed(1220)
n <- 200
concept_data <- tibble(
  x1 = runif(n, 0, 10),
  group = rep(c("Group A", "Group B"), each = n / 2)
)

# No interaction: same slope, different intercepts
no_int <- concept_data |>
  mutate(
    y = ifelse(group == "Group A", 2 + 1.5 * x1, 5 + 1.5 * x1) + rnorm(n, 0, 2)
  )

# Interaction: different slopes
with_int <- concept_data |>
  mutate(
    y = ifelse(group == "Group A", 2 + 1.5 * x1, 5 + 0.3 * x1) + rnorm(n, 0, 2)
  )

p1 <- ggplot(no_int, aes(x = x1, y = y, color = group)) +
  geom_point(alpha = 0.3, size = 1) +
  geom_smooth(method = "lm", se = FALSE, linewidth = 1.2) +
  labs(title = "No Interaction", subtitle = "Parallel lines: same slope",
       x = expression(X[1]), y = "Y") +
  theme_minimal(base_size = 12) +
  scale_color_brewer(palette = "Set1") +
  theme(legend.position = "bottom")

p2 <- ggplot(with_int, aes(x = x1, y = y, color = group)) +
  geom_point(alpha = 0.3, size = 1) +
  geom_smooth(method = "lm", se = FALSE, linewidth = 1.2) +
  labs(title = "Interaction Present", subtitle = "Non-parallel lines: different slopes",
       x = expression(X[1]), y = "Y") +
  theme_minimal(base_size = 12) +
  scale_color_brewer(palette = "Set1") +
  theme(legend.position = "bottom")

gridExtra::grid.arrange(p1, p2, ncol = 2)
No Interaction (Parallel Lines) vs. Interaction (Non-Parallel Lines)

No Interaction (Parallel Lines) vs. Interaction (Non-Parallel Lines)

2. Testing for Interaction: Stratified Analysis

2.1 The Stratified Approach

One way to assess interaction is to split the data by levels of the potential modifier and fit separate regression models in each stratum. If the slopes differ meaningfully, interaction may be present.

Let’s test whether the association between sleep and physical health days differs by sex.

# Fit separate models for males and females
mod_male <- lm(physhlth_days ~ sleep_hrs, data = brfss_ci |> filter(sex == "Male"))
mod_female <- lm(physhlth_days ~ sleep_hrs, data = brfss_ci |> filter(sex == "Female"))

# Compare coefficients
bind_rows(
  tidy(mod_male, conf.int = TRUE) |> mutate(Stratum = "Male"),
  tidy(mod_female, conf.int = TRUE) |> mutate(Stratum = "Female")
) |>
  filter(term == "sleep_hrs") |>
  select(Stratum, estimate, std.error, conf.low, conf.high, p.value) |>
  mutate(across(where(is.numeric), \(x) round(x, 4))) |>
  kable(
    caption = "Stratified Analysis: Sleep → Physical Health Days, by Sex",
    col.names = c("Stratum", "Estimate", "SE", "CI Lower", "CI Upper", "p-value")
  ) |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Stratified Analysis: Sleep → Physical Health Days, by Sex
Stratum Estimate SE CI Lower CI Upper p-value
Male -0.5243 0.1193 -0.7583 -0.2903 0.0000
Female -0.3752 0.1144 -0.5996 -0.1508 0.0011 
ggplot(brfss_ci, aes(x = sleep_hrs, y = physhlth_days, color = sex)) +
  geom_jitter(alpha = 0.08, width = 0.3, height = 0.3, size = 0.8) +
  geom_smooth(method = "lm", se = TRUE, linewidth = 1.2) +
  labs(
    title    = "Association Between Sleep and Physical Health Days, Stratified by Sex",
    subtitle = "Are the slopes parallel or different?",
    x        = "Sleep Hours per Night",
    y        = "Physically Unhealthy Days (Past 30)",
    color    = "Sex"
  ) +
  theme_minimal(base_size = 13) +
  scale_color_brewer(palette = "Set1")
Stratified Regression: Sleep vs. Physical Health Days by Sex

Stratified Regression: Sleep vs. Physical Health Days by Sex

2.2 Limitations of Stratified Analysis

The stratified approach is intuitive and visually compelling, but it has important limitations:

  1. No formal test: We can see the slopes are different, but we cannot compute a p-value for the difference without additional machinery
  2. Reduced power: Each stratum has fewer observations than the full dataset
  3. Multiple comparisons: With \(k\) strata, we would need \(k\) separate models
  4. Cannot adjust for covariates easily while testing the interaction

The solution is to use a single regression model with an interaction term.

3. Testing for Interaction: Regression with Interaction Terms

3.1 The Interaction Model in R

R provides two operators for specifying interactions:

  • X1:X2 — includes only the interaction term \(X_1 \times X_2\)
  • X1*X2 — shorthand for X1 + X2 + X1:X2 (main effects plus interaction)

Rule: Any model with an interaction term must also include the main effects that comprise the interaction. Always use * or explicitly include both main effects with :.

# Model without interaction (additive model)
mod_add <- lm(physhlth_days ~ sleep_hrs + sex, data = brfss_ci)

# Model with interaction
mod_int <- lm(physhlth_days ~ sleep_hrs * sex, data = brfss_ci)
# Equivalent to: lm(physhlth_days ~ sleep_hrs + sex + sleep_hrs:sex, data = brfss_ci)

tidy(mod_int, conf.int = TRUE) |>
  mutate(across(where(is.numeric), \(x) round(x, 4))) |>
  kable(
    caption = "Interaction Model: Sleep * Sex → Physical Health Days",
    col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
  ) |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Interaction Model: Sleep * Sex → Physical Health Days
Term Estimate SE t p-value CI Lower CI Upper
(Intercept) 6.8388 0.8723 7.8397 0.0000 5.1287 8.5490
sleep_hrs -0.5243 0.1222 -4.2898 0.0000 -0.7639 -0.2847
sexFemale -0.5786 1.1906 -0.4860 0.6270 -2.9127 1.7555
sleep_hrs:sexFemale 0.1491 0.1659 0.8986 0.3689 -0.1762 0.4744 
b_int <- round(coef(mod_int), 3)

3.2 Interpreting the Interaction Model

The fitted model is:

\[\widehat{\text{Phys Days}} = 6.839 + -0.524(\text{Sleep}) + -0.579(\text{Female}) + 0.149(\text{Sleep} \times \text{Female})\]

For males (Female = 0): \[\widehat{\text{Phys Days}} = 6.839 + -0.524(\text{Sleep})\]

For females (Female = 1): \[\widehat{\text{Phys Days}} = (6.839 + -0.579) + (-0.524 + 0.149)(\text{Sleep}) = 6.26 + -0.375(\text{Sleep})\]

Interpretation of each coefficient:

  • Intercept (6.839): Expected physical health days for a male with 0 hours of sleep (extrapolation, not meaningful)
  • Sleep (-0.524): The slope of sleep for males (the reference group for sex). Each additional hour of sleep is associated with 0.524 fewer physically unhealthy days among males.
  • Female (-0.579): The difference in intercept between females and males (the difference in expected physical health days at sleep = 0, which is an extrapolation)
  • Sleep:Female (0.149): The difference in slopes between females and males. This is the interaction term. It tells us how much more (or less) the sleep effect is for females compared to males.

3.3 Testing for Interaction (Parallelism)

The formal test for interaction is simply the t-test for the interaction coefficient:

\[H_0: \beta_3 = 0 \quad \text{(slopes are equal, lines are parallel, no interaction)}\] \[H_A: \beta_3 \neq 0 \quad \text{(slopes differ, interaction is present)}\]

int_term <- tidy(mod_int) |> filter(term == "sleep_hrs:sexFemale")
cat("Interaction term (sleep_hrs:sexFemale):\n")
## Interaction term (sleep_hrs:sexFemale):
cat("  Estimate:", round(int_term$estimate, 4), "\n")
##   Estimate: 0.1491
cat("  t-statistic:", round(int_term$statistic, 3), "\n")
##   t-statistic: 0.899
cat("  p-value:", round(int_term$p.value, 4), "\n")
##   p-value: 0.3689

3.4 Testing for Coincidence

A stronger test asks whether sex has any effect on the relationship (neither intercept nor slope differs):

\[H_0: \beta_2 = \beta_3 = 0 \quad \text{(the two lines are identical)}\] \[H_A: \text{At least one } \neq 0 \quad \text{(the lines differ in intercept and/or slope)}\]

This is a partial F-test comparing the interaction model to a model with only sleep:

mod_sleep_only <- lm(physhlth_days ~ sleep_hrs, data = brfss_ci)

anova(mod_sleep_only, mod_int) |>
  tidy() |>
  mutate(across(where(is.numeric), \(x) round(x, 4))) |>
  kable(caption = "Test for Coincidence: Does Sex Affect the Sleep-Physical Health Relationship at All?") |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Test for Coincidence: Does Sex Affect the Sleep-Physical Health Relationship at All?
term df.residual rss df sumsq statistic p.value
physhlth_days ~ sleep_hrs 4998 313611.5 NA NA NA NA
physhlth_days ~ sleep_hrs * sex 4996 313284.5 2 327.0047 2.6074 0.0738

3.5 Verifying Equivalence with Stratified Analysis

The interaction model recovers exactly the same stratum-specific equations as the stratified analysis:

tribble(
  ~Method, ~Stratum, ~Intercept, ~Slope,
  "Stratified", "Male",
    round(coef(mod_male)[1], 3), round(coef(mod_male)[2], 3),
  "Stratified", "Female",
    round(coef(mod_female)[1], 3), round(coef(mod_female)[2], 3),
  "Interaction model", "Male",
    round(b_int[1], 3), round(b_int[2], 3),
  "Interaction model", "Female",
    round(b_int[1] + b_int[3], 3), round(b_int[2] + b_int[4], 3)
) |>
  kable(caption = "Verification: Stratified Analysis = Interaction Model") |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Verification: Stratified Analysis = Interaction Model
Method Stratum Intercept Slope
Stratified Male 6.839 -0.524
Stratified Female 6.260 -0.375
Interaction model Male 6.839 -0.524
Interaction model Female 6.260 -0.375

The interaction model and the stratified analysis give identical stratum-specific equations. The advantage of the interaction model is that it provides a formal test of the difference between slopes and allows adjustment for additional covariates.

pred_int <- ggpredict(mod_int, terms = c("sleep_hrs [3:12]", "sex"))

ggplot(pred_int, aes(x = x, y = predicted, color = group, fill = group)) +
  geom_line(linewidth = 1.2) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.12, color = NA) +
  labs(
    title    = "Predicted Physical Health Days by Sleep Duration and Sex",
    subtitle = "From interaction model: sleep_hrs * sex",
    x        = "Sleep Hours per Night",
    y        = "Predicted Physically Unhealthy Days",
    color    = "Sex",
    fill     = "Sex"
  ) +
  theme_minimal(base_size = 13) +
  scale_color_brewer(palette = "Set1")
Interaction Model: Predicted Physical Health Days by Sleep and Sex

Interaction Model: Predicted Physical Health Days by Sleep and Sex

4. Interactions with Categorical Variables (k > 2)

4.1 Multiple Interaction Terms

When the modifier has \(k > 2\) categories (such as education with 4 levels), the interaction between a continuous exposure and the categorical modifier requires \(k - 1\) interaction terms, one for each dummy variable.

For sleep \(\times\) education (with “Less than HS” as reference):

\[Y = \beta_0 + \beta_1(\text{Sleep}) + \beta_2(\text{HS grad}) + \beta_3(\text{Some college}) + \beta_4(\text{College grad}) + \beta_5(\text{Sleep} \times \text{HS grad}) + \beta_6(\text{Sleep} \times \text{Some college}) + \beta_7(\text{Sleep} \times \text{College grad}) + \varepsilon\]

Each interaction coefficient \(\beta_5, \beta_6, \beta_7\) represents the difference in the sleep slope between that education group and the reference group.

mod_int_educ <- lm(physhlth_days ~ sleep_hrs * education, data = brfss_ci)

tidy(mod_int_educ, conf.int = TRUE) |>
  mutate(across(where(is.numeric), \(x) round(x, 4))) |>
  kable(
    caption = "Interaction Model: Sleep * Education → Physical Health Days",
    col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
  ) |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Interaction Model: Sleep * Education → Physical Health Days
Term Estimate SE t p-value CI Lower CI Upper
(Intercept) 11.9692 2.0406 5.8655 0.0000 7.9687 15.9697
sleep_hrs -0.7877 0.2919 -2.6986 0.0070 -1.3599 -0.2155
educationHS graduate -4.2823 2.3006 -1.8613 0.0628 -8.7925 0.2280
educationSome college -5.1543 2.3032 -2.2379 0.0253 -9.6695 -0.6391
educationCollege graduate -8.8728 2.3020 -3.8545 0.0001 -13.3857 -4.3600
sleep_hrs:educationHS graduate 0.2713 0.3274 0.8287 0.4073 -0.3705 0.9131
sleep_hrs:educationSome college 0.3386 0.3280 1.0322 0.3020 -0.3045 0.9816
sleep_hrs:educationCollege graduate 0.6889 0.3268 2.1079 0.0351 0.0482 1.3297

4.2 Testing the Overall Interaction with a Partial F-Test

Individual t-tests for each interaction term may be non-significant, yet the interaction as a whole could be meaningful. To test whether sleep \(\times\) education is significant overall, we use a partial F-test comparing the model with and without the interaction terms:

\[H_0: \beta_5 = \beta_6 = \beta_7 = 0 \quad \text{(the sleep slope is the same across all education levels)}\]

mod_no_int_educ <- lm(physhlth_days ~ sleep_hrs + education, data = brfss_ci)

anova(mod_no_int_educ, mod_int_educ) |>
  tidy() |>
  mutate(across(where(is.numeric), \(x) round(x, 4))) |>
  kable(caption = "Partial F-test: Is the Sleep x Education Interaction Significant?") |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Partial F-test: Is the Sleep x Education Interaction Significant?
term df.residual rss df sumsq statistic p.value
physhlth_days ~ sleep_hrs + education 4995 308368.1 NA NA NA NA
physhlth_days ~ sleep_hrs * education 4992 307957.2 3 410.9464 2.2205 0.0836 
pred_int_educ <- ggpredict(mod_int_educ, terms = c("sleep_hrs [3:12]", "education"))

ggplot(pred_int_educ, aes(x = x, y = predicted, color = group, fill = group)) +
  geom_line(linewidth = 1.1) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.1, color = NA) +
  labs(
    title    = "Predicted Physical Health Days by Sleep and Education",
    subtitle = "Are the lines parallel? Non-parallel lines indicate interaction.",
    x        = "Sleep Hours per Night",
    y        = "Predicted Physically Unhealthy Days",
    color    = "Education",
    fill     = "Education"
  ) +
  theme_minimal(base_size = 13) +
  scale_color_brewer(palette = "Set2")
Interaction: Sleep Effect by Education Level

Interaction: Sleep Effect by Education Level

5. Continuous × Continuous Interactions

5.1 When Both Variables Are Continuous

Interaction is not limited to categorical modifiers. We can also examine whether the effect of one continuous predictor changes across values of another continuous predictor. For example, does the sleep-physical health association differ by age?

\[Y = \beta_0 + \beta_1(\text{Sleep}) + \beta_2(\text{Age}) + \beta_3(\text{Sleep} \times \text{Age}) + \varepsilon\]

Here, \(\beta_3\) tells us how the slope of sleep changes for each one-year increase in age:

  • The slope of sleep for a person of age \(a\) is: \(\beta_1 + \beta_3 \cdot a\)
  • If \(\beta_3 < 0\), the protective effect of sleep weakens (becomes less negative) as age increases
  • If \(\beta_3 > 0\), the protective effect of sleep strengthens as age increases
mod_int_cont <- lm(physhlth_days ~ sleep_hrs * age, data = brfss_ci)

tidy(mod_int_cont, conf.int = TRUE) |>
  mutate(across(where(is.numeric), \(x) round(x, 4))) |>
  kable(
    caption = "Continuous Interaction: Sleep * Age → Physical Health Days",
    col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
  ) |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Continuous Interaction: Sleep * Age → Physical Health Days
Term Estimate SE t p-value CI Lower CI Upper
(Intercept) 9.6494 1.9752 4.8852 0.0000 5.7771 13.5217
sleep_hrs -1.2597 0.2758 -4.5668 0.0000 -1.8005 -0.7189
age -0.0490 0.0351 -1.3948 0.1631 -0.1178 0.0199
sleep_hrs:age 0.0138 0.0048 2.8444 0.0045 0.0043 0.0232

5.2 Visualizing Continuous Interactions

With continuous modifiers, we visualize the interaction by plotting the predicted relationship at specific values of the modifier (e.g., age = 30, 50, 70):

pred_cont <- ggpredict(mod_int_cont, terms = c("sleep_hrs [3:12]", "age [30, 50, 70]"))

ggplot(pred_cont, aes(x = x, y = predicted, color = group, fill = group)) +
  geom_line(linewidth = 1.2) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.12, color = NA) +
  labs(
    title    = "Predicted Physical Health Days by Sleep, at Different Ages",
    subtitle = "Does the sleep-health relationship change with age?",
    x        = "Sleep Hours per Night",
    y        = "Predicted Physically Unhealthy Days",
    color    = "Age",
    fill     = "Age"
  ) +
  theme_minimal(base_size = 13) +
  scale_color_brewer(palette = "Dark2")
Interaction: Sleep Effect at Different Ages

Interaction: Sleep Effect at Different Ages

If the lines are approximately parallel, age does not modify the sleep effect. If they fan out or converge, the interaction is meaningful.

6. Confounding

6.1 What Is Confounding?

Confounding exists when the estimated association between an exposure and an outcome is distorted because of a third variable that is related to both. When confounding is present, ignoring the confounder leads to a biased estimate of the exposure effect.

For a variable to be a confounder, it must satisfy three conditions:

  1. Associated with the exposure (sleep hours)
  2. Associated with the outcome (physical health days), even in the absence of the exposure
  3. Not on the causal pathway from exposure to outcome (i.e., not a mediator)
Confounding Structure: The Confounder Affects Both Exposure and Outcome

Confounding Structure: The Confounder Affects Both Exposure and Outcome

Age is a classic confounder of the sleep-physical health relationship:

  • Older adults tend to sleep fewer hours (associated with exposure)
  • Older adults have more physically unhealthy days (associated with outcome)
  • Age is not caused by sleep or physical health (not on the causal pathway)

6.2 Detecting Confounding: The 10% Change-in-Estimate Rule

The standard approach in epidemiology is to compare the crude (unadjusted) estimate to the adjusted estimate:

  1. Fit the crude model: \(Y = \beta_0 + \beta_1 X_1\)
  2. Fit the adjusted model: \(Y = \beta_0^* + \beta_1^* X_1 + \beta_2^* X_2\)
  3. Compute the percent change: \(\frac{|\beta_1 - \beta_1^*|}{|\beta_1|} \times 100\%\)
  4. If the change exceeds 10%, \(X_2\) is a confounder and should be included in the model

Important: This is a rule of thumb, not a rigid cutoff. The 10% threshold is conventional, not absolute. Some researchers use 5% or 15% depending on context.

6.3 Confounding Example: Is Age a Confounder?

# Crude model: sleep → physical health days
mod_crude <- lm(physhlth_days ~ sleep_hrs, data = brfss_ci)

# Adjusted model: adding age
mod_adj_age <- lm(physhlth_days ~ sleep_hrs + age, data = brfss_ci)

# Extract sleep coefficients
b_crude <- coef(mod_crude)["sleep_hrs"]
b_adj   <- coef(mod_adj_age)["sleep_hrs"]
pct_change <- abs(b_crude - b_adj) / abs(b_crude) * 100

tribble(
  ~Model, ~`Sleep β`, ~SE, ~`95% CI`,
  "Crude (sleep only)",
    round(b_crude, 4),
    round(tidy(mod_crude) |> filter(term == "sleep_hrs") |> pull(std.error), 4),
    paste0("(", round(confint(mod_crude)["sleep_hrs", 1], 4), ", ",
           round(confint(mod_crude)["sleep_hrs", 2], 4), ")"),
  "Adjusted (+ age)",
    round(b_adj, 4),
    round(tidy(mod_adj_age) |> filter(term == "sleep_hrs") |> pull(std.error), 4),
    paste0("(", round(confint(mod_adj_age)["sleep_hrs", 1], 4), ", ",
           round(confint(mod_adj_age)["sleep_hrs", 2], 4), ")")
) |>
  kable(caption = "Confounding Assessment: Does Age Confound the Sleep-Physical Health Association?") |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Confounding Assessment: Does Age Confound the Sleep-Physical Health Association?
Model Sleep β SE 95% CI
Crude (sleep only) -0.4381 0.0827 (-0.6002, -0.2761)
Adjusted (+ age) -0.5112 0.0828 (-0.6736, -0.3489) 
cat("\nPercent change in sleep coefficient when adding age:", round(pct_change, 1), "%\n")
## 
## Percent change in sleep coefficient when adding age: 16.7 %
cat("10% rule threshold: |", round(b_adj, 4), "| should fall within (",
    round(b_crude * 0.9, 4), ",", round(b_crude * 1.1, 4), ")\n")
## 10% rule threshold: | -0.5112 | should fall within ( -0.3943 , -0.4819 )

6.4 Systematic Confounding Assessment

In practice, we assess multiple potential confounders. Let’s evaluate age, exercise, education, and general health:

# Crude model
b_crude_val <- coef(mod_crude)["sleep_hrs"]

# One-at-a-time adjusted models
confounders <- list(
  "Age"             = lm(physhlth_days ~ sleep_hrs + age, data = brfss_ci),
  "Exercise"        = lm(physhlth_days ~ sleep_hrs + exercise, data = brfss_ci),
  "Education"       = lm(physhlth_days ~ sleep_hrs + education, data = brfss_ci),
  "General health"  = lm(physhlth_days ~ sleep_hrs + gen_health, data = brfss_ci),
  "Sex"             = lm(physhlth_days ~ sleep_hrs + sex, data = brfss_ci),
  "Income"          = lm(physhlth_days ~ sleep_hrs + income_cat, data = brfss_ci)
)

conf_table <- map_dfr(names(confounders), \(name) {
  mod <- confounders[[name]]
  b_adj_val <- coef(mod)["sleep_hrs"]
  tibble(
    Covariate = name,
    `Crude β (sleep)` = round(b_crude_val, 4),
    `Adjusted β (sleep)` = round(b_adj_val, 4),
    `% Change` = round(abs(b_crude_val - b_adj_val) / abs(b_crude_val) * 100, 1),
    Confounder = ifelse(abs(b_crude_val - b_adj_val) / abs(b_crude_val) * 100 > 10,
                        "Yes (>10%)", "No")
  )
})

conf_table |>
  kable(caption = "Systematic Confounding Assessment: One-at-a-Time Addition") |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) |>
  column_spec(5, bold = TRUE)
Systematic Confounding Assessment: One-at-a-Time Addition
Covariate Crude β (sleep) Adjusted β (sleep) % Change Confounder
Age -0.4381 -0.5112 16.7 Yes (>10%)
Exercise -0.4381 -0.4083 6.8 No
Education -0.4381 -0.3872 11.6 Yes (>10%)
General health -0.4381 -0.1948 55.5 Yes (>10%)
Sex -0.4381 -0.4434 1.2 No
Income -0.4381 -0.3739 14.7 Yes (>10%)

6.5 Important Caveats About Confounding Assessment

Identifying candidate confounders is not purely a statistical exercise. The three conditions for confounding (associated with exposure, associated with outcome, not on the causal pathway) require substantive knowledge from the literature and your understanding of the causal structure.

Caution about missing data: When you add a covariate to the model, observations with missing values on that covariate are dropped. This changes the analytic sample, which could alter \(\hat{\beta}\) for reasons unrelated to confounding. Always ensure that the crude and adjusted models are fit on the same set of observations.

# Verify our dataset has no missing values (we filtered earlier)
cat("Missing values in analytic dataset:", sum(!complete.cases(brfss_ci)), "\n")
## Missing values in analytic dataset: 0
cat("All models are fit on the same n =", nrow(brfss_ci), "observations.\n")
## All models are fit on the same n = 5000 observations.

Confounders vs. mediators: A variable on the causal pathway between exposure and outcome is a mediator, not a confounder. For example, if sleep affects exercise habits, which in turn affect physical health, then exercise is a mediator. Adjusting for a mediator would attenuate the exposure effect, but this attenuation is not bias; it reflects the removal of an indirect pathway. Whether to adjust depends on your research question.

General health status is a tricky case. It could be a confounder (poor overall health causes both poor sleep and more physically unhealthy days) or a mediator (poor sleep leads to poor general health, which leads to more physically unhealthy days). The direction depends on the assumed causal structure and should be guided by subject-matter knowledge.

7. Order of Operations: Interaction Before Confounding

7.1 Why Interaction Comes First

The standard epidemiological approach to model building follows this order:

  1. Assess interaction first — test whether the exposure-outcome relationship differs across subgroups
  2. If interaction is present — stratify by the modifier and assess confounding within each stratum
  3. If no interaction — assess confounding in the pooled (unstratified) data

The reason for this order is that confounding assessment assumes a single exposure effect. If the effect actually varies across subgroups (interaction), then a single “adjusted” coefficient is misleading. Reporting an average effect when the true effect differs by sex, for example, could mask important public health heterogeneity.

7.2 A Practical Workflow

Workflow: Interaction Before Confounding
Step Action
1 Specify the exposure-outcome relationship of interest
2 Identify potential effect modifiers (from literature, biological plausibility)
3 Test for interaction using interaction terms or stratified analysis
4 If interaction is present: report stratum-specific effects; assess confounding within strata
5 If no interaction: assess confounding in the full sample using the 10% change-in-estimate rule

7.3 Putting It All Together

Let’s build a final model that accounts for both potential interaction and confounding in our BRFSS data:

# Step 1: Test for interaction between sleep and sex
mod_final_int <- lm(physhlth_days ~ sleep_hrs * sex + age + education + exercise,
                    data = brfss_ci)

# Check interaction term
int_pval <- tidy(mod_final_int) |>
  filter(term == "sleep_hrs:sexFemale") |>
  pull(p.value)

cat("Interaction p-value (sleep x sex):", round(int_pval, 4), "\n")
## Interaction p-value (sleep x sex): 0.0417
# Step 2: If interaction is not significant, fit additive model with confounders
mod_final <- lm(physhlth_days ~ sleep_hrs + sex + age + education + exercise,
                data = brfss_ci)

tidy(mod_final, conf.int = TRUE) |>
  mutate(across(where(is.numeric), \(x) round(x, 4))) |>
  kable(
    caption = "Final Model: Sleep → Physical Health Days, Adjusted for Confounders",
    col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
  ) |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Final Model: Sleep → Physical Health Days, Adjusted for Confounders
Term Estimate SE t p-value CI Lower CI Upper
(Intercept) 9.9181 0.7955 12.4676 0.0000 8.3585 11.4776
sleep_hrs -0.4318 0.0805 -5.3662 0.0000 -0.5895 -0.2740
sexFemale 0.2862 0.2178 1.3140 0.1889 -0.1408 0.7132
age 0.0355 0.0065 5.4878 0.0000 0.0228 0.0482
educationHS graduate -1.9241 0.5085 -3.7838 0.0002 -2.9209 -0.9272
educationSome college -2.0609 0.5065 -4.0688 0.0000 -3.0539 -1.0679
educationCollege graduate -2.9504 0.4955 -5.9548 0.0000 -3.9217 -1.9791
exerciseYes -4.1551 0.2711 -15.3273 0.0000 -4.6866 -3.6236 
b_final <- coef(mod_final)["sleep_hrs"]

tribble(
  ~Model, ~`Sleep β`, ~`% Change from Crude`,
  "Crude", round(b_crude_val, 4), "—",
  "Adjusted (age only)", round(coef(mod_adj_age)["sleep_hrs"], 4),
    paste0(round(abs(b_crude_val - coef(mod_adj_age)["sleep_hrs"]) / abs(b_crude_val) * 100, 1), "%"),
  "Final (all confounders)", round(b_final, 4),
    paste0(round(abs(b_crude_val - b_final) / abs(b_crude_val) * 100, 1), "%")
) |>
  kable(caption = "Sleep Coefficient: Crude vs. Progressively Adjusted Models") |>
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Sleep Coefficient: Crude vs. Progressively Adjusted Models
Model Sleep β % Change from Crude
Crude -0.4381 —
Adjusted (age only) -0.5112 16.7%
Final (all confounders) -0.4318 1.5%

Summary of Key Concepts

Concept Key Point
Predictive vs. associative Confounding and interaction matter for associative (etiologic) modeling
Interaction The effect of the exposure differs across levels of a modifier
Stratified analysis Fit separate models per stratum; informative but no formal test
Interaction term \(X_1 \times X_2\) in the model; t-test on \(\beta_3\) tests parallelism
: vs. * in R : = interaction only; * = main effects + interaction
Must include main effects Never include \(X_1 X_2\) without also including \(X_1\) and \(X_2\)
Partial F-test for interaction Tests all \(k - 1\) interaction terms simultaneously
Continuous \(\times\) continuous The slope of one predictor changes linearly with the other
Confounding A third variable distorts the exposure-outcome association
Three conditions Associated with exposure, associated with outcome, not on causal pathway
10% change-in-estimate Compare crude and adjusted \(\hat{\beta}\); >10% change = confounder
Interaction before confounding Assess interaction first; if present, stratify before assessing confounding
Mediator vs. confounder Adjusting for a mediator removes an indirect effect, not bias

Part 2: In-Class Lab Activity

EPI 553 — Confounding and Interactions Lab Due: End of class, March 24, 2026

Instructions

In this lab, you will assess interaction and confounding in the BRFSS 2020 dataset. Work through each task systematically. You may discuss concepts with classmates, but your written answers and R code must be your own.

Submission: Knit your .Rmd to HTML and upload to Brightspace by end of class.

Data for the Lab

Use the saved analytic dataset from today’s lecture. It contains 5,000 randomly sampled BRFSS 2020 respondents.

Variable Description Type
physhlth_days Physically unhealthy days in past 30 Continuous (0–30)
sleep_hrs Sleep hours per night Continuous (1–14)
menthlth_days Mentally unhealthy days in past 30 Continuous (0–30)
age Age in years (capped at 80) Continuous
sex Sex (Male/Female) Factor
education Education level (4 categories) Factor
exercise Any physical activity (Yes/No) Factor
gen_health General health status (5 categories) Factor
income_cat Household income (1–8 ordinal) Numeric 
# Load the dataset
library(tidyverse)
library(broom)
library(knitr)
library(kableExtra)
library(car)
library(ggeffects)

brfss_ci <- readRDS(
  "/Users/mm992584/Library/CloudStorage/OneDrive-UniversityatAlbany-SUNY/Spring 2026/Epi 553/Lectures Notes/Confounding Interaction/brfss_ci_2020.rds"
)

Task 1: Exploratory Data Analysis (15 points)

1a. (5 pts) Create a scatterplot of physhlth_days (y-axis) vs. age (x-axis), colored by exercise status. Add separate regression lines for each group. Describe the pattern you observe.

1b. (5 pts) Compute the mean physhlth_days for each combination of sex and exercise. Present the results in a table. Does it appear that the association between exercise and physical health days might differ by sex?

1c. (5 pts) Create a scatterplot of physhlth_days vs. sleep_hrs, faceted by education level. Comment on whether the slopes appear similar or different across education groups.

Task 2: Stratified Analysis for Interaction (15 points)

2a. (5 pts) Fit separate simple linear regression models of physhlth_days ~ age for exercisers and non-exercisers. Report the slope, SE, and 95% CI for age in each stratum.

2b. (5 pts) Create a single plot showing the two fitted regression lines (one per exercise group) overlaid on the data. Are the lines approximately parallel?

2c. (5 pts) Can you formally test whether the two slopes are different using only the stratified results? Explain why or why not.

Task 3: Interaction via Regression (25 points)

3a. (5 pts) Fit the interaction model: physhlth_days ~ age * exercise. Write out the fitted equation.

3b. (5 pts) Using the fitted equation, derive the stratum-specific equations for exercisers and non-exercisers. Verify that these match the stratified analysis from Task 2.

3c. (5 pts) Conduct the t-test for the interaction term (age:exerciseYes). State the null and alternative hypotheses, report the test statistic and p-value, and state your conclusion about whether interaction is present.

3d. (5 pts) Now fit a model with an interaction between age and education: physhlth_days ~ age * education. How many interaction terms are produced? Use a partial F-test to test whether the age \(\times\) education interaction as a whole is significant.

3e. (5 pts) Create a visualization using ggpredict() showing the predicted physhlth_days by age for each education level. Do the lines appear parallel?

Task 4: Confounding Assessment (25 points)

For this task, the exposure is exercise and the outcome is physhlth_days.

4a. (5 pts) Fit the crude model: physhlth_days ~ exercise. Report the exercise coefficient. This is the unadjusted estimate.

4b. (10 pts) Systematically assess whether each of the following is a confounder of the exercise-physical health association: age, sex, sleep_hrs, education, and income_cat. For each:

  • Fit the model physhlth_days ~ exercise + [covariate]
  • Report the adjusted exercise coefficient
  • Compute the percent change from the crude estimate
  • Apply the 10% rule to determine if the variable is a confounder

Present your results in a single summary table.

4c. (5 pts) Fit a fully adjusted model including exercise and all identified confounders. Report the exercise coefficient and compare it to the crude estimate. How much did the estimate change overall?

4d. (5 pts) Is gen_health a confounder or a mediator of the exercise-physical health relationship? Could it be both? Explain your reasoning with reference to the three conditions for confounding and the concept of the causal pathway.

Task 5: Public Health Interpretation (20 points)

5a. (10 pts) Based on your analyses, write a 4–5 sentence paragraph for a public health audience summarizing:

  • Whether the association between exercise and physical health differs by any subgroup (interaction assessment)
  • Which variables confounded the exercise-physical health association
  • The direction and approximate magnitude of the adjusted exercise effect
  • Appropriate caveats about cross-sectional data and potential unmeasured confounding

Do not use statistical jargon.

5b. (10 pts) A colleague suggests including gen_health as a covariate in the final model because it changes the exercise coefficient by more than 10%. You disagree. Write a 3–4 sentence argument explaining why adjusting for general health may not be appropriate if the goal is to estimate the total effect of exercise on physical health days. Use the concept of mediation in your argument.

End of Lab Activity